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Optimal resource management in wireless access networks Mohsenian-Rad, Amir-Hamed 2008

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Optimal Resource Management in Wireless Access Networks by Amir-Hamed Mohsenian-Rad  B.Sc., Electrical Engineering, Amir-Kabir University of Technology, Iran, 2002 M.Sc., Electrical Engineering, Sharif University of Technology, Iran, 2004  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Electrical and Computer Engineering)  The University of British Columbia (Vancouver) August, 2008  ©  Amir-Hamed Mohsenian-Rad 2008  Abstract This thesis presents several simple, robust, and optimal resource management schemes for multihop wireless access networks with the main focus on multi-channel wireless mesh networks (MC WMNs). In this regard, various resource management optimization problems are formulated arid efficient algorithms are proposed to solve each problem. First, we consider the channel as signment problem in MC-WMNs and formulate different resource management problems within the general framework of network utility maximization (NUM). Unlike most of the previously proposed channel assignment schemes, our algorithms can not only assign the orthogonal (i.e., non-overlapped) channels, but also partially overlapped channels. This better utilizes the avail able frequency spectrum as a critical resource in MC-WMNs. Second, we propose two distributed random medium access control (MAC) algorithms to solve a non-convex NUM problem at the MAC layer. The first algorithm is fast, optimal, and robust to message loss and delay. It also only requires a limited message passing among the wireless nodes. Using distributed learning techniques, we then propose another NUM-based MAC algorithm which achieves the optimal performance without frequent message exchange. Third, based on our results on random MAC, we develop a distributed multi-interface multi-channel random access algorithm to solve the NUM problem in MC-WMNs. Different from most of the previous channel assignment schemes in the literature, where channel assignment is intuitively modeled in the form of combinatorial and dis  ii  Abstract crete optimization problems, our scheme is based on formulating a novel continuous optimization model. This makes the analysis and implementation significantly easier. Finally, we consider the problem of pricing and monetary exchange in multi-hop wireless access networks, where each intermediate node receives a payment to compensate for its offered packet forwarding service. In this regard, we propose a market-based wireless access network model with two-fold pricing. It uses relay-pricing to encourage collaboration among the access points. It also uses interference pricing to leverage optimal resource management. In general, this thesis widely benefits from several mathematical techniques as both modeling and solution tools to achieve simple, robust, optimal, and practical resource management strategies for future wireless access networks.  in  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  xi  List of Figures  xii  Abbreviations  xv  xvii  Acknowledgements  1  Introduction  1  1.1  Wireless Access Networks  2  1.1.1  Structure  2  1.1.2  Resource Management  3  1.2  Mathematical Foundation  4  1,2.1  Optimization Theory  5  1.2.2  Network Utility Maximization  12  1.2.3  Distributed Systems and Convergence Analysis  15  iv  Table of Contents  2  1.3  Summary of Results and Contributions  17  1.4  List of Publications  21  1.5  Thesis Organization  24  Topology Design, Interface Assignment, Channel Allocation, & Routing  .  .  Related Work  2.2  Joint Logical Topology Design, Interface Assignment, Channel Allocation, and Routing  2.4  25 27  2.1  2.3  .  29  2.2.1  Problem Formulation  29  2.2.2  Effective Capacity  32  2.2.3  Total Flows on a Logical Link  33  2,2.4  Flow Conservation at Each Node  34  2.2.5  Feasible Region and the Objective Function  35  2.2.6  Hop Count Constraint  37  2.2.7  Optimization Problem  38  2.2.8  Algorithm  39 40  Simulation Results 2.3.1  Optimal and Sub-optimal Solutions  42  2.3.2  Sample Logical Topology  42  2.3.3  Impact of the Network Traffic  44  2.3.4  Impact of the Number of NICs and Channels  47  2.3.5  Fairness  47 49  Summary  V  Table of Contents 2.5  3  Analytical Proofs  49  2.5.1  Proof of Lemma 1  49  2.5.2  Proof of Lemma 2  50  Cross-layer Fair Bandwidth Sharing  51  3.1  Problem Formulation  52  3.1.1  Channel Assignment Model  53  3.1.2  Interference Model  56  3.1.3  Cross-Layer Fair Bandwidth Sharing Problem  59  3.2  Design I: Exact Binary Linearization  59  3.3  Design II: Approximate Dual Decomposition  62  3.3.1  A Simple Heuristic Algorithm to Solve Aggregate Interference Cost Mini 67  mization Problem  4  3.4  Simulation Results  68  3.5  Summary  75  3.6  Analytical Proofs  76  3.6.1  Proof of Lemma 3  76  3.6.2  Proof of Theorem 9  76  3.6.3  Proof of Lemma 4  78  3.6.4  Proof of Lemma 5  78  Partially Overlapped Channel Assignment for MC-WMNs  79  4.1  80  System Model 4.1.1  Channel and Interface Assignment Model vi  80  Table of Contents  4.2  5  4.1.2  Channel Overlapping Matrix  81  4.1.3  Mutual Interference Model  82  4.1.4  Link-Layer Flow Rates  86  Joint Channel Assignment, Interface Assignment, and Scheduling Problem  .  .  .  .  86  4.2.1  Nonlinear Problem  87  4.2.2  Equivalent Linear Problem  88  4.3  Performance Evaluation  92  4.4  Summary  93  Simple, Robust, and Optimal Random Access  95  5,1  System Model  97  5.2  Algorithms  100  5.2.1  Single-Cell Topology  101  5.2.2  General Topology  107  5.3  5.4  5.5  Convergence, Optimality, and Robustness  .  110  5.3.1  Single-Cell Topology  110  5.3.2  General Topology  115 118  Simulation Results 5.4.1  Convergence and Optimality  119  5.4.2  Signalling Overhead  122  5.4.3  Robustness  124  5.4.4  Comparison with IEEE 802.11 DCF  126 127  Summary  vii  Table of Contents 5.6  6  5.6.1  Proof of Theorem 10  129  5.6.2  Proof of Theorem 12  135  5.6.3  Proof of Theorem 13  136  5.6.4  Proof of Theorem 14  138  5.6.5  Proof of Theorem 15  139  5.6.6  Proof of Theorem 18  140  Utility Optimal Random Access without Frequent Message Passing  142  6.1  System Model  144  6.2  Algorithm with No Explicit Message Passing  145  6.3  Convergence and Optimality  151  6.4  Simulation Results  153  6.5  Summary  155  6.6  Analytical Proofs  155  6.6.1  7  129  Analytical Proofs  155  Proof of Theorem 20  Multi-Interface Multi-Channel Random Access 7.1  7.2  161  System Model 7.1.1  NICs with Single-Channel Reception  7.1.2  NICs with Multi-Channel Reception  7.1.3  Partially Overlapped Frequency Channels  162 .  165 167 169  Network Utility Maximization 7.2.1  159  169  Problem Formulation viii  Table of Contents 7.2.2 7.3  7.4  8  172  Examples  DMMRA Algorithms  174  7.3.1  Local NUM Problems  174  7.3.2  Algorithms  177  7.3.3  Optimality and Convergence  179  Performance Evaluation  182  7.4.1  Convergence  183  7.4.2  Comparison with Combinatorial Channel Assignment  183  7.4.3  Impact of Utility Parameter a  183  7.4.4  Signalling Overhead  186  7.4.5  Partially Overlapped Channel Assignment  188  7.4.6  Impact of Delayed and Outdated Information  7.4.7  Optimality  .  190 191  7.5  Summary  192  7.6  Analytical Proofs  194  7.7  Proof of Theorem 22  194  7.8  Proof of Theorem 23  195  7.9  Proof of Theorem 24  196  7.10 Proof of Theorem 25  198  7.11 Proof of Theorem 26  199  7.12 Proof of Theorem 27  200  Two-Fold Pricing  201  ix  Table of Contents 8.1  9  Market-Based Wireless Access Network Model with Two-Fold Pricing  204  8.1.1  System Model  204  8.1.2  Two-Fold Relay and Interference Pricing  207  8.1.3  Resource Allocation  210  8.2  Guaranteed Positive Individual Profits and Maximum Social Welfare  212  8.3  Simulation Results  216  8.3.1  Performance Comparison with Single-Fold Pricing  8.3.2  Maximum Weight Matching  8.3.3  Impact of Fast-Fading  217 17 220  8.4  Summary  223  8.5  Analytical Proofs  223  8.5.1  Proof of Theorem 28  223  8.5.2  Proof of Theorem 29  225  8.5.3  Proof of Lemma 7  8.5.4  Proof of Theorem 30  26 227  Conclusions and Future Work  230  9.1  Summary of Work Accomplished  230  9.2  Future Work  233  Bibliography  235  x  List of Tables 2.1  Impact of varying the number of NICs and channels  48  2.2  Achieved fairness among different flows  48  xi  List of Figures 1.1  A sample multi-hop wireless access network  2.1  A sample multi-channel wireless mesh network  29  2.2  Average optimization error at versus the iteration number  41  2.3  Optimal topology for a sample multi-channel wireless mesh network  43  2.4  Comparison between TiMesh, Hyacinth, and GLICA MC-WMN architectures in 45  the presence of UDP traffic 2.5  3  Comparison between TiMesh, Hyacinth, and CLICA MC-WMN architectures in the presence of TCP traffic  46  3.1  A sample wireless mesh network and its corresponding contention graph  53  3.2  Scenario number 1 in our simulation model  68  3.3  Trend of network utility for scenario number 1  70  3.4  Network utility for ten different random scenarios  71  3.5  Impact of available network resources  72  3.6  Effects of varying the number of available frequency channels  73  3.7  Effects of varying the fairness parameter  74  4.1  Available eleven partially overlapped channels in 802.llb frequency band  82  xii  List of Figures 4.2  Different interference ranges depending on the frequency channel separation  4.3  Performance comparison within the IEEE 802.llb frequency band  94  5.1  A sample wireless ad-hoc network  98  5.2  Sufficient convergence conditions on different design parameters  114  5.3  A sample chain topology with one hop interference  116  5.4  Simulation results for a single-cell topology with three nodes and six links  120  5.5  Simulation results for a chain topology with three nodes and four links  121  5.6  Comparison between our algorithms and the subgradient-based algorithm in term  84  .  of signalling overhead 5.7  123  Comparison between Algorithm 6 and the subgradient-based algorithm in term of robustness with respect to communication delay and message loss  125  5.8  Convergence time in the presence of various communication delays  126  5.9  Comparison between DCF and Algorithm 6 in terms of throughput and fairness.  6.1  A sample single-hop single-cell wireless ad-hoc network with N  6.2  Simulation results for Algorithm 7  154  6.3  Comparison between Algorithm 7 and 802.11 DCF  158  7.1  An ad-hoc network with multi-channel links  162  7.2  Building blocks of the receiver in single-channel and multi-channel reception.  7.3  Trend of the network utility versus time slots using Algorithms 8 and 9  184  7.4  Comparison between Algorithms 8 and 9 with (UO-CIACA) algorithm  185  7.5  Comparison between Algorithm 8 and MMAC  187  xlii  =  .  3 users  128  145  .  .  166  List of Figures 7.6  Signalling overhead and network throughput for Algorithms 8 and 9.  189  7.7  Effect of assigning partially-overlapped channels  190  7.8  Trend of the network utility versus time slots using Algorithms 8.  7.9  Impact of delay on convergence time  192  .  193  7.10 Comparison between achieved network utility and its optimal value.  194  8.1  A sample multi-hop wireless access network with six wireless relay nodes.  205  8.2  Social welfare and throughput for different number of nodes  218  8.3  Results for each of the 10 simulated topologies for slow-fading channels.  219  8.4  Trend of the maximum weight matching objective  e(t) versus time slots  221  8.5  Results for each of the 10 simulated topologies for fast-fading channels  xiv  .  222  Abbreviations ACK  Acknowledgement  ACM  Association for Computing Machinery  CDMA  Code division multiple access  CLICA  Connected low interference channel assignment  CTS  Clear to send  DCF  Distributed coordination function  DMMRA  Distributed multi-interface multi-channel random access  EED  End-to-end delay  FF  Fast fading  IEEE  Institute of Electrical and Electronics Engineers  ILS  Iterated local search  KKT  Karush-Kuhn-Tucker (optimality conditions)  MAC  Medium access control  xv  Abbreviations MCR  Multiple channel reception  MCWMN  Multi-channel wireless mesh network  NIC  Network interface card  NUM  Network utility maximization  OCICA  Optimal combinatorial interface and channel assignment  PDR  Packet delivery ratio  RTS  Request to send  SCR  Single channel reception  SF  Slow fading  SFP  Single fold pricing  TCP  Transmission control protocol  TFP  Two fold pricing  UDP  User datagram protocol  WMN  Wireless mesh network  xvi  Acknowledgements It is difficult to overstate my gratitude to my supervisor Dr. Vincent Wong for his guidance, patience, and support. I sincerely appreciate the amount of time and effort he invested for teaching me technical writing and introducing me to several exciting research areas. I am very grateful to Dr. Mung Chiang, Dr. Jianwei Huang, and Dr. Victor Leung for their collaborations and insightful comments and suggestions that helped me during some research directions for my degree. I would also like to express my sincere thanks to Dr. Maurice Queyranne, Dr. Robert Schober, Dr. Vikram Krishnamurthy, Dr. Junshan Zhang, Dr. Lijun Chen, Dr. Dahai Xu, Dr. Yung Yi, Amir Nasri, Enrique Stevense-Navarro, and Shahab Moradi for their helpful suggestions. I also thank my colleagues and friends who assisted and encouraged me along the way. I am especially grateful to Mohammad Mahmoodi Ghidari, Ehsan Bayaki, Mohammad Hossein Bateni, MajidKhabbazian, Valid Shah-Mansouri, Keivan Ronasi, and Mani Malekesmaeili. Lastly, and most importantly, I wish to thank my lovely wife, Hoda, my parents, Mahdi and Alieh Mohsenian-Rad, my parents-in-law, Bagher and Fatemeh Mojazi-Amiri, my sister, Neda, my brother and his wife, Amir-Farzad and Lida, and my sister-in-law, Elham, for their love, support, and encouragement. To them I dedicate this thesis. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Bell Canada, Pacific Century Graduate Scholorship from the Ministry of Advanced  xvii  Acknowledgements Education of British Columbia Provincial Government, and the University of British Columbia Graduate Fellowship.  xviii  Chapter 1  Introduction Optimization-based approaches have been used extensively over the past several years to study various resource management problems in communications and computer networks. Example includes the Internet congestion control which is modeled in the general form of a distributed system to solve a network-wide optimization problem to maximize the aggregate network utility. These approaches can systematically model the complicated issues in the networking systems and result in deep understandings of the existing network protocols such as the transmission control protocol (TCP) [1]. The main focus of this thesis is on formulating various resource management schemes for wireless access networks in the form of tractable optimization problems within the general frame work of network utility maximization and also proposing simple and practical algorithms to solve the formulated problems. The rest of this chapter is organized as follows. We first provide an overview of the structure and the existing resource management problems in wireless access net works with the main focus on multi-channel wireless mesh networks in Section 1.1. We then list most of the mathematical and analytical tools that we use throughout this thesis in Section 1.2. Next, we summarize the our contributions and outline the results in Section 1.3. The list of related publications is given in Section 1.4. Finally, we describe the organization of the thesis in Section 1.5.  1  Chapter 1. Introduction  1.1 1.1.1  Wireless Access Networks Structure  Wireless access networks have recently received an increasing attention, especially under the context of wireless mesh networks (WMNs), where the wireless network is multi-hop and offers ubiquitous and inexpensive Internet access to various wireless users within a wide coverage area [2, 3]. A sample wireless mesh network is shown in Fig. 1.1. The network consists of a number of wireless stationary access points and several wireless users. In the context of WMNs, the access points are called wireless mesh routers while the wireless users are called the wireless mesh clients. The access points may also be referred to as wireless relay nodes. The wireless access points form a fully wireless backbone network to provide the connectivity service to the users. Some of them also act as gateways to the Internet via high-speed wired links. Users first transfer data to their associated access points, and these data are then transferred to the Internet or other users via the intermediate access points in a multi-hop manner. The research and development of wireless access and mesh networks are motivated by several applications including broadband home networking, community and neighborhood networking, enterprise networking, metropolitan area networking, and building automation [2]. Some vendors have recently begun to offer products in this area [4, 5, 6, 7]. The IEEE has also set up the 802.lls task group for mesh networking [8]. For the wireless access networks, interference is a major issue which can significantly limit the aggregate capacity of the network. As a practical solution, the capacity of the IEEE 802.11 a/b/g-based WMNs [9, 10] can be substantially improved via the use of multiple network interface cards (NICs) and multiple orthogonal frequency channels [11]. In this scenario, each access point  2  Chapter 1. Introduction High Speed Wired Internet  %%%  Wireless Link  / /  /  \ \  / / / ‘S...,..  —.-...  /  Wireless users  \  Wireless Access Point  Figure 1.1: A sample multi-hop wireless access network with six wireless access points (also called wireless relay nodes or wireless mesh routers), and 15 wireless users (also called wireless mesh clients).  is equipped with multiple NICs. Each NIC is then assigned to a distinct frequency channel. A pair of neighboring access points can communicate with each other as long as one of their NICs uses the same channel. On the other hand, those transmissions which take place over two different (orthogonal) frequency channels do not interfere with each other.  1.1.2  Resource Management  The number of available channels depends on the frequency band. For example, the 802.11 b/g standards have 11 channels within the 2.4 GHz frequency band, of which 3 channels are orthogonal (non-overlapping). The 802.lla standard has 79 operating channels in North America, of which 12 channels are orthogonal. Notice that the key benefit of using multiple NICs is to provide independent simultaneous transmnissiomis via different NICs. The number of NICs per each access point is usually limited to 2 or 3.  3  Chapter 1. Introduction The limited number of frequency channels implies that some transmissions may be required to operate over the same channels.  The limited number of NICs also avoids having several  simultaneous transmissions. These limit the performance gain of the multi-channel deployments and make it critical to design intelligent interface assignment and channel allocation (also called channel assignment) schemes. Interface assignment determines which wireless links should operate on each NIC. Channel allocation determines which frequency channel should be assigned to each wireless link. Other resource management problems include medium access control, transmission power control, routing, and congestion control. In this thesis, we apply optimization theory to solve the mentioned resource management problems. Our main focus is on cross-layer optimization where joint designs are given for multiple resource management problems which affect different layers in the network layering model [12]. A review of the work related to each resource management problem of interest is given later in each chapter. Tutorial papers on cross-layer design can be found in [13] and [14].  1.2  Mathematical Foundation  In this thesis, we extensively employed several powerful mathematical tools from optimization theory, theory of parallel and distributed computing, and utility theory.  Optimization theory  is a mature field that has experienced major development in the last twenty years [15, 16]. In particular, the theory of convex optimization has been widely accepted as both modeling language and solution tool in complex communication and networking problems [17]. The theory of parallel amid distributed computing is also a promising field that can be brought to bear on many important problems in both parallel computers and distributed data communication networks  4  Chapter 1. Introduction  [18). It can particularly be helpful in various convergence and robustness analysis studies. This thesis also benefits from the recent developments in using utility theory in networking problems which is indeed a promising research area towards establishing a mathematical theory for network architectures [19]. In this section, we provide an overview of each of these three areas.  1.2.1  Optimization Theory  Consider the following problem:  minimize  fo(x)  subject to  f(x)  a3EX  0, Vi  =  1,..  .  ,  (1.1)  N  h(x)=0, Vj1,...,Ne  which describes the problem of finding an x that minimizes function Jo (x) among all feasible  f(x)  points; i.e., all x e X that satisfy the conditions for all i  =  1,.  .  .  ,  N. Here x  variables where X  C  f(x)  the corresponding functions  f :  functions. The equations h(x) —*  =  1,  .  .  ,  .  RN. The function fo : RN  function. The inequalities  functions h : RN  (xi, V i  =  R for i  no constraints (i.e., N  =  Ne  =  <  RN =  —*  0 for i  1,..  =  0 for i  .  ,  =  1,  .  .  R for i =  1,..  =  .  ,  ,  0 for all i  =  1,... N, and h(x) ,  =  0,  N) e X denotes the vector of optimization R is called the objective function or cost  —>  .  <  1 are called the inequality constraints, and N  1,  .  .  .  ,  1 are called the inequality constraint N  N are called the equality constraints, and the  N are called the equality constraint functions. If there are  0), then we say that problem (1.1) is unconstrained. Generally  speaking, optimization theory provides the mathematical tools to model and solve optimization problems in the general form of problem (1.1). Notice that any maximization problem can also  5  Chapter 1. Introduction  be written in the form of (1.1) if we multiply its objective function by —1. Thus, we only focus on minimization problems in this section. The results can be extended to maximization problems accordingly. Next, we describe four important classes of optimization problems, namely linear, convex, integer, and mixed-integer optimization problems.  Linear Optimization An important class of optimization problems is the linear optimization problems. Problem (Li) reduces to a linear optimization problem if X  f(x) h(x)  where a, c  e  RN and b, d  e  =  RN and for each vector x  e X, we have:  =aTx+b, cTx  (1.2)  + d,  Vi  1,... Ne,  (1.3)  ,  R. Linear optimization problems can be solved in polynomial time.  That is, the run time is no greater than a polynomial function of the problem size N + N + Ne and the number of bits used to represent each variable. Various techniques can be used to solve linear problems such as simplex method [20] and interior point method [21]. Recent textbooks on linear optimization theory include [22, 23, 24].  Convex Optimization The optimization problem (1.1) is called a convex optimization problem if X  (1.3) holds, and for any choice of vectors x, y E X and any scalar 0  f(Ox + (1  —  O)y)  <Of(x) + (1  6  —  8)f(y),  Vi  =  =  RN, condition  1, we have:  0,  ...  ,  N.  (1.4)  Chapter 1. Introduction The above inequality implies that the objective function and all the inequality constraint functions are indeed convex functions [16, pp. 67]. Notice that any linear function is also a convex function. Thus, all linear optimization problems are also convex optimization problems. If problem (1.1) is a maximization problem, then it is a convex problem as long as (1.4) holds for —fo; rather than fo. That is, if the objective function fo is concave. Some example convex and concave functions are listed in [16, Chapter 3]. In a minimization problem as in (1.1), a vector x E X is called a local minimum if it is no worse (i.e., greater) than its neighbors feasible points. The vector x if it is less than all feasible points. That is, fj(x*) < < 0 for all i  =  fj(x)  for all x  1,... , N, and h(x) = 0, for all i = 1,  .  .  .  ,  e X is also global minimum  e X that satisfy the conditions  N. Recent textbooks on convex  optimization theory include [15, 16, 25].  Theorem 1 In a convex optimization problem, any locally optimal point x  e X is indeed globally  optimal /16, pp. 138].  We define the Lagrangian L : RN x RNi x RATe  —*  R associated with problem (1.1) as:  N  Ajf(x) +  L(x, A, ii) = fo(x) +  vh,(x),  where ), is defined as the Lagrange multiplier associated with the 0 for any i = 0,  .  .  .  ,  = (z, i = 0,  ...  ,  inequality constraint f(r)  N, and v is defined as the Lagrange multiplier associated with the  equality constraint h(x) = 0 for any i = 0, V  th  (1.5)  .  .  .  ,  ith  N. The vectors A = j, i = 0,... , N) and  N) are called the dual variables or Lagrange multiplier vectors associated  with the optimization problem in (1.1). We can also define the dual function g : RNI x R’e  7  ,‘  R  Chapter 1. Introduction as the minimum value of the Lagrangian L(x, A, v) over x. We have:  g(A,v)  =  inf L(x, A,v),  VA  aEX  e  RNi,  R’e.  (L6)  The dual problem associated with problem (1.1) is now defined as:  maximize  g(A, v)  .XERNi,ziERe  subject to  1 7 A  0,  Vi  =  1,  .  ..  ,  N.  Notice that since the dual function g(A, v) is the pointwise inflinum of a family of affine (La grangian) functions of (A, ii), it is a concave function. Since the inequality functions X, any i  =  1,  .  ..  ,  0 for  N are linear, the dual problem in (1.7) is indeed a convex optimization problem  even if the primal problem in (1.1) is not convex. We can show the following key result.  Theorem 2 Let p = fo(x*) denote the global minimum of the primal problem (1.1). Also let d* denote the global maximum of the dual problem (1. 7). We have [16, pp.  (1.8)  The inequality in (1.8) is called weak duality. It holds even if the primal problem in (1.1) is not convex. On the other hand, if problem (1.1) is convex, then we have /16, pp. 226]:  d*  =  8  (1.9)  Chapter 1. Introduction The equality in (1.9) is called strong duality. The difference  * —  d* is always non-negative and is called the duality gap. From Theorem 2,  the duality gap is zero for convex optimization problems. Theorems 1 and 2 show two fundamental properties of the convex optimization problems. Using these properties, various schemes can be used to solve convex optimization problems, including gradient projection methods, interior point method, and primal-dual method [16]. There are also some important optimization problems, called geometric optimization problems, which are not convex; but can be converted to equivalent convex optimization problems using logarithmic change of variables.  Details are given in [26] and [27].  Some other non-convex  optimization problems which can be converted to convex problems are described in [28, 29, 30].  Integer and Mixed-Integer Optimization Linear and convex optimization problems are two important classes of continuous optimization problems over real-valued optimization variables. In this section, we summarize two other im portant classes of optimization problems called integer optimization problems and mixed-integer optimization problems. Problem (1.1) reduces to an integer optimization problem if X  c  ZN  where ZN is the set of all N x 1 integer vectors; i.e., vectors with integer entries. Similarly, a mixed-integer optimization problem is an optimization problem in which some of the opti mization variables are integer-valued, while the rest are real-valued. Integer and mixed-integer optimization problems are linear if conditions (1.2) and (1.3) hold. Integer and mixed-integer optimization problems are usually difficult to solve. In particular, they cannot be solved within polynomial time. In fact, the complexity of integer and mixed-integer optimization problems is NP-hard. That is, finding the optimal solution may require examining all the feasible points 9  Chapter 1. Introduction using exhaustive search. Nevertheless, there are efficient commercial computer codes such as CPLEX [31] and MOSEK [32] that can solve integer and mixed-integer problems using branchand-bound [33] and branch-and-cut methods  [341.  In general, solving linear integer and linear  mixed-integer optimization problems is computationally easier compared to solving non-linear integer/mixed-integer problems.  Binary Linearization  An important class of integer problems is binary optimization problems where the optimization variables can only take the values of 0 and 1. That is,  {0,1},  Vi  =  1,...,N.  (1.10)  The binary optimization problems are especially important as they can be used to model various decision making problems: optimization variable x  =  1 if we should perform an action and x  =  0  if we should not perform the action. In this section, we explain how we can convert several non linear binary and mixed-binary optimization problems to the equivalent linear problems using the exact binary optimization schemes.  Theorem 3 Consider two binary variables x 1 and x . Their product (i.e., the non-linear quadratic 2  term xlx2) can be replaced by a new binary auxiliary variable ir, such that its value corresponds  10  Chapter 1. Introduction 2 as follows: to the values of xi and x  0,  if x =0, 1  X20,  0,  if xi=O,  X21,  (1.11)  =  0,  if xi=l,  1,  =1, X21. 1 if x  X2=0,  The desired correspondence is obtained by requiring that ir e {0, 1} and we have [35]:  1 +x x 2  —  ir <1, (1.12)  —X1  —  + 2’ir <0.  X  We can also show the following for mixed-binary products.  Theorem 4 Consider a binary variable rrnax  and a non-negative real variable  Xb  is an upper bound for the real variable  Xr.  The quadratic term  XbXr  Xr.  Assume that  can be replaced by a  new non-negative real auxiliary variable v, such that its value corresponds to the values of x and x as follows: if  0,  v  , 0 Xb  (1.13)  =  if Xb=l.  x,  The desired correspondence is obtained by simply requiring that [36]:  0  v  x (1.14)  —  rmax  (1  —  xb) <v <rmax  11  Xb.  Chapter 1. Introduction Theorem 5 Now consider two binary variables xj, and y and a non-negative real variable  Assume that  rmax is an  upper bound for the real variable  correspondence between u and variables  xb, Yb, and Xr  o  o-  o rmax(xb  +  Yb  —  2) +  Xr  Xr.  We define  CT  =  x. xbyb. The desired  is obtained by simply requiring that [37J:  rrnaxxb,  (1.15)  rmaxyb,  0  Xr.  rmax(2  —  Xb  —  Yb)  +  Xr.  Theorems 3, 4 and 5, together provide a toolkit of various linearization techniques which can help us to convert any non-linear binary or mixed-binary optimization problem with polyno mial non-linearity to the equivalent linear optimization problems. Further binary linearization techniques can be found in [38, 39, 40, 41, 42, 43].  1.2.2  Network Utility Maximization  Historically, most of the network protocols in the layered architectures have been designed based on a heuristic or ad-hoc basis, making the analysis sometimes difficult. A decade ago, Kelly et al. [44] proposed a new approach of optimization-based modeling and decomposition-based solutions using utility theory [45] to simplify the understanding of the complex interactions in network congestion control. Since then, this approach has been extended substantially in many ways, forming a promising direction towards a mathematical theory of network architectures. In this regard, the overall network is modeled as a network utility maximization (NUM) problem, where each layer corresponds to a decomposed subproblem, and the interfaces among layers are quantified as functions of the optimization variables coordinating the subproblems. An interesting  12  Chapter 1. Introduction survey on recent developments in formulating and solving various NUM problems is given in [19]. NUM problem is usually formulated either at the network layer (cf.  [44, 46, 47, 48, 49,  50, 51, 52] or at the link layer (cf. [53, 54, 55, 56, 57]). The former is mostly used to design congestion control algorithms while the latter is considered to design medium access control (MAC) protocols. Assume a network with .iV as the set of nodes and £ as the set of links. For each link I e  ,  we define c (x) as the capacity of link 1, where x denotes the vector of 1  all resource management variables. For example, x may include the vector of all transmission powers, persistent probabilities, routing settings, etc. Let S denote the set of all end-to-end flows. For each flow s e 5, let r (x) denote the transmission rate of flow s. We also define £ 8 (x) as 8 the set of links along the routing path of flow s. The NUM problem at the network layer can be defined as [44, 58]: maximize  8 ( 3 u ( x)) r  eX  (1.16) subject to  (x) < cj(x), 5 r  VI  e £,  s:lEI  where for each flow s e 5, the utility u is an increasing and concave function of the data rate r and indicates flow s’s degree of satisfaction on its data rate. Notice that since the data rates are functions of the resource management variables x, the utilities u (r(x)) are also functions of x 8 S. Different utility functions can be considered to achieve different design objectives.  for any s  In particular, the utility functions can be a-fair [50]:  (1 3 ( 8 u ( x)) r  —  1 r(x)’, a)  >  1, Vs  log r(x),  where a  if a  =  if a  =  5,  (1.17)  1,  0 is a design parameter, called fairness index. Using (1.17), a wide range of efficient  13  Chapter 1. Introduction and fair resource management objectives can be modeled. In particular, problem (1.16) reduces to throughput maximization with a mean fair allocation with a  =  —  0, to proportional fair allocation with a  2, and to max-mm fairness with a  —*  =  1, to harmonic  cc.  Similarly, we can define the NUM problem at the link layer as follows [53]:  maximize  uj(cj(x)).  (1.18)  let:  Assuming that the a-fair utility functions are being used, similar to problem (1.16), problem (1.18) can be solved to obtain various fair resource management objectives among the link-layer flows. Notice that both problems (1.16) and (1.18) can be written in the general form of the constrained optimization problem in (1.1). For example, considering the NUM problem in the network layer, we have:  fo(x)  =  f’(x)  = Zs:lECs  f Zsu ( 8 x)), r (x) 8 r  —  (1.19) cj(x).  (1.20)  Almost any network resource management problem can be formulated as a NUM problem in form of either (1.16) or (1.18). Various decomposition techniques can then be used to decom pose the formulated NUM problem into several small subproblems to be solved in a distributive fashion at each network node. The decomposition techniques include primal decomposition, dual decomposition,  and  indirect decomposition.  Survey papers on decomposition techniques can be  found in [59] and [60].  14  Chapter 1. Introduction  1.2.3  Distributed Systems and Convergence Analysis  The theory of parallel and distributed computing is also a promising field that can be brought to bear on many important problems in both parallel computers and distributed data communication networks [18]. In particular, it can be helpful in various convergence and robustness analysis studies in wired as well as wireless networking systems. As in Section 1.2.2, let x denote the vector of all resource management variables. Also let {t ,t 1 ,t 2 ,.. 3  .}  denote the set of time instances at  which the resource management variables are being updated. We consider the following general  update formulation: =  g(x(tk)),  (1.21) Vk=1,2,3,...,  where for any i  1,... , N, function gj is called the mapping function associated with resource  management variable x. In general, we are interested in those mappings g  =  (gj, Vi  =  1,... , N)  such that the update equations in (1.21) converge towards the optimal solution of the NUM problems in (1.16) or (1.18) or any other resource management optimization problem in the general form of problem (1.1).  Important Mappings A mapping function g: X  —  X is a contraction mapping if we have [18, pp. 181]:  IIg(x) ()II —  711x yII —  15  Vx,y e X,  (1.22)  Chapter 1. Introduction where  is some norm and scalar  E [0, 1) is called modulus of mapping g. On the other hand,  mapping g is a monotone increasing mapping if we have [18, pp. 191]: x-y  =  g(x)-<g(y),  Vx,yéX,  (1.23)  g(x)g(y),  Vx,yeX.  (1.24)  and is a monotone decreasing mapping if, xy  =  Here -< and succeq are interpreted coordinatewise.  Convergence Theorems For each mapping g: X  —*  X, the vector x’ E X is a fixed point if we have [18, pp. 181]:  g(x*)  =  x.  (1.25)  In general, we are interested to know (a) whether there exists any fixed point (existence), (b) whether the fixed point is unique (uniqueness), and (c) whether the mapping can reach its fixed point starting from any arbitrary initial point (convergence). It is then ideal if we have: x where x is the optimal solution for the optimization problem of interest.  Theorem 6 Assume that g : X  —*  X is a contraction mapping.  Then, the mapping g has a  unique fixed point and for every initial vector x(0) E X, it synchronously converges to its unique fixed point x geometrically fast [18, Proposition 1.1, Chapter 3].  Theorem 6 provides results on synchronous convergence of contraction mapping. We can also show the following results on asynchronous convergence.  Theorem 7 Assume that g : X  —*  X is a contraction mapping with respect to any weighted 16  Chapter 1. Introduction maximum norm:  IIxII, where w  e  =  max--, i  (1.26)  Wi  RN is an N x 1 vector with positive entries. Then, the mapping g has a unique fixed  point and for every initial vector x(O) x geometrically fast /18, pp.  e X,  it asynchronously converges to its unique fixed point  43J.  The asynchronous convergence is of special importance as it implies robustness and less co ordination overhead. Finally, we have the following results on the asynchronous convergence of monotone mappings.  Theorem 8 Assume that g X  —*  X is a monotone mapping (i.e., either increasing or decreas  ing). Set X is assumed to be bounded. Then, for every initial vector x(O)  e  X, the mapping g  will asynchronously converge to one of its fixed points /18, pp. 15-146J.  Similar to Theorem 7, Theorem 8 provides convergence results for asynchronous updates. However, monotone mappings may have multiple fixed points. Depending on the initial point x(O), the mapping may converge to different fixed points.  Combining Theorems 6 and 8, a  monotone mapping which is also a contraction mapping with respect to an arbitrary norm is guaranteed to asynchronously converge to its unique fixed point.  1.3  Summary of Results and Contributions  This thesis covers several resource management problems in wireless access networks with special focus on MC-WMNs. The results are divided into seven chapters. The contributions in each chapter are as follows. 17  Chapter 1. Introduction • Chapter 2 considers the problem of joint optimal design of topology formation, interface assignment, channel allocation, and routing for MC-WMNs as a unified a cross-layer linear optimization problem to maximize the network performance in a centralized and perma nent fashion. Structural results and numerical algorithms are derived. Extensive simula tions show that the proposed joint design results in significantly higher aggregate network throughput and lower transmission delays compared to two recently proposed algorithms in the literature. • Chapter 3 formulates the interface channel assignment problems jointly with MAC in the NUM framework. The formulated problems, with a-fair utility functions, achieve various fair and efficient resource management objectives. An optimal design, based on exact binary linearization techniques, is proposed which leads to a global maximum. A near-optimal design, based on approximate dual decomposition techniques, is also proposed which is practical for distributed deployment. Numerical results show that our proposed designs can lead to MC-WMNs which are more efficient and fair compared to their single-channel counterparts. The performance gain on both efficiency and fairness increase as the number of available NICs per router or the number of available channels increases. • Chapter 4 addresses the problem of assigning partially-overlapped channels in MC-WMNs and provides the required modeling tools accordingly. The proposed modeling is different from the previous channel assignment models in the literature, where only the orthogonal (i.e., non-overlapped) channels are being used. Simulation results show that the network capacity can increase up to 90% when all partially overlapped channels are being used. • Chapter 5 is different from the previous chapters and addresses the problem of designing 18  Chapter 1. Introduction simple, robust, and optimal random access for multi-hop wireless networks. In this chapter, we propose two distributed contention-based MAC algorithms to solve a NUM problem. Our algorithms overcome four important performance bottlenecks in the previous NUM based random access algorithms in the literature.  First, only limited message passing  among nodes is required. The complexity reduction is in the order of 10. Second, fully asynchronous updates of contention probabilities are allowed. Furthermore, our algorithms are robust to arbitrary large message passing delays and loss. Third, we do not utilize any stepsize during updates, thus our algorithms can achieve faster convergence. Finally, our proposed algorithms have provable convergence, optimality, and robustness properties under a wider range of utility functions, even if the NUM problem is non-convex. The analysis techniques we propose in this chapter are general and can be used to tackle other nonconvex optimization problems in communications and networking. Simulation results show the optimality and fast convergence of our algorithms, performance improvements compared with the subgradient-based MAC, and better efficiency-fairness tradeoff compared with the IEEE 802.11 distributed coordination function. Chapter 6 gives a positive answer to the following question: Is it possible to design a NUM based MA C algorithm that can achieve the optimal performance without frequent explicit message passing? Compared with the related algorithms in the literature, our proposed al gorithm in this chapter achieves the optimal performance without frequent explicit message passing among wireless users. This is of critical importance in practice, since any explicit message passing among wireless users will lead to further contentions in the network and reduce the network performance. We prove the convergence of our algorithm under the as  19  Chapter 1. Introduction sumption that the users can estimate the required information through local observation of the shared wireless medium with asymptotically converging estimation errors. This includes the important case where the underlying channel is lossy and thus not every transmission can be correctly decoded. When the channel is perfect, our algorithm converges to the global optimal solution of the NUM problem. Simulation results show the optimality and fast convergence of our algorithm, arid better efficiency-fairness tradeoff compared with the IEEE 802.11 distributed coordination function. Chapter 7 considers the problem of designing a distributed and optimal multi-interface multi-channel random access algorithm based on the results from the previous chapters. In general, most of the recently proposed channel and interface assignment algorithms are based on formulating combinatorial optimization problems and discrete optimization. The key is to assign exactly one channel to each NIC. However, as we show in this chapter, combinatorial channel assignment models may result in computationally complicated algo rithms and inefficient utilization of the available frequency spectrum. In this chapter, we revisit channel assignment problem by formulating a novel continuous multi-interface multichannel random access model. This includes elaborate modeling of the link data rates for various multi-interface multi-channel networking scenarios. We then propose a fast, fully distributed and easy to implement multi-interface multi-channel random access algorithm. Simulation results show that our proposed algorithm significantly outperforms combinato rial channel assignment algorithms in terms of achieved network utility and throughput. • Chapter 8 considers the problem of pricing in multi-hop wireless access networks. In a multi-hop wireless access network, where each node is an independent self-interested corn  20  Chapter 1. Introduction mercial entity, pricing is helpful not only to encourage collaboration but also to utilize the network resources efficiently. In this chapter, we propose a market-based model with two  fold pricing (TFP) for wireless access networks. In our model, the relay-pricing is used to encourage nodes to forward each other’s packets. That is, each node receives a payment as a compensation for the relay service it provides. We also consider interference-pricing to leverage optimal resource allocation. Together, the relay and the interference prices in corporate both cooperative and competitive interactions among the nodes. We prove that TFP guarantees positive profit for each individual wireless node for a wide range of pricing functions. The profit increases as the node forwards more packets. Thus, the cooperative nodes are well rewarded. We then determine the relay and the interference pricing func tions such that the network social welfare and the aggregate network utility are maximized. Simulation results show that, compared to a recently proposed single-fold pricing (SFP) model where only the relay prices are considered, our proposed TFP scheme significantly increases the total network profit and the network throughput. TFP also leads to more fair revenue sharing and profit distribution among the wireless nodes.  1.4  List of Publications  The following publications have been completed based on the work in this thesis. In some cases the conference papers contain materials overlapping with the journal papers. Journal Papers  • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Joint Logical Topology Design, In terface Assignment, Channel Allocation, and Routing for Multi-Channel Wireless Mesh 21  Chapter 1. Introduction Networks,” IEEE Trans. on Wireless Communications, vol. 6, no. 12, PP. 4432-4440, December 2007. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Cross-layer Fair Bandwidth Sharing for Multi-Channel Wireless Mesh Networks,” accepted for publication in IEEE Trans. on Wireless Communications, 2008. • Amir-Hamed Mohsenian-Rad, Jianwei Huang, Mung Chiang, Vincent W.S Wong, “Utility Optimal Random Access: Reduced Complexity, Fast Convergence, and Robust Perfor mance,” accepted (pending to minor revision) for publication in IEEE Trans. on Wireless Communications, 2008. • Amir-Hamed Mohsenian-Rad, Jianwei Huang, Mung Chiang, Vincent W.S Wong, “Utility Optimal Random Access: Optimal Performance without Frequent Explicit Message Pass ing,” accepted (pending to minor revision) for publication in IEEE Trans. on Wireless Communications, 2008. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, Victor C.M. Leung, “Two-Fold Pricing to Guarantee Individual Profits and Maximum Social Welfare in Wireless Access Networks,” submitted to IEEE Trans. on Wireless Communications, 2008. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Distributed Multi-Interface Multi Channel Random Access Using Convex Optimization,” submitted to IEEE Trans.  on  Mobile Computing, 2008. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Congestion-Aware Channel Assign ment for Multi-Channel Wireless Mesh Networks,” submitted to Computer Networks, 2008. 22  Chapter 1. Introduction Conference Papers • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Joint Optimal Channel Assignment and Congestion Control in Multi-Channel Wireless Mesh Networks,” in Proc. of IEEE International Conference on Communications (ICC), Istanbul, Turkey, June 2006. • A. Hamed Mohsenian-Rad, Vincent W.S. Wong, “Logical Topology Design and Interface Assignment for Multi-Channel Wireless Mesh Networks,” in Proc. of IEEE Global Telecom munication Conference (GLOBECOM), San Francisco, CA, November 2006. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Joint Channel Allocation, Interface Assignment, and MAC Design for Multi-Channel Wireless Mesh Networks,” in Proc. of IEEE Conference on Computer Communications (INFO COM), Anchorage, AK, May 2007. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Partially Overlapped Channel As signment for Multi-Channel Wireless Mesh Networks,” in Proc.  of IEEE International  Conference on Communications (ICC), Glasgow, UK, June 2007. • Amir-Hamed Mohsenian-Rad, Jianwei Huang, Mung Chiang, Vincent W.S Wong, “Sim ple, Optimal, and Robust Random Access,” accepted for publication in Proc. of Military Communications Conference (MILCOM), San Diego, CA, November 2008. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, “Optimal Multi-Interface Multi-Channel Random Access,” accepted for publication in Proc. of IEEE Global Telecommunications Conference (GLOBECOM), New Orleans, LA, November 2008. • Amir-Hamed Mohsenian-Rad, Vincent W.S. Wong, Victor C.M. Leung, “Two-Fold Pric ing to Guarantee Individual Profits and Maximum Social Welfare in Wireless Access Net23  Chapter 1. Introduction works,” accepted for publication in Proc. of IEEE Global Telecommunications Conference (GLOBECOM), New Orleans, LA, November 2008.  1.5  Thesis Organization  The remainder of the thesis is organizes as follows. Our proposed static and centralized joint logical topology design, interface assignment channel allocation, and routing algorithm to achieve max-mm resource allocation in MC-WMNs is described in Chapter 2. We then extend our design to dynamic and distributed channel allocation in Chapter 3, where we consider some design objectives such as proportional fair and harmonic fair resource allocations. The formulations for assigning partially overlapped frequency channels are given in Chapter 4. Our low-complexity, robust, fast, and optimal distributed random access algorithm is described in Chapter 5. We then extend our model and propose a random access algorithm which does not require any explicit message passing among the wireless nodes in Chapter 6. In Chapter 7, we combine the results from Chapters 2 to 6 and propose a distributed multi-interface multi-channel random access using convex optimization techniques. Our proposed optimal two-fold pricing scheme for multihop wireless access networks in described in Chapter 8, where we also analytically prove its key properties. Finally, Chapter 9 contains discussions of main results, conclusions and proposals of future research directions.  Each of the main chapters in this thesis is self-contained and  included in separate journal articles and conference papers. A review of the related work for each optimization context is given for each chapter accordingly. The notations are separately defined for each chapter, but have been consistent throughout the thesis.  24  Chapter 2  Joint Logical Topology Design, Interface Assignment, Channel Allocation, and Routing for MC-WMNs The aggregate capacity of WMNs can be increased by the use of multiple frequency channels as explained in Section 1.1.1. Within the IEEE 802.11 a/b/g frequency bands, the number of available channels is limited. The 802.llb/g bands and the 802.lla band provide 3 and 12 nonoverlapping frequency channels, respectively. This implies that some logical links may operate on the same channel. In addition, the number of NICs is also limited. In the experimental MC WMN test-beds in [11] and [61], each router is equipped with two NICs. A small number of NICs implies that some logical links in a router may need to share an NIC to transmit and receive data packets. Two nearby links that operate on the same channel or share the same NIC cannot be active simultaneously. In general, given the physical topology (i.e., the physical locations of the wireless mesh routers), four key issues should be addressed in MC-WMNs:  25  Chapter 2. Topology Design, Interface Assignment, Channel Allocation,  Routing  1. Logical Topology Formation: Given the physical topology, how many logical links (if any) should be assigned between a pair of neighboring routers? 2. Interface Assignment: Given the logical topology, how should the logical links be assigned to each NIC in a wireless mesh router? 3. Channel Allocation: Given the logical topology and interface assignment, how should a frequency channel be allocated on each logical link? 4. Routing: Given the logical topology, interface assignment, and channel allocation, through which logical links should the packets be forwarded?  In this chapter, we mathematically formulate the logical topology formation, interface assign ment, channel allocation, and routing as a joint mixed-integer linear optimization problem. Our goal is to better understand the channel and interface assignment problems and evaluate the performance of MC-WMNs at optimal operation. We call our proposed MC-WMN architecture TiMesh. Our contributions are as follows:  • Our model formulation takes into account several design parameters such as the number of available NICs in each wireless mesh router, the number of available frequency channels, the communication range and the interference range of routers, and the expected traffic load between different source and destination pairs. • Unlike most of the previously proposed channel and interface assignment strategies, our model formulation allows having multiple logical links between the same pair of routers. This further increases the effective data transmission rate between the two routers and more efficiently utilizes the available frequency channels. 26  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing • Our proposed algorithm guarantees the network connectivity. It also supports both internal traffic among the wireless routers and external traffic to the Internet. • Simulation results show that TiMesh achieves higher throughput and lower end-to-end delay than the recently proposed Hyacinth [61] and CLICA [62] MC-WMN architectures.  The rest of this section is organized as follows. We present related work in Section 2.1. Our proposed joint design is described in Section 2.2. Performance evaluation and comparison are given in Section 2.3. A summary of the chapter is given in Section 2.4.  2.1  Related Work  Several logical topology formation, interface assignment, channel allocation, and routing algo rithms have been recently proposed for MC-WMNs.  Raniwala et al. [61] proposed a logical  tree topology architecture for MC-WMNs, called Hyacinth. The tree construction mechanism is similar to the IEEE 802.1D [63] spanning tree formation. The gateways are the roots. Each router uses an up-NIC to exclusively connect to its parent and uses several (probably shared) down-NICs to connect to its children. Each parent router provides the Internet connectivity to its children routers. As a result, each wireless mesh router can access the Internet through the shortest available routing path. In the Hyacinth architecture, each router allocates the channels that are less used by its neighboring routers to its down-NICs. Marina et al. [62] also proposed the Connected Low Interference Channel Assignment (CLICA) algorithm for topology formation in MC-WMNs. The interference among logical links is modeled by a weighted conflict graph. The weight of the edge between two vertices in the weighted conflict graph indicates the extent of interference between their corresponding logical links using the protocol interference model 27  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing [64]. The proposed heuristic algorithm determines the logical links and assigns their channels permanently so that the average interference weight in the resultant conflict graph is minimized while the network connectivity is maintained. In [65], the channels are allocated so as to minimize the maximum number of interfering links within each neighborhood, subject to the connectivity constraint of the logical topology graph. A bandwidth-aware routing algorithm is also proposed to facilitate the path finding operation. Various centralized and distributed channel allocation algorithms have also recently been proposed for MC-WMNs. The centralized schemes (e.g., [66, 67]) require a network controller to collect the topology information and assign the channels. In the distributed schemes (e.g., [68, 69]), some of the routers are responsible for channel assignment for a subset of interfaces. The channel allocation algorithms can also be classified as static and dynamic. The static algorithms (e.g., [70, 71, 72, 72, 73]) assign a frequency channel to each NIC permanently, while dynamic algorithms allow each NIC to change its channel either in a short-term (e.g., packet-by-packet [69, 74, 75]) or a long-term basis (e.g., every several minutes [66, 67, 76, 77]). Unlike the static algorithms, the dynamic channel allocation requires a coordination mechanism to ensure that the sending and the receiving routers/NICs use the same channel at the same time. Various joint designs for MC-WMNs have also been proposed. Some recent work include joint channel assignment and routing [67, 71, 78], joint routing and interface assignment [79], and joint channel assignment and congestion control [77].  28  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  Q  Mesh Router — interface Card Wireless Link Mobile Device  —  1 5  ‘  Operaling Chann  Figure 2.1: An MC-WMN with six wireless mesh routers, five frequency channels, and three NICs per router. The number on each link indicates the operating channel number.  2.2  Joint Logical Topology Design, Interface Assignment, Channel Allocation, and Routing  In this section, we formulate the logical topology formation, interface assignment, channel allo cation, and routing as a joint linear optimization problem. For the rest of this chapter, the terms routers and nodes are used interchangeably.  2.2.1  Problem Formulation  We first model an MC-WMN by a physical topology graph G(Jf, é) where N denotes the set of all vertices and é’ denotes the set of all unidirectional edges. Each vertex n  e  stationary wireless mesh router. For simplicity, we assume that N  ,  29  =  Al represents a  {1, 2,... N}. Notice that  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing N  =  I.A/. Similarly, we define E  For any two nodes m, n  =  .  For the network in Fig. 2.1, we have: N  emn  enrn  E  6 and E  =  16.  e /, if node n is within the communication range of node m, then there  is an edge or link from node m to node n in set by  =  .  The link from node m to node n is denoted  e é if and only if  E. We assume the connectivity to be symmetric. That is, link  e S. Each mesh router is equipped with I network interface cards. There are C orthogonal  frequency channels available. For any two nodes m and n such that emn E 5, and any channel i  In the logical topology, if node m communicates with  a link channel allocation variable  node n over the ith frequency channel, then the MC-WMN in Fig. 2.1 with C  e {1,... , C}, we define  =  is equal to 1; otherwise, it is equal to zero. For  5, we have xb  =  1 and xb  =  =  Xab  =  Xab  =  0. In  general, two nodes may communicate with each other over multiple distinct frequency channels. For example, consider the MC-WMN in Fig. 2.1, we have  Xad  Xad  =  1 and  Zad  =  =  Xad  =  0.  As a result, the neighboring nodes a and d can communicate with each other over both channels 3 and 4 at the same time. To establish the logical links, nodes m and n should assign the same frequency channels to communicate with each other. This requires that,  =  V m,n  E  Ar, emn eS, V i  =  (2.1)  1,..., C.  The link channel allocation variables implicitly provide the required information to create the logical topology. Due to traffic and interference constraints, it is possible that there is a link between nodes m and n in the physical topology graph (i.e., ernm  e 5), but there is no logical link  between them in the logical topology. In that case, we have x  30  =  0 for all i  =  1,... , C. Note  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing that we allow multiple logical links between the same pair of nodes in the logical topology. They operate independently over distinct channels and can significantly increase the effective capacity between two neighboring nodes. For any node m e J’f and any channel i  1,  ifn  e  {1,... , C}, we define y to be as follows:  ej’f and  emn E  ‘,  such that x  =  1 (2.2)  0,  otherwise.  We refer to y as the node channel allocation variable corresponding to node m and channel i. For node a in Fig. 2.1 with C (2.2),  =  5 and I  =  3, we have  =  y  =  y  =  1 and y  =  y  =  0. From  y indicates the total number of channels that are being used by node m to establish  logical links with its neighboring nodes. Since each NIC operates on a distinct frequency channel, y cannot be larger than the total number of available NICs on node m. That is,  (2.3)  VmE.  y<I,  The link and node channel allocation variables implicitly provide the required information  for interface assignment. For example, given y  =  y  =  y  =  1, we assign channel 1 to the first  NIC, channel 3 to the second NIC, and channel 4 to the third NIC of node a. Since xb Xd  =  =  Xd  =  1, node a uses its first NIC to communicate with node b and its second and third NICs to  communicate with node d.  Lemma 1 The desired correspondence in (2.2) is obtained by having y be a continuous real  31  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing variable for all nodes m  e Jf and all channels i  0 <  E  {1,.  .  ,  C} and also requiring that:  (2.4a)  < flEJ\I,emnb  and x <y < 1,  2.2.2  Vn  e .Af,  emn E  (2.4b)  .  Effective Capacity  Let c 0 denote the nominal link-layer data rate in the corresponding 802.11 standard (e.g., 54 0 denote the effective capacity of the logical link (m, n) c  Mbps in 802.lla). Also let 0 <  in the direction from node m to node n over frequency channel i. We have:  c<xc°,  Vm,nEjV, emnEE, Vi=1,...,C.  (2.5)  From (2.5), if node m does not allocate frequency channel i to communicate with node n (i.e., channel allocation variable x = 0), then node m cannot transmit any packet to node n over channel i (i.e.,  =  0).  For any two nodes m and n such that emn .Fmn  C E.  Fm,,,  includes all epq  e  e  ,  we define a set of potential interfering links  such that nodes p or q (or both) are within the interference  range of nodes m or n (or both). Note that we always have enm E  32  Fmn.  Considering the IEEE  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing 802.11 based RTS-CTS-DATA-ACK model, we have [67, 76]:  ff +  < 1,  Vm, n E N, emn E E,  Vj  =  1,... C  (2.6)  ,  p,q, epqEFmn  where c/c° denotes the fraction of time that logical link (m, n) can be active in the direction from node m to node n over frequency channel i.  2.2.3  Total Flows on a Logical Link  For efficient network planning, a statistical model for network traffic needs to be available. Let sd denote the expected traffic rate to be delivered between source and destination pair (s, d), 7  where s,d E N. We assume that the information  for all source and destination pairs is given.  For any source and destination pair (s, d), any nodes m, n channel i  e {1,  .  .  .  ,  e N such that emn  é  ,  71 The variable a . 1 C}, we define a binary routing variable ,  and any is equal  to 1 if the traffic from source s to destination d is being routed via link (m, n) in the direction from node m to node n over channel i, and is equal to 0 otherwise. Note that  $  in  general. Multiple links between a pair of nodes can provide more than one path between them. Since each of the multiple links is operating over a distinct channel, packets that are forwarded on different links experience different latencies. Thus, if packets that belong to the same flow use parallel links between a pair of neighboring nodes, this can cause packets to arrive out of order. To avoid this issue, only one of the available logical links between each pair of neighboring nodes is used to route packets of each flow. That is,  Vs,d,m,neN, emne.  33  (2.7)  Chapter 2. Let  Topology Design, Interface Assignment, Channel Allocation, & Routing  denote the aggregate traffic from  all source and destination pairs that is routed on  link (m, n) in the direction from node m to node n over channel i. We have:  V m, m  =  e Af, emn E  ,  j  =  1,... , C.  (2.8)  s,de.Af  The aggregate traffic such that emn E  ‘,  cannot be more than the effective capacity  and all channels i  for all nodes m, n E Ai’  e {1,. . . C}. Consider the following constraint: ,  Vm,nef, emnE, Vi=1,...,C.  where A  <  (2.9)  1 is a positive parameter. From (2.9), the parameter A imposes an upper bound on the The higher the link utilization, the higher the queueing delay  expected link utilization  [80]. In the current Internet, an access link is considered overloaded when its average utilization is greater than 80% [81, 82]. Thus, we set A  2.2.4  =  0.8.  Flow Conservation at Each Node  The flow conservation requires that for s, d, m e f,  =  —  nJ’f,  nEJV,  emnE&  enmE  sd, 7  if s  =  m,  sd, 7 _  if d  =  m,  0,  otherwise.  (2.10)  In (2.10), the term on the left-hand side is the net flow out of node m for the flow from source s to destination d. The net flow is the difference between the outgoing flow and the incoming flow.  34  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing The term on the right-hand side is equal to 0 if node m is neither the source nor the destination for that specific flow. If node m is the source (i.e., s node m is the destination (i.e., d  =  =  m), then the net flow is equal to  m), then the net flow is equal to  in (2.10) can be divided by the common factor  sd• 7  sd 7  j  Note that both sides  We also notice that (2.10) implies (2.7).  The constraint in (2.10) also guarantees that there is at least one routing path available between each source and destination pair (s, d). In practice, all nodes have traffic to or from the Internet; thus we can make the valid assumption that  sd 7  >  0, if either s or d is a gateway  node. The constraint in (2.10) and the aforementioned assumption guarantee that the obtained topology is connected. That is, there is neither an isolated node nor an isolated group of nodes.  2.2.5  Feasible Region and the Objective Function  Given the expected traffic demand y and the network resources C, I, and c°, the constraints in (2.1)-(2.10) form the feasible region for all logical topologies that can properly support the expected traffic demand y. The feasible region could be empty. We can enlarge the feasible region by choosing a higher value for parameter A; however, even for A  =  1, the feasible region  can be empty if and only if the network resources cannot support the expected traffic demand. In that case, we need to either increase the available resources or limit the traffic demand. From constraint (2.9), the difference (Ac  the difference (Ac prone to congestion. Let  —  ömjfl  —  is always non-negative. As  approaches  )) tends to 0 and the corresponding logical link becomes more denote the minimum difference (Ac  35  —  A) across all channels  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing and all links that exist in the logical topology. That is,  Smin  mm  =  (Ac  —  (2.11)  .  rn,rzE.A1, emn EE,  iC{1,..,C}, x,—1  Note that  Smin  corresponds to the most congested (i.e., the bottleneck) logical link across the  network. Our objective is to maximize the variable  Smln.  It can be achieved by decreasing the  aggregate traffic load or increasing the effective capacity (or both) on the network’s bottleneck link. The former implies load balancing: balancing the traffic load among different logical links using proper logical topology formation and routing schemes; while the latter implies congestionaware capacity planning: providing higher effective capacity for more congested logical links using proper logical topology formation, interface assignment, and channel allocation schemes. Load balancing is shown to be a proper objective for joint topology control and routing algorithms in optical networks [83]. Congestion aware capacity planning is also proposed for cross-layer congestion control designs in wireless ad-hoc [49] and mesh networks Maximizing  [771.  can also be justified in terms of providing fairness among the existing logical  links. In fact, it leads to achieving max-mm  or bottleneck-optimal fairness [50, 80, 84]. This is  equivalent to solving the NUM problem in (1.18) for cr-fair utility functions with o  —*  cc. Note  that the system is fair in the sense that all the links experience similar level of congestion.  Lemma 2 The desired correspondence in (2.11) can be obtained by requiring that:  m’in < 8  + Ac°(1  —  Vm, n E f,  36  emn E E,  Vi  =  1,..  .  ,  C.  (2.12)  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  2.2.6  Hop Count Constraint  Load balancing avoids highly loaded links and prevents congestion; however, it may lead to assigning long routing paths. For each source and destination pair (s, d), the hop count along the assigned routing path is obtained as  Zm,nEN, emnee  Z  Let h denote the hop count  for the minimum hop path between source and destination pair (s, d) in the physical topology graph G(N, S). The ratio  is always greater than or equal to 1,  Zi  and is defined as the stretch factor for the routing path from the source node s to the destination node d. We define the hop count constraint to be:  V s, d e  F h,  (2.13)  .  m,nEJ\1, emnE i=1  where F  1 is a tunable parameter to set an upper bound on the routing stretch factor. Note  that there is always a trade off between load balancing and shortest path routing [85]. This trade off can be controlled by using the tunable parameter F. By assigning F the algorithm becomes the shortest path routing. If F  =  =  1, the routing part of  cc, the hop-count constraint (2.13) is  relaxed. In general, the greater the tunable parameter F, the larger the feasible region.  37  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  2.2.7  Optimization Problem  Given the physical topology graph G(JV, maximize  ‘)  and all parameters C, I, F, A,  , 0 C  .Fmm,  sd, h, 7  ömir,  x,y,c,a,A,ömjn  subject to x, Xmn  Ym  =  —  Ym’  /-néJ.f,emnEE’ m 1 n’  —  -C L.,j=1 Ym  —  i 0 Cmn <i XmnC —  Cmn  +  Zp,q, epqEFmn 4 q  ZnEJf,emmEE  —  mn —  Smin  C°,  Z  1 an,jflEW,enmE  sd s,deJr”mn,i  7 sd<A —  sd  — —  ifs=m,  —1,  if d  0,  otherwise,  =  (2.14)  m,  —  p ‘  a.j e {0, 1}, y, c, A, 5 mjfl ,Vm,n,s,dEJ\f, 0 c<c  1,  Crnn,  < (Ac — A) + Ac°(1  ‘çC L.im,nJV, emnE L.j1  where  <  emnee,  0, y < 1,  Vi=1,...,C.  Let W denote the number of source and destination pairs. The linear mixed-integer problem (2.14) has EC(1+W) integer variables and C(2E+N)+1 real variables. It also has 1.5EC+NW equality and F(5C + W) + N(C + 1) + W inequality constraints.  38  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, Algorithm 1 Iterated Local Search Algorithm to Solve Problem (2.14) 1: set K = the maximum number of iterations. 2: set x[1]=1, Vm,neN, emnee. 3: set x[1j = =x[1] =0, Vm,n eAr, emn C 4: for k = 1 to K do 5: Randomly choose p, q e Ar such that emn C 6: solve problem (2.14) subject to 1, V s,d,m,n eJ\f, emn C , ViE {1, 0 ,C} x = x[k], Vm,n e Ar, em, m,’n eE, . 7 {p,q}. 7: setx[k+1]=4 , Vm,nejV, emméVie{1,... ,C}. 7. 8: end 9: for each source and destination pair (s, d) do 10: setm=s. 11: while m d do 12: set = 1 where {i,n} = arg max 13: set m = n. 14: end 15: for all m, n e Ar and all channels i e {1,... , C} do 16: if 1, then set = 0. 17: end 18: end .  2.2.8  .  -  Routing  Centralized  .  Algorithm  There are efficient commercial software (e.g., CPLEX [31]) to solve linear mixed-integer programs. Most of them use the branch-and-cut algorithm [86]. Problem (2.14) can easily be solved for small-scale MC-WMNs. However, finding the optimal solutions are not trivial for large-scale networks. An alternative is to use some simple arid efficient metaheuristic methods to find the sub-optimal solutions [87]. In this chapter, we use the Iterated Local Search (ILS) [88] which is a powerful metaheuristic algorithm. We will investigate the sub-optimality of the ILS algorithm in comparison with the optimal branch-and-cut algorithm in Section 2.4. The pseudo-code for the proposed ILS algorithm is provided in Algorithm 1. In line 2, a fully connected single-channel logical topology is selected as the starting point. At each iteration, lines 4 and 5 are used to randomly select a pair of nodes p, q C Ar (e.g., with probability that is  39  Chapter 2. Topology Design, Interface Assignment, Channel Allocation,  Routing  proportional to the worst congestion status among the current logical links between them). From the two additional constraints in line 6, the integer constraint on the routing variable a is relaxed. Most of the variables are also set as constants in the current iteration. Therefore, the modified problem only has a few integer variables and can be solved easily. Given the sub-optimal topology formation, interface assignment and channel allocation solutions, the routing path from source s to destination d is assigned by traversing the logical topology from source s to destination d, and by choosing the next hop based on the maximum observed value for routing variable a (lines 10-18). The intuitive justification is that if the relaxed  is close to 1, it indicates that it  is better to forward the packets from source s to destination d on the logical link (m, n) over channel i. On the other hand, if the relaxed  is close to 0, it implies that it is better to avoid  forwarding packets on logical link (m, n) over channel i.  2.3  Simulation Results  In this section, we evaluate the performance of our proposed TiMesh MC-WMN architecture and compare it with the Hyacinth [61] and CLICA [62] architectures using ns-2 simulations. We consider both UDP (User Datagram Protocol) and TCP (Transmission Control Protocol) traffic. The default simulation model is as follows. The size of the network field is 1000 m x 800 m. Ten sample MC-WMNs are generated. Each MC-WMN consists of 30 routers. Four of them serve as gateways. The gateways are located at the four corners of the field. The communication and the interference ranges are 250 m and 450 m, respectively. Each router is equipped with three NICs. Six channels are available. The parameter F is set to two. The IEEE 802.lla standard with 54 Mbps data rate is being used. In each topology, there are 30 flows: 15 flows are internal, and 15  40  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  1  80  60  40  20  0  10  20  30  40  50 60 Iteration  70  80  90  100  Figure 2.2: Average optimization error versus the iteration number using the ILS algorithm  when the number of nodes/flows vary from 10 to 30. flows are external. For each internal flow, two non-gateway nodes are randomly selected to be the source and destination nodes. Each external flow is established between a randomly selected node and a gateway. For UDP traffic, the packet size is 1000 bytes and the transmission rate is 500 kbps. For TCP traffic, the packet size is 1020 bytes and the transmission rate is set by the TCP Vegas. The simulation time (i.e., the duration of the simulation) is 300 sec. For UDP traffic, the performance metrics are: 1) packet delivery ratio: the total number of packets received by all destinations divided by the total number of packets transmitted by all sources; 2) end-to-end delay: the time takes for a packet to traverse the network from a source to a destination. For TCP traffic, the metrics are: 1) aggregate throughput: the total number of correctly received packets (in bits) divided by the simulation time; 2) round-trip time: the time delay between sending a TCP segment and receiving its acknowledgement.  41  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  2.3.1  Optimal and Sub-optimal Solutions  In this section, we compare the solutions for problem (2.14) obtained from the ILS algorithm with those obtained from the optimal branch-and-cut solver. Let S denote the optimal value. Also let  Smin  —  [k] denote the value obtained from the ILS algorithm after the kth iteration. We define Smin[k])/Snjn as the optimization error. Fig. 2.2 shows the average optimization error  across all ten topologies versus the iteration number when the number of nodes and flows vary from 10 to 30. We can see that the optimization error decreases when the number of iterations in ILS algorithm increases. After 50 iterations, the error is 1.4%, 1.9%, and 4% for 10, 20, and 30 iiodes/fiows, respectively. These results show that a near optimal solution can be achieved within a limited number of iterations. For the results presented in the subsequent sections, the near optimal solutions of problem (2.14) are obtained using ILS with 50 iterations.  2.3.2  Sample Logical Topology  Fig. 2.3(a) shows a sample physical topology.  The corresponding logical topology, interface  assignment, and channel allocation are shown in Fig. 2.3(b). For the physical topology graph in Fig. 2.3(a), 72 pairs of neighboring nodes are within the communication range of each other. The logical topology in Fig. 2.3(b) includes at least one logical link between 43 pairs of neighboring nodes. There are two links between nodes C 1 and a, G 3 and i, G 4 and w, as well as s and n. These links operate over distinct channels. In Fig. 2.3(b), there are 12 logical links that share an NIC with some others. The sharing of the logical links do not happen often as it reduces the corresponding link capacities. In the obtained routing solution, the 2-hop route {o, t, G } is 2 replaced by the 3-hop route {o, t, x, G } to take the advantage of the unused logical link (t, x). 2  42  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  (a)  (b)  Figure 2.3: A random topology with 30 routers. Each router is equipped with 3 NICs. (a) Physical topology, (b) Logical topology, interface assignment, and channel alloca tion. Solid lines are wireless links that use only exclusive (not shared) NICs. Dashed lines are the links that share an interface with some other links. The number on each link indicates the channel number.  43  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  2.3.3  Impact of the Network Traffic  In this section, we first investigate the performance with UDP traffic. The number of active flows varies from 6 to 30. Fig. 2.4 shows the results of the packet delivery ratio and the average end-toend delay. In this figure, each point is the average of measurements for all 10 simulated topologies. When the number of UDP flows increases, the network becomes congested. Since UDP does not have any congestion control mechanism, there is a reduction of the packet delivery ratio and an increase of the end-to-end delay. When there are 30 UDP flows, Fig. 2.4(a) shows that the packet delivery ratio obtained from TiMesh is 7% and 18% higher than CLICA and Hyacinth, respectively. Fig. 2.4(b) shows that the average end-to-end delay obtained from TiMesh is 28% and 52% lower than CLICA and Hyacinth, respectively. Fig. 2.5 shows the results of the aggregated throughput and the average round-trip time when there are different number of TCP flows established in the network. In this figure, each point is the average of measurements for all 10 simulated topologies. If only a few flows are established (e.g., less than 6 flows), the TiMesh, CLICA, and Hyacinth architectures achieve almost the same performance. By increasing the number of flows, the network becomes congested and the round-trip time increases significantly. When there are 30 flows, Fig. 2.5(a) shows that the aggregated throughput obtained from TiMesh is 11% and 32% higher than CLICA and Hyacinth, respectively. Fig. 2.5(b) shows that the average round-trip time obtained from TiMesh is 29% and 52% lower than CLICA and Hyacinth, respectively. The better performance of TiMesh can be explained based on the features of the three ar chitectures. Unlike Hyacinth that concentrates the traffic on long routing paths with highly loaded links (especially the links connected to the gateways), TiMesh distributes and balances  44  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  (a)  0  ci) > ci)  U ci) C.)  0  6  12  18 Number of UDP Flows  24  30  (b) C.)  a) Cl, E >..  a)  U  w -D  w 18 Number of UDP Flows  Figure 2.4: Comparison between TiMesh, Hyacinth [61], and CLICA [62] MC-WMN archi tectures in the presence of fixed-rate UDP traffic, (a) Packet delivery ratio, (b) End-to-end delay.  45  Chapter 2. Topology Design, Interface Assignment, Channel Allocation,  Routing  (a) (I)  2 I— I  0 .3)  z  Number of TCP Flows (b)  C)  1) U)  E ci)  E I— 0 S.  0  18 Number of TCP Flows  Figure 2.5: Comparison between TiMesh, Hyacinth [61], and CLICA [62] MC-WMN archi  tectures in the presence of TCP traffic, (a) Aggregated throughput, (b) Average round-trip time.  46  Chapter 2. Topology Design, Interface Assignment, Channel Allocation,  Routing  the traffic among different links. It also assigns shorter routing paths. TiMesh has two distinct advantages when it is compared to CLICA. The logical topology created by TiMesh depends on the expected traffic demand. TiMesh also allows having multiple links between the routers. This further increases the effective data transmission rate between two neighboring routers.  2.3.4  Impact of the Number of NICs and Channels  In this section, we compare the performance by varying the number of interfaces and the available channels. Thirty TCP traffic flows are generated in the network.  Results for the aggregate  throughput and average round-trip time are shown in Table 2.1. We can see that both TiMesh and CLICA improve the performance significantly when the number of channels is increased from 6 to 9. The performance gain is less for Hyacinth as it has fewer logical links in its topology and cannot efficiently use the available resources. When the number of NICs increases from 3 to 4, the aggregate network throughput increases by 36%, 29%, and 37% for TiMesh, CLICA, and Hyacinth, respectively. The average round-trip time also decreases by 92%, 17%, and 48%, respectively. The observed high performance gain for TiMesh is due to the fact that it uses the extra available NICs to assign multiple links between the routers.  2.3.5  Fairness  Recall from Section 2.2.5 that TiMesh can achieve max-mm  fairness among logical links. To  quantitatively measure the fairness that is attained among different flows, we let  ’PDR 1  and  PEED  denote Jam’s fairness indices [89] for packet delivery ratio and end-to-end delay, respectively: \2  /  8do 7 (Zs,deji, PDR VV  PDR(s, d))  rni I 2’ o r i1J:-LjS, U) 7 2_ds,deJ%/,  47  (  .  )  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing Table 2.1: Impact of varying the number of NICs and channels. Throughput (Mbps) Round-Trip Time (msec) Architecture I = 3, I = 4, I = 4, I = 3, I = 4, I = 4, C=6 C=9 C=12 C=6 C=9 C=12 TiMesh 57.9 92.0 25.5 68.3 21.8 11.6 CLICA 51.7 61.5 79.1 36.8 32.7 24.4 Hyacinth 43.1 48.2 66.4 52.7 49.9 41.0  Table 2.2: Achieved fairness among different flows. Fairness Index Architecture Packet Delivery Ratio End-to-End Delay  TiMesh CLICA Hyacinth  (‘PpDa)  (‘PEED)  0.914 0.867 0.712  0.903 0.836 0.678  where PDR(s, d) denotes the packet delivery ratio for the flow from source s to destination d. The fairness index  PEED  can be expressed similarly. The measured  ‘PPDR  and  ‘PEED  for TiMesh,  CLICA, and Hyacinth architectures are shown in Table 2.2. We can see that TiMesh offers better fairness among the flows. The lower fairness indices in CLICA and Hyacinth are the result of several highly congested bottlenecks. Those flows that traverse the bottleneck links experience higher delays and more packet loss compared to the rest of the flows. This is also the case for Hyacinth where the links connected to the gateways are the bottlenecks. The proposed topology control, routing, interface assignment, and channel allocation algorithms in TiMesh manage to avoid different links experiencing different congestion levels. Thus, different flows that are traversing different links achieve similar performance.  48  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing  2.4  Summary  In this chapter, we proposed the TiMesh MC-WMN architecture by formulating the logical topol ogy design, interface assignment, channel allocation, and routing as a joint linear mixed-integer optimization problem. Our model formulation takes into account the number of available NICs in routers, the number of available orthogonal frequency channels, expected traffic load between different source and destination pairs, and the effective capacity of the logical links. The proposed scheme balances the load among logical links and provides higher effective capacity for the bot tleneck link(s). We conducted extensive ns-2 simulation experiments to evaluate our algorithm and compared it with Hyacinth and CLICA MC-WMN architectures. Simulation results show that our proposed TiMesh architecture provides a higher aggregated network throughput and a lower end-to-end delay for both TCP and UDP traffic. The available NICs and channels are better utilized. The TiMesh also offers better fairness among different flows. In this chapter, we mainly focused on centralized and static channel and interface assignment problem by formulating a linear mixed-integer problem in (2.14). In the next chapter, we will propose a dynamic and distributed channel assignment strategy. We will also study the possibility of formulating the channel and interface assignment problem as a convex optimization problem in Chapter 7.  2.5 2.5.1  Analytical Proofs Proof of Lemma 1  Assume that node m is assigned to communicate with K neighboring nodes over channel i. Thus, constraint (2.4a) can be written as 0 < y  K. If K 49  =  0, then constraints (2.4a) and (2.4b)  Chapter 2. Topology Design, Interface Assignment, Channel Allocation, & Routing become 0 < y <0 and 0 if K> 0 (i.e., if K  1, respectively. This implies that y  y  0. On the other hand,  1), then constraints (2.4a) and (2.4b) become 0  respectively. This implies that y  2.5.2  =  =  K and 1  1.  Proof of Lemma 2  If there exists a logical link between nodes m and n over frequency channel i (i.e., A c°(1  1,  y  =  —  0 and constraint (2.12) simply becomes  min 8  <  (A  —  XZflrn  =  1), then  A). On the other  hand, if there is no logical link between nodes m and n over frequency channel i (i.e., x then Ac°(1  —  x)  =  Ac°. From eqs. (2.5) and (2.9) we also have (Ac  constraint (2.12) becomes  min 5  —  . Note that Ac° is an upper bound for variable 0 <Ac  50  =  =  0),  0. Thus,  Smin.  •  Chapter 3  Cross-layer Fair Bandwidth Sharing for MC-WMNs Most of the previously proposed channel assignment algorithms for MC-WMNs mainly focus on network efficiency (i.e., increasing the network throughput) while the issue of fairness remains less studied. Although several test-bed and simulation studies have shown that various channel and interface assignment algorithms can provide a higher throughput in MC-WMNs compared to their single-channel counterparts [61, 62, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 90, 91, 92, 93], it is not clear whether the same statement is true for the case of fairness. Note that, an efficient but unfair channel allocation may cause some flows to starve. In this chapter, we formulate a cross-layer bandwidth sharing problem in MC-WMNs as a NUM problem [19, 44] (see also Section 1.2.2). We then use the a-fair utility functions [50] to model a wide range of well-known fairness allocations. The contributions of our work are as follows.  • We mathematically model the channel and interface assignment problems by introducing link and node channel assignment binary vectors. Using these vectors, we also model the feasible region for the link-layer flow rates. • We present a formulation for cross-layer fair bandwidth sharing problem as a non-linear  51  Chapter 3. Cross-layer Fair Bandwidth Sharing mixed-integer NUM. It takes into account the number of NICs at each router, the number of channels, and the interference constraints. • We solve the NUM problem via both exact and approximate design schemes. The exact design results in an optimal static algorithm while the approximate design results in a near-optimal long-term basis dynamic and distributed algorithm. • Our proposed designs take into account both network efficiency and fairness. In particu lar, some of the well-known fairness criteria, such as proportional fairness, harmonic-mean fairness, and max-mm fairness, can be modeled using a tunable design parameter. The rest of this chapter is organized as follows. The problem formulation is described in Section 3.1. The first design scheme (using exact binary linearization) is presented in Section 3.2. The second design scheme (using approximate dual decomposition) is described in Section 3.3. The performance of our algorithms is assessed through numerical examples in Section 3.4. A summary of the chapter is given in Section 3.5.  3.1  Problem Formulation  In this section, we describe the mathematical model to formulate a cross-layer fair bandwidth sharing problem in MC-WMNs. The terms wireless mesh routers and nodes will be used inter changeably. Consider an MC-WMN and let N Each node m  e  =  {1, 2,... , N} denote the set of stationary nodes.  N is equipped with Im NICs. Different nodes carl be equipped with different  number of NICs. There are C orthogonal frequency channels available. We assume that the logical topology of the network has been pre-determined. Let £ denote the set of all unidirec tional logical links. The cardinality of set £ is denoted by L. Notice that the logical topology 52  Chapter 3.  Cross-layer Fair Bandwidth Sharing  High Speed Internet  o —  Wired Link  —  J  Gateway$  Mesh Router Wireless Link Mobile Device  2 /  /1/  c±K  I  4”. 3  NIC  Operating Channel  (a) Physical and logical topologies (a,c)  (cd)  (a, b)  (d,e)  (b, a)  (e, d) (ca) (dc) (b) Single-channel contention graph CGs  Figure 3.1: A sample MC-WMN with five routers, eight unidirectional logical links, and three frequency channels.  graph G(N, £) is always a subgraph of the physical topology graph G(JV, é) where E is defined in Chapter 2. The logical link from node m to node n is denoted by (m, n) connectivity to be symmetric. That is, link (m, n)  3.1.1  e  e £. We assume the  £ if and only if (n, m) E £.  Channel Assignment Model  For any two nodes m, n E N such that there exists a logical link (m, n) E £, we define a C x 1 link channel assignment vector  Xmn.  The  jth  entry of x, is denoted by x. Jf i frequency  channel is assigned to unidirectional logical link (m, n), then x  53  =  1; otherwise, x  =  0. As  Chapter 3. Cross-layer Fair Bandwidth Sharing an example, for the MC-WMN in Fig.3.1(a) with C  =  Xab=[1  o o  Xba=[1  o  4, we have’:  (3.1) xac=[O  1  0  T 01  xca=[0 0 0  Since all logical links need to be assigned to a frequency channel, it is required that  lTXmn  =  1,  Vm,m éA/,  (m,n)  (3.2)  EL,  where 1 denotes a C x 1 vector with all entries equal to 1. Notice that for any m, ii XTmnXnm  =  e .A/, if  1, then both logical links (m, n) and (n, m) are assigned to the same frequency channel  (e.g., links (a, b) and (b, a) in Fig.3.1(a)). On the other hand, if  XTmnXnm  =  0, then logical links  (m, n) and (n, m) are assigned to two different channels (e.g., links (a, c) and (c, a) in Fig. 3.1(a)). In fact, for any pair of unidirectional logical links (m, n), (p, q) e L, we have:  T Xmn Xpq  1,  if (m, n) and (p, q) use the same channel,  0,  otherwise.  —  For any node m E .Af, we also define a C x 1 node channel assignment vector Ym. The entry of Ym is denoted by y. Jf  jth  frequency channel is assigned to one of the NICs of node m,  ‘The link channel assignment model in this chapter is exactly the same as that in Chapter 2. However, here we consider the channel assignment variables in vector form.  54  Chapter 3. Cross-layer Fair Bandwidth Sharing then y,.  =  1; otherwise, y  =  : 2 0. Consider Fig. 3.1(a) as an example, we have  1  0  1  0  0 0  yc[0  1  1  Ya[1  Yb  =  [  (3.4)  By definition, lTYm indicates the total number of channels that are being used by node m to establish outgoing and incoming logical links with its neighboring nodes. Since each NIC operates on a distinct frequency channel, lTyrn cannot be larger than the total number of available NICs on node m. That is, lTYm < Im,  Vm  N.  (3.5)  The link and node channel assignment vectors are related. For each node m e N, we have =  1 if and only if there exists n e N such that either x  =  1 or  =  1; otherwise, y  =  0.  The following Lemma, proved in Section 3.6.1, mathematically models the desired correspondence between link and node channel assignment vectors:  Lemma 3 For eachm eN, and anyi e {1,...,C},  0  Yn  x< y<1, Xm  (3.6)  ZnEN,(m,n)E Xn+ZnEJf ,(n,rn)ED :lm,  VneN, (m,n)e,  (3.7)  eL.  (3.8)  VneN, (n,m)  < y <1,  The node channel assignment variables here are slightly different from those in Chapter 2. Compare Lemma 3 2 with Lemma 1.  55  Chapter 3. Cross-layer Fair Bandwidth Sharing The link and node channel assignment vectors together provide all the required information to assign channels. They also implicity show how the interfaces should be assigned. For the example of Fig. (3.1), given Ya  =  [i 1 0 1  T, 1  we assign channel 1 to the first NIC, channel 2  to the second NIC, and channel 4 to the third NIC of node a. Since Xac  =  [o  1 0 0  T 1  and  Xca  =  { oo  1  T, 1  Xab  =  Xba  =  [1 0 0 0  node a uses its first NIC to establish both links (a, b)  and (b, a), its second NIC to establish (a, c), and its third NIC to establish (c, a). We stack up all link channel assignment vectors and denote the obtained vector by x. Sim ilarly, we stack up all node channel assignment vectors and denote the obtained vector by y. A channel assignment strategy, denoted by (x, y), is defined as determining vector  Xmn  for all  links (m, n) E £, and vector Ym for all nodes m E Jf. Given an MC-WMN logical topology, a channel assignment strategy (x, y) is feasible if conditions (3.2) and (3.5)-(3.8) hold. The set of all feasible channel assignment strategies is denoted by ‘P.  3.1.2  Interference Model  In an MC-WMN, two logical links (m, n), (p, q)  e  £ are defined to mutually interfere with each  other whenever both of the following conditions hold:  1. The logical links operate over the same frequency channel (i.e.,  =  1), and  2. The sender/receiver of one logical link is within the interference range of the sender/receiver of the other logical link.  To model the interference, we construct a link-layer flow contention graph (or simply con tention graph [53]). In a contention graph, vertices correspond to the logical links. There is an edge between two vertices if the corresponding logical links mutually interfere with each other  56  Chapter 3. Cross-layer Fair Bandwidth Sharing and cannot be active simultaneously. The contention graph depends on the assigned channels. Given (x, y), the corresponding contention graph is denoted by CG.?,).  As a special case,  consider a feasible channel assignment strategy that assigns all links to the first channel (i.e.,  Xmn = Ym = [ 1  o  ...  o ]T  for all m, n  e )v such that  (m, n)  e £).  The corresponding contention  graph is a single-channel contention graph and is denoted by CG. The single-channel contention graph for the MC-WMN in Fig. 3.1(a) is shown in Fig. 3.1(b). Although the vertices in CG(,) and CG are the same, for a general channel assignment strategy (x, y), CG(,,) may have fewer edges than CGs. Thus,  CG(,) ç CGs,  V (x,y)  e  ‘I’.  (3.9)  . The links which correspond to 3 we can identify all of its maximal cliques  Given  the vertices of a maximal clique cannot be active simultaneously [53, 64, 94]. Let  denote  the set of all maximal cliques in CG(,). The number of maximal cliques is denoted by For notation simplicity, we enumerate the maximal cliques. The The set of vertices that form  denoted by Let  mm f  >  IQ(a,y)  maximal clique of CG(,) is  is denoted by V)  c £.  0 denote the normalized link-layer flow rate on logical link (m, n)  e  £ (i.e., the  proportion of time that link (m, n) is active). For notation simplicity, we stack up all link-layer flow rates and denote the obtained vector by  f. Since flows within the same maximal clique  cannot transmit simultaneously, we have the following clique capacity constraint [53, 64, 94]:  fpq  1,  Vi:Q> EQ(a,y).  (3.10)  p,q: (p,q)EV) A clique of a graph is a complete subgraph of the graph. Each clique is either a maximal clique or a subgraph 3 of a maximal clique.  57  Chapter 3. Cross-layer Fair Bandwidth Sharing Recall that the contention graph  depends on selected channel assignment strategy.  Any changes in (x, y) will cause changes in CG() and the set of its maximal cliques  Q().  This  will result in changes in structure and number of inequalities in (3.10). Therefore, the current form of clique capacity constraint in (3.10) cannot be used to formulate an optimization-based channel assignment problem where x and y are optimization variables. The following theorem can overcome this problem.  e ‘I’, the feasible region formed by constraint (3.10) is equivalent to  Theorem 9 Given (x, y)  the feasible region formed by the following constraint,  >  xyjxpqfpq  1,  Vi:  Q  C Qs, Vm,n esV,(m,n)e14  (3.11)  p,q: (p,q)c)4  where Qs,  Q,  vertices in the  and ‘14 denote the set of maximal cliques, the  ith  maximal clique, and the set of  maximal clique of the single-channel contention graph CGs, respectively.  The proof of the above theorem is given in Section 3.6.1. Note that the number of constraints in (3.10) and (3.11) are not the same. Depending on (x,y), the number of inequalities in (3.10) can vary from  Z!2’  !QsI  to  .  However, the number of inequalities in (3.11) is fixed and is equal to  In addition, all the inequalities in (3.10) are maximal clique constraints; while there  may be some inequalities in (3.11) that are just clique (but not maximal clique) constraints. As an example, consider CGs in Fig. 3.1(b). Two maximal cliques are recognized:  ‘14  =  {(a,  b), (b, a), (a, c), (c, a), (c, d), (d, c)}  (3.12)  =  {(a, c), (c, a), (c, d), (d, c), (d, e), (e, d)}  (3.13)  58  Chapter 3. Cross-layer Fair Bandwidth Sharing They form  IVI + IVI  =  12 inequalities in (3.11). If we assign the frequency channels as shown  in Fig. 3.1(a), then we have:  3.1.3  fac  +  fde  1,  1, fai +  fca  fdc  1, and  fed  1.  Cross-Layer Fair Bandwidth Sharing Problem  The model in (3.2)-(3.11) can be used in various cross-layer designs. In this chapter, we extend the fair bandwidth sharing framework in [53] to obtain two cross-layer fair bandwidth sharing algorithms for MC-WMNs. Given an MC-WMN logical topology with N nodes and L links, C orthogonal channels,  ‘m  NICs per each router m é .Af, CG and the set of its maximal cliques Qs,  our objective is to choose the normalized link-layer flow rates, and assign channels and interfaces, so as to solve the following NUM problem:  maximize f>-O, (r,y)e’  umn(i’.fmn)  m,n: (rn,n) EC  subject to  (3.14)  xxpqfpq <  1,  e Qs, m, n e f, (m, n)  Vi :  é  V  p,q: (p,q)EV  where  ic  denotes the nominal link-layer data rate in bits per second, and  Umm  is a continuously  differentiable, increasing, and strictly concave utility function. The utility functions are assumed to be a-fair [50]. Recall from Section 1.2.2 that, if a layer flows is obtained; a  =  =  1, then proportional fairness among link-  2 corresponds to harmonic mean fairness; and a  —*  cc  corresponds  to max-mm fairness.  3.2  Design I: Exact Binary Linearization  Problem (3.14) is a non-linear mixed-integer problem and is not easy to solve. Note that:  59  Chapter 3. Cross-layer Fair Bandwidth Sharing 1. It has real variables  f and binary variables x and y.  2. It has mixed binary-real cubic constraints. After relaxing the binary constraints, problem (3.14) is still non-convex. Thus, even the relaxed problem cannot be easily solved. In this section, we present some binary linearization techniques to obtain the global optimal solution of problem (3.14) in a static and centralized manner. Let Es denote the set of all edges in CGs. We denote eT e Es if there is an edge between vertices (m, n) and (p, q). The cubic constraint in (3.14) can be linearized as follows: Step 1: For each pair of logical links (m, n), (p, q) e £ such that eT e Es, we define a C x 1 auxiliary link channel assignment vector  as follows:  mn  Vpq  —  Xmn  Xpq,  . From (3.15) we have, 4 where o denotes the Hadamard product Xpq  is quadratic while  T 1  v7 is linear. Since Xmn,  Xpq  =  X  T 1  v7.  Notice that  and v are C x 1 binary, (3.15)  is equivalent to the following linear constraints (see Theorem 3):  Xmn m Xpq  mm  —  Vpq  —  (3.16) Xmn  —  Xpq  mn Vpq  For notation simplicity, we stack up all vectors strategy, denoted by (x, y, v), is defined as determining  —  as v. A linearized channel assignment Xmfl  for all links (m, n) e £, vj for all  links (m, n), (p, q) e £ such that e m e E , and Ym for all nodes m e .A/. A linearized strategy 5 4 T he Hadamard product of two C xl vectors a and b is a C x 1 vector whose entry of a and the entry of b.  60  th  entry is the product of the  jth  Chapter 3. Cross-layer Fair Bandwidth Sharing (x, y, v) is feasible if (x, y)  e 1’ and condition (3.16) holds. The set of all feasible linearized  channel assignment strategies is denoted by 1.  Step 2: For each pair of logical links (m, n, (p, q) e £ such that epT e CG, we define an auxiliary real scalar variable z7 as follows:  =  Since  lTrnn  Xpq fpq  (iT  Th v )fpq.  (3.17)  is a binary scalar and the normalized link-layer flow fpq is upper bounded by  one, equation (3.17) is equivalent to the following linear constraints (see Theorem 4):  0 fpq_1+1Tv  We stack up all scalars  z7  Zq’  fpq,  (3.18)  471 <  lTmn  (3.19)  and denote the obtained vector by z. Combining steps 1 and 2,  problem (3.14) is equivalent (cf. [16, pp. 130]) to the following problem:  maximize f  >.-  o, z  >-  o,  m,n: (m,n)EC  (x,y,v) E  subjectto  4711,  Vi:QeQs,Vm,nEJ’f,(m,n)EVg,  (3.20)  p,q: (p,q)EV  vm,n,p,q:eees,  zfpq, fpq  —  1±  T 1  v < z, V m, n,p, q : e e  zlTv  Vm,n,p,q:ees.  61  Chapter 3. Cross-layer Fair Bandwidth Sharing By relaxing the binary constraints on x, y, and v, problem (3.20) becomes a convex optimiza tion problem. There exist several efficient algorithms to solve convex problems [16]. By solving the relaxed problem, we can obtain the upper and lower bounds that are required in branch and bound algorithm [15, pp. 577-580]. By using branch and bound, we can find the global optimal solution of the mixed-integer problem in (3.20). Since problems (3.14) and (3.20) are equivalent, the global optimal solution of the mixed-integer problem in (3.14) is also readily found.  3.3  Design II: Approximate Dual Decomposition  The exact binary linearization scheme in Section 3.4 helps us to find the optimal solution of problem (3.14) in a static and centralized manner. In this section, we propose an alternative but approximate design which is more practical. Consider the dual problem of the primal problem (3.14):  (3.21)  minimize D(p) PO  with partial dual function  /  IQsI D(p)  =  maximize  p (i  )+ f m u m,m: (m,n)E  i1 m,n: (m,n)eV  -  p,q: (p,q)EV  where we relaxed the clique capacity constraint in (3.14). The Lagrange multiplier for the clique capacity constraint associated with clique  Q e  Qs and vertex (m, n)  e  V is denoted by  p. For notation simplicity, we stacked up all Lagrange multipliers and denoted the obtained vector by p. Our proposed joint algorithms are shown in Algorithm 2 and Algorithm 3, where 62  Chapter 3. Cross-layer Fair Bandwidth Sharing Algorithm 2 Executed by each node m e N 1: For Vn e N, (m, n) e £ do 2:  (Zi:(m,n)éV Zp,q:(p,q)eV 4qxxmn)]  fmn  End 4: For Vn e N, (m, n) £ and Vi: (m, ii) e V, do 0 5: (i p,q:(p,q)cV xpTqxmnfpq)] Pmn := 6: End 7: Inform the updated values to all nodes p e N such that 3:  —  —  [x]  =  n, q e N, e Th é  max (mm (x, b), a). We make the following assumptions:  1. The normalized link-layer flow rates arid the Lagrangian multipliers are updated distribu tively and asynchronously every TMAC time units using Algorithm 2. 2. The channels are updated every Ti time units using Algorithm 3. . 01 3. The time interval TMAC <<T Consider the time interval between two consecutive channel updates (i.e., the period of length 01 time units right after any channel assignment performed by Algorithm 3). During this period, T Algorithm 2 is just a fair MAC [53] over fixed channels and fixed interfaces. Given x and p as constants, line 2 of Algorithm 2 selects  , m f  to maximize the dual objective function in (3.22).  Line 5 of Algorithm 2 also updates Lagrange multiplier where parameter  using a sub-gradient method [15],  is a constant stepsize. We can interpret the Lagrange multipliers as clique  contention prices to regulate between the supply and the demand. From line 5 in Algorithm 2, if the demand  Z(pq)Ev  Xpq fpq  exceeds the supply (that is 1), the price  will increase.  The prices are then used to adjust the flow rates in the next iteration. If Ti is large enough, stepsize  , then the convergence of 5 is small enough, and the asynchronism measure is bounded  The asynchronism measure of distributed Algorithm 2 is bounded if there exists a positive finite B such that 5 1) each node m E iV executes Algorithm 2 at least once during any interval of length B time units, and 2) the information is used by each node for executing Algorithm 2 is outdated by at most B time units [18, pp. 481].  63  Chapter 3. Cross-layer Fair Bandwidth Sharing Algorithm 3 Executed by a pre-authorized gateway 1: Gather the information on f and p from all nodes m 2: For Vm,n,p,q: (m,n),(p,q) e £ do 3: If e Th E ‘f-s then 4:  Wpq  :=  (frnn  +  e f.  fpq) (Zi:(rn,n),(p,q)ev pq)  else  5:  6:  7:  End if 8: End i._v’ U’ ‘-•  10:  rnrif T  L.im,ri,p,q:(m,rz),(p,q)ef”pq  (lTi3n) 7 (, th ‘o) := argrnin Z m,n,p,q:(m,n),(p,q)eL w (a,y,v)’I’  ii: IL_7.  .—  12: 13: 14: 15: 16:  mnf-T Z_,m,n,p,q:(m,n),(p,q)JJpq iXrnnXpq  ‘  With probability  -  [o (d/ci— 1)]’  do  (x,y):=(,’)  Inform (x, y) to all nodes m E .iV. End S := 5/2.  Algorithm 2 is guaranteed [18, pp 527-535]. That is, before the new channels are being assigned by Algorithm 3 in its next iteration, Algorithm 2 will reach its fixed point. The fair MAC in Algorithm 2 can be implemented by modifying the contention window size adjustment mechanism within the IEEE 802.11 distributed coordination function as in [57). Now consider the channel assignment scheme in Algorithm 3. We first gather all information on flow rates and Lagrangian multipliers every Ti time units in a pre-authorized node (e.g., one of the gateways). Then, we select the linearized channel assignment strategy to minimize =  3-7.  Zflpq.( n) (p q)E  7(xxpq),  where  is as in lines  Recall that the Lagrange multipliers can be interpreted as clique contention prices and  xnxpq indicates whether links (m, m) and (p, q) mutually interfere with each other. Thus, we  can interpret w as the interference cost of having the logical links (m, n) and (p, q) operate over the same frequency channel. By definition, the interference cost is high if the interfering  64  Chapter 3. Cross-layer Fair Bandwidth Sharing links are highly loaded and belong to highly contended maximal cliques. In the linear binary optimization problem in line 10, we minimize the aggregate interference cost across the network. The optimal solution of aggregate interference cost minimization problem (i.e.,  ,  ii) in line  10) is taken into account with probability (d/d— 1) bounded between 0 and 1. Notice that d (in line 9) is always greater than or equal to d (in line 11). We initially set  =  1. Since parameter S  is decreasing (see line 16), the probability of switching to new channel assignment strategies will gradually decrease through iterations. That is, Algorithm 3 becomes less willing to make changes in the assigned channels as time goes by. This will guarantee the convergence. To further explain how Algorithm 3 works, we present the following two Lemmas. Lemma 4 Let  Q and Q be two arbitrary maximal cliques in single-channel contention graph  CGs. For any links (m, n), (p, q)  e £,  we have: (m, n)  eV  and (p, q)  e 4.  Given the assigned  channels (x,y), if  V\ {(l, k) :  mn1k  =  o}  C  V\ {(l, k) :  XpqXlk  =  o},  (3.23)  then *nXlk fik  <  l,k:(1,k)eV and,  if (9.24) holds  for  all t >  ‘qXlk fik,  (3.24)  1,k:(l,k)eV  to for some to, then  lim p(t)  t—oo  =  (3.25)  0.  The proof of Lemma 4 is given in Section 3.6.3. From (3.25), if Tj is large enough, then the contention prices converge to zero for those cliques that are 65  not  maximal cliques of the  Chapter 3. Cross-layer Fair Bandwidth Sharing single-channel contention graph  Lemma 5 For arbitrary links (m, n), (p, q)  e £ such that e%’ e Es, if  Xpq  =  1 and p(0)  =  then =  pq(t),  e  V i: (m, n), (p, q)  V, V t  (3.26)  0.  The proof of Lemma 5 is given in Section 3.6.4. For arbitrary logical links (m, n), (p, q) E such that e  e  Es, we have:  (t4,qfmn + p;qfpq) i:(m,ri),(p,q)eV =  (3.27) (%fpq  +  pqfrnn),  i:(m,n),(p,q)eV  where the first equality comes from line 3 of Algorithm 2 and the second equality results from Lemma 5. From (3.15), (3.27), and the fact that  xTmnxpq  =  m,n,p,q:(m,n),(p,q)E  we have  /  IQsI w (1Tv)  xpTqxmn,  p  =  m,n: (m,n)EV  (  xnxpqfpq)  \p,q: (p,q)EV  (3.28)  .  /  This implies that solving the interference cost minimization problem in line 10 of Algorithm 3 is the same as selecting a feasible channel assignment strategy which maximizes the dual objective function in (3.22). In summary, both Algorithms 2 and 3 try to solve the dual problem of the primal NUM problem in (3.14). Algorithm 2 selects optimal  f  and p while x and y are assumed  to be fixed. On the other hand, Algorithm 3 selects optimal x and y while assuming  f  and p  are fixed. The optimality of joint Algorithms 2 and 3 is not guaranteed. We will investigate the sub-optimality of the solutions and their effects on network performance in Section 3.4. 66  Chapter 3. Cross-layer Fair Bandwidth Sharing  Algorithm 4 To replaced line 10 of Algorithm S 1: K := number 2: mn := 1 0 3: rn := 1 0 4: ‘:=[ 10 5: For k 1 to  of iterations T, Vm, n e Al, (m, n) E 0 1 [ T, 1 V m E iV 0 [ T 01 Vm,m,p,qeAl, eEes K do Randomly choose nodes v, w Al such that (v, w) Using branch-and-bound [33], solve  6: 7:  •..  minimize subject to  e £.  mn) 3 wmn (lT pq  mn  =  mm, V m, n  Ym=!m,  if  =  e Al\{v, w}, (m, n)  VmeAl\{v,w} Vm,n,p,q éAl\{v,w}, e  é  8: 9:  End  3.3.1  A Simple Heuristic Algorithm to Solve Aggregate Interference Cost Minimization Problem  In line 10 of Algorithm 3, we need to solve a linear binary problem to minimize the interference cost across the network. There are effective commercial software packages (such as CPLEX [31] or MOSEK [32]) that can solve linear binary problems. However, the process can be time consuming for large scale MC-WMNs. An alternative is to use some simple and efficient metaheuristic methods to find suboptimal solutions [87].  Here we use the iterated local search [88].  Our  algorithm is shown in Algorithm 4. We first assign all links to the first channel (lines 2-4). Then, at each iteration, we randomly choose a small neighborhood in the network (line 6) and locally solve the mixed-integer interference cost minimization problem for that neighborhood assUming the channels are fixed for the rest of the network (line 7). The iterations continue until a termination condition is met (e.g., we finish K iterations). Using heuristic Algorithm 4, Design II can easily be applied to large-scale MC-WMNs.  67  Chapter 3. Cross-layer Fair Bandwidth Sharing  3.4  Simulation Results  In this section, we evaluate the performance of our proposed cross-layer designs. In the model, the size of the network field is 500 m x 500 m. Ten different random scenarios are considered. In each scenario, the WMN consists of twenty wireless mesh routers that are arbitrarily located in the field. Unless stated otherwise, the routers are equipped with four NICs (i.e., I there are five orthogonal frequency channels available (i.e., C  =  =  4) and  5). The communication and  interference ranges are 100 m and 150 m, respectively. For each scenario, there is a logical link between each pair of nodes if they are within the communication range of each other. One of the network scenarios (scenario number 1) that we used in our analysis is shown in Fig. 3.4. 500 450 400 350 -  300 250 200  U-  150 100 50 0  50  100  150  200  250  300  350  400  450  500  Field Length (m)  Figure 3.2: Scenario number 1: A wireless mesh network with 20 nodes (i.e., wireless mesh routers), 46 unidirectional logical links.  The utility functions are selected as in (1.17). Unless stated otherwise, we set c  =  1. Recall  that the logarithmic utility functions lead to proportional fairness among the link-layer flow rates. 68  Chapter 3. Cross-layer Fair Bandwidth Sharing For the second design, we have: TMAC  =  1, T 01  =  1500,  Note that depending on the selected value for the stepsize  =  0.01,  k  =  11 Mbps, and B  =  5.  and the value of the asynchronism  measure B, the channel update interval Ti should be large enough to let the fair MAC (i.e., Algorithm 2) reach its steady state. At time t  =  0, we set S  =  10. Later, we reduce S by half  every Ti time units (as shown in line 16 of Algorithm 3). We use Algorithm 4 with K =25 to solve the aggregate network interference cost minimization problem (see Section 3.3). All NICs initially (i.e., at t  =  0) are assigned to a single channel.  Fig. 3.3 shows the evolution of the network utility for scenario number 1. We see that, after only four channel/interface update intervals, the utility reaches 99.46% of its optimal value. Later on, there is only one more slight channel/interface adjustment (at t  =  9000) before the system  reaches a steady state. We notice that right after each channel/interface assignment update, the utility experiences an overshoot. In fact, for the duration of overshooting, the assigned rates are not feasible as they were set accoridng to the previously assigned channels/interfaces. Thus, they result in higher utility until they settle down in their optimal feasible values. The achieved network utilities for the ten different random scenarios are shown in Fig. 3.4. We see that the proposed MC-WMN deployments significantly increase the network utility in all scenarios, On average, the second (i.e., approximate) design scheme is able to find near optimal solutions with 99.6 % optimality. Recall that the second design scheme is simple to implement and its signalling overhead is not significant. To evaluate the network performance, two metrics are considered: 1) network throughput, and 2) fairness index. The network throughput is the aggregate actual link-layer flow rate across all logical links in bits per second. That is,  m,n:(rn,ri)EI. ‘fmn.  69  The fairness index is a dimensionless  Chapter 3. Cross-layer Fair Bandwidth Sharing  620  I  /  I  I  I  Design I, Optimal  600/JJ  _  580  560  Design II, Near—Optimal  540  520o  Single Channel  3000  9d00  6000  12000  15000  Iteration Number  Figure 3.3: Evolution of network utility for scenario number 1.  metric between zero and one. It is defined as [891: (Zm,n:(m,n)  fmn)2  /  (Lm,n:(m,fl)E(fmn)2).  (3.29)  The higher the fairness index, the more fair the rate allocation is. Fig. 3.5 shows the throughput and fairness index when the number of NICs varies between 2 and 4 and the number of orthogonal channels varies from 1 to 5. Each point is the average of the measurements for all ten scenarios. We can see that when each router is equipped with 3 NICs and there are 5 channels available, the throughput and fairness index increase by 2429 and 3.4%, respectively, compared to the single-channel case. If each router is equipped with four NICs, then the throughput and fairness index further increase by 5% and 0.4%, respectively.  70  Chapter 3. Cross-layer Fair Bandwidth Sharing  b’J  I 600  Design I IDesign II Single channel  550 D 0  500  450  400  1  2  3  7 4 5 6 Scenario Number  8  9  10  Figure 3.4: Network utility for ten different random scenarios. Each router is equipped with 4 NICs and there are 5 orthogonal channels available. On average, the second (i.e., approximate) design is able to find near optimal solutions with 99.6 % optimality. The average utility improvement compared to single-channel case is 12.5 %.  Results from Fig. 3.5 show that our proposed designs can lead to MC-WMN deployments which are not only more efficient but also more fair compared to their single-channel counterparts. Fig. 3.6 clarifies this issue in more details. In this figure, the average network throughput and the average fairness index across all ten topologies are shown when the number of channels varies from 1 to 12. Each mesh router is equipped with enough NICs so that the only resource limitation is the number of channels. We can see that the fairness index increases smoothly as the number of channels increases. To examine whether there is a similar trend for every channel and interface assignment algorithm, we consider the load-aware algorithm [66] which is a centralized long-term dynamic channel and interface assignment scheme. By monitoring the amount of traffic being transmitted over each frequency channel, the load-aware algorithm assigns 71  Chapter 3. Cross-layer Fair Bandwidth Sharing  a)x 0 0  a)  I  U-  Number of Available Frequency Channels, C (a)  250 D  r)  2 I—  I .  100  1  2  3  4  5  Number of Available Frequency Channels, C  Figure 3.5: Impact of available network resources (i.e., available channels and NICs): (a) Aggregate network throughput, (b) Fairness index.  72  Chapter 3. Cross-layer Fair Bandwidth Sharing  (a) 500 *  450  .i  Design ll Load—Aware I  400 350 2 300  -c 1.  636%  250  0 ci)  200  z  ci) c,)  150 100  Single Channel  <50 n  V  1  2  4 7 10 11 3 5 6 8 9 Number of Available Frequency Channels, C  12  (b)  x  ci) U) U)  ci)  C  U  1  2  3  4  5  6  7  8  9  10  11  12  Number of Available Frequency Channels, C Figure 3.6: Effects of varying the number of available frequency channels with enough NICs:  (a) Aggregate network throughput, (b) Fairness index.  73  Chapter 3. Cross-layer Fair Bandwidth Sharing  (a) 300 C’)  250  -  6.8%  200  2  332%  I 1.  324%  150  0  ci)  z 100  ci)  C)  ci) I.  50  C) C) 4:  *  0  0— 0.2  0.4  0.6  0.8  1  1.2  1.4  Design II Single Channel  1.6  1.8  2  Parameter a (b)  0.9 0.85 x  a)  0  0  ci)  0.8 0.  I-  Cu  U-  0.7 0.65 0.6  *  0  0.6  0.8  1 1.2 Parameter a  1.4  Design II Single Channel 1.6  1.8  2  Figure 3.7: Effects of varying the fairness parameter: (a) Aggregate network throughput, (b)  Fairness index.  74  Chapter 3. Cross-layer Fair Bandwidth Sharing the channel with minimum usage within the neighborhood of each link. We implemented the loadaware algorithm jointly with fair MAC [53]. To make the comparison consistent, channels and interfaces are updated every 1500 intervals as in our second design scheme. We can see that the average throughput is almost the same for both load-aware and our proposed schemes (see Fig. 3.6(a)); however, our proposed design is more fair (see Fig. 3.6(b)). In some cases (i.e., for 2,  C  ..  .  ,  6), the MC-WMN is even less fair than the single-channel WMN when the load-aware 7), achieving  algorithm is being used. Note that by increasing the number of channels, (e.g., C fairness becomes trivial due to the availability of sufficient resources.  As stated in Section 1.2.2, different fairness criteria can be taken into account by tuning fairness parameter a. Fig. 3.7 shows the throughput and fairness index when C  =  5, I  =  4, and  utility parameter a varies from 0.2 to 2. We can see that by increasing a, the system becomes more fair but less efficient. As an example, we can achieve 24% higher fairness index by setting a  =  2 (instead of a  =  0.2), at the expense of reducing the network throughput by 6.8%. From  the results in Fig. 3.7, we can also conclude that as a decreases, the performance gain on fairness index becomes higher compared to the single-channel case.  3.5  Summary  In this chapter, we presented a formulation for cross-layer fair bandwidth sharing in MC-WMNs. We first modeled the channel and interface assignment problems by introducing binary channel assignment and binary interface assignment vectors. We then obtained the feasible region of the link-layer flow rates as a function of the channel and interface assignment vectors. A cross-layer fair bandwidth sharing problem was then formulated as a non-linear mixed-integer network utility maximization problem. An optimal design, based on exact binary linearization techniques, was 75  Chapter 3. Cross-layer Fair Bandwidth Sharing proposed which leads to a global maximum. A near-optimal design, based on approximate dual decomposition techniques, was also proposed which is more practical for implementation. Our proposed designs take into account both network efficiency and fairness. Some of the well-known fairness criteria, such as proportional fairness, harmonic-mean fairness, and max-mm  fairness,  can also be modeled using a tunable design parameter.  3.6 3.6.1  Analytical Proofs Proof of Lemma 3  Assume that node m  e .N  incoming and  is assigned to establish  outgoing logical links  with its neighboring nodes over frequency channel i. Thus, constraint (3.6) can be re-written as o <  < (K + KLt). If K  =  =  0, then (3.6) becomes 0 < y  and (3.8) become 0 < y < 1. This implies that y  =  0. If K  (3.6) or either one of (3.7) and (3.8) becomes 0 <y < K +  3.6.2  0 or  0 and constraints (3.7) 0, then constraint  and 1 <y < 1.  Proof of Theorem 9  Since (3.10) inc]udes all maximal clique capacity constraints for the single-channel contention graph CG) and each inequality in (3.11) is a clique (not necessarily a maximal clique) capacity constraint for CG(,), then the feasible region formed by (3.10) is a subset of or equal to the feasible set formed by (3.11). We only need to prove that the reverse is also true. That is, the feasible region formed by (3.11) is a subset of or is equal to the feasible region formed by (3.10).  76  Chapter 3. Cross-layer Fair Bandwidth Sharing From (3.9) we have:  V  We refer to set set  E  Q(,)  as the parent of set  Q’  E  Qs:  That is, (m, n)  X Xpql,  (3.30)  In general, there may be more than one parent for and one of its parents  Consider an arbitrary maximal clique  a logical link in  çQ . 3  Q. Let (m, n) be  We can show (by contradiction) that,  Vp,q:  (p,q)  e (3.31)  X  XpqO,  Vp, q: (p, q) e V\V)  Thus, we have:  fpq  Oxfpq  lxfpq+  =  p,q: (p,q)EV  p,q: (p,q)EV)  =  p,q: (p,q)EV\V  xxpqfpq  xxpqfpq+  p,q:  (pq)EV,y)  (3.32)  p,q: (p,q)EV\Vy,  XmnXpqJpq, ZTc  p,q: (p,q)EV  where the second equality follows from (3.31). Eq. (3.32) implies that for every inequality in (3.10), there is an equivalent inequality in (3.11). Therefore, the feasible region formed by (3.11) is a subset of or is equal to the one formed by (3.10).  77  Chapter 3. Cross-layer Fair Bandwidth Sharing  3.6.3  Proof of Lemma 4  Inequality (3.24) is obtained from (3.23) as follows,  XjjXjj fik  fiic  =  1,k:(1,k)EV  l,k:(1,k)EV\{(l,k):,flak=O}  fu  <  (3.33)  l,k:(l,k)EV\{(l,k):x’q xlk_O} =  XpqXflk. 1k: (1,k) EV.  From (3.11) and (3.33), we have:  jflX 7 X J 1 çfJ  (3.34)  < 1.  l,k:(l,k)EV  Eq. (3.25) results from replacing (3.34) in the update equation of Lagrange multipliers in line 5 of Algorithm 2, assuming that TMAC <<Tci.  3.6.4  Proof of Lemma 5  Consider a maximal clique  Q e Qs  Xrnn Xlk  so that (m, n), (p, q)  =  e V.  Since  Xpq  =  1,  V1,k: (1,k) E  Xpq Xlk,  (3.35)  Thus, Xpq Xlk.  =  l,k:(1,k)EV  1,k:(l,k)E’14  Since %(O)  =  pq(O), the equality (3.26) follows from (3.36) and line 5 of Algorithm 2.  78  (3.36)  Chapter 4  Partially Overlapped Channel Assignment for MC-WMNs As we showed in Chapters 2 and 3, the aggregate capacity of wireless mesh networks can be increased by the use of multiple frequency channels and multiple network interface cards in each router. However, so far we limited our studies to the case where only the orthogonal (i.e., nonoverlapped) frequency channels are being used. Recent results have shown that the performance can further be increased when both non-overlapped and partially overlapped channels are used. In this chapter, we systematically model a joint channel assignment, interface assignment, and scheduling design problem. The contributions of our work are as follows: • Our proposed model takes into account various parameters including the number of available frequency channels, the number of available NICs in each wireless router, transmission power, path loss information, signal to interference plus noise ratio, expected traffic load, and frequency response of each channel filter. • Since the model is formulated as a linear mixed-integer program with a few integer variables, the computation complexity is low and it is feasible for implementation. • We propose the channel overlapping matrix and mutual interference matrices to model the  79  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs non-overlapped and partially overlapped channels. • Simulation results show that the aggregate capacity increases by 90% when all partially overlapped channels within the IEEE 802.llb frequency band are used. The rest of the chapter is organized as follows. The problem formulation is described in Section 4.1. Our proposed joint design is presented in Section 4.2. Performance evaluations are given in Section 4.3. The chapter is concluded in Section 4.4.  4.1  System Model  The system model in this chapter is very similar to the one in Chapter 3. Consider an MC-WMN and assume that .Af  =  {1, 2,... , N} denotes the set of stationary wireless mesh routers. Each  router is equipped with I NICs. There are C frequency channels available. We assume that the network’s logical topology has been pre-determined. Let £ denote the set of all unidirectional logical links. The cardinality of set £ is denoted by L. The logical link from router a to router b is denoted by (a, b) and only if (b, a)  4.1.1  e £.  We assume the connectivity to be symmetric. That is, link (a, b) E £ if  e £.  Channel and Interface Assignment Model  For any two routers a, b  e  channel assignment vector channel, then the  jth  f such that there exists a logical link (a, b) Xab.  element in  e £,  we define a C x 1  If router a communicates with router b over the Xab  is equal to 1; otherwise, it is equal to zero.  80  frequency  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs Same as Chapters 2 and 3, we have:  Xab  =  Xba,  V a, b E J\f,  (a, b) E £  (4.1)  e .Af, (a, b) e £  (4.2)  and Xab  =  1,  V a, b  1 where 1 denotes a C x 1 vector with all entries equal to 1. The term T Xab is equal to 1 if router  a assigns one of the available frequency channels to communicate with router b. Similar to Chapter 3, for any two routers a, b  interface assignment vector  Yab.  e N  such that (a, b)  e L,  we define an I x 1  If the jth network interface in router a is used to communicate  with router b, then the jth element in Yab is equal to 1; otherwise, it is equal to zero. We have:  lTyabl,  Va,beJV,  (a,b)1  (4.3)  1 where 1 denotes an I x 1 vector with all entries equal to 1. The term T Yab is equal to 1 if router  a assigns one of its NICs to communicate with router b. Notice that,  IbXacYaYac,  4.1.2  Va,b,cel’f,  (a,b),(a,c) i.  (4.4)  Channel Overlapping Matrix  Assume that m and m are two of the available channels within the 802.11 frequency band (i.e.,  m, n E {1,... C}). Let ,  Fm()  and F(w) denote the power spectral density (PSD) functions of  the band—pass filters for channels m and n, respectively. The PSD functions can be obtained from  81  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs  1  2  3  4  5  6  7  8  9  10  11  (J) 0  0  z  2430 2440 Frequency (GHz)  Figure 4.1: Available eleven partially overlapped channels in 802.llb frequency band. The number on each curve indicates the corresponding channel number. Channels 1, 6, and 11 are non-overlapped (orthogonal). the channels’ frequency responses. Without loss of generality, we assume the use of raised cosine filters [95]. Fig. 4.1 shows the frequency responses of the channel filters in the IEEE 802.llb frequency band. To model the overlapping among channels, we define a symmetric C x C channel overlapping matrix W. The entry in the Wrnn  m’  row and the  column of W is denoted by scalar  and is defined as: Wmn  dw f°°c,Fm(w) roo (w)dw 2 F J-m  (4.5)  Now assume that channels m and n are assigned to arbitrary links (a, b) and (c, d), respectively. Let  Pa  denote the transmission power of router a. Also let  denote the path loss from router  a to router d. The interference power from link (a, b) on link (c, d) can be modeled as:  W  4.1.3  Xcd 9ad Pa  Wrnn ad Pa.  (4.6)  Mutual Interference Model  We first consider an MC-WMN where only non-overlapped channels are being used, two links (a, b), (c, d)  e AD are defined to be mutually interfered with each other whenever they are assigned 82  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs to the same channel (i.e.,  X  1) and the sender or receiver of one link is within the  =  interference range of the sender or receiver of the other link. The interference range is then defined as the region where a given receiver cannot decode the signal correctly if there is another transmission within that range. Given the modulation scheme, the interference range depends on the minimum required signal to interference plus noise ratio SINRmin. Now consider an MC-WMN where both non-overlapped and partially-overlapped channels are  e £ are assigned to channels m and n, respectively.  being used. Two neighboring links (a, b), (c, d)  If the interference power of the transmission on link (a, b) causes the signal to interference plus noise ratio on link (c, d) to be below SINRmim, then the transmitter of link (a, b) is within the interference range of the receiver of link (c, d):  cd Pc < ad Pa + 7 1d  Wrnrj  where  ‘/d  C’T1TD kJlIVlL f 7 ljfl,  denotes the thermal noise power at the receiver router d. Without loss of generality, we  model the path loss  ad  using the Friis free space model [95j:  a Yad  where  rad  =  (4.8)  (rad)  is the distance between routers a and d, ic is the path loss exponent, and a is a  constant which depends on the transmitter and receiver antenna gains and signal wavelength. By substituting (4.8) into (4.7), link (a, b) interferes with link (c, d) if  rad  <  iI(  \9cd  apa p/SINR  83  I  —  TldJ  Wmn.  (4.9)  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs  ib  af  /C  t--  ,.-d-..  )  ‘•  / --‘S  -.  5%-S  -‘S  ,,  II 0  D II  II F’.)  -  II ()  Figure 4.2: Different interference ranges depending on the frequency channel separation m Logical links (a, b) and (c, d) use channels m and n, respectively.  —  n.  The importance of (4.9) is that we now have different interference ranges depending on the assigned channels to the neighboring links. The less the frequency overlapped, the shorter the interference range is. Given that the bandwidth and roll-off factor are the same in all raised cosine channel filters, the interference range only depends on the frequency channel separation  (Im  —  n). This fact is illustrated in Fig. 4.2, where rn is the channel assigned to link (a, b) and  n is the channel assigned to link (c, d). The outermost circle indicates the interference range of receiver router d when rn  —  n  =  0 (i.e., the same channel is being assigned to links (a, b) and  (c, d). The next circle shows the interference range when rn corresponds to the interference range when rn  —  n  =  —  n  3. When rn  =  —  1. The innermost circle  nl > 3, there is no overlap  between frequency channels rn and n for IEEE 802.llb (see Fig. 4.1). Thus, the corresponding interference ranges are equal to zero.  Note that in this example, transmission on link (a, b)  interferes with transmission on link (c, d) only when either rn For any two links (a, b), (c, d)  e  —  nI  =  0 or rn  —  nI  =  1.  £, we define a symmetric C x C mutual interference  Mj. If we have either  84  matrix  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs  Tad  <  Tcb  <  flth  column of M,j is equal to 1; otherwise, it is equal to 0. If the  (gcdpc/SRmin_d)  Wmn,  or  then the entry in mthi row and  (Yab pa/SINRmin_b)  Wnm,  transmission powers are fixed, the mutual interference matrices are constant. For the scenario in Fig. 4.2, the corresponding mutual interference matrices are tridiagonal with all diagonal, subdiagonal, and superdiagonal entries equal to 1:  11  00  11  1  ...  0”.0  01  M  =  M  (4.10)  =  10 0•••  0  1  1  1  0••  0  0  1  1 lix ii  Note that if logical links (a, b) and (c, d) are far enough from each other, then all entries of  become zero. According to the definitions of channel assignment vectors and mutual  interference matrices, we have:  1,  if links (a, b) and (c, d) are mutually interfered, (4.11)  0,  otherwise.  85  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs  4.1.4 Let 0  Link-Layer Flow Rates < fab  1 denote the normalized link-layer flow rate on logical link (a, b)  proportion of time that link (b, a) is active). If two logical links (a, b), (c, d) interfered (i.e., x M,j  Xcd  =  e  E  L (i.e., the  £ are mutually  1), they cannot be active simultaneously. We can extend the  interference constraint in [67] to have:  fab  1.  +  (4.12)  c,d:(c,d)EC  Let c 0 denote the nominal data transmission rate (e.g., 11 Mbps data rate in IEEE 802.llb). The effective link-layer data rate on link (a, b) is equal to  4.2  fab C°.  Joint Channel Assignment, Interface Assignment, and Scheduling Problem  The mathematical formulation introduced in (4.1)-(4.12) models the channel assignment, inter face assignment, mutual interference among the links, and link-layer flow rates when all non overlapped and partially overlapped channels within the IEEE 802.11 frequency band are being used. The model can be employed to develop different cross-layer design schemes for MC-WMNs. In this chapter, we focus on a joint channel assignment, interface assignment, and scheduling design problem. We assume that the network logical topology and routing paths have been pre-determined.  86  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs 4.2.1  Nonlinear Problem  Given the expected end-to-end traffic rates and routing paths, we can determine the expected aggregated traffic load on each logical link (a, b)  e £. It is denoted by  on logical link (a, b) is defined as the total traffic load rate  fab c°.  7ab  ‘Yab.  The link utilization  divided by the effective link-layer data  Based on the results from queuing theory, when the link utilization is close to 1,  the queueing delay tends to be very large [80]. On the other hand, a small value of the link utilization tends to provide a small queueing delay. It also implies that the network is less prone to congestion. Thus, our objective function is to minimize the maximum (i.e., bottleneck) link utilization in an attempt to manage the network capacity according to the expected traffic load. , and I1/1j for all a, b, c, d 0 Given the parameters I, C, L, 7ab, c  minimize a,y,f  subject to  7ab  maximum a,b:(a,b)E  Xab  e N such that (a, b), (c, d) e £,  fab C°  Xba,  =  Xab  =  1,  Yab  =  1,  T Xab Xac  —  —  (4.13)  T Yb Yac,  x M  fab +  Xcdfcd  1,  c,d:(c,d)EIL  where  Xab E  {0, 1}  Yab E  {0, 1}’,  fab >  0, V a, b: (a, b) eL  Intuitively, if a particular link is heavily loaded, its effective link-layer data rate should be in creased by reducing the interfering links in its neighborhood. Note that problem (4.13) is a non-linear mixed-integer program and is not easy to solve. The terms x  87  Xac  and y  Yac  are  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs quadratic and the term x Mj zj fcd is cubical.  4.2.2  Equivalent Linear Problem  Following the binary linearization techniques (see Section 1.2.1), the non-linear problem (4.13) can be converted to its equivalent linear problem using these steps: Step .1. For each pair of logical links (a, b), (c, d)  e  ,  we define a C x 1 auxiliary vector u  to be as follows: (4.14)  cd  =  where o denotes the Hadamard product [96]. From (4.14) we have, x x  =  T 1  u. Since  Zab  and a are binary vectors, (4.14) is equivalent to the following linear constraint [39]:  Xab  +  o o  Xcd  —  U  < Uj  <  Xab,  1,  (4.15)  UCCL  Step 2. For each pair of logical links (a, b), (a, c)  e £, we define an I x 1 auxiliary vector v  to be as follows: V  =  Yab  88  Yac.  (4.16)  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs From (4.16),  Ya Yac  =  lTvab. Eq. (4.16) is equivalent to the following constraint [39]:  Yab + Yac  —  0  <  1, (4.17)  Yab,  —  ab, Vac Yac. —  Step 3. For each pair of logical links (a, b), (c, d)  e L,  we define a C x C auxiliary matrix  Q  to be as follows: =  The entry in the entry in vector  Xab,  row and the  the  flth  e {1,  ...  ,  Xd,  (4.18)  fcd.  column of matrix  entry in vector  linear constraint for all m, m  Xab  Q  is equal to the product of the m  and fcd. Eq. (4.18) is equivalent to the following  C} (see Theorem 5):  ,T imab. -iT rn ‘‘cd -n :: ImXab, 1 ‘T_—iab cd 4 m’  n D lnXcd,  (4.19) 1Xab+1Xcd2+fcd < m < 21Xab1Xcd+fcd, 1Q1  where  ‘m  denotes the standard basis vector (i.e., a C x 1 constant vector with all entries equal  to zero, except the  m’  entry which is equal to 1). The terms 1 Xab,  simply denote the m entry of vector  Xab,  the  r  89  entry of vector  Xcd,  x, and 1  Q  and the entry in the  i  m’  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs row and  th  column of matrix  Q, respectively. From (4.18), we have:  x  Note that  T (Qab 1  Mj) 1.  (4.20)  is a constant matrix, thus eq. (4.20) is a linear equality in terms of  addition, we have, tr(Q) Step  Xdfd  Q. In  (lTu) fed.  4. We first note that:  7ab  minimize maximum ,y,f  a,b:(a,b)C  .  =  maximize minimum x,y,f  0 fabC  a,b:(a,b).C  faiC°  Here equality indicates equivalence of the two optimization problems. By defining for all (a, b)  (4.21)  ‘Yab  =  min(fab C /7ab) 0  e £, solving the right hand side in (4.21) is equivalent to maximizing S subject to  the constraint S  for all links (a, b)  e  Problem (4.13) can now be replaced by its equivalent linear mixed-integer program. Given  90  ‘ob’oa ‘on  16 3 (p’)’(q’v) ‘0< qvf  ‘.tv p’’q’v  1 { q 3 vx 0 o 1 { Q 3 v,1 } }  3J8I1M  (vL/p) qvj 5 Pf+PxuT _QVXULT  3’••  uT bT >  5  Ui  PXT+(WXT  qv UL  3’•• ‘1=’”-’- A  ‘PX T>  3’••  5  UT  UT  OT bT : ;a  ‘]  + qvj  —  qv  ‘I ‘1 > T (iv  fl  b)i  PX  —  j(p’3):p’  qv  +x  + qvf  —  q’o  j  ‘1  j;  =  =  vq  =  qv  9  (p ‘) ‘(q ‘v)  NIIM3I’V  qons pj 3 p ‘  .1OJ UtUUS jUtItJ3  ‘  ‘  o poqns OZIUJIXgUI  j p” 0 4  i  qvj  • JacItI3  pcIcIjJAQ idffrn1JcI  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs Note that the optimal solutions obtained from (4.13) and (4.22) are the same. There are effective commercial software (such as CPLEX  [311)  of them use branch-and-bound algorithm [33].  to solve linear mixed-integer programs. Most  The linear mixed-integer problem (4.22) has  L(C + I) binary variables and 2 L ( C+I+2 C + L + 1 real variables, respectively. In general, ) problem (4.22) can be solved in practice for only small-scale and medium-scale MC-WMNs. Using some heuristic algorithms similar to Algorithm 4 in Section 3.3.1, we can also approximately solve problem (4.22) for large-scale MC-WMNs.  4.3  Performance Evaluation  In this section, we evaluate the performance gain when not only the non-overlapped channels, but also the partially overlapped channels are being used. In the simulation model, the size of the network field is 1 km x 1 km. Five different random scenarios are simulated. In each scenario, the WMN consists of 15 wireless mesh routers that are randomly located in the field. The routers are equipped with 3 NICs. (i.e., I data rate (i.e., c°  =  =  3). The IEEE 802.llb with 11 Mbps nominal  11 x 106) is being simulated. Thus, up to 11 partially overlapped frequency  channels are available (see Fig. 4.1). Three of them (i.e., channels 1, 6, 11) are non-overlapped. The SINRmjn is set to be 13 dB. The value of  ic  is equal to 2. The transmission power is the  same for all routers. For each scenario, 30 source and destination pairs are randomly selected to generate UDP traffic. We obtain the global optimal solution for linear mixed-integer program in (4.22) by using branch-and-bound algorithm [33]. To evaluate the performance, two metrics are considered: 1) aggregate network capacity, and 2) bottleneck link utilization (i.e., 1/S). The simulation results are shown in Fig. 4.3. In this figure, each point is the average of  92  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs measurements for all five simulated scenarios. The dashed lines correspond to the measured performance metrics when a single channel (i.e., channel 1), two non-overlapped channels (i.e., channels 1, 6), and three non-overlapped channels (i.e., channels 1, 6, 11) are being used, respec tively. It is observed that, by using the partially overlapped channels 1, 2, 3, 4, 5, and 6 instead of using only non-overlapped channels 1 and 6, the aggregate network capacity increases by 96% and the bottleneck link utilization decreases by 20%. On the other hand, by using all partially overlapped channels 1, 2,  ...,  10, and 11 instead of using only non-overlapped channels 1, 6, and  11, the aggregate network capacity increases by 93% and the bottleneck link utilization decreases by 50%. Note that the performance improvements are achieved without using extra resources (frequency spectrum). Thus, the spectrum is utilized more efficiently when partially overlapped channels are being used.  4.4  Summary  In this chapter, we proposed a joint channel assignment, interface assignment, and scheduling algorithm for MC-WMNs when all non-overlapped and partially-overlapped channels are being used. The joint problem is formulated as a linear mixed-integer program with a few integer variables. The computational complexity is low and is feasible for implementation in practical networks. Simulation results show that there is a significant performance improvement in terms of a higher aggregate network capacity and a lower bottleneck link utilization when all the partially overlapped channels within the IEEE 802.llb frequency band are being used.  93  Chapter 4. Partially Overlapped Channel Assignment for MC-WMNs  I  I  Using 11 partially overlapped channelsj  *  40 35  93%  30  :‘  0  0  25  3 non—overlapped channels -  (channelsl6,11) . 20 z  96%  i 15  2 non—overlapped channels  U)  -  (channels 1, 6)  10  Single channel  —  —  —  —  5  (channell)  U  1  2  3  4  5  6  7  8  9  10  11  Number of Channels (a) Aggregate network capacity  1  I  I  Using 11 partially overlapped channelsi  *  0.9  —  — —  Single channel  —  —  (channel 1)  0.8  I  C,,  .2 u.7 N -  D  0.6 0.5 2 non overlapped channels  0  0.4  —  —  —  e  —  6)  3 non—overlapped  0.3  (channels 1,6, 11)  -  -  -  0.2 0.1 C  1  2  3  4  5  6  7  8  9  10  11  Number of Channels (b) Bottleneck link utilization  Figure 4.3: Performance comparison within the IEEE 802.llb frequency band.  94  Chapter 5  Utility Optimal Random Access: Reduced Complexity, Fast Convergence, and Robust Performance Different from the cross-layer designs in the previous chapters, in this chapter, we narrowly focus on the MAC problem in wireless ad hoc networks. In general, there are two major types of wireless MAC protocols: scheduling-based (e.g., in cellular systems) and contention-based (e.g., in wireless local area networks). In this chapter, we focus on the study of contention-based MAC, where wireless nodes randomly and distributively access the shared communication channel with certain transmission (persistent) probabilities. The contention-based protocols are scalable and inherently flexible, but they typically have poor performance due to insufficient feedback. For example, in IEEE 802.11 distributed coor dination function (DCF) [9], a node updates its transmission probability based on the binary feedback of its data transmission: success (no collision) or failure. This leads to low throughput  95  Chapter 5. Simple, Robust, and Optimal Random Access and unfair resource allocation. Such mechanism also cannot achieve a stable equilibrium [97]. In this chapter, we design distributed contention-based random MAC algorithms through the general framework of NUM [44]. Several related algorithms that are also proposed based on the same NUM framework include [47, 51, 56, 98, 99]. They have various performance bottlenecks due to one or more of the following: (1) extensive message passing among nodes to achieve semi-distributed implementation, (2) synchronous updates of contention probabilities that require homogeneous computational capabilities and software implementations among wireless nodes, (3) small update stepsizes to guarantee convergence with typically slow speed, and (4) supporting only a limited range of utility functions due to non-convexity. Our proposed algorithms overcome the performance bottlenecks of previous proposed NUM based random access algorithms in all four aspects. First, they only require limited message passing (i.e., signalling) among nodes. Based on the messages from other nodes, each node updates its persistent probabilities by solving a local and myopic optimization problem in an attempt to maximize the total network utility. Compared to the NUM-based random access algorithm in  [981, our algorithms can reduce the total signalling overhead by more than a factor of 10. Second, our algorithms allow fully asynchronous updates of messages and contention probabilities. They can tolerate arbitrary large and finite asynchronism and message delays and are also robust to message losses. For example, even when the packet loss rate of the underlying communication channel is down to 0.5 (i.e., on average, half of the messages are lost), our algorithms can still achieve the optimal performance within a short time. Third, in our algorithms, nodes update their contention probabilities through best response updates, thus no small stepsizes are needed in the update. This enables our algorithms to achieve a much faster convergence compared with the previously proposed subgradient-based update methods (e.g., in [47, 51, 56, 96  981). Finally,  Chapter 5. Simple, Robust, and Optimal Random Access our algorithms have provable convergence property under a wider range of utility functions, even if the NUM problem cannot be transformed into a convex optimization problem. The analysis techniques we use here are quite general and can be used to tackle other non-convex optimization problems in communications and networking. Besides the NUM-based approach, another related thread of research focuses on the analysis of random MAC algorithms using game theory (e.g., [100, 101, 102, 103]). The focus is on noncooperative interaction among nodes, while here, we focus on global network optimization. The rest of this chapter is organized as follows. The system model is described in Section 5.1. Our proposed algorithms are presented in Section 5.2. The convergence, optimality, and robustness of our proposed algorithms are analytically proved in Section 5.3. Simulation results are shown in Section 5.4. Conclusions are discussed in Section 5.5.  5.1  System Model  Consider a wireless ad-hoc network. Let A[= {1,... N} denote the set of nodes and £ = {1, ,  .  ..  ,  L}  denote the set of unidirectional wireless links. For each node n E .N, we denote the set of its outgoing links by AD 7 CL, with size L =  I.C,I.  Each node n has L separate queues, each queue  holds the packets for one of its outgoing links of node n (see Fig. 5.1). Time is divided into equal length slots. At each time slot, node n may choose to transmit on one of its outgoing links i with a persistent probability I-N. The probabilities need to satisfy ZiE prnax  <  prnax  e  < 1, where  denotes the given maximum total persistent probability. That is, node n may remain silent  in some slots. For the network in Fig. 5.1, node a has La  =  2 outgoing links, where La  =  {1, 2}.  In this node, those packets which are destined to node b are enqueued in queue 1. Similarly, the  97  Chapter 5. Simple, Robust, and Optimal Random Access — —.-  t:  7 link ,  ‘r  qucuel  / Pi (rEfjWi  nodeb  Iink2\  node a  node c  Figure 5.1: A sample wireless ad-hoc network. We have, La = {1, 2}, Lb = {3, 4, 5}, and 4 = {6, 7}. In node a, those packets which are assigned to be sent to node b (over link 1) are enqueued in queue 1. Similarly, those packets that are assigned to be sent to node c (over link 2) are enqueued in queue 2. R and T boxes represent receiver and transmitter units, respectively.  packets which are destined to node c are enqueued in queue 2. At each time slot, a packet from queue 1 is sent over wireless link 1 with probability P1, and a packet from queue 2 is sent over wireless link 2 with probability For each node n another node s  P2.  Notice that links 1 and 2 will not be active at the same time.  e N, if the receiver node of link i C 4 is within the interference range of  ) 8 e N\{n}, then any transmission by node s (i.e., transmission on any link j eL  interferes with transmissions of link i. Those nodes which interfere with transmissions of link i are denoted by set .½. For each node n  e N, let r denote the average data rate for link i e 4,  which is a function of the persistent probability vector p  rj(p)  where  ,  =  7iPifleK(l  —  =  (p,, Vi C L) of all links. We have [80):  (5.1)  denotes the fixed peak data rate for link i (i.e., the rate achieved by link i if no node 98  Chapter 5. Simple, Robust, and Optimal Random Access in set AI is active). To ensure that no link encounter starvation, for each node n and any link i E £,, we require  <P’. Here pi1 denotes the given minimum  P” > 0 and  i  persistent probability. Thus,  rj(p) >  minpmin 7  (1  —  pmax)N—1  >  o,  vi  ,  (5.2)  where  pmifl  prnax  min 7  =  7 minejP  (5.3)  =  ma  (5.4)  =  max 7  Each link i  e £ has  nEJV  P  minie%7i  (5.5)  maxjEyj.  (5.6)  a utility which is an increasing and concave function of r and indicates  link i’s degree of satisfaction on its average data rate. The utility of link i is denoted by u(rj(p)), which is also a function of the persistent probabilities p of all links. We are interested in finding the value of p that solves the following NUM problem [53]:  u(rj(p)),  max  where the feasible persistent probability region is  99  (5.7)  Chapter 5. Simple, Robust, and Optimal Random Access and the utility function is a-fair (see Section 1.2.2). That is, [50]:  (1  u(rj)  —  a)—’  if a  e  (0, 1) U (1, oo),  Vi  =  logr,  e £.  (5.8)  ifa=l,  Notice that problem (5.7) has the form of the NUM problem in (1.18) where the optimization variables are the persistent probabilities. Also recall from Section 1.2.2 that using (5.8), a wide range of efficient and fair allocations can be modeled. In particular, problem (5.7) reduces to throughput maximization with a mean fair allocation with a  =  —  0, to proportional fair allocation with a  2, and to max-mm fairness with a  =  1, to harmonic  —*  Although the objective function in problem (5.7) is concave in link rates r  =  (ri, V i  e £),  it  is not concave in persistent probabilities p due to the product form of the average rate in (5.1). Thus, finding the optimal solution for this non-convex and constrained optimization problem is difficult even in a centralized fashion. In this chapter, we propose two algorithms which are able to find the optimal solution of problem (5.7) in a distributed fashion under easily verifiable sufficient technical conditions. In comparison with the existing algorithms, our proposed algorithms do not require any synchronization, converge much faster, are more robust to message delay and message loss, and support a wider range of a values in the utility function.  5.2  Algorithms  In this section, we propose two distributed algorithms to solve problem (5.7), one for singlecell topologies in Section 5.2.1 and another one for general topologies in Section 5.2.2. In both algorithms, each node n performs a myopic and local optimization, i.e., optimizing the total  100  Chapter 5. Simple, Robust, and Optimal Random Access network utility by choosing the persistent probabilities of its own outgoing links, assuming others do not change theirs. Despite the complexity of the problem, we show that the solution of this local optimization problem can be obtained in closed form, facilitated by limited message passing among nodes and a simple local sorting procedure. Various properties of the algorithms, including convergence, optimality, and robustness, will be proved in Section 5.3.  5.2.1  Single-Cell Topology  We begin by considering a single-cell topology, where all links interfere with each other. That is, for each n  e 1V and any i e £, the interference node set JV  =  Jf\{n}. This models some  important practical wireless networks including wireless personal area networks where multiple wireless devices interact with each other over short distances in a piconet, as well as indoor wireless local area networks where several wireless devices communicate with an access point and each other (e.g., in a large conference room). We extend our work to the general topology case in Section 5.2.2.  Node n’s Local Optimization Problem We begin by considering a single-cell topology, where all links interfere with each other. That is, for each n  e A( and any i e £,, the interference node set AI  =  JV\{n}. This models some  important practical wireless networks including wireless personal area networks where multiple wireless devices interact with each other over short distances, as well as indoor wireless local area networks where several wireless devices communicate with an access point and each other.  101  Chapter 5. Simple, Robust, and Optimal Random Access Node n’s Local Optimization Problem For each node n, let p,-, Also let p  = (p,  = (p,  Vi e ) denote the persistent probabilities of its outgoing links.  Vj £\E) denote the persistent probabilities of all links other than the  outgoing links of node n. Consider the following local and myopic optimization problem: max eP  ZiEJ  pn  u(rj(pn,p_n)),  (LOCALNUM)  where the feasible persistent probability region for node n is Pn  {pn :  > O,V i  < 1,p,  e £},  (5.9)  By solving problem (LOCAL-NUM), node n can select p such that the total network utility is maximized assuming that p is fixed (i.e., none of the other nodes change their persistent probabilities). It is clear that nodes are  not selfish in this case, and they cooperate with each  other. This is necessary for achieving the optimal network performance in a distributed fashion. Although problem (LOCAL-NUM) is difficult to solve, we can convert it to an equivalent instructive representation. Its objective function in the single-cell case can be written as:  u(rj(pn,p_n))  =  (1  —  (1—  a)  ‘(flCew\{fl}(1  pi))’ 1 Z  Ze  p,, problem (LOCAL-NUM) can be  1 [z (P)  ZEJ(7iPi)  Since the multiplicative term (flCE\{fl}(1  max  —  a+  / (1— ZIEJL PI)] 3  ))‘ does not depend on the vector variable  —  equivalently written as: u Qyjp) + v(p_)  (1 —  pi),  (5.10)  where vm(p_m)  =  sE\{fl}  (1—  (Z(7iPi)’).  102  (5.11)  Chapter 5. Simple, Robust, and Optimal Random Access Since  uQyjpj)  vn(p_m)  and u(1—Z pi) are strictly concave functions with respect to p, and  is independent of Pn, problem (5.10) is strictly concave in local variable vector p. In  other words, there exists a unique optimal solution of problem (5.10) and thus problem (LOCAL NUM).  Closed-Form Solution of Problem (LOCAL-NUM) Next, we show how to obtain a closed-form optimal solution for problem (5.10). Consider a node n  e  .Af and the set of its outgoing links £. We define a permutation, j ,... , i, of the link 1  indices in set  %  such that for any  Thus, in the case of a 1 > 7i  ZL, 7  4, 2 i  Let  ci  j  and I that satisfy 1  1, we have  =  12, i 2  =  4, and i 3  =  7. On the other hand, if a  denote the smallest number in the set {0,.. . L, ,  —  e (0, 1),  L+  (/7j)_1  ci <  —  then we have  1} such that  + 1. We define the set l3  =  ,... 1 {i,-+  if condition (5.12) does not hold for any  ci  ,iL,},  with its size B  e {0,... L, ,  —  =  —  113n1  —  then it also holds  ci,  =  cr.  —  1}, then we set l3  Similarly, let c denote the smallest number in the set {0, . . . , L,prnax/prnin  (5.12)  v,(p_n).  +  We can show that (see (5.43) in Section 5.6.1), if condition (5.12) holds for ci  (0, 1), we have  and i =12. 3  /prnifl 1  for  e  and in the case of a  <  1 j  L,, we have  I  For example, let ={4,7, 12}, 74=18 Mbps, 77=24 Mbps, and 712=6 Mbps.  If a> 1, then we have i 1 =7, 1 i  j  =  {}  Notice that  and B  =  0.  1} such that  L+c  (5.13)  Again, we can show that (see (5.44) in Section 5.6.1), if condition (5.13) holds for c, then it also holds for c + 1. We define C (5.13) does not hold for any c  =  {ic+i,.  .  . , i}, and its size C,,  e {0,... L, ,  —  1}, then we set C,,  103  =  =  {}  CrLI  =  L  with C,,  —  =  c. If condition  0.  Chapter 5. Simple, Robust, and Optimal Random Access We now define An A?-,, = {ik+1,  ...  }.  =  B U C?,, with its size A,-,,  =  IATh = L  —  ,c where ,c  =  min{u, c}. In fact,  Depending on the value of v (p), either A,-,, = 13,-,, or An = C,-,, (see Section  5.6.1). Using An, the closed-form solution of problem (5.10) can be obtained as follows.  Theorem 10 For each node n e J\f, the  unique  optimal solution of problem (5.10) is p,,(p—n)  =  (fi(p—n), V j E er,,), where for each link i e £,, the mapping f(p_) is defined as  fn(p—n)  ifieA, ) m f(P— A  =  pmin  (5.14)  (prnax_Aprnin)/( /‘wn) ,  (wnnpn))]prnjn  where [x]  =  if i e  max [mm [x, a] , b] and we have: Wn =  Zje\A (1/7).  (5.15)  The proof of Theorem 10 is available in Section 5.6.1. The key is to show that fn(p—n) satisfies the  necessary  and  sufficient  Karush-Kuhn-Tucker (KKT) optimality conditions [16, pp. 244j.  Since problems (LOCAL-NUM) and (5.10) are equivalent, p,,(p—n) is also the unique optimal solution of problem (LOCAL-NUM). We notice that, regardless of the selected system parameters and the value of ti, for each i e £n\An, the upper bound is always greater than or equal to the lower bound P,-  /‘n) in (5.14) (see (5.63) and (5.75) in Section 5.6.1).  It is clear that to compute fj(pn) in (5.14), the only information node n needs from other nodes is  Vn(p_n).  If each node 8 m  node  n  =  (i  —  s  announces a message m where (jEs  (7iPi)’)  Vs  e\{n},  (5.16)  can compute vn(p_n) = Zjv\{fl}ms. This motivates us to propose our first algorithm.  104  Chapter 5. Simple, Robust, and Optimal Random Access Algorithm 5 Executed by each node n E Jf in a single-cell topology. 1: 2: 3:  Allocate memory for messages m = (mi,... mpr). Randomly choose p > 0 for each link i e L such that Randomly choose m 8 > 0 for all s e J’f. ,  p  1.  4: repeat  5:  6: 7: 8:  Transmit on outgoing link i é £,-, with probability Pi. if t e then Set A = getA(n, vn(p_n), L, 71,. 7Lj Update j = P,” for all i E A. ..  ,  (pmax_AprnirI)/(  9:  Update Pi =  10:  [-‘  /‘w)  for all i e (Wfl±Vfl;fl))]  if t E Tn,qm then  Update m = (1— ZjEPi)’ (ZE (1/7) (i/p)’). 12: Broadcast m. if a message is received then Update m. 13: 14: until node n decides to leave the network.  11:  A Distributed MAC Algorithm  Our distributed random MAC algorithm is given in Algorithm 5. In this algorithm, each node n  .Af, regardless of how many outgoing links it has, announces only a single message m. All  nodes choose the persistent probabilities of their outgoing links based on the received messages from other nodes. The persistent probabilities and messages are asynchronously updated. Let and Tn,m be two unbounded sets of time slots at which node n updates Pn and m, respectively. We assume that the asynchronism of the updates is bounded; i.e., there exists a finite H (called asynchronism measure [18]) such that:  Vti e  ,  2 t  e  1 such that t 2 —t  (5.17)  < H,  4 —t 3 E Tn,m , 4 Vt t E Tn,m such that (t ) +D 3  < H,  (5.18)  where D denotes an upper bound on communication delay (e.g., queueing or propagation delay).  105  Chapter 5. Simple, Robust, and Optimal Random Access function getA(n,v(p_), L,7 ,. ,7j) 1 SetB={}andC={}. 3: if a 1 then Set il,. Lm such that ‘y 1 4: else Set ‘11,.. ,iL such that y 1 5: forcr=0,...,L—1do 1:  .  .  2:  ..  6: 7:  8: 9: 10: 11: 12:  /prnifl 1  L+ Set Bfl={iU+1,...,iLfl}. Goto Line 10. end for for c = 0, , L, 1 do if  .  ..  ...  then  +  —  if PaX/P,in  7i2  2 y  —  —  L+c  then  Set  Goto Line 15. end for 15: Set A = 13 U C. 16: return A. 13: 14:  From (5.17), each node updates the persistent probabilities of its outgoing links at least once  during any time interval of length H time slots. From (5.18), the information used by each node is outdated by at most H time slots. We notice that H can be arbitrarily large as long as it is bounded. The exact value of H is not important and needs not be known by all nodes. Compared with the distributed MAC algorithms proposed in the literature, Algorithm 5 has several distinct features: (i) less explicit message passing is needed (e.g., in the subgradient algorithm proposed in [98], each node needs to announce two messages), (ii) asynchronous updates with arbitrarily finite delay, which minimizes the coordination overhead and allows maximum heterogeneity among wireless nodes, and (iii) does not use any stepsizes, which avoids the slow convergence problem due to small stepsizes in the commonly used subgradient methods. We note that if Algorithm 5 has a fixed point, then every node n achieves the optimal solution of problem (LOCAL-NUM) and each node will not change its persistent probability vector. In Section 5.3.1, we show that this fixed point indeed corresponds to the unique global optimal  106  Chapter 5. Simple, Robust, and Optimal Random Access  solution of the non-convex problem (5.7) under proper sufficient conditions, and Algorithm 5 globally converges to such a fixed point with fast and robust performance.  5.2.2  General Topology  Now let us consider the general case, where each node is within the interference range of an  e .Af and any of its outgoing links  i E £,  the set of nodes that interfere with link i is an arbitrary subset of all nodes, i.e., .N ç  .Af\{n}.  arbitrary subset of the other nodes. For each node n  In this case, the objective function of problem (LOCAL-NUM) can be written as:  u(r(p,p_))  =  (1—  a ‘ [iE  EJV\{n} 8 Z  ((7j fl(1  :nEJrj 5 Zjet  ZsN\{n} Zje ;ngj 8  —  (iPi flCe\{}(1  (7jPj  1 flce(  —  + (1— +  Zicii))’  —  Since the last term in the bracket does not depend on vector variable Pn, problem (LOCAL-NUM) can be equivalently written as: Ziec u (y(p)pj) + v(p,) u  (i —  ZiEJn  (5.19)  pi),  where  =  FISEJ/ 1  —  11  —  —  V  p1),  :nEArj’JiPi llcaNj\{n}’ 8 L..seJV\{r} /_.djEI  E  (5.20)  £,  V’ —  L...leIP1))  .  Notice that 7(P—n) does not represent the peak data rate of link i. We can show that problem (5.19) is strictly concave in p and has a unique optimal solution. The closed-form solution of problem (5.19) can be obtained similarly as that of problem (5.10)  107  Chapter 5. Simple, Robust, and Optimal Random Access in the single-cell case. For the outgoing link set £ of node n, we can define a permutation of  j  ,••. i, such that for any 1 link indices in this set, i ,  < /%(p_n)’.  cE(O, 1), we have y /pin 1 —  L  1, we have  In the case of c  7Z  Let odenote the smallest value in {O,.  “  j I L,,  and I that satisfy 1  .  .  In the case of  L—1} such that  ,  ,÷j(P-n)/7 P_n))’ + ‘(p_n)  +‘ <  we have  V(p_n).  (5.22)  Similarly, let c’denote the smallest value in {O,... , L— 1} such that pmax/pmin —  We define Z3  .  =  not hold for any  C  =  0  ,  . .  ,  —  with its size A’ 7  {O,... , L,  —  1}, then we set C  =  IAI  =  3 L,  —  Theorem 11 For each node n  f(p)  =  (f’(p), V i e  (5.23)  i}, with its size B  =  1}, then we set 13,,  {ic’+i,... , i}, with its size C  {0,.  z  L + c’  =  =  =  {}  ,cV where  L  with C, it’ =  =  L  =  {}  —  u’. If condition (5.22) does  with B  =  0. Similarly, we define  c’. If condition (5.23) does not hold for any  —  =  BJ  0. Given Z3 and C, we define A  min{u’,c’}.  e J\f, the unique optimal solution of problem (5.19) is p(p)  £), where for each link i  ifieA’, max  /yi(p_)a_1  with w(p_n)  =  f, the mapping f(p_) is defined as  , m1 P,’ fj(p)=  13 UC  I I  1—A’‘ pmin  ‘\  i/-,(1_,)  /  mm  (5.24)  w(p_) ,  \w(p_)+  J  otherunse  ZjEC\A  The proof of Theorem 11 is similar to that of Theorem 10. It is clear that (5.24) provides the optimal solution for problem (LOCAL-NUM) in the general topology case, which includes the  108  Chapter 5. Simple, Robust, and Optimal Random Access Algorithm 6 Executed by each node n E .Af in a general topology. 1: Allocate memory for m = (mi,,.. . , mr,,.,,) and q = (qi,... , qir). 2: Randomly choose pj > > 0 for all i e £ such that pj 3: Randomly choose m , > 0 and q 8 8 e (0, 1) for all s e J\f. 4: repeat 5: Transmit on outgoing link i e L with probability Pi. 6: if t e T, then 7: Set A’ = getA(n, v(p_), L, 7’i’• . . , 8: Set pj = P,” for all i e A’,.  1.  pmax_A! pnl  9: 10: 11:  Set  [‘‘) (W(Pfl)V(Pn))] ;:‘‘--  for all ie\A’.  if t E Tn,m then Update q = 1 p. Update m, 8= for any s n. /(7jPj flcEN\{s} 1 Inform m, 5 to all s e Inform q to all s e A/\{n} such that E j e £ to have n e J\f. if a message is received then Update m and q. until node n decides to leave the network. —  12:  13: 14: 15: 16:  single-cell case as a special case. We can define node s’ messages as  q  =  1—  Then v(p_)  and  =  ZsEN\{fl}  , 8 m  =  ZjEJS:nEJ1j  and ‘y,(p)  1/(7jpj11\{fl}qC)’.  5 for all i e £,. Message q  =  (5.25)  qs  simply  denotes the probability that node s remains silent at a time slot. Also note that for each node n  s, if there does not exist any  j e £ such that  n e  then  =  0 (i.e., node n does not  cause interference to any outgoing link of node s). Our second proposed algorithm works for any general topology and is shown in Algorithm to all nodes s whose transmissions interfere  6. In this algorithm, each node n e Jf informs  with transmissions of at least one of the outgoing links of node n. It also informs q to all nodes s whose outgoing transmissions is interfered by transmissions from node n. All nodes then choose the persistent probabilities of their outgoing links based on the received messages from 109  Chapter 5. Simple, Robust, and Optimal Random Access other nodes. In Algorithm 6,  and Tn,qm are two unbounded sets of time slots at which  node n updates p and announces q, and  for all s  n, respectively. The assumptions on  asynchronism measure are the same as those in Algorithm 5. We will show in Section 5.3.2 that for any general topology, the fixed point of Algorithm 6 also corresponds to the global optimal solution of the non-convex problem (5.7) under proper conditions. In comparison with the prior algorithms in the literature (e.g., [98]), Algorithms 5 and 6 are more robust, converge faster, and require less signalling. We further discuss the properties of our proposed algorithms in Sections 5.3 and 5.4.  5.3  Convergence, Optimality, and Robustness  In this section, we prove convergence, optimality and robustness of Algorithms 5 and 6 by using recent developments in the theory of parallel and distributed computation [18] (also see Section 1.2.3) and the theory of non-linear optimization [15, 16] (also see Section 1.2.1).  5.3.1  Single-Cell Topology  Here we study Algorithm 5 which was proposed to solve problem (LOCAL-NUM) in a single-cell topology. We first show that if Algorithm 5 has a unique fixed point, then it will globally converge to that fixed point. After that, we provide the conditions under which the uniqueness of the fixed point of Algorithm 5 is guaranteed. We also show that such unique fixed point corresponds to the unique global optimal solution of non-convex problem (5.7). We define f(p) where  f(p)  =  (f(p),  ‘v/n  e  is as in Theorem 10 for each node n. Recall that a fixed point of mapping f(p) is  also a fixed point of Algorithm 5. We can show that:  110  Chapter 5. Simple, Robust, and Optimal Random Access Theorem 12 Mapping f(p) is monotone increasing (see Section L2.3) if a  tone decreasing if a  1, and is mono  1.  The proof of Theorem 12 is given in Section 5.6.2. This enables us to show the following: Theorem 13 Assume that f(p) has a unique fixed point p*• Starting from any initial point p  e P, Algorithm 5 globally converges to The proof of Theorem 13 is given in Section 5.6.3. The key idea is to show that the monotone  mapping f(p) satisfies synchronous convergence and box conditions; thus, the asynchronous convergence theorem [18, pp. 431] is applicable (see Theorem 8). Theorem 13 is general and applies to any choice of system parameters. It only requires that mapping  f(.) has a unique  fixed point. Next, we will show that not only Algorithm 5 has a unique fixed point under mild technical conditions, the fixed point is indeed the global optimal solution of problem (5.7). Let F denote the set of fixed points of Algorithm 5. For each p we have p  =  f(p*,).  e F and any link i  E  We also let S denote the set of stationary points [15, pp. 194] of problem  (5.7). Note that all local (and global) optimal solutions of problem (5.7) belong to set S. Theorem 14 F  =  S.  The proof of Theorem 14 is given in Section 5.6.4. From Theorems 13 and 14, Corollary 1 If either S or F is a singleton set (i.e., it has one element), then Algorithm 5  globally and asynchronously converges to the unique global optimal solution of problem (5. 7). In [98], it has been shown that the set of stationary points S is a singleton set for all a  1.  They used logarithmic mapping and transformed problem (5.7) to an equivalent convex problem and showed that it has a unique stationary point. However, this transformation does not work if 111  Chapter 5. Simple, Robust, and Optimal Random Access a  e (0, 1). That is the reason the algorithm proposed in [98] does not support the a-fair utility  functions with a  e (0, 1). Here we are able to provide sufficient conditions under which the e (0, 1).  non-convex problem (5.7) has a unique optimal solution with a  Theorem 15 Consider the case where the fairness index parameter a E (0, 1).  Set F is a  singleton if the following holds:  (1  —  a  a  max \7min  ) m V  It  J  (  )  —  L/L  <1,  (5.26)  —  where  (v”, v ) m  =  {  =  mm L,  (5.27)  =  maxL,  (5.28)  nE.N  (Vmax)l/a (1+(Vm) ‘/)2  ,  <1, m if V  ,  if vmmn> 1,  (Vmin)’ (1+(Vmin)h/a)2  0.25,  r  (i m (P  ‘I’  =  pmin))  /(1 m L  —  / (pmin(l  otherwise,  —  )), m P  /pmin ) +1 m P  112  (5.29)  (5.30)  (5.31)  Chapter 5. Simple, Robust, and Optimal Random Access  i/(L/L  ZEJV  max(i/pmin 7 vmmn = (N—i) (  vrnax  =  min(i/pmin 7 (N—i) (  —  —  —  1),  (5.32)  min), 7 i)/  (5.33)  i)/7max).  (5.34)  The proof of Theorem 15 is given in Section 5.6.5. The key is to show that if (5.26) holds, then mapping  f  -norm contraction mapping (see 2 is not only a monotone mapping, but also an 1  Section 1.2.3 for the definition of contraction mapping). The sufficient condition in Theorem  15 can further be simplified in some cases. For example,  if all nodes have the same number of outgoing links, then L”  L/L = L/Lmmn = N. Thus, 2— 1/(L/Lm —1)  (  i—o  a  =  =  L for each node n, and  N/(N— 1)— i/(N— 1)= 1 and (5.26) becomes  2  (v’, vmax))  max  (-r)  1—a  <1.  (5.35)  In general, all the terms in (5.26) and (5.35), except 1, are independent of the number of nodes N. The value of Corollary 2 For any a  can be arbitrarily close to 0 if N is large enough. Thus,  e  (0, 1) and any choice of other system parameters, there exists a positive  integer N such that Algorithm 5 has a unique .fixed point if the number of wireless nodes N> N. Theorem 15 provides practical bounds on system parameters that guarantee the uniqueness of the fixed point. For example, consider the IEEE 802.lla standard where  113  7mm  =  6 Mbps and  Chapter 5. Simple, Robust, and Optimal Random Access  0.9 0.8 0.7 0.6  /  8  ,/,/  a=0.9  /  /  I  ,/  0.5 0.4 a  =  0.8  0.3 0.2  0.1 20  30  40  50  60  70  80  90  100  Number of users, N  Figure 5.2: Sufficient conditions on the upper bounds of P and lower bounds of  pmn  for a e [0.1,0.9] and N e [2, 100] when Algorithm 5 is being used and each node has one outgoing link. Solid lines represent lower bounds on pmuT and dashed lines represent upper bounds on pmax  7max  =  m and 54 Mbps . In Fig. 5.2, we plot the sufficient conditions on the upper bounds of P 6  lower bounds of  fl m  for utility parameter a  e [0.1,0.9] and number of nodes N  E [2, 100], where  each node has one outgoing link. As we caii see, the difference between the lower and upper bounds increases as a or N increases, indicating the convergence condition is less restrictive. In many cases, convergence of Algorithm 5 can be obtained even when the sufficient condition (5.26) is not satisfied. For example, it is easy to numerically verify that for N  =  2, problem (5.7) with  a E (0.5, 1) has a unique global optimal solution with any choice of system parameters. Theorems 13 to 15 together show that Algorithm 5 globally and asynchronously converges to the unique global optimal solution of the problem (5.7) when either a IEEE 802.lla supports 6,9, 12, 18,24,36,48, and 54Mbps data rates [9]. 6  114  e (0, 1) (under condition  Chapter 5. Simple, Robust, and Optimal Random Access (5.26)) or a  1 (with any system parameters). In particular, Algorithm 5 works properly under  delayed or even occasionally lost messages. To have a better understanding on how the system behaves with message loss, consider M consecutive messages announced by an arbitrary node n. The first M  —  1 messages are lost (e.g., because of collision) while the last message is properly  received by all other nodes s e .Af\{n}. In this case, all derived results will go through with a redefined asynchronism measure. Let H bounded. Considering  MH. Since H and M are bounded, H is also  =  as the new asynchronism measure, Theorems 13 to 15 can still be  applied. Thus, convergence and optimality of Algorithm 5 are still guaranteed. Interestingly, this robust behavior is accompanied with fast convergence speed as shown in Section 5.4.  5.3.2  General Topology  Consider vector mapping f’(p)  =  ç(p) is defined in Theorem 11. We 7 (f,(p), V i e hf), where f  denote the set of fixed points of mapping f’(p) by F’, which is the set of fixed points of Algorithm 6. We also denote the set of stationary points of problem (5.7) by S’in this case. Theorem 16 F’  =  8’.  The proof of Theorem 16 is similar to that of Theorem 14. If a  1, then from [98, Lemma  1] we know that stationary point set S’is a singleton set. Together with Theorem 16, we have: Corollary 3 If a  1 and Algorithrrz 6 converges to some fixed point, then the fixed point is the  unique global optimal solution of problem (5. 7). If a e (0, 1), we can use the same idea of Theorem 15 and obtain sufficient conditions to assure that the stationary point set 8’ is a singleton set. We first notice that since not all links interfere with each other, for each node n ef and any link i e £, function 115  f  may only depend on a small  Chapter 5. Simple, Robust, and Optimal Random Access  j+2 --  -  ‘•‘  i_S.  s-i  i—i  i—si—s  “  Figure 5.3: A chain topology with one hop interference.  Arrows indicate the direc tion of the corresponding unidirectional wireless links. Function f’ only de pends on , 1 ,Pi—1,Pi+2, ,pj+). Notice that -y,, -y and mfl+2,fl depend on (Pi+1,...,Pi+4), (pj5,...,pj2), (pi—4,Pi—3,pi—2), (Pi—2, Pi+1, Pi+2), (p—, Pi—2, pi+i), and (pi+i, Pi+2,Pi+3), respectively. .  subset of entries in vector  p.  .  .  For example, consider the chain topology in Fig. 5.3, where the in  terferences are within one hop. For each node i, AI on  ,Pi—1,Pi+2,  ,pj+). Notice that  {n+1, n+2}. In this figure,  =  depends on (Pi+i,  on (pj5.... ,Pj2). Thus w depends on and  depend on  =  ,i’j+) and y, depends  In addition,  (pi—4,pi—3,pi—2), (Pi—2,Pi+1,Pi+2), (pi—3,pi—2,pi+1),  .  .  .  ,Pi—1,Pi+2,  .  .  .  and  ,Pi+4). We de  {i—5,. .,i—1,i+2,... ,i + 4} as the dependency set for link i. Similarly, we can .  only if Pj appears in the formulation of =  .  .  define X for all i E £ in any arbitrary topology. That is, for any i, j  xmax  .  ,Pi—1,Pi+2,.. ,pj+).  (Pi+1,pi+2,Pi+3), respectively. Thus, v also depends on  fine set Xj  .  ft only depends  ft.  Let X  =  e £,  we have  denote the size of set  max X. As an example, for the chain topology in Fig. 5.3, xmax  =  je2  .  if and  We define  8.  Theorem 17 For any general topology, the fixed point set F’ is a singleton if c é (0, 1) and  1 where  a  ) m m A (Zmmn, Z x  <1,  (5.36)  is as in (5.29) and we have:  A  =  /pmin + 1/(1 1  116  —  ), m P  (5.37)  Chapter 5. Simple, Robust, and Optimal Random Access  zm= )‘(Lmh1  m z  =  ()‘  (mmfl —  .ymin = min( 1  —  —  1),  (5.38)  1)+ /(L— Lmin)Pmax)  (539)  pmax)N_,  (5.40)  7!max7max  (5.41)  Theorem 17 guarantees that Algorithm 6 has a unique fixed point which is indeed the global optimal solution of problem (5.7). Notice that 7Il(Wn(p_n)+/Vn(p_n))a  Zmu1  and zmax are lower and upper bounds on The proof of Theorem 17 is similar to  for any ne.Af and any  -norm 2 that of Theorem 15. The key is to show that condition (5.36) is sufficient to make f’ an 1 contraction mapping. Theorem 17 is general and applies to any topology. Given the particular topology of interest, we can further refine (5.36), e.g., as in (5.26) for single-cell topologies. We note that condition (5.36) is only sufficient as in the single-cell case. For example,we can numerically verify that for many practical topologies (e.g., chain topologies), problem (5.7) with a(0.5, 1) has a unique local (thus global) optimal solution for any choice of system parameters. Unlike mapping  f  in the single cell case, mapping  f’  here may not always be a monotone  mapping. Thus, the convergence results in the single cell case (i.e., Theorem 13) do not apply in the general topology case. On the other hand, we can also find sufficient conditions under which the convergence to the unique fixed point is guaranteed. 117  Chapter 5. Simple, Robust, and Optimal Random Access Theorem 18 For any general topology, Algorithm 6 globally and asynchronously converges to  the unique global optimal solution of problem (5.7) if  Ii aI  ) m ,Z m (z  <1  (5.42)  where A is as in (5.37) and Zmh and Z m are defined in (5.38) and (5.39,), respectively.  The proof of Theorem 18 is given in Section 5.6.6. The idea is to use the relationship between 12 and l norms to obtain a sufficient condition under which f’(p) is a weighted maximum norm contraction mapping with unit weights (see Section 1.2.3). Thus, Algorithm 6 asynchronously converges to its unique fixed point. From Theorem 16, the convergence is indeed towards the unique global optimal solution of problem (5.7). Notice that condition (5.42) is a sufficient (but not necessary) condition for asynchronous convergence. Simulation results in Section 5.4 verify that Algorithm 6 converges under a wide range of system parameters. We also notice that if (5.42) holds for some a  e  (0, 1), then (5.36) also holds for the same a. Thus, Theorem 18 also  implies Theorem 17. The exact value of asynchronism measure H is not important for any of the proofs. Following the same argument in Section 5.3.1, Algorithm 6 works properly under delayed or occasionally lost messages. In Section 5.4, we assess the optimality, robustness, and convergence speed of Algorithm 6 for several randomly selected topologies and under different channel conditions.  5.4  Simulation Results  In this section, we assess the optimality, convergence and robust performance of our proposed algorithms. In particular, we show the advantages of our algorithms compared with the previously proposed subgradient-based algorithm [98] and IEEE 802.11 DCF. 118  Chapter 5. Simple, Robust, and Optimal Random Access  5.4.1  Convergence and Optimality  We first consider a single-cell topology with N =3 nodes and L =6 links. In this network, each node has two outgoing links, one to each of the other two nodes. For all nodes n  e f, we set the  minimum persistent probability P,’’ = 0.01 and the maximum total probability P,!” = 0.99. We also set asynchronism measure H = 10. The peak rates of 6 links are ‘y  6 Mbps, 72=36 Mbps,  73=9 Mbps, ‘y =12 Mbps, 75=18 Mbps, and 76=54 Mbps. Communication delays among nodes  are up to 9 slots and the packet error rate is 0.1 (i.e., on average, 10% of the messages are lost). Fig.5.4(a) shows the trajectories of adjusted persistent probabilities and their optimal values when a  =  2 (which is greater than 1). In this case, the global optimal persistent probabilities  0260 *[ 2 1 . . 0 , , 6 ]T. 09 o. l 1 8  We see that Algorithm 5 converges to  *  within less than  300 slots, even with communication delay and message loss. Similar results when a is less than 1) are shown in Fig.5.4(b). In this case, p= [0.06,0.21,0.07,0.09,0.18,  =  0.6 (which  381 • 0 T  Again,  we see that Algorithm 5 asynchronously converges to p within 320 time slots. Next, consider a chain topology with N  =  3 nodes and L  =  4 links (see Fig. 5.3). Note that  this belongs to the general topology case, where each node only interferes with a subset of other links. For all nodes n =  36 Mbps, 73  =  m e .A/, we set P,”  9 Mbps, and  4  =  =  0.01 and  prnax  =  0.99. We also set,  ‘  =  6 Mbps,  12 Mbps. Communication delays, asynchronism measure,  and packet loss rate are the same as the previous experiment. Simulation results when a a  =  72  =  2 and  0.6 are shown in Fig. 5.5(a) and Fig. 5.5(b), respectively. We see that the optimal persistent  probabilities are achieved very fast and Algorithm 6 can tolerate both delay and message loss.  119  Chapter 5. Simple, Robust, and Optimal Random Access  (a)  >  2  0  a) C,)  a)  Time Slot (b)  U)  a) Cu  2 ci) Co U) I-  ci)  0  0  100  300 200 Time Slot  400  500  Figure 5.4: Simulation results for a single-cell topology with three nodes and six links. Algo rithm 5 is being used. (a) a = 2, (b) a = 0.6.  120  Chapter 5. Simple, Robust, and Optimal Random Access  (a)  U)  a) .0 Cu .0  2  0  C  a)  U)  a)  0  Time Slot (b) 0.9 0.8  0.5 0.4 0.3 0.2 0.1 500 Time Slot  Figure 5.5: Simulation results for a chain topology with three nodes and four links. Algorithm 6 is being used. (a) Utility parameter a = 2, (b) Utility parameter a = 0.6.  121  Chapter 5. Simple, Robust, and Optimal Random Access  5.4.2  Signalling Overhead  High signalling overhead is a critical problem for algorithms which require cooperation among nodes in a wireless ad-hoc network.  In this section, we compare the signalling overhead in  our proposed algorithms with the subgradient-based algorithm [98]. In the simulation, wireless nodes are randomly located in 100 m x 100 m and 1 km x 1 kin fields for single-cell and general topologies, respectively. The communication and interference ranges are 150 m and 300 m, respectively.  Each node has an outgoing link to any of its neighboring nodes within its  communication range. The peak transmission rates (i.e.,  for all i  e  are selected randomly  between 6 Mbps to 54 Mbps. Utility parameter c is set to 2 which models harmonic mean fair allocation. We assume that each message value requires two bytes. Thus, the signalling overhead for each algorithm is defined as the total required message exchange (in KBytes) that the algorithm needs before it reaches the corresponding optimal solution of problem (5.7). Results for single-cell and general topologies when number of nodes N varies from 5 to 30 are shown in Fig. 5.6(a) and Fig. 5.6(b), respectively. We see that increasing the number of nodes increases the signalling overhead. However, Algorithm 5 and 6 manage to reach the optimal solutions via much less signalling. Compared to the subgradient-based algorithm and when N  =  30, Algorithm 5 and  Algorithm 6 reduce the signalling overhead by 1120% (from 55.2 KByte to 4.5 KByte) and 810% (from 111.3 KByte to 10.8 KByte), respectively. Notice that one reason for the superiority of our algorithms is their faster convergence. In addition, Algorithm 5 reduces the message amount by half, which also contributes to reducing the signalling overhead.  122  Chapter 5. Simple, Robust, and Optimal Random Access  (a) 60 ci)  .4-’  -o  0 ci) -C 4— ci) >  C  0) C 0 C 0) Cl)  Number of Nodes N (b) 100 ci)  4-  >‘  80 -o  0 ci) -C ci) >  C  60 40  0) C 0 C 0) Cl)  20 15 20 Number of Nodes N  Figure 5.6: Comparison between our proposed algorithms and the subgradient-based algorithm [98] in term of signalling overhead when the number of wireless nodes varies from 5 to 30. Each point represents the average results from simulating 10 random topologies. (a) Results when the simulated topologies are single-cell and Algorithm 5 is being used, (b) Results when the simulated topologies are general and Algorithm 6 is being used.  123  Chapter 5. Simple, Robust, and Optimal Random Access  5.4.3  Robustness  Since the underlying communication channels are not ideal, transmitted messages by MAC pro tocols may be delayed or even be lost. In this section, we show that our proposed algorithms are robust with respect to both message delay and loss. The simulation model is the same as that in Section 5.4.2. We only consider general topologies with N  =  30 nodes. Results for single-cell  topologies are similar. The simulation time is set to 50, 000 time slots. For each algorithm, we measure the optimality of the achieved network utility at the end of each simulation run. First, we assume that the communication delay varies from 10 to 50 time slots. Results are shown in Fig. 5.7(a). We see that by increasing delay up to 50 time slots, the subgradient-based algorithm leads to 8.4% optimality loss while Algorithm 6 can always find the exact optimal solutions. Notice that although Algorithm 6 is robust to communication delay, higher delays can cause more iterations for the algorithm to converge. In our simulation model, Algorithm 6 converges to the optimal persistent probabilities on average after 421, 1581, 3641, 6472, and 9923 time slots when communication delay is 10, 20, 30, 40, and 50 time slots, respectively. Convergence time versus the communication delay is illustrated in Fig. 5.8. Next, we consider the effect of message delay when the packet error rate varies from 0.1 (i.e. 10% of the messages are lost) to 0.5 (i.e., 50% of the messages are lost). Results are shown in Fig. 5.7(b). We see that Algorithm 6 is robust to message loss. On average, it converges to the optimal persistent probabilities after 312, 473, 531, 629, and 727 time slots when packet error rate is 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. Notice that both Algorithm 6 and the subgradient-based algorithm are less sensitive to message loss compared to long delays.  124  Chapter 5. Simple, Robust, and Optimal Random Access  (a)  > Cu  2  ci  0  90 0  10 20 40 30 Communication Delay (Time Slots)  50  (b)  >.  cu 2 ci 0  0.2 0.3 Packet Error Rate  Figure 5.7: Comparison between Algorithm 6 and the subgradient-based algorithm [98j in term of robustness with respect to communication delay and message loss. Each point represents the average results from simulating 10 random general topologies, each including 30 nodes. (a) Optimality in percentage when maximum communication delay varies from 10 to 50 time slots, (b) Optimality in percentage when packet error rate varies from 0.1 to 0.5.  125  Chapter 5. Simple, Robust, and Optimal Random Access  1)  I 0 C ci) 0) ci) > C  0  0  30 Communication Delay  Figure 5.8: Convergence time in presence of various communication delays.  5.4.4  Comparison with IEEE 802.11 DCF  Since our proposed algorithms achieve the global optimal solution of problem (5.7), they establish a performance upper-bound for all MAC algorithms that are designed to solve the same problem. On the other hand, they can resolve some of the existing problems in the current 802.11 DCF, e.g., its well-known short-term fairness problem due to binary exponential backoff. Next, we compare Algorithm 6 with DCF in terms of both system throughput and Jam’s fairness index [89] when N = 30. The short-term fairness is obtained using a sliding window of size 200 slots. Results when utility parameter a varies from 0.5 to 5 are shown in Fig. 5.9. Each point represents the average results from simulating 10 random general topologies. We see that, parameter a acts as a knob in Algorithm 6 to control the tradeoff between efficiency and fairness. By increasing a 126  Chapter 5. Simple, Robust, and Optimal Random Access we can make the system more fair but less efficient (and vice versa). If a  =  0.5, then the system  throughput is 54.9% higher than DCF (see Fig. 5.9(a)). Besides, for any choice of a  e  [0.5, 5],  the fairness is much better than DCF (see Fig. 5.9(b)).  5.5  Summary  In this chapter, we designed two distributed contention-based MAC algorithms to solve a NUM problem at link-layer in wireless ad hoc networks. Both algorithms globally converge to the unique global optimal solution of the NUM problem under mild technical conditions on the system parameters. In particular, it is true for a-fair utility functions with a  e  (0, 1), in which case  the NUM problem cannot be transformed into a convex optimization problem using logarithmic change of variables as in the previous literature, and thus, it is already challenging to prove the uniqueness of the optimal solution before designing any algorithm. Besides supporting a wider range of utility functions, our algorithms have several other ad vantages over previous algorithms, including less message passing, fully asynchronous updates, robustness to message delays and losses, and fast convergence. Simulation results show that our algorithms converge faster than the recently proposed algorithm  iii  [981 with significantly less  signaling overhead. They are also robust to message delays and losses due to channel errors. Moreover, they achieve better efficiency-fairness trade-off compared to the IEEE 802.11 DCF.  127  Chapter 5. Simple, Robust, and Optimal Random Access  (a) .  80 Algorithm6  <20 0.5 0.6 0.70.80.91  2 Utility Parameter a  3  4  5  3  4  5  (b)  xa) -  U) U)  Cu U-  0 0.5 0.6 0.7 0.80.91  2 Utihty Parameter a  Figure 5.9: Comparison between DCF and Algorithm 6 in terms of system throughput and fairness index when utility parameter a varies from 0.5 to 5. Each point represents the average results from simulating 10 random general topologies, each including 30 nodes. (a) Aggregate network throughput, (b) Jam’s fairness index.  128  Chapter 5. Simple, Robust, and Optimal Random Access  Analytical Proofs  5.6  Proof of Theorem 10  5.6.1  Lemma 6 For each node n E /, we have:  /pflifl 1 —  Bn  —  Jjmax / ,min  ‘n  fin  VjEBn,  (5.43)  /lWn,  ViéC,  (5.44)  V i E £n\Bn,  (5.45)  ViEn\Cn.  (5.46)  —  /prnifl 1 —  pmax / pmin I’n  ‘ri  -‘(w+ vm(P_n)),  B>  (w + Vvn(P-n))  /‘Wn,  n  —  /‘  Proof: From (5.12) and knowing that £n\An  n  —  B <Zje\A =  (7i+i/7j)’  ) 1 (///F  +  . 15 {i  .  .  , i,}, for each link i  e  13,. we have:  7Vn(p_n)  (iEfl\Afl  yi/yj)  a—I  (5.47)  +  (w +  where the last inequality comes from the fact that that  i  =  min{J, c} and  k+l  Assume that there exists i  E ç/ for all i E L3.  Recall  Proof of (5.44) is similar. We prove (5.45) by contradiction.  e £n’ytn  such that (5.45) does not hold. It is clear that  129  6  u. We  Chapter 5. Simple, Robust, and Optimal Random Access have:  /prnifl 1  1 (z=  L+a  —  -L+(- 1)+S- ( 1)  1/prnifl  —  L+e  —  1  (1/i)’ + vn(p_n))  + Vn(Pn))  <(  <  a  —  +Ze()’  1 for all I  =  (549) (5.50)  (/yl)’ + /ijvn(p_n),  <  where the last inequality results from the fact that /‘(-yj/’yj)’  (5.48)  e,.  . .  ,S which implies  e + 1. Comparing with (5.12), inequality (5.50) implies that e C  which contradicts the assumption that i e  Proof of (5.46) is similar.  Since the objective function of problem (LOCAL-NUM) is strictly concave in p,,, its solution is unique [16, Pp. 137] and satisfies the following necessary and sufficient KKT conditions 7 [16, pp. 244]:  Vie,  (5.51)  <prnax  (5.52)  vn(p_n)  i 7  z’ )‘.  (  —  (Z  —  s, (prnin 0,  where  =  •  —  V i é £n,  prnax)  pj)  =  =  (5.54)  0,  0,  Si > 0,  Vie £,,,,  (5.55)  e £,  (5.56)  Vi  denotes the Lagrange multiplier corresponding to constraint  denotes the Lagrange multiplier corresponding to constraint pj 1t is clear that Slater’s condition [16, 7  (5.53)  —iEPt)  pp.  226] always holds as long as  130  pj  p,m for each i <prnax  P’ and S £,. Now we  for all nodes n E  N.  Chapter 5. Simple, Robust, and Optimal Random Access  need to show that (5.14) leads to (5.51) to (5.56). Condition (5.51) directly results from (5.14). We also have:  z  f()  A  pmin  ri1 71 11  —  + ZiE\A prnax  A  =  17mm  n  +  n  —  A  —  pmin  n  wn  AThP,’)  ) 71 /( /w  (5.57) by (&15) prnax /(1/7)_1)  (z  Thus, condition (5.52) also holds. Two cases are possible: Case I:  Z f()  Thus, there exists 1  (1/  <prnax  which only happens if Zie\A f(p ) <Pmax 71 71  —  A  17mm  71 such that £ 71 e \A  ç/’) (i (i  —  —  + vn(P_n)) < 71 Pmmn)/(w A nfl ) (p_ 71 71 /v A + 71 P’)/(w  )  <  prnax  n  prnax  —  A  Pmin/(  fl ‘ 7 1 j  —  I  A  —  71 Vtn(P_n)) m A + 71 P)/(w <(P n  (5.58)  71 pmin/ ‘ f 1 lri I  Multiplying both sides of this inequality by /(i/7)’, for any link i  (1/ ‘) (i  K’wn),  A  e  71 we have £ \A,  min)/( 1w) 17  —  Hence,  ) —max (Prnmn (1/ _1) (1_A 71 f(p_ 71 ,Vi é \A v ))) 71 + (p_ P)/(w 71 71 £ . 71  In this case, we have c  <ç  and  i  =  min{, c}  =  o. In other words, A 71  131  =  71 l3 and A  =  (5.59)  71 We B .  Chapter 5. Simple, Robust, and Optimal Random Access prove this by contradiction. Assume that u > ç and  1  —  (L  —  ç)prnin  by (5.58) prnax  + /Vn(p_n)  i =  —  c. We have:  (L  —  c)Pmm  /7ii)’  >=i  by (5 13) < ç/Eiprnin —  Thus, /pin 1 —  L+c  )’ + 1 (z (1/7j 1  <  Comparing (5.60) and (5.12), we have counter assumption u> c. Thus,  k  = cT  (5.60)  e 13,. This implies that o <ç which contradicts our and we have:  A  = t3,  =  B.  (5,61)  We can verify that conditions (5.53)-(5.55) hold for =f(p) Th if we set A,=0, and p  =  {  /  vn(p_n)  (1_AflPflm1n_Efl\Afl f(p_))  —  0,  132  1  (5.62)  Chapter 5. Simple, Robust, and Optimal Random Access Next, we need to show that (5.56) holds. For each i  (1  by (5.59) >  v.,,, (p) 0mm  —  A ni  Ti  —  e A, ,, we have: 7  Zje\A  vn(P_n)  f(_))  1/7’  (1—APin)  jE\A  Wn+ \/Vn(p_n)  a by (5.15)  by (5.43) > and (5.61)  Thus, S  >  0 for all i  (  1/P_A /‘(i—  0min’  A  TL 1 .Ti  .‘  e C,., and (5.56) holds. Finally, for each i e £,.,,\A we have:  prnax  71 A P,  a!  c/F1  7 V  A  1  by (5.58) >  0mm  and (5.45)  prnin  (5.63)  by (5.61)  (w+ /vn(P_n))  which guarantees that the upper bound in (5.14) is indeed greater than the lower bound. Case II: ZiE  f()  —  pmax  _-‘n  which happens only if  ZiEL\A  f(p_)  =  pmax Ti  —  rmin A fl 1  (5.64)  We already know that  (prnax Ti  —  AP)/( /F’w)  133  by (5J5) prnax Ti  —  A fl pmin ii  (5.65)  Chapter 5. Simple, Robust, and Optimal Random Access From (5.64) and (5.65) and since  (P  f(p)  f(p_)=(P  —  AP’)/( ç/F’w) for all  e  AP)/( ‘w), Vie  —  (5.66)  From (5.14) and (5.66) we have  (i  —  AThP,in)  In this case, we have o-  >  /(w +  c and  t  (P’  c/vn(p_n))  min{a,c}  =  =  1 Z  (L  —  a)Pifl by (5.67) <  ‘(l/7j)’  1  1 Z_  —  —  (L  —  , =  +  =  C,, and A  =  C.  a. We have:  a)Pmifl  /(1/7i)’  (5.67)  AnP,”)/wn.  c. In other words, A  We prove this by contradiction. Assume that a <c and  —  —  by (5.12) <  /vn(p_n)  (5.68)  —  Thus,  prnax/prnin —  )’). 1 (z (i/7 1  L+a  Comparing (5.69) and (5.13), we have i 1 e C. This implies that a counter assumption a < c. Thus,  i  =  ç which contradicts our  c and we have:  A=C.  We can verify that conditions (5.53)-(5.55) hold for each p  A=  max  (Pn  (5.69)  ‘a w \ Pmin An ri /) 1  —  134  —  (5.70)  =  ) if we set 7 f(p_  prnax (/))a  1  —  (5.71)  Chapter 5. Simple, Robust, arid Optimal Random Access and \  /  =  I  i\p_Anp)  /  _I  1 Tprnin)  ‘feA. (5.72)  0,  ifiEL\A.  Next, we need to show that condition (5.56) holds. From (5.67) and by reordering,  (w, + /vn(p_n))(P  w(]. — —  AP,”)  /v(p_)/(l  —  —  AP”)  ). m P  (5.73)  On the other hand, from (5.44) and (5.70) and by reordering,  —  ) > 1 AP  Replacing (5.73) in (5.71), we have ) have 5  0 for all i  1/(  /ipmin)  (5.74)  0. On the other hand, replacing (5.74) in (5.72), we  e £,. Thus, condition (5.56) holds. Finally, for each i e r  (prnax — AP,’ )/( /7’wn) 1T1  we have:  and (5.46)  (5.75)  tm1 P ’, by (5.70)  which guarantees that the upper bound in (5.14) is no less than the lower bound.  5.6.2 If c  Proof of Theorem 12 1, then for any i, i  e P such that  —Zjecsj 1  j3 —< j5, we have: 1 j 3  —jEpj, 1  135  , for all 3 j  j e £. Thus,  eJf,  (5.76)  VjL  (5.77)  Vs  Chapter 5. Simple, Robust, and Optimal Random Access  From (5.76) and (5.77),  (j3_) 8 m  é .A,  ms(j5_),  V  v(j5_),  V i E £,  (5.79)  w + /v(_)  V j  1,  (5.80)  fQj-),  V i e iZ.  (5.81)  Thus, vector function f(p) is a monotone increasing mapping. If a  1, then the sign of the  v(j5_) w + /v(_)  S  (5.78)  inequalities in (5.78)-(5.81) is reversed and f(p) becomes monotone decreasing.  5.6.3  Proof of Theorem 13  Since H is bounded, the local memory of each node n e Al is updated infinitely often as t  —*  Thus, the total asynchronism assumption [18, pp. 430] holds. First consider the case with a Let  pmin  and  Since f(p*)  pmax  z*  denote two N x 1 vectors withall entries equal to  pmin  .<  <  plflX,  fl m p  Note that, for each n  f  for k  and  pmax  1.  respectively.  and f(.) is monotone increasing, we have:  ifl) m f(p  e Al and j e ,  Let fk(p) denote the composition of and applying  pmin  00.  <  <  X) .< m f(p  f(prnin) > prnifl  >  prnaX•  pmin  and  f with itself k times, where f°(p)  (5.82)  f(prnax)  =  <  prnax  < prnax  p. Starting from (5.82)  0 times, we have:  fk(pmin)  fk+l(pmax) < fk( max). 13  fk+l(pmin)  136  (5.83)  Chapter 5. Simple, Robust, and Optimal Random Access We also define: (5.83), pk+2  pi  e  pk  pk+1 ç pk  pk pk,  =  for all k  That is, then  {p: fk(pmin) <  pk--2  >  c  < fk(,,max)}  If  e  pk,  then f(p)  0. We can also show (by contradiction) that if  pk  Thus, lim  (pi, ,Pi—1,Ji,Pi+1,” ,pL) E  fk(pmin)  pk  =  for any k  lim÷ >  fk(pmaX)  e  pk+1  pk  =  p*•  From  {p*}, then Besides, if  0. Therefore, both synchronous  convergence and box conditions [18, PP. 431] hold. From asynchronous convergence theorem [18, , Algorithm 5 will converge to p. Since P ç P°, the proof 0 pp. 431], starting from any point in p for c  1 is complete. Now consider the case with c  <  1, from Theorem 12, the mapping  f(.)  is  monotone decreasing. We have:  fl m p  fl) m f(p  f(prnax)  Comparing with (5.82), the orders of  f(prnifl)  prnax  (5.84)  and f(p ) is exchanged in (5.84). Applying m  once more, we have:  f(prnax)  (pffix) 2 f  (pmin) 2 f  f(pmin)  By mathematical induction, we can show that  fk(pmax)  fk(pmin)  < fk+l(pmin) .< .<  fk+l(pmax)  For each (either odd or even) k  pk  =  .<  *  .< fk+l(pmax)  fk(pmin)  .< fk+l(pmin) ...< fk(pmax),  if k is odd, if k is even.  0, we define:  fk(pmax)] {p: min[fk(pm), - p -<  137  max[fk(pmmn),  fk(pmax)]}  f(.)  Chapter 5. Simple, Robust, and Optimal Random Access The rest of the proof follows the case of a  1.  Proof of Theorem 14  5.6.4  For each 3*  e F and each link i e f, Theorem 10 states that j3 is the unique global optimal  solution of convex problem (LOCAL-NUM), and satisfies the KKT conditions in (5.51)-(5.56). We denote the corresponding Lagrange multipliers by =  (,  V n E f) and  *  (Sr, V i  =  and  for all i  £,. We also define  e L). On the other hand, since any local optimum j3  E S is  a regular point [15, pp. 315] of non-convex problem (5.7), and so must satisfy the KKT necessary conditions [15, Proposition 3.3.1], for any i E 1V,  Pn,  i:  Vme/, ieI,  -* pmax LdiEJPi < n  V’  Vn  —  1  (  v(p)  1  ‘  (5.86)  V n E J\f, i E £,,  —  —  e .Af,  —  (5.85)  (5.87)  —  * (ç’ “n V—diEL  Pi  —  pmax’\ j ‘n  —  ii  V  —  *(pmin —jj—v, n  ?7  e .N,  V n E .Af, i  vi  Vi  A0, S0,  (5.88)  e £,,  e £,  (5.89)  (5.90)  where n(-n)  We define, A*  =  (,  =  (flSEN\{fl}(1  V n E .A/) and  >0.  —  *  ‘v/i  =  138  £) Jf  (13*A**)  (5.91)  satisfies the KKT  Chapter 5. Simple, Robust, and Optimal Random Access conditions in (5.51)-(5.56) for all wireless nodes n  e Al, then the tuple:  (13* (/y(13),Vn eAl), (&/y(13),Vm e Al, i e  (5.92)  satisfies (5.85)-(5.90). On the other hand, if tuple (3*, A”, o*) satisfies the KKT conditions in (5.85)-(5.90), then the tuple  (13* (*yn(13*)VTh cAl), (y(13),Vn cAl, i e  (5.93)  satisfies conditions (5.51)-(5.56) for all n E Al. The former implies that F C S while the later implies $  5.6.5  c F.  Thus, set F  =  S.  Proof of Theorem 15  For any p  e P, the Jacobian J(p) is defined as an L x L matrix whose entry in row i and column  . Consider node m e Al and each i e £,, we have: 3 j is c9f/8p ifje,  0,  J(p)  =  ) m i(1—AP  (5.94)  m ‘  PjYj  )  Pj  pmax  This is under the assumption that f E  —  )  if k  AThP)/  c £.., s  m. If f is chosen  to be one of the boundary points, then the corresponding entry is 0. Thus, (5.94) provides  139  Chapter 5. Simple, Robust, and Optimal Random Access upper-bounds (in absolute values) on the entries of the Jacobian. We can show that:  I1JII  =  max  zE,nEN  I )  —  pminV’  Pk’’ 3 (1—k  )  2 Pj7  18\{n} I  1—a iEinEf  1—  A  + (7ivn(p_n))  (1+Qyjv(p_))’)2  tm + f’_L Jj_pmaic  I  £  \1kEPk  I + Pj  vn(p_n))2  pmin 1  tm v (v’’, a). 0 (5.95)  Note that 1  —  <  1 and  >  1 for all i E £\A. Function xh/a/(1 +  is always non-negative and less than 0.25. It has a unique maximum at x is monotonically increasing for 0 approaches zero as either x  IJ(p) Iii  —*  < x <  0 or x  —*  a  1. The function  1 and monotonically decreasing for x  00.  >  1. Its value  We can similarly show that:  tm v ax)  1—a  =  2 xh/a)  max  1—cr  (  (p)  1 —  L/L  (5.96) —  From (5.26), (5.95), and (5.96), we have [18, pp. 635]:  IIJ(p)112  0 IJ(p)IIi \/IIJ(p)I1D  (5.97)  < 1.  Let j3, j5 E P. From (5.97) and by Cauchy Schwarz inequality,  IIf(i3)  —  ) 112 3 f(i  2 lIP 13112 IIJ(p)11 —  where p is a convex combination of j5 and 3. Thus,  5.6.6  f  <  lIP — P112,  is 1 -norm contraction mapping. 2  Proof of Theorem 18  Following the same argument as in Section 5.6.5, condition (5.36) implies that for each p E 2, we have:  iIJ’(p)II < 7 v’ 2  1 where J’(p) denotes the Jacobian matrix of f’(p). From linear algebra,  we also know that for each N x 1 vector a, we have  140  00 hall  hall2  . 00 v’NIlalI  Let  e  P.  Chapter 5. Simple, Robust, and Optimal Random Access Using the Cauchy-Schwarz inequality,  IIf’(i)  —  f’(i)II  IIf’(i5)  —  f’(13) 12  2 Il IIJ’(p)11  Condition (5.36) guarantees that function  f’(•)  —  is an  P112 <  lIP pII2/V’ —  II iII. —  norm contraction. Thus, it asyn  chronously converges to its unique fixed point. From Theorem 16, this implies that Algorithm 6  converges to the unique global optimal solution of problem (5.7).  141  Chapter 6  Utility Optimal Random Access: Optimal Performance without Frequent Explicit Message Passing In the existing contention-based MAC protocols, there is a tradeoff between system performance (e.g., throughput and fairness) and the amount of explicit message passing required among wire less users. One example is the IEEE 802.11 DCF, where users do not explicitly exchange any message related to their transmission probabilities and adapt their transmission probabilities only based on the binary implicit feedback from the network (e.g., collision or not). In this chapter, we use “messages” to denote control signals that are explicitly related to users’ transmission probabilities. IEEE 802.11 DCF does not have any explicit message passing, although it has various other control signals (e.g., RTS/CTS/ACK). Lack of message exchange typically leads to low throughput and unfair resource allocation [97]. On the other hand, several MAC algorithms (e.g., [47, 51, 98]) have been designed based on the framework of NUM which lead to the op timal system performance without taking the signalling overhead into account. However, these algorithms require extensive frequent message passing among users. Considering the fact that any message transmission leads to additional contention in a random access network, this chapter 142  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  aims to address the following question: Is it possible to design a MA C algorithm that can achieve the optimal performance without frequent explicit message passing? We provide a positive answer to the above question in some special but important cases, based on the NUM-based MAC algorithms we proposed in [104]. Compared with the previous algorithms (e.g., [47, 51, 98]), the algorithms in [104] support a wider range of utility functions, converge faster, and allow fully asynchronous operations among users. However, frequent explicit message passings are still needed in [104]. In this chapter, we show that in the simple case of a single-cell interference topology (e.g., as in wireless personal and local area networks), we can completely eliminate the need for frequent message passing. Users will be able to estimate the required information through local observation of the channel contention history. We prove the convergence of our algorithm under various channel conditions. If the channel is perfect and the estimations are asymptotically accurate, then the optimality of the algorithm is also guaranteed. The estimation techniques we use here are related to [100, 105]. However, our estimation model is more elaborate and captures more information (i.e., each user’s transmission probability). Simulation results show that our algorithm is robust to changes in user populations and channel conditions. These encouraging results provide important insights and useful hints to design fully distributed utility optimal MAC algorithms without frequent explicit message passing for more general topologies. The rest of this chapter is organized as follows. The system model is described in Section 6.1. Our algorithm is presented in Section 6.2. Convergence and optimality are proved in Section 6.3. Simulation results are shown in Section 6.4. A summary is given in Section 6.5.  143  Chapter 6.  6.1  Utility Optimal Random Access without Frequent Message Passing  System Model  Consider a single-hop wireless ad-hoc network with J\f  =  {1,..  .  ,  N} as the set of wireless links.  Each link, together with its dedicated transmitter and receiver nodes, is called a user. A sample network with 3 users is shown in Fig. 6.1. We assume that each user’s receiver node can hear every other user’s transmissions. Thus, each user interferes with all other users. This models some important wireless networks including wireless personal area networks where wireless devices interact with each other (e.g., in an office) and indoor wireless local area networks where the nodes interact with each other and an access point (e.g., in a large conference room). Time is divided into equal-length slots. At each slot, user i transmits with probability  i  e  =  X] m P  with  0< Pmin <Jm < 1. A transmission is successful only if it is the only transmission in the current slot. Let r denote the average data rate for user i. We have [80]: rj(p) where p  =  (p,, Vi  e £)  =  ‘yjpj fJje-\{j}(l  —  (6.1)  Vi é J\1,  pj),  is the vector of all users’ transmission probabilities and  fixed peak data rate for user i. Each link i  denotes the  £ maintains a utility which is an increasing and  concave function of r and indicates link i’s level of satisfaction on its average data rate. The utility of link i is denoted by uj(rj(p)) which is also a function of p. We are interested in finding the value of p that solves the following NUM problem [44, 53]: max  where P (1  —  =  {p : pj  e  P, Vi  e .Af},  (6.2)  uj(rj(p)),  and the utility functions are a-fair [50]. That is, uj(rj(p))  1 if a é (0, 1) U (1, co), and uj(rj(p)) a)’rj(p)  =  1ogr(p), if a  =  1. In [104], we have  shown that the a-fair utility functions can model a wide range of efficient and fair allocations.  144  =  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  Figure 6.1: A single-hop wireless ad-hoc network with N = 3 users. Each user includes a wireless link and its dedicated transmitter and receiver nodes.  6.2  Algorithm with No Explicit Message Passing  Local Optimization For each user i, consider the following local optimization problem:  uj(rj(pi,p_j)),  max where pj  =  (pd, Vj  e .A/\{i})  (6.3)  denotes the transmission probabilities of all users other than user  i. To solve problem (6.3), user i will choose pj to maximize the total network utility,  assuming  that none of the other users change their transmission probabilities.  Theorem 19 For each user i  e .A/,  the unique global optimal solution of problem (6.3,) is p(pj)  =  ), where the mapping function f(p) is defined as 1 f(p pmax  f(pj) Here [x]  =  =  [1/ (1+ vj(pj))]  max [mm [x, a] , b] and vj(p_j)  =  1 Yi  ..  ZjEN\{i} (1/7)’ (l/p  (6.4)  —  1)’.  The proof of Theorem 19 is similar to that of Theorem 13 and is omitted for brevity. It is clear that if user i wants to compute (6.4), the only information it needs from other users is vj(p_).  145  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  If each user i can estimate the value of m then it can compute  =  (1/7)’ (l/p  vj(p_j)=7j’  1)’,  —  Vj  umax  mifl)a_1(l/pmifl 7 / 1 (  7=minjEf7, and =  —  1)_1 where  max=mjeN j 7 .  max)a_l(1/Pma 7 (1/  its transmission probability  —  i  (6.5)  Zjeg\{} m and set p=f(p). Notice that for each je/,  3 is bounded between Mmifl and MrnaX. If a> 1, then m and  e f,  =  pmin =  m)_(1/Pmax 7 (1/  ’, m11 minéj-P  pmax  —  =  mi1_1(l/Pmi11 7 If a<1, then Mmin=(l/  —  1)1  and  1)—’. As shown in [104, Section Tv-A], if each user i updates  accordingly to (6.4), then the whole system will converge to the  optimal solution of problem (6.2). The key question is how to obtain the values of m for all  j  i. Next, we answer this question through local observations of the shared channel.  Learning from Contention History From (6.5), we see that only the values of ‘yj and Pi are required to calculate the value of m. Notice that a is the same for all users. The value of the peak rate yj depends on the channel gain between the transmitter and receiver of user  j; thus, it can only be measured by user i and  then announced to the whole network once user i joins the network. The remaining task is to determine how to obtain the value of p . 3 From user i’s viewpoint, any time slot falls into one of the following possible states: idle (no user transmits), busy (at least one other user transmits), success (user i transmits successfully), le, 4 and failure (user i transmits but it fails). Let p  JUSY  of experiencing these four states, respectively. Also let  146  pucc,  and  denote the probabilities  denote the packet error rate of the  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  channel from the transmitter node of user pdle  busy  j to the receiver node of user  i. We have: (6.6)  =fl.El—Pj) =  (1  fljev\{i}(1  —  (1  pj))  —  —  pj)  flj(1  —  —  (6.7) (1  —  dle,  —  pUCC  (6.8) idle (1  —  —  1_pi  Pi  pfail =  —  err 1’  ‘.  FIjE.Ar\{i}(1  —  pd))  ±  pr —  (1  —  fljer\{i}(1  —  p))pr)  (6.9) Pi  idle)(l —  p[).  —  From (6.6), the channel is idle if all users are silent. We note that for each user probability of being silent is (1  —  j  E .N, the  ps). From (6.7), the channel is busy from user i’s viewpoint if  user i is silent (so that it can sense the channel) and at least one other user  j  i is transmitting  packets. The former has probability (1 —p) while the latter has probability (1— flje\{i} (1 pj)). Multiplying the two terms and after reordering, the probability in (6.7) is resulted. From (6.8), the transmission from the transmitter node of user i to the receiver node of user i is successful if the transmitter node of user i transmits the packet, no other user  j  i transmits any packet at  the same time, and the transmitted packet is not corrupted. Notice that the latter happens with probability (1  —  p). Finally, from (6.9), user i observes a failure slot if it transmits a packet  and the transmitted packet either collides with some other transmission(s), or gets corrupted, or both. We notice that  pidle  and  busy  do not depend on the value of  Since user i knows the local transmission probability  j,  r[ while  it can estimate  pdle  pucc  and pl do.  using either (6.6) or  any of the expressions in (6.7)-(6.9). However, it is clear that user i is still unable to estimate the individual transmission probability Pj for any  j  i even if it can accurately estimate all the state  147  Chapter 6.  probabilities  Utility Optimal Random Access without Frequent Message Passing  pi4le,  pUCC,  and p”. In fact, finding the values of individual transmission  probabilities requires gathering more individual information from other users as we explain next. Recall that, at a busy slot seen by user i  e .N,  at least one other user transmits. Since users  can hear each other, user i may successfully decode the transmission of user j  pdecd  Let  pj(jJlEA,.\{.}(1 —))(l  —pf)  =  (/(1 —h))  i with probability;  (1i-(1 —pi)) (1 —pJ).  (6.10)  denote the number of slots between any two consecutive successful decoding of trans  missions of user  j by user  i. We have: 1ecd  where  =  1/(1 +  n’),  (6.11)  is the mean value of n’7 and can be locally estimated by user i through observation  of the channel contention history. Notice that in practice, the transmitted signal by user  j can  be decoded by the network interface of user i’s receiver node; however, as its destination MAC address is not the same as the one in user i, the packet is simply discarded. Now, user i needs to obtain the sender’s MAC address from the packet header before discarding the packet. Similarly, let  kdle  denote the number of non-idle slots that user i observes between any two  consecutive idle time slots. User i can estimate le 4 pi  where  j  E  dle  is the mean value of  dIe  =  pdle  as follows [100]:  e), 1 1/(1 + n  (6.12)  Substituting (6.6), (6.11), and (6.12) into (6.10), for each  /\{i}, we have:  l/p -1  Let  11e 7  and  J,id1e  =  ((1 +  decd)/(1 +ñ1e))  (1  -m•  (6.13)  denote the set of time slots at which user i observes an idle slot and  148  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  decodes the transmissions of user nclle(t  + 1)  decd  the estimation of  ñd,  Dd iteratively as: 7 i, respectively. We estimate ñldle and ñ  (6.14)  —  (1—  =  ñd1e (t), id1e (t)  where d1e  (1  +1)  j  (t) and  (t) I{t e decd}’  2d (t) 7 ,j (t))ñ ndecd (t)  denote the estimation of  and the measurement of  is an indicator function. Here pj and  id1e  (6.15)  the measurement of  at time slot t, respectively, and I{.}  are tapering stepsizes. Based on the asynchronous  Qij  stochastic approximation theory [106), we know that the estimation error decreases to zero when users do not change their transmission probabilities. For each user i and any other user ml(t)  j  i, given 7j, ñ’r’ and ñd1e, we define:  (1/)’ ((1 + n(t))/(1 +  =  d1e(t))),  Vj  \{i},  (6.16)  where ml(t) denotes the estimation of m made by user i at time slot t. In general, m(t) where 3(t)  >  (6.17)  ,6(t) mj(t),  0 is the estimation gain, which can represent accurate estimation (i.e., /3(t)  1), over-estimation (i.e., /3(t) estimations on rate), then  =  /31(t)  and =  >  nhle  1) or under-estimation (i.e.,  /31(t)  =  < 1). From (6.13), if the  are accurate and the channel is perfect (with zero packet error  1 and we have ml(t)  =  mj(t) for all  j E J\f\{i}. Notice that if the value of  the existing packet error rate pJ is known (e.g., via measurements at the physical layer), then we can redefine ml(t)  =  1 ((1 + ñd(t))/(1 + (1/7j)  ñd1e(t)))’/(1 —  p)’ and obtain a  more accurate estimation by canceling out the effect of channel imperfections. However, in this chapter, we consider the general case and assume that the packet error rates are not known. For each user i minimum difference  e  .N and for all  between  i e A[\{i}, we set T,m such that as time goes by, the  any two consecutive time slots in the union of sets {,m’ 149  ‘  .i  E  Utility Optimal Random Access without Frequent Message Passing  Chapter 6.  .Af\{i}} increases. This implies that for each  more samples of  ndle  and  j, we update m less frequently to be able to collect  Thus, the estimations of mean values  ñdle  and  improve  gradually and become asymptotically accurate. We also reset the tapering stepsizes p 2 and p.j, to 1 after each t  e  Tim so that the errors  in  previous estimations do iiot affect new estimations.  Based on these assumptions, there exists a  >  0 such that limt,c, i3(t)  From (6.16) and  =  (6.17), =  1/(1  If the channel is perfect, then lossy channel, if a < 1, then  the other hand, if a  >  1, then  —pJY’, =  <  Vi,j e f, i  j.  1 and all estimations are asymptotically accurate. For a  1 and m is asymptotically under-estimated for all  >  j  i. On  1 and rn is asymptotically over-estimated.  Distributed Algorithm Our proposed distributed MAC algorithm with no explicit message passing (except when each user joins or leaves the network) is shown in Algorithm 7. In this algorithm, each user i continuously updates  nule  =  and  2 the shared channel to estimate m with v  =  Zje\{i} m. Sets  =  (cd,  (m, Vj  eN  Vj eN\{i}) based on its local observations from  e .Af\{i}). Then, it chooses  j  according to (6.4)  and Ti,rn are two unbounded sets of time slots at which user i  updates IN and m , respectively. Notice that the updates are asynchronous across users which 2 includes synchronous updates as a special case.  150  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  Algorithm 7 Executed by each user i e A. 1: Allocate memory for pj and m = (mi,.. , mjr). 2: Allocate memory for ñecd and ñecd = (ncicd,... , 3: Randomly choose p, e [pmin, pmax]. 4: Randomly choose m é [Mm, M’] for all j E /. 5: Choose ñdIe = 1 and decd = 1 for all j E JV. 6: Broadcast the fixed data rate 7j to all other users. 7: repeat 8: Transmit with probability Pi. 9: 1 according to Eqs. (6.14) and (6.15). Update nle and ñ’ 10: if t E T then  Update  11: 12: 13: 14: 15: 16: 17:  i=  [1/ (1+a_1  Z  jE\{i} m  end if if t e T,m then Update m according to Eq. (6.16). end if until the user decides to leave the network. Broadcast termination message.  6.3  ;min  Convergence and Optimality  For each i e4 and at any time t p(t+ 1)  where v(pj, t) (f(pj, t), Vi as t  )]  prnax  IIf’(p,t)  as t  e /).  —  —÷  Algorithm 7 updates  f(p_,t)  =  1 (l/p ZjeN\{i} (i/yj)  [1/ (1 + /v(p_,t) —  1)a_1  I(t).  e .A/,  the sequence of mapping  For any t  0, we define f’(p, t)  =  {f’ (p, t)} converges to a unique mapping  cc. That is, for any p E P and any e’  f’(p,cc)II  p max )]jmin,  Notice that f’(p, t) is a time-varying vector mapping. Since 3(t) approaches  cc for all i, j  —*  f’ (p, cc)  =  =  e  < e’ for all t  “  Ømln(t)  /  \  1  —  a  0, there exists t, > 0 such that  ti.  Theorem 20 Assume there exists t amaxj’t  >  a  0 such that for all t \2  , V/rna)j m1 (V’  151  /  1 1  t and any p max  —F 7mm  E  P. we have:  \ 1  I J  < 1,  (6.18)  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  where /3rnin(t)  =  mm i3(t),  (6.19)  max j3(t),  (6.20)  ‘L,EJV  I3rnax(t)  1 =Inax{p.(  =  Pmin)’Pm(l  1 pmax(  F— pmin(  —  —  —  pmin  (+(Vmax)l)2’ =  622  P)’  I //max\1/c  ) m (v’’, v”  (6.21)  pm)}’  ” (V!mh) / 1 (1+(VFmrn)’)2’  0.25,  1 ifV’ < m .‘ —  if v/mm  >  (6.23)  1  —  otherwise.  Then, Algorithm 7 globally and asynchronously converges to the unique fixed point of  f’ (p, oo).  Notice that V’m and v’m are the lower and upper bounds on v(p, t) for each i e N and at any time t. If c  1, then V’ =(N mm  If c < 1, then V!mi=(N  —  —  min)_1 and Vlax=(N 7 l)Mmmn(  m)a_l and Vfrnax=(N 7 1)Mmmn(  —  m)_l. 7 1)Mmax(  —  The proof of Theorem 20 is given in Section 6.6.1. Notice that, at any time t (Mm1/Mm) </ min(t) 3  (t) m 3  (Mmc/MmuI).  0, (6.24)  We notice that all the terms in (6.18), except F, are bounded and independent of the number of users N. Thus,  can be arbitrarily close to 0 if N is large enough:  152  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  Corollary 4 For any choice of system parameters, there exists an integer N > 0, such that  Algorithm 7 globally and asynchronously converges to the unique fixed point of mapping  f’ (p, oo),  if the number of users N> IJ, i.e., there are enough users competing for the channel.  Theorem 20 is general and does not depend on the exact values of the estimation errors as t  oo; however, the performance at the asymptotic fixed point still depends on the accuracy of  —÷  the estimations. The following Theorem can be shown for perfect channel case.  Theorem 21 If the channel is perfect such that 1imt_  3min(t)  =  limt_  3rnax(t)  =  1, then the  unique ,fixed point of Algorithm. 7 is the unique global optimal solution of problem (6.2.  The proof of Theorem (21) is similar to that of Theorem 15 and is omitted. Note that since limt, /3(t)  =  1, f’(p, oo)  =  f(p)  =  (fj(p), Vi  e .Af)  where f(p) is as in (6.4).  From Theorems 20 and 21, if the channel is perfect and (6.18) holds, Algorithm 7 asyn chronously converges to the unique optimal solution of non-convex problem (6.2). If the channel is not perfect, although the algorithm still converges, optimality is not always guaranteed.  6.4  Simulation Results  To evaluate the performance of our proposed distributed algorithm, we develop a discrete-event simulator that implements Algorithms 7 and the IEEE 802.11 DCF access method. We first consider a network with N =  18, -y  =  36, and -y  =  =  4,  pmin  =  0.01, and  54, all in Mbps. Utility parameter a  pmax  =  =  0.99. We set ‘y  =  6, 72  0.5 < 1. Notice that none of  the previous NUM-based MAC algorithms (e.g, [47, 51, 98]) support a-fair utility functions with a  e  (0, 1) because of non-convexity (see [104, Sections II and IV-A]). Each slot is 20 s (as in  153  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  I 0.9 0  0.8  w  0.7 -Q -Q 0 L.. 0..  0.5  I  2  4  8  6  12  10  14  16  18  20  Time (sec)  Figure 6.2: Simulation results for Algorithm 7 when a = 0.6. The number of users and the fea tures of the communication channel change after t = lOs. The optimal probabilities before t = lOs (i.e., dashed lines) and after t = lOs (i.e., dotted lines) are accurate and obtained using Algorithm 5.  802.lla) and the simulation time is 20s. We assume that from time t is perfect and N  =  4. Then, from t  =  lOs to t  =  =  0 to t  =  lOs. the channel  20s, the channel is lossy and N  =  3 (i.e.,  user 4 leaves the network). Packet error rates are randomly selected between 0 and 0.01 (i.e., the maximum allowed packet error rate in 8O2.lla) at t  =  lOs and theii become fixed until t  =  20s.  Results are shown in Fig. 6.2. We see that Algorithm 7 converges to a small neighborhood of the optimal values very fast. It is also robust to the change of user population and channel conditions. Similar results have also been obtained for a  1.  It is well-known that 802.11 DCF has a short-term fairness problem, due to binary exponential  154  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  backoff. Next, we compare 802.11 DCF with Algorithm 7 in terms of both system throughput and Jam’s fairness index [89]. The short-term fairness is obtained using sliding windows with size of 200 slots. There are N  10 users in the network and their fixed peak rates are randomly selected  between 6 and 54 Mbps. Simulation time is lOOs. The results when a varies between 0.5 to 5 are shown in Fig. 6.3. We see that, parameter a acts as a knob to control the tradeoff between efficiency and fairness. By increasing a we can make the system more fair but less efficient (and vice versa). If a any choice of a  6.5  =  0.5, then throughput is 29.7% higher than DCF (see Fig. 6.3(a)). Besides, for  e [0.5,5], the fairness is much better than DCF (Fig. 6.3(b)).  Summary  In this chapter, we designed a distributed contention-based MAC algorithm to solve a NUM without frequent explicit message passing among users. Our algorithm is fully asynchronous problem, enjoys fast convergence, and supports a wider range of utility functions compared to previously proposed NUM-based MAC algorithms. Simulation results show that our algorithm achieves a better efficiency-fairness trade-off compared with the IEEE 802.11 DCF. It is also robust to the changes of user population and channel conditions.  6.6 6.6.1  Analytical Proofs Proof of Theorem 20  For any p  e  P and t  i and column  t, the Jacobian J(p, t) is defined as an N x N matrix whose entry in row  . We can show that, 3 j is 8f(p, t)/0p  155  Chapter 6.  Utility Optimal Random Access without equent Message Passing  (Ii cl/a)  IIJ’(p,t)IIoo  ‘P  —  IIJ’(p, t)IIi  (I’  —  (6.25)  ,  (max(t)/min(t))  al/a)  ‘P  (Qrnax/rnin) p)l_a  (6.26)  P. From (6.18), (6.25), (6.26), and by Cauchy Schwarz inequality, we have:  Let j3,j3  IIf’( t)  f’(j3, t)11 2  —  jJ’(p,  2 lli t)11  —  /jIJ’(p, t)IIllJ’(p, t)lIi lip P112  P112  —  <  lIP P112, —  Thus, for any t > t, vector function f’(p, t) is a  where p is any convex combination of i and  contraction mapping [18, PP. 181] and has a unique fixed point [18, PP. 183], denoted by p. We also denote the unique fixed point of mapping f’  Wf’(p t) where ?7t Since  =  iIJ’(p, t)li,  f’ (p, t)  =  maxt>ti  —  ,  2 I  and  other words, Ve > 0, to > t, such that Vt  Together with (6.27), we have  by p. Thus,  lip pIi 2 —  ll pIJ . 2 —  f’ (p, t)  =  (6.27)  < 77,  Note that  f’ (p, oc),  < 1 and  is bounded.  we have limt p  =  p. In  to,  *  Pt  1t  =  is continuous at p and limt, 0  (p, oc)  *  —  00 P  2  <  —  +iip— p,lI 2 2 hf’ (p, t)— pii 2 If’ (p, t)— pJI  Similarly,  ± II’+i —pII 2 —pII If’ (f’(p,t) ,t+ 1) —i+iW 2 2 (jf’(p,t) —pII (+e+e)+e=(i+2e)+e. + W+i —p 2 ) +e 2  f’ (f’(p,t) ,t+ 1)  For any k  0, we recursively define  fC(p, t)  =  f’(f’’(p, t), t +  (6.29), and by mathematical induction, we can show that for any k  156  k  —  1) where f’°  0,  =  (6.29) p. From  Chapter 6. For any e  >  Utility Optimal Random Access without Frequent Message Passing  0, there exist k such that if k  , then 6 k  .  By choosing e =  f!k(pt) -p<  fm(p,t)  For all t>to, define e=maxk>o Pt  Ct  max[e,  2(1+)  max [,  2(1-j-)  =  =  I  —*  maxk>o,PFE2  el,  all I  IIfIk±t_to(p/,to)  —pII  cc. Construct a new  0. Since x (C) < 1, sequence =  0. Furthermore,  {p:  {Et}  li  and  if t < to+C, (6.31) 1  ,  x (C) Ct_C], otherwise,  ($f+i), integer constant C  For each I > 0, define P  (6.30)  _p<+ =e,  the ceiling function. From (6.28) and (6.30), {e} and converge to zero as t  ,  =  Let,  where function (C)  e  ()  =  {Et}  rlog()/log  (17)1 +1, and 1  denotes  are infinite decreasing sequences and  time sequence {} where  =  to ± IC for all integer  is also decreasing and in particular, we have lim _e 1 e}. It is clear that  —  p  e P and P÷ 1  =  cP  0.  for  C P for some finite li’. For any p e  1 iIP—PJ± p  From (6.30), we know that  f” (p )  On the other hand, if e  p*W =  <  <eEi ±E.  k  2(1+i)  +  or if e  + =  If  =  X(C)EI_C, then e-  e, then e and  Thus, for all three possibilities in (6.31), we have  If’°(p,ti) -pI <f Thus, V  p  e P, 1  C 1  (p,  t) e P÷’.  =x(C)e Since synchronous convergence and box conditions hold,  Algorithm 7 globally and asynchronously converges to  157  p  [18, pp. 431].  Chapter 6.  Utility Optimal Random Access without Frequent Message Passing  (a) 16  I  I  *  0.5 0.6 0.70.80.91  2  Algorithm 7  3  4  5  3  4  5  (b)  0.5 0.6 0.7 0.80.91  2 Utility Parameter oe.  Figure 6.3: Comparison between Algorithm 7 and 802.11 DCF when the number of users N = 10. Notice that, neither Algorithm 7 nor 802.11 DCF use any frequent explicit message passing. However, Algorithm 7 results in significantly better throughput fairness tradeoff.  158  Chapter 7  Distributed Optimal Multi-Interface Multi-Channel Random Access Most of the recently proposed channel and interface assignment algorithms for MC-WMNs (e.g., see Chapters 2, 3, and 4) are based on formulating combinatorial channel assignment problems and discrete optimization.  Each NIC is assumed to be assigned to exactly one fixed chan  nel. Examples include graph coloring problems [78], integer optimization problems [61, 107], and mixed-integer optimization problems [70, 108]. It is known that the combinatorial problems are NP-hard [34]. That is, finding the optimal solutions may require examining all the possible combinations within the search space. Thus, the combinatorial channel assignment algorithms are usually computationally complicated. In addition, they display poor performance gain as the ratio between the number of channels and the number of NICs at each node increases [61, 108]. Finally, the distributed implementation of the combinatorial channel assignment algorithms is difficult because of several design challenges such as the ripple-effect problem [61]. In this chapter, we overcome the performance bottlenecks of the previous combinatorial chan nel and interface assignment algorithms in all the aforementioned aspects. We first derive the mathematical models for average link data rates in single-channel reception and multi-channel re  159  Chapter 7. Multi-Interface Multi-Channel Random Access 8 scenarios. We also consider the case when not only the orthogonal (i.e., non-overlapped) ception channels, but also the partially-overlapped frequency channels are being used. The NUM problem is then formulated for each scenario. Our models are extensions of the results on NUM-based single-interface single-channel random access in Chapter 5. Finally, we propose two fast, fully distributed, and easy to implement algorithms, called distributed multi-interface multi-channel random access (DMMRA), to solve the formulated NUM problem for each scenario. We prove that DMMRA always outperforms combinatorial channel assignment. We also analytically study the optimality and convergence properties of our proposed algorithms. Simulation results show that DMMRA algorithm with single-channel reception leads to 36% and 23% higher network util ity and aggregate throughput compared to the utility-optimal combinatorial interface assignment and channel allocation algorithm in [107] (i.e., a modified version of the algorithm in Chaper 3. When multi-channel reception model is implemented, the utility and throughput are further increased by 57% and 71%, respectively. On the other hand, in both single-channel reception and multi-channel reception scenarios, using all available partially overlapped channels result in sig nificant performance improvement compared to the case where only the non-overlapped channels are being used. Our proposed algorithms are also robust to communication delay and delayed information exchange. The rest of this paper is organized as follows. The data rates for various scenarios are modeled in Section 7.1. The NUM problems are formulated in Section 7.2. The DMMRA algorithms are proposed in Section 7.3. Simulation results are presented in Section 7.4. A summary of the chapter is given in Section 7.5. Analytical proofs of all theorems are given in Section 7.6. 1n this paper, multi-channel reception refers to a hardware implementation scenario, where each NIC can 8 correctly decode multiple simultaneously received packets if they are transmitted over orthogonal channels. This differs from the receiver diversity model in MIMO systems (cf. [109]). Multi-channel reception is studied in Section 7.1.2.  160  Chapter 7. Multi-Interface Multi-Channel Random Access  7.1  System Model  Consider a multi-interface multi-channel wireless ad-hoc network with N of wireless nodes and L  =  with size LoI1t  as the set of in-neighbors and Nut  =  =  • 1 ut  {m : (n, m)  e  eN  =  We also define N  I,  I =  and the set of  {m : (m, n) e  as the set of out-neighbors of node n,  respectively. The set of available frequency channels is denoted by C NICs for each node n  N} as the set  ,  ,  C £,  £U  { 1,...  {1,. .. L} as the set of unidirectional wireless links. For each node  n E N, we denote the set of incoming links by £‘ C £, with size L outgoing links by  =  is denoted by I,, with size I,  =  =  {1,... , C}. The set of  . Each node n  E  N has Lt  separate queues, where each queue stores the packets for one of the outgoing links of node n (see Fig. 7.1). Time is divided into equal-length slots . At each time slot, node n may choose to 9 transmit packets to each of its out-neighbors m e Nt using its NIC i e I over channel c with a link persistent probability L  =  2 outgoing links, where 2  For the network in Fig. 7.1, node n has =  {i, j} and Nt  =  {m, s}. We also have: C  =  =  e  C  2 NICs and {1, 2, 3}. In  node n, those packets which are destined to node m are enqueued in queue [n, m]. Similarly, the packets which are destined to node s are enqueued in queue [n, s]. At each time slot, a packet from queue [n, m] is sent to node m (i.e., through link (n, m)) using NIC i over channels 1, 2, or 3, with probabilities  and  respectively. Each NIC may only transmit one  packet at a time. Next, we obtain the average data rate model for each wireless link for various multi-interface multi-channel wireless networking scenarios. The length of the time slots should be large enough, compared to the channel switching delay at NICs, i.e., 9 the time it takes for an NIC to switch from one channel to another. In the current commercial NICs, the channel switching delay is between 100 us to 224 ts [9].  161  Chapter 7. Multi-Interface Multi-Channel Random Access  Wireless links  node m  V  A  /  node s  node n  Figure 7.1: An ad-hoc network with N = {n, m, s} as the set of nodes. Each node has 2 NICs and there are C = 3 channels available, denoted by 3 colors. Nodes m and s are the out-neighbors of node n. We have: I={i,j}.  7.1.1  NICs with Single-Channel Reception  In this section, we consider the case where each NIC can decode the received packets over only one channel at a time. We assume that all the available channels are orthogonaP°. For each node n e N, let  denote the probability that node n listens to channel c e C using its NIC i e I.  To be able to listen to channel c, NIC i on node n needs to be in the receive mode (i.e., does not transmit any packet) and also operates over channel c. The key feature of the single-channel reception model is that if node ri is in the receive mode, and operates over channel d  c, then  it cannot decode the signals transmitted over channel c. Most of the existing commercial NICs : 11 implement such a single-channel reception model. In this case, for each node n eN, we have (p,c) ZcEC  +  Q$)  =  1,  Vi e In,  (7.1)  ‘°The case where some of the available channels are partially-overlapped is studied in Section 7.1.3. “Here we assume that each NIC operates either in transmit or receive mode. If an NIC also operates in idle mode, then the equality in (7.1) is replaced with non-strict inequality “<“ and NIC i E 2 would be operating in (p,)(c) idle mode with probability 1— E +Q). 5  162  Chapter 7. Multi-Interface Multi-Channel Random Access where  pi)(t  denotes the probability that node n transmits some data from NIC i  channel cC to one of its out-neighbors. We call  e  over  the node persistent probability for NIC i  of node n over channel c. We have: p(i)(c)  —  —  For each link (n, m)  2t 1 mEJf  (i)(c) Pnm  e £, we first consider the case where there is no interference in the  network (Le., assuming that there are only two nodes). Let where at least one NIC  j  denote the action set for all cases  1 transmits packets over frequency channel c. The probability of  this happening is:  (A)) Let  =  1  —  [I jEIm  (i  (7.3)  —  .  denote the action set for all cases where no NIC on node m transmits packets over  channel c, and no NIC listens to channel c either. The probability of this happening is: iF’  Since the sets  and  (A))  = ]JjEIm  (i  —  n  are two disjoint sets (i.e.,  (A u A)) =i (A2) The transmission from NIC i  e lEfl,  Q()  —  +i  on sending node n e  (7.4)  .  is an empty set), we have:  (A?).  (7.5)  over channel c e C can be received  correctly by the receiving node m e J*qut only if at least one NIC  j Elm  is listening to channel c  and none of the other NICs on node m are transmitting packets over frequency channel c. From (7.3)- (7.5), this happens with probability: 1  —  i  (A) u A2)  =  j  (i  —  p(i)(c))  j  (i  —  pU)(c)  —  .  (7.6)  —  Next, we model the effect of interference in a network with N > 2 nodes. For each pair of nodes s, mE.Af, we define S= 1 if node s is within the interference range of node m, and 163  sm 5 ‘  0  Chapter 7. Multi-Interface Multi-Channel Random Access  otherwise. Since the interference range is at least as large as the communication range, Ssm = 1 if s  e  A transmission from NIC i on node n to node m via link (n, m)  e £ over channel  c  does not encounter collision if there is no simultaneous transmission over channel c from any NIC jEIn\{i}  on node m, and any NIC keI on nodes with Ssm=1. This happens with probability:  (F1jEIn\{j} (i  —  For each wireless link (n, m)  p4i)(c)))  e  £, let  (FIsEg\{n,m} flkEZ 8 (i  (7.7)  5smPs)).  —  denote the average data rate, which is a function of  Tnm  the following persistent and listening probability vectors:  Q  =  (pC)  =  (Q)(C)  vn  e /,  e c),  m é jVout, j é in, c  (7.8)  VnE/, i eI, cEC).  (7.9)  From (7.2), (7.6) and (7.7), we have [80]12: rnm(p,  Q)  7c) ZCEC  =  (fIseJ.r\{n,m} EIkE2  (n where  (1  k m jE  (fl.\{}  (i.  (j)(c)’ m )  (i  —  pU)(c)))  p(k)(c)))  (7.10)  —  (1 2rnk  (j)(c)  m  ,—(j)(c)’’\  ‘dm  denotes the fixed peak data rate 13 for link (n, m) over frequency channel c (i.e., the  data rate achieved by link (n, m) over channel c if there is no other transmission in the network at the same time). The data rate model in (7.10) sums up all the average data rates that can be achieved by transmitting packets from each NIC i E I and over each channel  C E  C.  The node persistent probability vector P = 12 Vn E f, i El,,., CE C) can always be constructed from the link persistent probability vector p using (7.2). Thus, for each link (n, m) E we can denote the average data rate for the single-channel reception scenario as mm (p, Q), rather than ran,. (p, P, Q), to avoid redundancy. 1n general, -y -y, for any c d as the channel properties are different at different frequency bands. 3 ‘ ,  164  Chapter 7. Multi-Interface Multi-Channel Random Access  7.1.2  NICs with Multi-Channel Reception  Next, consider the case where each NIC can decode multiple simultaneously received packets as long as they are transmitted over different orthogonal channels. That is, each NIC does not listen to only one channel while it is in the receive mode. Instead, it listens to all frequency channels arid applies the band-pass channel filters to all the received signals. The output of each filter is then processed separately (i.e., in parallel) to distinguish transmissions over different channels [95]. Figs. 7.2 (a) and (b) show the basic building blocks of the receiver device when the single-channel reception and multi-channel reception models are being implemented, respectively. We notice that, most of the existing commercial NICs do not yet implement multi-channel reception model. However, we will show in Section 7.4 that it can significantly improve the network performance. Thus, it is an attractive and promising candidate for future deployment. As in Section 7.1.1, we first assume that there is rio iiterference in the network. Let denote the action set where all NICs on node m transmit packets on some channels other than channel c, and no NIC is in the receive mode. We have:  ip’ (c)  (A) .  =  F1jm ZdeC\{c}  .  Since the sets Am (defined in Section 7.1.1) and Am  (i)(d)  (7.11) .  .  (c)  ‘(—c)  are disjoint (i.e., Am fl Am  is an  empty set), we have:  IP (A) u A))  =  IP  (A))  +  (A)).  In the multi-channel reception model, for any link (n, m) i  e1  on node n  one NIC  j  e .iV  e  (7.12)  £, the transmission from NIC  over channel c e C can be received correctly by node m  e J\/ut  if at least  2 is in the receive mode and none of the other NICs on node m are transmitting  165  Chapter 7. Multi-Interface Multi-Channel Random Access  Estimated Message Signal  Received Signal  (a) Single-Channel Reception  Estimated Message Signal  Estimated Message Signal  Received Signal  Estimated Message Signal (b) Multi-Channel Reception  Figure 7.2: Building blocks of the receiver unit in single-channel reception and multi-channel reception models. BPF stands for band-pass filter. For each channel c e C, the central frequency of the channel filter is shown by f. packets over channel c. From (7.3), (7.11), and (7.12), this happens with probability: 1  —  (A) u A(_c))  (i.  =  —  pCi)(c)) —-  fTjEln ZdEC\{c} p$1)@).  (7.13)  When the interference is taken into account, from (7.2), (7.7), and (7.13), for each link (n, m) E we have:  rnm(p)  7(c)()(c) =  (fliEz\{} (i  iEZ ZccC  (flseiv\{n,rn}  (rr  kh1jEIm  ( 1  —  (i  3 FlkeI  DO)(c)’\ m  4  )  —  (7.14)  sm  —  1—f —  p(i)(c)))  lljerm L.dEC\{c}  (j)(d) m  I  Note that since a node can listen to all channels when it is in the receive mode, the data rate model in (7.14) does not depend on the listening probability vector 166  Q.  Chapter 7. Multi-Interface Multi-Channel Random Access  7.1.3  Partially Overlapped Frequency Channels  In this section, we extend the data rate models in (7.10) and (7.14) to the general cases, where both orthogonal (i.e., non-overlapped) and partially overlapped channels are being used. In this regard, we borrow the concept of multiple interference ranges from our recent work in [110, Section II-D]. For each pair of nodes s, m e A, we define  = 1 if node s is within the interference  range of node m, while node s is operating over channel c and node m is operating over channel d; otherwise,  = 0. In general, the smaller the frequency spectrum overlapping between two  channels c and d, the shorter the corresponding interference range (cf. [110, Fig. 3]). In fact, as two wireless links use lower overlapped channels, the less is the interference power that they cause on each other’s transmissions. Thus, the interfering transmissions need to be in shorter distance to corrupt each other’s packets. We first assume that NICs use single-channel reception model and there is no interference in the network. For each link (n, m) one NIC  j  e  denote the action set for all cases where at least  £, let  é Im transmits packets on some channel d e C such that  = 1. Also, let  denote the action set for all cases where no NIC on node m transmits packets on any channel (ed) deC such that ömm =1, and no NIC listens to channel c either. We have:  P (Bin) = 1  ip Since the sets i_p  and  (E))  —  ]JjEIm (i  —  ZdEC  = jj (1—ZdEC  PV(d)) ,  (cd) pCi)(cl)  —  (7.15)  QCJ)(c))  (7.16)  are disjoint sets, we have:  () U n) =fljalm(1_ZdaC  $°)  —  fljEZm  p)(d)  (i —  —  QO)(c))  Zdec  (7.17) To model the interference, we can modify the collision avoidance probability model in (7.7) 167  Chapter 7. Multi-Interface Multi-Channel Random Access as: ItT  11jeZ\{i}  -  —  (  (  ,ç(cd),(j)(d)’\’ llsEAf\{n,m} 11kE2 5 /—deC vrjm  ))  —  s L.dEC usm 1  7.18  .  From (7.17) and (7.18), when the NICs implement single-channel reception and all partially  overlapped channels are available, the average data rate of link (n, m) mm (p,  Q)  = ZiEInZ cEC7flmPflm  (FIsJsf\{n,m} Flke1 3 ItT  (fIjI\{} (i  ZdECöThm  —  e .C is:  p,)()))  (cd) p(k)(d)))  (i  (7.19)  ZdEC  x(cd)p(i)(d)’\ L d 4 Cvmm m  )  i—i —  11jeZm  11  x x(cd)p(i)(d) LddEC’mm m  r)(j)(c) ‘m  We now consider the multi-channel reception scenario. For any link (n, m)  e  £, let  denote the set of actions for all cases where no NIC on node m transmits packets over channel c or any other channel d  E  C\{c} such that  = 1, and no NIC is silent either. In other words,  all NICs on node m transmit packets on some channels other than those channels that have (full or partial) overlapping with channel c. We have: p  (E(_c))  Notice that for any node m overlapped, then 1  —  =  flIm  ZdEC  (i  i)(d) 7 —  (7.20)  e N, if frequency channels c, d e C are either fully or partially  = 0. From (7.15), (7.18), and (7.20), when the NICs use multi-channel  reception and all non-overlapped as well as partially overlapped channels are being available, the average data rate of link (n, m) Tnm (P)  £ is: (fljEI\{} (i  =  (FIsEJ’..f\{n,m} fIkE2  (cd) p(i)(d))) 8 —  dEC  (k)(d))) 8 P  (i —  (7.21)  ZdEC  j(j)(d)’\, (cd) 7 (1—1 (1 i11jEXm V/_sdeC Vmml m  )  —  i_i 11EIm Z—’dEC  Ii  —  x(cd)’\ (i)(d) vmm) m  It can be verified that, if all the channels are orthogonal (i.e., 6 = 0 for all s, m  168  e N and  Chapter 7. Multi-Interface Multi-Channel Random Access any c  d), then the rates in (7.19) and (7.21) reduce to (7.10) and (7.14), respectively.  7.2  Network Utility Maximization  The formulation in (7. 1)-(7.21) models the average link data rates for different multi-interface multi-channel random access scenarios. In this section, we formulate the random access problems in those scenarios according to the NUM framework (cf. [44]).  7.2.1  Problem Formulation  Within the NUM framework, the resource allocation problem can be formulated either at link layer [98, 104] or at transport layer [44]. Here, for the ease of exposition, we limit our study to the link-layer NUM’ . In this regard, each link (n, m) e L is assumed to maintain a utility 4 u(r), which is an increasing and concave function of its rate mm and indicates the degree of satisfaction of link (n, m) on its data rate. The utility of link (n, m) is also a function of all the persistent and listening probabilities p and  Assuming that the single-channel reception model  Q.  is used, we are interested in finding the optimal solution of the following NUM problem:  maximize  ZnEji  ZmEj0t’t u(rnm(p, Q)),  (NUMS)  where the data rates are as in (7.10) if only the orthogonal channels are used, and as in (7.19) if both orthogonal and partially overlapped channels are used. We also have: 4  ={ (p, Q): (i)(c) n  —  —  c) p(i)(c) Q)(c)  e  (i)(c)  mCJV?t Pam  ‘  dEC  [0,11,  (  (i)(d)  n  (i)Q1)’\  + r  )  _  1,  —  VnEf, mENot EIa, cC}. We can extend the model to a transport-layer NUM similar to the joint congestion control and medium access 14 control design in [111]. 169  Chapter 7. Multi-Interface Multi-Channel Random Access On the other hand, if multi-channel reception is used, then the following NUM problem is being solved: maximize >ZnEJ\1 ZmE,ut u(rnm(p)),  (NUMM)  where the data rates are as in (7.14) if only the orthogonal channels are used, and as in (7.21) if both orthogonal and partially overlapped channels are used. We have: — —  (i)(c)  (i)(c) E n  7 Pmm .  in  1 j  (i)(c)  (i)(c)  — —  mENt  Pnm  mEut, IEIn, cEC}.  Notice that the sets  and ‘P are formed by linear constraints.  Thus,  and ‘I’ are convex  sets [16]. Various concave utility functions can also be considered to achieve different network design objectives. A popular class of utility functions are a-fair utilities [50], where for each link (n, m) e £, we have: (1 u(rnm)  —  a)’r, if a e (0,1) U (1, co), (7.22)  =  log Tnm,  if a  =  1.  Using (7.22), a wide range of efficient and fair resource allocations among the link-layer flows can be modeled. In particular, optimization problems (NUM-S) and (NUM-M) reduce to throughput maximization with a — 0, to proportional fair allocation with a = 1, to harmonic mean fairness with a = 2, and to max-mm fairness with a —* oc. Unlike most of the previously proposed optimization-based channel assignment models, where the formulated optimization problems are combinatorial and discrete-valued (cf. [61, 72, 107, 108, 112, 113]), problems (NUM-S) and (NUM-M) are continuous-valued optimization problems. They make the analysis of our models substantially easier. Notice that, within the NUM framework at the link-layer, the combinatorial channel assignment problem can be formulated as the following  170  Chapter 7. Multi-Interface Multi-Channel Random Access  mixed-integer optimization problem [107]: maximize (P,Q)E, ‘eT  ZnE/ ZmE.A10ut  u(rnm(p,  Q)),  subjectto ZCECXfl_1, p(i)(c)  VnEJ’f, iEI , 1  (NUM-C)  < Vn  Q)(c)< )(c)  where for each node  n  E/, i ei, cE C,  VThE/, iE1, cEC,  E f, any NIC i  e I, and each channel  c  e C, the  integer  variable  is defined as: 1,  if NIC i operates over channel c, (7.23)  =  0, We also have x  =  otherwise.  VneAr, iI céC) and T={x:x)(c)E{O,1}, VnE,iEIncEC}.  From the first constraint in (NUM-C), each NIC can be assigned to exactly one channel. In the context of combinatorial channel assignment, the selection of the operating frequency channel for each NIC is called interface-to-channel binding [61]. From the second and the third constraints, NIC i cannot transmit over or listen to channel c E C if it is not operating on channel c. By solving the mixed-integer optimization problem (NUM-C), we can select not only the operating channel for each NIC, but also the persistent and listening probabilities corresponding to the operating channel of each NIC to achieve utility-optimal network performance within the combinatorial channel and interface assignment framework.  Theorem 22 Let UcR, U7GR, and flb 07 denote the optimal solution of problems (NUM-S), U (NUM-MJ, and (NUM-C), respectively. We can show the following:  171  Chapter 7. Multi-Interface Multi-Channel Random Access (a) Random access with multi-channel reception outperforms random access with single-channel reception: TT* SCR < UMCR.  Uk  —  (b) Random access with single-channel reception outperforms NUM-based combinatorial channel assignment: TT*  (  ._-rr*  ‘Comb  —  ‘-‘SCR  The proof of Theorem 22 is given in Section 7.7. For better understanding of the inequalities in (7.24) and (7.25), we consider two examples and compare the optimal utilities an-1  UCR,  Tr*  MCR  7.2.2  Examples  First, consider a unidirectional ring topology with N =3 nodes, L =3 links, and C =3 channels. The utilities are logarithmic (i.e., c  =  1). Each node has one NIC. We have: N  (c (c) ={(m,m),(m,s),(s,n)}, and C={1,2,3}. For any céC, y=y  (c) =  =  =  {n, m,  11Mbps. In  this scenario, combinatorial channel assignment strategies can only assign the same channel to . 15 all NICs in the network. Otherwise, at least two nodes cannot communicate with each other Thus, we have:  Uomb  active with probability w h ere (1)(1)  rn  (1)(1)  n —  —  —  —  (1)(3)  m  Qm(1)(3) —  —  =  .  —  —  ,-(1)(2)  “em  3 log(11 x  x (1  —  )  x (1— ))  =  1.465, where each link is optimally x (1—0) x (1—))=3.035,  On the other hand, U= log(ll x 3 1 ,  —  —  (1)(2)  —  n  —(1)(3)  —  —  —  (1)(3)  —  n  —  (1)(3)  U,  s  —  —  (1)(1)  Qn ,.-(1)(2)  (1)(2)  —  —  —  —  Qn 1 ,  — —  an ,  0, (1)(1)  (1)(2)  m —  —  —  f—)(1)(1)  —  (1)(2)  —  1  —  —  —  ,._)(1)(1)  —  —  !SThis is one of the key limitations of the combinatorial channel assignment models. In fact, as shown in [61, 108], combinatorial channel assignment algorithms display poor performance gain as the ratio between the number of channels and the number of NICs at each node increases. In this example, there are multiple channels available, while each node has only one NIC.  172  Chapter 7. Multi-Interface Multi-Channel Random Access QX3)  =  0. That is, on average, each wireless node transmits to its out-neighbor on one channel  during half of the time slots and listens to its in-neighbor on a different channel during the other half of the time slots. As a result, each of the 3 wireless links in the network is active on a distinct channel and the available frequency spectrum is fully utilized. Performance improvement is in the factor of  =  2.1. In this example, we have: rr  rr  Comb  ‘-SCR  —  —  rr  MCR  Next, consider a bidirectional ring topology, where .N  =  {n, m, s} and C  =  {(n, m), (m, n)  (m, s), (s, m), (s, n), (n, s)}. The rest of the parameters are the same as the previous example. Again, any combinatorial channel assignment assigns the same channel to all NICs. We have: Uomb  =  6log(11 x  probability  .  x (1  —  x (1  = —  —1.229, where each link is optimally active with  Each node is also optimally active with probability 2 x  links. On the other hand, UCR  =  6log(11 x 0.2113 x (1  —  =  as it has two outgoing  0.2113) x 0.5774)  =  0.341, where  each node listens to one distinct channel with probability 0.5774. Each node also transmits to its out-neighbors using two different channels, other than the channel that it listens to. Finally, U1CR  =  6log(11 x  x (1 —0) x (1  = —  1.9107, where each node transmits to both of its out-  neighbors using one of the three channels with probability  .  Each NIC is silent with probability  In this example, we have: rr* rr* rr* UComb < USCR < UMCR.  Theorem 22 and the above examples show that our proposed random access models can outperform combinatorial channel and interface assignment. We will investigate this issue further in Section 7.4.2.  173  Chapter 7. Multi-Interface Multi-Channel Random Access  7.3  DMMRA Algorithms  Although the objective functions in problems (NUM-S) and (NUM-M) are concave in link rates r  =  (mm, V (n, m)  e £),  they are not concave in persistent and listening probabilities p and Q  due to the product forms in (7.10), (7.14), (7.19), and (7.21). Thus, finding the optimal solutions of these non-convex optimization problems are not easy in general. In this section, we discuss some of the features of problems (NUM-S) and (NUM-M) which will help us to develop our distributed multi-interface multi-channel random access (DMMRA) algorithms.  7.3.1  Local NUM Problems  For each node ne]’.1 and any NIC i E (i)  Q  =  =  I7,  we define:  (c) (c  jout,  Vm  c  c),  (7.28)  c) ,  Vc e  (7.29)  to be the persistent and listening probabilities of NIC i, respectively. Consider the following local and myopic optimization problem in NIC i  e  I, when the single-channel reception model is  being used: maximize  ZmJ\I  ZSEJV0Ut  (LOCALNUMS)  u(rms(p, Q)).  Here, the average data rates are as in (7.10) and we have: (i) n —1  ........  (0 ,  \Pn  p(O(c)  (0  n, —  —  YZdC  .  /  (0(c) Prim ‘  (0(c)  n  ‘  (0(c) C r ri  ‘  1  (i)(c) rriE.A101tPThTI  (phi d)+Q(d)) =  174  (7.30) 1, Vm eJ’10t, cEC}.  Chapter 7. Multi-Interface Multi-Channel Random Access Similarly, consider the following local and myopic problem when the multi-channel reception model is used: (LOCALNUMM)  jout U(rms(p)). p  ec  Here, the average data rates are as in (7.14) and we have: j(i)  ={ p():  c) p(O(c)  e [0, 1],  p(i)(d) dC  < 1,  (7.31) Zmejvut  P?r Vm eJ\f0t, cC  }.  The objective functions in (LOCAL-NUM-S) and (LOCAL-NUM-M) are the same as the objec tive functions in (NUM-S) and (NUM-M), respectively. However, the optimization variables in (LOCAL-NUM-S) and (LOCAL-NUM-M) are local to NIC i in node ri. Consider the case where the single-channel reception model is being used. We define: p)  (i)  =  Vj eI\{i},  Vk Elm, m EN\{n}),  (Q Vj eI\{i}, Q, Vk Elm, m f\{n}).  (7.32) (7.33)  The above are the persistent and listening probabilities corresponding to all NICs in the network (i other than NIC z in node n. By solving problem (LOCAL-NUM-S), we can select p’ and  such that the total utility is maximized assuming that  (i)  Q  and Qare fixed (i.e., the persistent  and the listening probabilities of the other NICs in the network do not change). Solving problem (LOCAL-NUM-M) leads to similar results if multi-channel reception model is being used. We can show the following key theorem:  Theorem 23 Problems (LOCAL-NUM-S) and (LOCAL-NUM-M) are convex optimization prob  lems.  The proof of Theorem 23 is given in Section 7.8. From Theorem 23, we can use various convex 175  Chapter 7. Multi-Interface Multi-Channel Random Access programming techniques (cf. [16]) to solve problems (LOCAL-NUM-S) and (LOCAL-NUM-M). The optimal solutions of problems (LOCAL-NUM-S) and (LOCAL-NUM-M) can also be obtained in closed-form for some simple scenarios:  Theorem 24 For a two-node single-interface multi-channel network (i.e., when N  and I,  =  1 for all n e  2, C > 1,  N) with a-fair utilities, the optimal solution of problem (LOCAL-NUM-M)  can be obtained as: (j)*  =  M  where 1 is a unit C x 1 vector and  (p)  M(pZ))  Vn eN, i e I,  1,  (7.34)  is a C x C matrix with its entry in the ctl row and  dth column is: (ps_i))  =  (d) (c) 7nm/7nm  1+  I(  (7.35)  (c) eECPm (e) (j)(e)7nm 1/ 1-eC 7mnPmn V \ -,  Here j denotes the (only) NIC of node m.  The proof of Theorem 24 is given in Section 7.9. Although the solution in (7.34) always exists, it may not always be unique. In general, when the network has a large number of nodes, for any arbitrary topology, we can use the interior-point method (1PM) [16, Chapter 11] to solve the convex problems in (LOCAL-NUM-S) and (LOCAL-NUM-M) via local iterations. 1PM can find the optimal solution of convex optimization problems in polynomial time [16]. Finally, we notice that for each node n e N and any of its NICs i e I,-, we can rewrite the objective function of problem (LOCAL-NUM-S) to be as follows:  SEns1ZmEN(rm8(Th  where F only depends on  (—i)  p  and  +  f(i),  (i)  (—i)  Q  Q))  ,  but not p 176  and  Q(_i))  (‘  Q’’.  Recall that 6,  (7.36) =  1 if  Chapter 7. Multi-Interface Multi-Channel Random Access node  ii  is within the interference range of node s. By writing the objective function of prob  lem (LOCAL-NUM-S) in the form of (7.36), we divide it into two parts. The first part (i.e., ZSE,V,Ssn=1 ZmEJf  u(rms(p, Q))) depends on the optimization variables p and  ond part (i.e., P(p),  Q))  Q.  The sec  is a constant. Thus, the value of I’ needs not be known to solve  problem (LOCAL-NUM-S). On the other hand, for any link (m, s), where s e f, S and m e AI’, the data rate  8 Tm  =  1,  is fully modeled if the persistent and listening probabilities of  nodes within 2R distance of node n are known. Here, R 1  =  maxme Rm and for each  node m e /, Rm is the interference range of node m. Thus, the convex optimization problem (LOCAL-NUM-S) can easily be solved locally, as long as all nodes within 2R m distance of node n can inform their persistent and listening probabilities to node n. The same statement is true for the convex optimization problem (LOCAL-NUM-M).  7.3.2  Algorithms  Our proposed DMMRA algorithms, when the single-channel reception and multi-channel recep tion models are being used, are shown in Algorithm 8 (DMMRA-S) and Algorithm 9 (DMMRA M), respectively. For each node n E JV and any of its NICs i e I,, let  be an unbounded  set of time slots at which node n updates NIC i’s persistent and listening probabilities. We assume that the updates are asynchronous across the network. That is,  j e I\{i} and  -k) 7 =  {}  for all m E A1\{n} and any k E  Im.  fl  =  {}  for all  In line 2 of Algorithm  8, node n randomly initiates all of its persistent and listening probabilities. Lines 4 to 14 are then executed repeatedly at every time slot until node n leaves the network or switches off. In lines 4 to 6, node n either transmits or receives packets according to its persistent and listening probabilities. On the other hand, lines 8 to 10 are executed only if there exists an NIC i e 2 such 177  Chapter 7. Multi-Interface Multi-Channel Random Access Algorithm 8 DMMRA-S: Executed by each node n e N when the NICs implement singlechannel reception. 1: Allocate memory for p and Q. 2: Randomly choose p and Q such that (p, Q) e 3: repeat 4: for each NIC i e Z, do 5: Either transmit to node m e jfout on channel c E C or listen to channel c e C (i)(c) (i)(c) with probabilities Prim and Q 7 , respectively. 6: end for 7: If t TJ’ for some i I then 8: Solve problem (LOCAL-NUM-S) using 1PM [16]. 9: Update p and Q according to the solution. (i) (i) 10: Inform p and Q to nodes in 2R max distance. 11: end if 12: if a message is received from another node then 13: Update p and Q accordingly. 14: end if 15: until node n leaves the network. -  .  that t  TT. That is, the current time slot is a time slot at which the persistent and listening  probabilities of NIC i need to be updated by solving problem (LOCAL-NUM-S). Recall from Theorem 23 that problem (LOCAL-NUM-S) is convex. Thus, it can easily be solved using 1PM. In line 10, node n announces its updated persistent and listening probabilities to all nodes within its 2R distance, where umax is defined in Section 7.3.1. This can be done using limited-scope message  6 [12, pp. 408]. Upon reception of the new probability values from other nodes, .flooding’  in line 13, node  n  updates its local memory accordingly. Finally, for distributed implementation  of Algorithm 8, we need to slightly modify the feasible sets in (7.30) such that for each any NIC ieI, and each design parameter (e.g., c  we  =  have:  E[e,  1—c], where 0< c <<  n  eN,  is a small  10_6). This requires all NICs to listen to all channels and transmit  over all channels with arbitrarily small but non-zero probabilities. Similar assumptions are made to avoid node starvation in the single-channel random access algorithms in [98] and [104]. We measure the signalling overhead for each of our proposed DMMRA algorithms in Section 7.4.4. 6 ‘  178  Chapter 7. Multi-Interface Multi-Channel Random Access Algorithm 9 DMMRA-M: Executed by each node n e .A/ when the NICs implement multichannel reception. 1: Allocate memory for p. 2: Randomly choose p such that p e II’. 3: repeat 4: for each NIC i E 1 do (i)(c) 5: Transmit to node méNout on channel c eC with probability p 6: end for 7: if t for somei eI then 8: Solve problem (LOCAL-NUM-M) using 1PM. 9: Update p according to the solution. 10: Inform to nodes within 2Rrna distance. 11: end if 12: if a message is received from another node then 13: Update p accordingly. 14: end if 15: until node n leaves the network. -  .  .  .  Algorithm 9 works similarly. The persistent probabilities are adjusted by solving the convex optimization problem in (LOCAL-NUM-M). Notice that, in general, Algorithm 9 is computationally less complicated compared to Algorithm 8 as problem (LOCAL-NUM-M) has fewer variables and also fewer constraints compared to (LOCAL-NUM-S). Both Algorithms 8 and 9 are fully distributed and allow each node to adjust its operation based on a few simple local tasks and some limited-scope message exchange with other nodes.  7.3.3  Optimality and Convergence  In this section, we analytically investigate the optimality and convergence properties of DMMRA algorithms. For each node n eJV and any NIC i eI , we define: 7 (i)  1’  (—i)  f,scR Pn  (—i)  Q  (7.37) =  arg max ZrnE  ZsENt  179  u(rrns(p,  Q)),  Chapter 7. Multi-Interface Multi-Channel Random Access and (i) f,A1cR  We also define  fscR  =  /  (—i)\  )  (fscR,  =  arg max ZmEjVsCJVt  (7.38)  U(Tms(p)).  pe’-Ti  Vi  E  1, n  e N)  and  fMCR  =  (fwcR’  V Th  the mapping functions corresponding to Algorithms 8 and 9, respectively. Let denote the set of fixed points (cf. That is, if (p* fAcR(P)  =  p*  Q*) e  .FSCR,  Also let  SSCR  [18, pp.  then and  181]) of mappings  fscR(p, SMCR  Q)  =  (,*  Q*).  fscR  and  eN, j é Z)  .FSCR  fp1cR,  Similarly, if p  e  and  FMCR  respectively. FMCR,  then  denote the set of all stationary points (cf. [15, pp.  194]) of optimization problems (NUM-S) and (NUM-M), respectively. Note that all local (and thus global) optimal solutions of problems (NUM-S) and (NUM-M) belong to the sets MCR, 8  SCR 8  and  respectively. We can show the following key results:  Theorem  25  ..‘FSCR  =  SCR 8  and  ‘MCR  =  SSICR•  The proof of Theorem 25 is given in Section 7.10. From Theorem 25, any fixed point of Algorithm 8 is indeed a stationary point of optimization problem (NUM-S) and vice versa. The same statement is true for Algorithm 9 and optimization problem (NUM-M).  Theorem 26  (a)  For any choice of system parameters, the fixed point of Algorithm 8 (Algorithm  9) always exists. (b) If the number of channels C  =  1, then Algorithm 8 (Algorithm 9) has a  unique fixed point. (c) If C> 1, then Algorithm 8 (Algorithm 9) has at least two (i.e., non-unique) fixed points.  The proof of Theorem 26 is given in Section 7.11. From Theorem 25 and Theorem 26(b), if the number of channels C  =  1, then the unique fixed point of Algorithm 8 (Algorithm 9) is the  180  Chapter 7. Multi-Interface Multi-Channel Random Access unique global optimal solution (i.e., the only stationary point) of problem (NUM-S) (problem (NUM-M)). On the other hand, from Theorem 25 and Theorem 26(c), if C> 1, then any fixed  point of Algorithm 8 (Algorithm 9) is at least a local maximum of problem (NUM-S) (problem (NUM-M)). Next, we discuss convergence. Let USCR (t) and UMCR (t) denote the aggregate network utili ties at time slot t, while running Algorithms 8 and 9, respectively. We can show that:  Theorem 27 For any choice of system parameters, (a) At each time slot t, the instantaneous  aggregate network utilities USCR(t) and UMCR(t) are upper bounded: USCR(t), UMCR(t)  where  Jm  USCR(t)  =  maxnEv I,- and  max 7  =  <L u(C i  7max)  max(n,m)e1,ccc ‘YL.  V(n, m)  (7.39)  (b) The instantaneous network utilities  and Uj1CR(t) are non-decreasing. That is, for any T> 2, USCR(1)  ) 2 USCR(  USCR(T),  (7.40)  UMQR(l)  UMCR(2)  UMCR(T).  (7.41)  (c) Starting from any arbitrary initial point (p,  Q) and p, Algorithms 8 and 9 asynchronously  converge to one of their fixed points, respectively. That is, there exist UcR and UJ4CR such that:  UCR  =  UJ,ICR  =  lim USCR(t),  (7.42)  lim UAJGR(t).  (7.43)  t—*oo  and t-+oo  The proof of Theorem 27 is given in Section 7.12. Notice that, from Theorem 25, UCR and  181  Chapter 7. Multi-Interface Multi-Channel Random Access  are the local maxima of optimization problems (NUM-S) and (NUM-M), respectively. In many cases, the achieved fixed points are not only locally optimal, but also globally optimal. For example, we can verify that for the sample topologies in Section 7.2.2, we have: UcR and U7cR  7.4  =  =  U7cR. We further discuss optimality in Section 7.4.7.  Performance Evaluation  In this section, we evaluate the performance of our proposed DMMRA algorithms. We study the convergence, robustness, and optimality properties, evaluate the performance gain of assign ing partially overlapped channels, and measure the signalling overhead for both Algorithms 8 and 9. We also compare DMMRA with utility-optimal combinatorial interface assignment and channel allocation (UO-CIACA) [107] and multi-channel medium access control (MMAC) [114] algorithms.  In our simulation model, we consider ten different random topologies. Unless otherwise is stated, each topology includes N =10 nodes, randomly located in a 500 m x 500 m square field. Communication and interference ranges are 150 m and 250 m, respectively. The peak data rates (i.e., -y for all (n, m) E  and c e C) are selected randomly between 6 and 54 Mbps, as in the  IEEE 802.lla standard. Except in Section 7.4.3, where we study the impact of utility parameter c, the utility functions are assumed to be logarithmic. Each time slot is 1 sec. The simulation time is 1000 time slots.  182  Chapter 7. Multi-Interface Multi-Channel Random Access  7.4.1  Convergence  In this experiment, we set the number of channels C all n  =  6 and the number of NICs I  =  2 for  e .A/. The trends of the network utilities for the first simulated topology (i.e., topology  number 1) when the DMMRA algorithms are being used are shown in Fig. 7.3. We can see that both Algorithms 8 and 9 converge to their fixed-points very fast, i.e., within 152 and 146 time slots, respectively. We can also observe that the utility values are non-decreasing and bounded, which confirm the results in Theorem 27(b). At the steady state, Algorithm 8 results in 34% higher utility, compared to UO-CIACA. Using Algorithm 9, the utility is further increased by 40%. Thus, equations (7.40) and (7.41) hold as strict inequalities in this case. Similar results are observed for other topologies.  7.4.2  Comparison with Combinatorial Channel Assignment  Next, we compare DMMRA with UO-CIACA [107] in terms of both utility and throughput. Simulation setting is the same as in Section 7.4.1. Results are shown in Fig. 7.4. In this figure, each point is the average of the measurements for all ten simulated topologies. We can see that both utility and throughput increase as there are more channels available. Algorithm 8 results in 36% and 23% higher utility and throughput, compared to UO-CIACA, respectively (see Theorem 22(b)). Using Algorithm 9 leads to further 57% and 71% increase in utility and throughout, respectively (see Theorem 22(a)).  7.4.3  Impact of Utility Parameter a  Recall from Section 7.2.1 that by changing the utility parameter a, various design objectives can be modeled. In particular, a can act as a knob in Algorithms 8 and 9 to control the trade-off 183  Chapter 7. Multi-Interface Multi-Channel Random Access  60 50 onvergenceThme 40  Aloithni (DMMRA-M Algorithm 8 (DMMRA—S) 40 %  30 UO—CIACA Algorithm D  20  ...r.  -  .  r.  .  .  -—  °4  —  .  L  .  0  10 231,%  0 —10  Single—lntèrfáce Single—Channel  —20 —30  0  100  200 300 Time Slot  400  500  Figure 7.3: Trend of the network utility versus time slots using Algorithms 8 and 9 for the first simulated topology.  184  Chapter 7. Multi-Interface Multi-Channel Random Access  (a) 50  I  I  0 *  30  z  4 6%  20 -  I  Algorithm 9 (DMMRA—M) Algorithm 8 (DMMRA—S) UO-CIACA Algorithm .  r’  *  0 -10  210%  Sir gle-Interface Single-Channel  —20 —30  .  -  1  2  3 4 5 Number of Frequency Channels C  6  (b) 160 140 120 C)  2 I—  I  0 *  Algorithm 9 (DMMRA-M) Algorithm 8 (DMMRAS) UO-CIACA Algorithm  100  .. .,,,  .  .  .  80 ••J’J  .  -  3% .  a)  344% .-  20  SkgIeHnterfaceSine-Ch:nnei_-__6  -  Number of Frequency Channels C  Figure 7.4: Comparison between Algorithms 8 and 9 with (UO-CIACA) algorithm [107] in terms of both utility and throughput. The number of available channels varies from 1 to 6. (a) Network utility, (b) Aggregate throughput.  185  Chapter 7. Multi-Interface Multi-Channel Random Access between efficiency and fairness. In this section, we compare DMMRA-S algorithm with MMAC [114]. MMAC is the multi-channel extension of the IEEE 802.11 distributed coordination func tion. For each NIC, it assigns the channel which has the least scheduled transmission within the neighborhood. MMAC is designed for single-interface multi-channel networks. It also assumes that each NIC can listen to only one channel at a time. Thus, MMAC is most comparable with Algorithm 8, where  =  1 for all n e jV. We assume that the number of channels C  =  6. Running  both Algorithm 8 and MMAC for all ten topologies, the network throughput and fairness index, when a varies between 0.5 and 5, are shown in Fig. 7.5 (a) and (b), respectively. The fairness index is calculated among the data rates of all links as in [89]. We can see that, by increasing a, we can make the system more fair, but less efficient (and vice versa). If a  0.5, then Algorithm 8  results in 74% higher throughput, compared to MMAC. If a = 5, then Algorithm 8 results in 97% higher fairness index. Thus, Algorithm 8 can be set to achieve efficiency or fairness (or both). Notice that for a  e (0.5, 2), Algorithm 8 makes the system both more fair and also more efficient.  Similar results are observed for Algorithm 9.  7.4.4  Signalling Overhead  Both Algorithms 8 and 9 require message exchange among the neighboring nodes. In this section, we measure the signalling overhead for each algorithm and compare it with the aggregate network throughput. We assume that each probability value occupies two bytes. Thus, for each node n  e /, the message size is 2C(L,h1t + 2) bytes and 2C(Lt + 1) bytes for Algorithms 8 and 9,  respectively. For each NIC i  e I,, we assume the use of limited-scope message flooding [12, pp.  408] to distribute (p, Q) (for single-channel reception scenario) and p (for multi-channel reception scenario) to all nodes within 2Rrnax distance. The signalling overhead (in Kbps) and 186  Chapter 7. Multi-Interface Multi-Channel Random Access  (a) 100 Algorithm 8 (DMMRA—S) MMAC  *  C’)  80  60  40  20 0.5 0.6 0.7 0.80.91 I  I  I  I  I  4  3  2  5  Utility Parameter, a. (b) 1  x 0.8  a) C Cl) (I)  a)  C I-  Cu U-  0.4  .  .  rIMMAC .  ..  .  .  .  .  Algorithm 8 (DMMRA—S)  0.2 0.5 0.6 0.7 0.80.91 I  I  I  I  I  2 Utility Parameter, a.  3  I  4  5  Figure 7.5: Comparison between Algorithm 8 and MMAC [114]. Each node has one NIC and there are 6 channels available. For Algorithm 8, the parameter a is varied from 0.5 to 5. (a) Throughput, (b) Fairness index.  187  Chapter 7. Multi-Interface Multi-Channel Random Access throughput (in Mbps), when the number of nodes N varies from 10 to 50, are shown in Fig. 7.6 (a) and (b), respectively. The signalling overhead increases as the number of nodes increases. However, it is always negligible compared to the throughput (i.e., less than 0.002%). We also notice that Algorithm 9 always incurs lower signalling overhead as it has smaller messages and converges faster. When N  =  50, Algorithm 9 results in 37% lower signalling overhead and 91%  higher throughput, compared to Algorithm 8.  7.4.5  Partially Overlapped Channel Assignment  Given the data rate models in (7.19) and (7.21), Algorithms 8 and 9 can be used to assign not only the non-overlapped channels, but also the partially overlapped channels. This is particularly important when the number of orthogonal channels is limited; e.g., as in IEEE 802.llb standard, where only 3 out of 11 channels are non-overlapped. The throughput, when the IEEE 802.llb standard is used, is shown in Fig. 7.7. The channel filters are assumed to be raised cosine with roll-off factors equal to 1 (cf. [95]). The dashed lines correspond to the measured throughput when either single channel (i.e., channel 1), two non-overlapped channels (i.e., channels 1 and 6), or three non-overlapped channels (i.e., channels 1, 6, and 11) are used. We see that, using Algorithm 9, assigning the partially overlapped channels 1,  ...,  6, instead of assigning only  non-overlapped channels 1 and 6, results in 11% higher throughput. By assigning all partially overlapped channels 1,  ...,  11, instead of assigning only the non-overlapped channels 1, 6, and  11, the throughput is increased by 13%. Similar results are observed for Algorithm 8. Here, the improvements are achieved without using extra resources. Thus, the available spectrum is utilized more efficiently.  188  Chapter 7. Multi-Interface Multi-Channel Random Access  (a) 1.6 *  ..  1.4  °  Algorithm 8 (DMMRA—S) Algorithm 9 (DMMRA-M)  —  —  7%  .  12  06  20  30 Number of Nodes, N  40  50  (b)  I Number of Nodes, N  Figure 7.6: Signalling overhead and aggregate network throughput for Algorithms 8 and 9 when the number of nodes N varies from 10 to 50. Each node is equipped with 2 NICs and there are 6 orthogonal channels available.  189  Chapter 7. Multi-Interface Multi-Channel Random Access  j3%7 3 non—overlapped channels 60  50  .  :  ..  40  2non-overlapped channe  11%  :  2 i-30  :  -  (channel 1, 6) ..  20  Siriglèchàhñe Algorithm 9 (DMMRA—M)  0  101 1  _,-‘j  (chàrihel1,:6, I  I  2  3  I  4  I  :  (channel 1)  I  5 6 7 Number of Channels  8  9  10  11  Figure 7.7: Performance improvement when both orthogonal and partially overlapped channels are being used. The number of available channels is varied from 1 to 11.  7.4.6  Impact of Delayed and Outdated Information  In some practical scenarios, the nodes may receive outdated information about the persistent and listening probabilities of other nodes. This can be due to communication delay (e.g., queueing or propagation delay) or message loss. The latter can occur due to channel imperfections (e.g., fading) or packet collision. In this section, we study the effect of outdated information exchange on the performance of DMMRA algorithms. In particular, we consider the case where the com munication medium imposes random delay on the exchanged messages. The trend of the network utility for the first simulated topology, when Algorithm 8 is being used and the messages experi ence delay up to 10 time slots (i.e., 10 seconds), is shown in Fig. 7.8. We can see that Algorithm 190  Chapter 7. Multi-Interface Multi-Channel Random Access 8 still converges to its fixed-point, even though the information used by the nodes is outdated. However, communication delay may cause utility fluctuation (compare Fig. 7.8 and Fig. 7.3). That is, at some time instances, since the node, which executes DMMRA, may not have accurate information about the persistent and listing probabilities of its neighboring nodes, the value of the utility may be decreased. Nevertheless, Algorithm 8 still (asynchronously) converges to its fixed point, but with lower convergence speed. Similar results are obtained for Algorithm 9. From the results in Fig. 7.9, we can also see that as the delay increases from 0 to 10 time slots, the convergence time increases by 98% and 83% for Algorithms 8 and 9, respectively. We notice that Theorem 25 still holds. In fact, the fixed-points of DMMRA algorithms are independent of delay. Thus, the fixed points of Algorithms 8 and 9 are still guaranteed to be the stationary points of problems (NUM-S) and (NUM-M), respectively.  7.4.7  Optimality  From Theorem 25, every fixed point of DMMRA algorithms is a stationary point of the formulated NUM problems. That is, each fixed-point is at least a local maximum. However, the fixed-points  may not always be globally maximum. In this section, we investigate the optimality of Algorithms 8 and 9. Results are shown in Fig. 7.10. In this figure, the average utility is compared with the optimal utility for each topology. To obtain the results, we calculate the utility at all fixed-points. This is done by partitioning the search space and running DMMRA with the initial points varying among the partitions. Specifically, for each topology, we obtain both mean utility value and maximum utility value among all fixed-points. The former indicates the average performance of our proposed DMMRA algorithms, while the latter indicates the optimal performance. From the results in Fig. 7.10, DMMRA achieves near-optimal solutions for all ten simulated topologies, On 191  Chapter 7. Multi-Interface Multi-Channel Random Access  40 30 20  Utility Fluctuation  >.,  D 0  ci)  z —10 —20  —30  0  100  200 300 Time Slot  400  500  Figure 7.8: Trend of the network utility versus time slots using Algorithms 8 for the first sim  ulated topology in the presence of communication delay. average, Algorithms 8 and 9 result in 96.5% and 97.4% optimality, respectively. In fact, although the fixed-points are not always globally optimal, they lead to near-optimal performance.  7.5  Summary  In this chapter, we formulated a novel continuous multi-interface multi-channel random access model. Both single-channel reception and multi-channel reception scenarios were considered and the data rate models were obtained accordingly. We then formulated a multi-interface multi channel NUM problem and proved that its optimal solution outperforms most of the previously proposed combinatorial channel assignment strategies. We also proposed a simple, fast, and 192  Chapter 7. Multi-Interface Multi-Channel Random Access  *  °  450  Algorithm 8 (DMMRA—S) Algorithm 9 (DMMRA—M  )  I  Cl)  .  400k  150 100  I  0  2  10  4 6 8 Communication Delay (Time Slots)  Figure 7.9: Impact of delay on convergence time. distributed algorithm, called DMMRA, to solve the formulated NUM problem. DMMRA re quires each node to only iteratively solve a local, myopic, and convex optimization problem. The convergence and optimality of the proposed algorithm were proved.  Simulation results show  that DMMRA with single-channel reception results in 36% and 23% higher network utility and throughput compared to combinatorial channel assignment. If multiple-channel reception is being used, then the utility and the throughput further increase by 57% and 71%, respectively.  193  Chapter 7. Multi-Interface Multi-Channel Random Access  70  I  I  I  I  I  Algorithm 8 (DMMRA—S) 1 Optimal Utility (Single—Channel Reception) Algorithm 9 (DMMRA—M) I Optimal Utility (Multi—Channel Reception)  60 I 50 > 4-’  V  D -  I  0  .30 1)  z  20 10  0—  -  1  2  -  -  -  -  -  -  4 7 5 6 Topology Number  3  8  9  10  Figure 7.10: Comparison between achieved aggregate network utility and its optimal value for  each proposed algorithm and each simulated topology.  7.6  Analytical Proofs  7.7  Proof of Theorem 22  Part (a): For each wireless node m é A , any of its NICs 1 c  e C,  j  é  m, 1  and each frequency channel  we have: 1_P)(c) _q(i)(c)  by(7.1) —dEC\{c}  L.dEC\{c}  p(j)(d)+  Q(3)(d) dEC\{c}  (i)(d) m  Replacing the above inequality in (7.4) and (7.11), we have:  ‘P (A) (-), iil  194  (744)  Chapter 7. Multi-Interface Multi-Channel Random Access  A))  which also implies that (i—IP(A U  (1—IP(A  A)). Hence mm  u  greater than or equal to the one in (7.10). Since the utility function  jfl  U(mnm)  (7.14) is always is an increasing  function of mm, the inequality (7.24) is resulted when we sum up the utilities of all links. Part (b): Let A denote the feasible set of problem (NUM-C). It is clear that, A 1. Thus, any  (p,  Q) e A is also a feasible solution of problem (NUM-S). Hence, no  (, )  A can lead  to an aggregate network utility which is greater than the optimal utility UCR over the set  .  Therefore, the inequality in (7.25) always holds.  7.8  Proof of Theorem 23  For each node n  e .Af  and any NIC i  e  I,, the objective function of problem (LOCAL-NUM-S)  can be written as:  Z meKt U  cEC  (i)(c) (i)(c) (kn,m Pnm +  + mE.Af (ZcEc(@?$ + Z mEJf\{n} where for each node m  e  .Af0t  1’  jeI\{i}  sei’.f\{n,rn}  (f1im(1 (i)(c)  (n,m  (  E  —  x u,  (i)(c)”  C,  p(i)(C)  p(k)(c) S  Q(t)(C))) F1iim(1 p4_ —  —  (c) (i)(c)I  sEN\{n,m}  (1  —  jeZ\{i} Ynm Pnm  (n (i 31 (n  (i)(c) —  (c))+ Q)(C)))  i)(c) eeC /—‘n,ms  I fl 57 5 lV kI  —  —  —  (  I\  and any channel c  —  (n  ?$) (ip(i)  seK,ut\{n}  (i)(c)_ (e) n,m 7nm  ( 1  (i)(c) (,irn  keX\{i,j}  p(k)(c)  j 1 —  (k)(e)  Il  5 VUsm kEZ  —  —  ,(i 1 n 195  —  P$V(c)_ Q(a)(c)))  Chapter 7. Multi-Interface Multi-Channel Random Access For any node mEJV  and any channel eEC, (c) (k)(c) 7mnPmn (FIsEJV\{m,n} FuEls  (i)(c) 8n,m =  (i  (flvEIm\{k}  (flI\{} (i  —  (c) (k)(c) (i)(c) c—, n,m = kEIm7mnPmn (fls{m,n}  (FlvEIra\{k}  Q  —  (i  1n 18 (i  p(vXc)V\  m  j (flex\{} 1  (i  —  —  —  (j)(c)\ —  fl  \  )  (l)(c))\  Pfl Q?c2)(C))).  For any link (rn,n)E\( Ut) and each cEC, (i)(c) = 8 , fl,TTIS  (c) (j)Q) L.kEIm 7ms Pms  (lJjEIm\{k} (i (i  (HvEW\{m,s,n} fl1EI —  pQ)(c))  (nIEIS(1  Notice that  (i)(c) cn,m  can be treated as  ,  ,(i)(c) ,n,rn  ,  O)(c) n,m  constants  ,  —  ri,m  and  —  p(i)(c)))  —  fl•  (1_P1)(c)_Q1)(cD)))  (—i) ()(c) (—i) In fact, they 13n,ms only depend on p and Q .  as far as problem (LOCAL-NUM-S) is concerned. Since the utility  functions are concave, the objective function of problem (LOCAL-NUM-S) is a summation of concave-affine  (z)  compositions over p and  Q.  Therefore, it is concave [16, pp. 84]. Clearly,  is also a convex set. Together, these imply that problem (LOCAL-NUM-S) is a convex optimization problem. The proof for (LOCAL-NUM-M) is similar and is omitted.  7.9 Let N  Proof of Theorem 24 = {n, m}, £r, =  {i}, and  rnm(p)  £m =  {j}. For wireless link (n, m) e £, we have: (c)  =  (i)(c)  ZcC7nmPnm  196  —  dECPmfl)  (745)  Chapter 7. Multi-Interface Multi-Channel Random Access The data rate  rmn(p)  can also be obtained similarly. We can re-write optimization problem  (LOCAL-NUM-M) as: maximize u p C  (c)(c)))  ((1  eC  (zC  (7.46) (‘—Ccp)).  From Theorem 23, problem (7.46) is a convex optimization problem. By solving the Karush Kuhn-Tucker (KKT) sufficient and necessary optimality conditions (cf. rearrangement the terms, we can show that for any fairness index  (- (i_ z (>  Q  [16, pp.  244]) and  and for any channel c E C,  ((d/(C))d) *  (e) (j)(e)* eec 7mnPmri  )  I  (Zec —  (i)(d)* deC Prim  (7.47)  c.  (7.48)  / From (7.35) and (7.47), the optimal solution of convex optimization problem (7.46) can be ob —  tained by solving the following system of linear equations: Zdec  (i)(d)*  ((_i))  Vc e  =  In vector representation, (7.48) is equivalent to: M (p))  (j)* =  (7.49)  1  The solution of the system of linear equations in (7.49) is obtained as in (7.34). In general, the closed-form solution can also be obtained using substitution method (cf. [115]). Notice that, any optimal solution of problem (7.46) should satisfy (7.49). Since problem (7.46) is always a feasible optimization problem (cf. [15, pp. 9]), the solution in (7.34) always exists, regardless of the choice of system parameters. However, the solution may not be unique. For example, if o any persistent probability vector  is globally optimal as long as Zc  197  c)  =  =  1,  .  Chapter 7. Multi-Interface Multi-Channel Random Access  7.10  Proof  of  Theorem  25  Consider the case where the NICs implement single-channel reception. (p*  (j)*  Q*) e  the tuple (pn  FSCR,  (z) , Q  lem in (LOCAL-NUM-S) for any i  j  e  e  ,  For any fixed point  is an optimal solution of the convex optimization prob  2, for all n  e  N; otherwise, there would exist some NIC  2 for some node n e N which deviates its persistent or listening probabilities from the fixed  point to achieve a better solution for its local optimization problem (see line 10 in Algorithm 8). Thus, for any node n.  e  N and each of its NICs  (j)*  i  e  I, the tuple (pn  ()* , Q  )  needs to satisfy  the following KKT conditions [16, pp. 244]. For each channel c e C, (i)(c) n,m ()(d)*  jd  ( EdEc sn,m Pnm  —  —  ZvEJ’f  (  +  (i)(c) (n,m  $4)  ((i)(d) dEC  (  (i)(c)  _p(i)(d)* 1 ( ))  ,(i)(d)) (l_P,  )  —  vEN0ut  (i)(d)  (i)(c)  =  VEUPflV n  where (7.30).  .,.$  +Q$—1)  —  fl  rim  ,  Vm  (i)(c)  (7.51)  —  (i)(c) (i)(c)* = 0 m 1 Qri Q(i)(C)  0,  S(i)(c)  rim  (7.52)  0,  Z  n  0,  (i)(c)* , Priv (i)(c)  ,  )  0,  =  *  =  ))  (7.50)  1_6Sp*)a =  (i)(d) (i)(d) ZvEJffl(((SOfl,V -1fl,V  Pnm  _pi)(d)* 1 (  *)+,,(d) Q(i)(d)\  (i)  (8 i)(c) (i)(c)* nm Pnm  Pnv  —  ZveJ1\{n}Zse.A/t\{n} (EdEc  (z  (i)(d)*  and u,  (7.53) (7.54) (7.55)  are the Lagrange multipliers corresponding to the constraints in  On the other hand, by definition (cf.  [15, pp.  1941), any stationary point in  SCR 8  satisfies all KKT conditions of problem (NUM-S). Since the objective functions in (NUM-S) and (LOCAL-NUM-S) are the same and the set of constraints in (NUM-S) is the union of the sets of constraints in (LOCAL-NUM-S) for all NICs, the KKT conditions for non-convex optimization 198  Chapter 7. Multi-Interface Multi-Channel Random Access problem (NUM-S) are indeed the union of the conditions in (7.50)-(7.55) for all i  e N.  any n that,  FSCR  Thus, any fixed point (p*,  c  SSCR.  Q*) e  FSCR  c SSCR and SSCR c .FSCR, we have:  7.11  Proof of Theorem 26  and PI  = Qi)(c)  =  FSCR.  SSCR  for all c  e N e  and any NIC i  C and any m  e  e I, we can select  ut  as a feasible (not  necessarily optimal) solution for problem (NUM-S). Similarly, we can select p for all c and  I’I  >  e  C and m  e  £‘  Since  FSCR = 8 SCR•  Part (a): It is easy to verify that for any node n =  is also a stationary point. This implies  Following a similar argument, we can show that,  .FSCR  e 2 and  =  and  L  as a feasible solution for problem (NUM-M). Thus,  1  1. Since both problems (NUM-S) and (NUM-M) are feasible problems, they have  at least one stationary point [15, pp. 194. From this, together with Theorem 25, we have: FSCRI  =  ISSCRI >  1 and IFMcR  =  1.  ISMCRI  Part (b): From [98, Lemma 2], in a single-channel network, problems (NUM-S) and (NUM-M) can be transformed to equivalent convex optimization problems using the logarithmic change of variables. Since a convex problem has a unique stationary point, we have: From this, together with Theorem 25, IFSCRI  =  IFMcRI  ISSCRI  =  SMcl?I  =  1.  1 and the fixed points are unique.  =  Part (c): We prove this part by contradiction. Consider Algorithm 8 and assume that it has a unique fixed point (p, Q). From Theorem 25, (p, problem (NUM-S). Now consider another point i  e 1, any channel c e C and any node —(i)(c)  Prim  —-  —  m  (i)(C-c+1) Pnm  £ut,  ,  Q)  is also a stationary point of optimization  (, Q),  where for all nodes n E N, any NIC  we have: —  —  199  (—)(i)(C——c+1) n “  (7  Chapter 7. Multi-Interface Multi-Channel Random Access For example, if the number of frequency channels C (i)(3)  r(p,  (i)(2)  _(i)(3)  Pnm , Prim  =  _(i)(4)  Prim , and Prim  Q) = r(j5, ).  Thus,  (, Q)  (i)(i)  =  Pnm  =  From (7.10) and (7.19),  .  it  ) 2  =  4, then we have:  =  is easy to verify that:  is a stationary point of optimization problem (NUM-S). From  Theorem 25, it is also a fixed point of Algorithm 8. However, this contradicts the assumption that (p,  7.12  Q)  is a unique fixed point. The proof for Algorithm 9 is similar.  Proof of Theorem 27  Part (a): From (7.10), (7.14), (7.19), and (7.21), for each (n,m)E, mm  —  —  ‘ç’  ç-’  C  nm LdjeXn /dcEC 7  —  Thus, the utility of each wireless link (n, m)  rmax max V’ 7 L.dCEC  e  — —  rmax max  7  £ is upper bounded by u(C  aggregate network utility is upper bounded by L u(C Part (b): We prove this part by contradiction.  irnax 7maX)  irnax 7max).  First consider the case where the NICs  implement multiple-channel reception model and assume that at some time slot t USCR(t  t E  —  1) >  USCR(t).  and the  In that case, there exist a node n and an NIC i  e  I7  e  [2, T],  such that  and solving problem (LOCAL-NUM-S) at NIC i reduces the network utility (i.e., the  objective function in problem (NUM-S)). However, this is impossible as the objective functions in problems (LOCAL-NUM-S) and (NUM-S) are the same. Thus, USCR(t  —  1)  < USCR(t)  and  (7.40) holds. The proof for (7.41) is similar. Part (c): The existence of the limitations in (7.42) and (7.43) is directly resulted from parts (a) and (b). Notice that any bounded non-decreasing sequence of real numbers always converges to a fixed point.  200  Chapter 8  Two-Fold Pricing to Guarantee Individual Profits and Maximum Social Welfare in Multi-Hop Wireless Access Networks Various pricing schemes have recently been proposed either to encourage collaboration among the network elements or to utilize the network resources efficiently. Pricing as a tool for resource allocation was first proposed in [44, 46] for congestion control among elastic traffic sources. In this regard, the network is designed to solve a network utility maximization (NUM) problem across all traffic sources, subject to the link capacity constraints. The corresponding Lagrange multipliers are interpreted as the congestion prices. Each source which uses a link resource is charged with the link’s congestion price. The transmission rates and the congestion prices are iteratively updated using the gradient projection method until the global optimal network utility is achieved. The work in (44] has been extended to other resource allocation problems such as medium access control, power control, frequency channel assignment, and spectrum sharing  201  Chapter 8. Two-Fold Pricing [30, 47, 116, 117, 118, 119, 1201. Recently, it has also been shown that the gradient updates can be replaced by the best-response updates to achieve faster convergence and more robust performance [54]. Another thread of research focuses on using pricing to encourage collaboration among the nodes [121, 122, 123, 124, 125, 126, 127, 128, 129]. In a multi-hop network, where the nodes need to forward packets for other nodes, the optimal network performance might be at the cost of performance degradation for some intermediate relay nodes. When the intermediate nodes have no incentive to collaborate, the well-known forwarder’s dilemma (cf.  [130]) can occur,  where no node forwards the packets for other nodes. To resolve this problem, incentives can be offered to the relay nodes in the form of payments or rewards in turn for their help in forwarding other nodes’s traffic. In general, achieving the optimal network performance may not be always guaranteed in the incentive-based strategies as they mainly take the individual profit objectives into consideration. The problem of designing pricing models for Internet service providers (ISPs)  in a fixed wired network has been studied in [122, 123, 124, 125]. In [122], He and Wairand proposed a pricing scheme which encourages the ISPs to collaborate to achieve a fair revenue sharing. In [123], Shen and Basar studied the pricing problem in wired access networks using game theory. The ISPs are modeled as strategic players to maximize their revenue and the users are modeled as natural players to maximize their utilities. The interaction between each ISP and its associated users is modeled as a Stackelberg game. In [124], Davoli et al. considered the pricing problem where the ISPs do not have any knowledge about users’ utility functions. Pricing for multicast wired Internet has also been studied in [125]. The pricing models for wired networks cannot be easily extended to wireless access networks. In general, there are two main challenging issues that need to be addressed in wireless access 202  Chapter 8. Two-Fold Pricing networks: channel imperfection (e.g., wireless fading), and interference. In [126], Neely proposed an economic model for wireless ad-hoc networks, with stochastic channel states, within the general framework of back-pressure algorithms [131, 132]. The relay prices are used to encourage packet forwarding. However, it is essentially assumed that the network is interference-free  That is,  each node can transmit with an arbitrary high transmission power without interfering with other nodes. Interference-free pricing has also been addressed in [127] and [128]. In general, most of the previously proposed pricing models in the literature have one or more of the following performance bottlenecks: (1) network resources are not efficiently (i.e., optimally) allocated, (2) individual profits are not taken into consideration, and (3) interference among the wireless transmissions is not taken into account. In this chapter, we address these performance bottlenecks in all three aspects. In particular, we extend the work by Neely [126] and propose a market-based network model with two-fold pricing (TFP) which fully incorporates the effect of interference. Our model uses relay-pricing to encourage nodes to collaborate and forward each other’s packets. We also use interference-pricing to encourage the wireless relay nodes to properly share the common network resources. Together, the relay and interference prices incorporate both cooperative and competitive interactions among the nodes. We analytically prove that for a wide range of pricing functions, our proposed TFP scheme leads to a guaranteed positive profit for each individual node. The profit increases as the node forwards more packets. This better pays off the collaborative nodes. Finally, assuming the presence of slow-fading channels, we obtain the relay and interference pricing functions in a code division multiple access (CDMA) network such that not only the positive individual profits are guaranteed, but also the network social welfare and the network utility are maximized. Compared with the single-fold pricing (SFP) model in [126], simulation results show that our TFP scheme can increase the social welfare and the network 203  Chapter 8. Two-Fold Pricing throughput by 24.6% and 14.7%, respectively. It also leads to more fair revenue sharing among the nodes as it results in 18.3% higher fairness index. The rest of this chapter is organized as follows.  Our proposed two-fold pricing model is  described in Section 8.1. The key properties of our model are analytically proved in Section 8.2. Simulation results are presented in Section 8.3. A summary is given in Section 8.4.  8.1  Market-Based Wireless Access Network Model with Two-Fold Pricing  8.1.1  System Model  Consider a stationary wireless access network. Let J’f, with size wireless relay nodes and £, with size I £  =  IA/i  =  N, denote the set of  L, denote the set of unidirectional wireless links. For  each node nEV, the set of all incoming and outgoing links are denoted by £ C £ and respectively. We also define JV,  =  {m: (m, ii)  e £}  and  ut 4 j.  =  {m: (n, m) E  £t  £t}  CL,  as the  set of in-neighbors and the set of out-neighbors of node n, respectively. Wireless relay nodes are assumed to be independent commercial entities. Together, they form a wireless backbone to provide connectivity among wireless users in a multi-hop manner. The set of users is denoted by D, with size  ID!  =  D. Each relay node n eJV offers connectivity only to a subset of users, denoted  by D c V. Each user is offered connectivity from exactly one wireless relay node. All users 3 wants i, j e D are able to communicate directly with each other. However, if any user i e V, to send data to another user k e V, where c E f\{n}, it should first transfer the data to node n, and the data are then transferred to node c via the intermediate wireless relay nodes before delivering to user k. In turn, node n charges user i for its offered connectivity service. We assume 204  Chapter 8. Two-Fold Pricing  /  (( ))  (( ))  N N  /  Wireless Link  /  N  /  b  / \  1  ()  /  N N  Wireless Relay Node  /  N J  \  N  / ()  Wireless users  Figure 8.1: A sample multi-hop wireless access network with six wireless relay nodes, labeled as n,m,s,a,b,c, and fifteen wireless users. Here D = {i,j} and k e DC. Users i and j can directly communicate with each other. However, if user i (or user j) wants to send data to user k, it should first transfer data to its associated wireless relay node (i.e., node n), and the data are then transferred to wireless relay node c via the intermediate nodes (e.g., wireless relay nodes s and a) in a multi--hop manner before being delivered to wireless user k. In turn for the Drovided connectivity service, wireless relay node n and all the intermediate wireless relay nodes are paid according to their offered relay prices.  that all wireless relay nodes communicate over the same frequency band which is different from those frequency bands used by the users to communicate with each other and their associated wireless relay nodes. This avoids interference between access and relay transmissions. However, the transmissions among the wireless relay nodes can still interfere with each other. A sample wireless access network is shown in Fig. 8.1. In this figure, there are N labeled as n, rn, s, a, b, and c. There are also D Each wireless relay node n  =  =  6 wireless relay nodes,  15 wireless users.  e .A( is assumed to have N—i separate queues to store the incoming  data according to their final destination. All data that are destined to any of the users of relay  205  Chapter 8. Two-Fold Pricing node ceN\{n} are stored in the  queue. The contents of the  queue are called commodity ç A1t, which includes its  c data. For each commodity c data, node n maintains a set  neighboring relay nodes with minimum hop-counts to node c and can relay commodity c data (c) towards node c. For example, h = {m, s},  (c) l1m  (c)  =  {a}, fl  Time is divided into equal-length slots T = {O, 1, 2,.  .  .  =  }.  {a}, and  (c)  fla  =  {c} in Fig. 8.1.  For each link (n, m) e £, let  2nm  denote the set of all possible channel states. Channel states can vary (e.g., due to wireless fading). At each time slot t e T, the current channel state is denoted by  wrjm(t)  e  flnm.  We stack up  the channel states of all links at time t and denote the obtained L x 1 vector by w(t). That is, =  (wnm(t),  V n, m E N, (n, m) E  £).  Let T C T denote the set of time slots at which the  channel state vector w changes. We assume that  ‘  has an independent and identical distribution  (i.i.d.) over time slots 7.,. We also assume the slow-fading scenario such that:  2 It  —  j > A, 1 t  (8.1)  ,t e Ta,, 1 Vt 2  where A>> 1. That is, there are at least A time slots between any two consecutive changes in channel states. We will consider the fast fading case (i.e., when A  —*  1) in Section 8.3. let p (p(t), w(t)) 4  For each wireless relay node n e N and any of its neighboring nodes m e  o  denote the transmission rate offered to commodity c data over link (n, m) during time slot t.  Here, p(t)  =  (p(t), Vn, meN, VceN\{n}, (n, m) ei) denotes the L(N  —  1) x 1 vector of  transmission powers for all links and all commodities. The scalar pAL(t) > 0 denotes the trans  mission power corresponding to the transmission of commodity c data over link (n, m). In this chapter, we assume that all nodes use CDMA technology. At each time slot t e T and for each wireless relay node n e N, the commodity c e .Af\{n} data transmission rate over wireless link  206  Chapter 8.  (n,m) e  Lout  Two-Fold Pricing  can be modeled as [133]: p(t),w(t))=Aslog(l+nmm(t)PThrn(t)), nm(p_n(t)) + Tim  (8.2)  where A 3 denotes the channel symbol rate, K is the processing gain, Tim denotes the noise power at the receiver node m, hnm is the channel power gain from relay node n to relay node m, p_(t)  =  (p(t), Vme.Af\{n}, sE.Nt, dEAr\{m}) denotes the transmission power of all nodes other  than node n, and Inm(p_n(t)) is the aggregate interference power on link (n, m). Notice that the term K hnm  wmm(t)  p)n(t)/(Inm(p_n(t)) +  Tim)  is the signal to interference plus noise ratio  (SINR) for commodity c data transmissions over link (n, m). We have: Inm(P_n(t))=>ZaEJ\{n} ham  Each node n eN limits its total transmission power such that where  prnax  (8.3)  (ZdEJr\{n} ZbEJVUtPab(t)).  ZCEv\{fl}  ZmEut  <prnax  > 0 is fixed. Thus, the transmission rates are always bounded. We can define: in  (p, w), 4  =  max, Zc6f\{n}  ZmE6f  =  max, ZcN\{n}  ZmEut  (8.4)  and out  44 w), i (p,  (8.5)  as the maximum data rate on any incoming and any outgoing link of node n eN, respectively.  8.1.2  Two-Fold Relay and Interference Pricing  Pricing among the wireless relay nodes In our market-based model, at any time slot t e T, if wireless relay node n e N transmits commodity c data with rate t(p(t), w(t)) to its neighboring wireless relay node meN,t, then it pays p$(p(t), w (t)) q (t) units of currency to node m as relay service charge. Here  207  (t)  0  Chapter 8.  Two-Fold Pricing  denotes the relay price corresponding to commodity c data, advertised by wireless relay node m. In total, at time slot t, node n pays: (p(t), w(t)) $(t) Zc/\{n} 4  (8.6)  units of currency to any neighboring wireless relay node m  as relay service charge.  4(p(t), w(t))) (Z-\ 1  (t) units of currency from any  Similarly, in total, node n receives wireless relay node m  e N,’ for its  offered relay service.  Besides the mutual relay services that the neighboring wireless relay nodes offer to each other, relay nodes also affect each other’s operation through interference power as shown in (8.2) and (8.3). From (8.3), for each wireless relay node n ZcEJf\{n} ZmEJV1t  e .N, the higher the total transmission power  p(t), the greater is the interference power that wireless relay node n causes  on other nodes. In our pricing model, at each time slot t, wireless relay node n pays: (ci\{n} ZmENit  p(t))  (t)  (8.7)  units of currency to each wireless relay node a e J’f\{n} as interference compensation charge. Here n)  (t)  0 denotes the interference price informed by relay node a to node n. Unlike the relay  prices which vary depending on the commodity data, the interference prices are the same for all commodities as the contents of the transmissions do not affect their interference level. Instead, the interference prices may vary depending on the node locations. The closer two relay nodes are located, the higher is the corresponding channel power gain. This results in higher interference power and consequently higher interference price. Similar to (8.7), at each time slot t n receives  (d\{a} bEJ\1a01t  (t))  e  T, node  (t) units of currency from node a as the compensation  for the interference node a causes on the transmissions of node n. For each wireless relay node n e A!’ and at any time slot t e 7”, let U (t) denote the current 208  Chapter 8. Two-Fold Pricing commodity cA/\{n} queue backlog. We define U(t)  =  (u(t),  Vnf, VcV\{n}) as the  vector of queue backlogs in all wireless relay nodes at time slot t. The relay and the interference prices are assumed to be set as follows: =  U(t—T),... U(t),p(t—T),... ,p(t)),  V c1\{n},  (8.8)  =  4)(U(t—T),... U(t),p(t—T),... ,p(t)),  V aA(\{n},  (8.9)  ,  ,  where I > 1 is a design parameter and i  (.)  and  ()  are two non-negative real scalar  pricing functions of all queue backlogs and all transmission powers at time slots {t  1,.  .  .  ,  —  I, t  —  I+  t}. The above pricing functions are general and can model various relay and interference  adjustment schemes. We only make a few technical assumptions. First, if U, (t) > 0, then  (.)  > 0.  That is, if relay node n already has some backlogged commodity c data, it will  not offer free relay service. Second, if  then  ‘]‘  (.) >0.  ZCEJr\{fl}  U, (t) > 0 and  ZcE,v\{n} ZmE.A4ut  p(t) > 0,  That is, if relay node n has any backlog and it is currently transmitting some  data on at least one of its outgoing links, it will not set its advertised interference prices to  ()  zero. Third,  is an increasing function of U,(t). Finally, there exists a large enough  tm such that for any commodity c é f\{n} and any time slot t but bounded constant V ) (t) < mU) (t). relay prices (t) ($(t), Vn  =  e T,  In general, the unbounded sets of time slots at which the vector of  ((t), Vn  e N, ceAr\{n}) and the vector of interference prices  &(t)  =  e N, a eN\{n}) are being updated are denoted by T cT and T CT, respectively.  Pricing between each wireless relay node and its users In our market-based model, each wireless relay node n E N provides relay service for its associated wireless users according to its relay prices. At each time slot t E T, if wireless user i E D wants  209  Two-Fold Pricing  Chapter 8.  e DC  to send data to another user k  of currency to wireless relay node  n  (for c  n) at rate r (t), it needs to pay  RX  ri  increasing, and strictly concave utility  its rates r  =  of user  (r(t), Vk  i’s  max  (8.10)  r(t)  Each user i E D also maintains a non-negative,  function g (r (t))  for any k  level of satisfaction from sending rate  e D\D)  (t) units  assigns all its users with a maximum allowed sending rate  according to its processing capacity.  monetary measure  (t)  as relay service charge. At time slot t, in total, user i pays:  CE\{fl} ZkEVC  We assume that wireless relay node  r  e D\D  r(t).  which indicates a  Thus, user i adjusts  by solving the following local optimization problem: (k)((k)(t))  ZCEJV\{fl} ZkeD  —  r(t) )(t))  “ )_  5. t .  (8.11) (k)(t<Rm \ L.iJf\{n} L..’keD r —  Notice that the objective function of the above optimization problem is always at least for r=O, it is equal to zero. We define user t9(t)  =  (ZCEJV\{fl}  ZkEv grt)  —  i’s profit  at each time slot t  non-negative  as  e T as:  (CEw\{fl}5ZkEv r(t))b)(t)).  (8.12)  From (8.11), user i adjusts r(t) to maximize its profit subject to the total rate constraint. Unlike the network model in [126], where each relay node can only support at most allow each relay node to offer the connectivity service to an  8.1.3  arbitrary  one  user, here we  number of users.  Resource Allocation  At each time slot t e T, given the advertised relay prices from all its out-neighbors, node  n  eiV  t and each cN\{n} as [126]: 7 can compute dfferential relay price for any mE.AI  =  4)(t)  —  210  )(t)  —  max  (8.13)  Chapter 8. Two-Fold Pricing where  max =  vm umax denotes the largest possible change in any relay price during one  time slot. Here, vm  =  max  /m8 1  and um  =  max{4’  the largest possible change in any queue backlog, where  1ax, in 1  (8.4) and (8.5), respectively. At the beginning of each time slot t Wnm (t)  for all of its outgoing wireless links (n, rn)  p, (t) = (p (t), V c e N\{n}, p(t)?O  £ut  and e  ±  fl1aX,  RX}  out  represent  are defined in  T, relay node n measures  and adjusts its transmission powers  mE Nout) by solving the following local optimization problem:  ZcEK\{n}Zm.Aft  t(p(t), w(t)) ö(t)  (Ze1\{} Zm’tp}n(t)) (aéi\{n}  —  kt)  (8.14)  pmax L.scW\{ri} L.rne.A4Ut Pnm. I < n  5. .  —  p(t)=0, V cE J’f\{n}, m where  e  in  out  or c  crn(t)  or o(t)  0,  is as in (8.2), i- is defined in Section 8.1.1, and we have: c(t)=arg max S(t),  Vnf,  (8.15)  rne41t.  c:mE’t4f  The optimal objective function in (8.14) is always non-negative since at least when  p(t)=O,  the  objective function is equal to zero. Comparing to the resource allocation problem in [126], the objective function in (8.14) has an extra negative term:  —  (Zc\{n}rnEgt p(t)) (aEN\{n}  (t)),  (8.16)  which denotes the total interference compensation charge that wireless relay node n should pay to other relay nodes. By solving (8.14), node n finds the trade-off between maximizing ZcEJf\{n}ZmEjVm1t  i(p(t), w(t))6(t) (i.e., the original objective function in [126]) and mini  mizing its interference compensation cost. Each node then implements the same routing strategy as in [126]. That is, node n transmit.s commodity cm(t) data on link (n, m) as long as  211  öm(t))  >0  Chapter 8. No commodity  CCm(t)  Two-Fold Pricing  data is sent on link (n, m) at time t.  Theorem 28 Let p(t) denote the optimal solution of problem (8.L4). Assuming that K >> 1 and all links operate in the high SINR regime (cf. [30, 133]), for each neighboring relay node mEJVout and any commodity c  p) (t)  =  e J\f\{n},  mm  1 i.  otherwise, p,? (t)  =  if c  =  cm(t), S(t) > 0, and m S(m(t))  nm (n) aEJ’f\{n} /)a  ij  prnax  e  then:  ‘1 (8.17)  (t)  S(a(t)) ZaEJf0ut  na  )  0.  Theorem 28 provides a closed-form solution for the constrained optimization problem in (8.14). The proof of Theorem 28 is given in Section 8.5.1. We show that (8.17) satisfies the necessary and sufficient Karush-Kuhn-Tucker (KKT) optimality conditions (cf. [16]).  8.2  Guaranteed Positive Individual Profits and Maximum Social Welfare  Recall from Section 8.1.1 that each wireless relay node n  eN  is a self-interested independent  commercial entity. Thus, it is willing to make money out of its offered relay and connectivity T we define node n’s profit as:  services. In this regard, at each time slot t  x(t)  =  ± I  ‘  ZcE\{n}  Ziev  ZkeD  ZCev\{}  /  r(t)  (t)  4 (p(t), w (t)) (t) ZmEjvt ZCEJr\{fl} —  (d)  (a)  Z aJ’f\{n} Y”  V’)  di\f\{a}  bEJfa0Ut  Pab  V)  —  / (n) aEV\{n} ya V’I  (p(t), w (t))} (t) (c) cJ1\{n}  (8.18) The first term in (8.18) is the total relay charges from all wireless users i E D. The second and the third terms denote the total relay charges from and to all the neighboring wireless relay 212  Chapter 8. Two-Fold Pricing nodes, respectively. The fourth and the fifth terms denote the total interference charges from and to all other nodes a E N\{n}, respectively. We are now ready to present our first key result: Theorem 29 For each T>> 1 and any wireless relay node n  0 x(t) Z=  e N,  we have: (8.19)  Z=oZaEAr\{n}(t) (ZdEN\{a}ZbENoutP(t)).  The proof of Theorem 29 is given in Section 8.5.2. From Theorem 29, each relay node is guaranteed to obtain a profit which is at least as high as the right-hand side (RHS) of (8.19). All the terms in the RHS of (8.19) are non-negative. From the assumptions on the pricing functions in Section 8.1.2, the RHS of (8.19) is zero only if for the duration from time t  =  relay node in set N\{n} transmits any data and there is no data in any of the N node n. This happens only if either N  =  0 to t —  =  T, no  1 queues in  1 and there is no other relay node in the network or  node n has set its relay prices too high so that none of its users and neighboring relay nodes are interested in transferring their data to node n. The former is the case when there is no need to relay node m as all users in set D  =  V can communicate with each other directly. The latter is  the case when node n is reluctant to contribute as a part of the wireless access network.  Corollary 5 Each wireless relay node that contributes in relaying data is guaranteed to receive a positive-valued profit. The profit increases as the node forwards more packets.  Theorem 29 and Corollary 5 are general and apply to any choice of user utilities and pricing functions. Next, we determine the pricing functions any commodity c  e .iV\{n},  for all relay nodes n  and  and any neighboring relay node m  e N,t  e N,  to maximize the network  social welfare; i.e., the aggregate profit across all wireless relay nodes and users: Zne.N  xn(t) + z;= 1 piED t9(t). 213  (8.20)  Chapter 8. Two-Fold Pricing Lemma 7 The social welfare model in (8.20,.) is equal to the following:  t=i ZnjvZin ZcEJf\{n}  ZkDg(r(t)).  (8.21)  The proof of Lemma 7 is given in Section 8.5.3. From Lemma 7, the monetary exchanges among the relay nodes and the users cancel out each other. Eq. (8.21) is the aggregate network  utility across all users. Thus, maximizing the social welfare in our TFP model is equivalent to maximizing the network utility. This helps us to present our second key result as follows. Theorem 30 Given T (i.e., the set of time slots), T (i.e., the set of time slots at which the  vector of channel states w changes), and A> 1 (i.e., the fading parameter), let:  where T,., T,,, and  r  T  T,  =  =  T,  and  T  =  A,  (8.22)  are defined in Section 8.1.2. The aggregate network utility and the network  social welfare are maximized if each wireless relay node n E .Af at each time slot t’ E T sets: =  and also at each time slot t  e  {t’,.  .  .  ,  V U,(t’),  Vc.A[\{n},  t’ + T} C 7, each node n E .N sets: 1S(Cm(t’))t1\  ham  =  (8.23)  nmP—n\(t” ii  mEJ/otLt  O  VaeJ\f\{n},  (8.24)  ‘7m  where V> 0 is an arbitrary design parameter.  The proof of Theorem 30 is given in Section 8.5.4. The key is to show that our proposed two fold pricing functions result in solving the well-known maximum weight matching problem (cf. [131, 132]) periodically (i.e., every T  =  A time slots). Together, Theorems 29 and 30 show that if  the transmission powers, relay prices, and interference prices are set according to (8.17), (8.23), and (8.24), respectively, then not only each relay node receives a guaranteed positive profit, but also the social welfare and the network utility are maximized. The pseudocode of the pricing 214  Chapter 8.  Two-Fold Pricing  algorithms that each node and each user need to execute are given in Algorithms 10 and 11, respectively. In lines 5 to 7 and lines 17 to 19 of Algorithm 10, the relay and interference prices are adjusted according to Theorem 30, respectively. On the other hand, in lines 21 to 28, the transmission powers are set according to Theorem 28. Notice that Algorithm 11 simply adjusts the transmission rates of the users based on the optimal solutions of the profit maximization problem in (8.11). Algorithm 10 Executed by each wireless relay node n e JV 1: Randomly choose the relay prices, interference prices, and the transmission powers. 2: repeat 3: Transmit commodity c E .IV\{n} data to out-neighbor node m E with power p. 4: if tT then 5: 6: 7 8:  9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:  u. 21: 22: 23: 24: 25: 26: 27: 28: 29: 30:  for all commodity c E A1\{n} do Set = V u. end Inform relay prices çb = Vc E .N\{n}) to all its in-neighbors and users. for all out-neighbors m e do = max. for all commodity c E .Af\{n} do Set end Set Cm = argmax E(c) Snm(c). end end if then for all relay nodes a E /\{n} do jm), Set 0}. = Zrnjqout (harn/(Inm + 7m)) max{S end Inform interference prices ‘çb = Va e .Af\{n}) to all other nodes. Set p = 0. for all out-neighxr m E do an)) if Zm ,ut Thm < then (ZaI\{m} (n) (c) £(cm) / (‘c-’ Jeb Pmm vnm Ya  (a),  —  —  —  —  i i,L.daEJ\f\{n}  else (c) = (om)Prnax) Set / (ZaEN,,ut Sna). end end end Charge any node meJV and any iEV for relaying commodity cE.Af\{n} data at price ut for relaying commodity c 7 Pay any node m E Jf e N\{n} data at price Charge any other node a e .Af\{n} for the interference it causes on node n at price Pay any other node a e A1\{n} for the interference node n causes on it at price gin) until the wireless relay node n switches off. 215  Chapter 8. Two-Fold Pricing Algorithm 11 Executed by each wireless user i e D, 1: repeat Set the transmission rates r by solving the convex optimization problem in (8.11). 2: 3: Pay wireless relay node n for relaying any commodity c e ..A/\{n} data at price 4: until the wireless user i switches off or leaves the network.  8.3  Simulation Results  In this section, we evaluate the performance of our proposed TFP scheme and compare it with a recently proposed SFP scheme in [126], where only the relay prices are taken into account and the network is assumed to be interference-free. We consider three performance metrics: 1) network social welfare, 2) fairness index, and 3) aggregate throughput. The fairness index is calculated among the profits that the wireless relay nodes achieve [89]:  NflEN  x(t)) 2 where T x(t)) 1 (  is the simulation time. Each wireless relay node n provides the connectivity for D!  =  =  5000  5 wireless  users. Each wireless user is interested in sending data to two other (randomly selected) users inside the network. We consider ten different random topologies. In each topology, the wireless relay nodes are randomly located in a 1 km x 1 km square field and the communication range is 200 m. There is a link between any two neighboring wireless relay nodes if they are within the communication range of each other. For each wireless relay node n and  R&X  =  e N,  we have:  prnax  =  20 W  100 kbps. The transmission power, relay prices, and the interference prices are set  according to (8.17), (8.23), and (8.24), respectively. The unit of currency is selected such that for a unit quieue backlog, relaying 1 Mbps data costs 1 cent, i.e., 0.01 dollar. Unless stated otherwise, we assume the presence of slow-fading channels with the fading parameter A of fast-fading is also studied in Section 8.3.3.  216  =  10. The impact  Chapter 8. Two-Fold Pricing  8.3.1  Performance Comparison with Single-Fold Pricing  The network social welfare and throughput, when the number of relay nodes N varies from 10 to 50, are shown in Fig. 8.2 (a) and (b), respectively. Each point is the average of the measurements for all 10 simulated topologies. We can see that our proposed TFP always outperforms the SFP scheme in [126], in terms of both throughput and social welfare. The performance gain increases as N increases. It is because our interference pricing scheme better leverages optimal resource allocation. As the number of nodes increases, more nodes interfere with each other’s transmissions and optimal interference control becomes crucial. In fact, as Theorem 30 shows, TFP leads to maximum social welfare. Considering the case where N = 50, TFP results in 24.6% higher social  welfare and 14.7% higher throughput compared to SFP. The exact value of the social welfare, fairness index, and throughput for each of the 10 simulated topologies, where N = 50, are shown in Fig. 8.3 (a), (b), and (c), respectively. From Fig. 8.3 (a) and (c), TFP always results in higher social welfare as well as higher throughput compared to SFP. From Fig. 8.3 (b), TFP also always acts more fair. Recall from Theorem 29 that TFP guarantees high positive profits for all relay nodes. In fact, having the interference prices helps those relay nodes that do not experience high traffic demand. Instead, they make some money out of the interference charges. This results in a more fair revenue sharing among the nodes. On average, TFP leads to 18.3% higher fairness index compared to SFP.  8.3.2  Maximum Weight Matching  At each time slot t E T, we define: 0(t)  =  nE\iZcEN\{n}  mEj\[ut  217  L?n(p(t),W(t))  (t’), 1 S  (8.25)  Chapter 8.  Two-Fold Pricing  (a) 4500  TFP —e-—SFP  24.6%  4000 -  3500  0  P. 3000 G) .  2500 2000  1500 1000  1  10  20  30 Number of Nodes N  40  50  (b)  C’, 0. .0  0. -C 0) 0  I.  30 Number of Nodes N  Figure 8.2: Network social welfare and aggregate throughput when the number of wireless relay  nodes N varies from 10 to 50. Each wireless relay node provides network connec tivity for 5 wireless users. Each point is the average of the measurements for all ten topologies: (a) Network social welfare, (b) Aggregate network throughput.  218  Chapter 8.  Two-Fold Pricing  I Topology Number (b)  Topology Number (c)  i::Eii b b b b  b  Topology Number  ! !‘  Figure 8.3: Simulation results for each of the 10 random simulated topologies when N = 50 and the communication channels experience slow-fading (i.e., A = 10): (a) Network social welfare, (b) Fairness index among the profits achieved by wireless relay nodes, and (c) Aggregate network throughput.  where t’ is the smallest time slot in set 7 such that: t  t’. In other words, t’ is the most  recent time slot at which the vector of channel states w has changed. We notice that, 9(t) is indeed the same as the objective function in the maximum weight matching problem in (8.46). From Theorem 30, TFP results in maximum network social welfare and maximum network utility by periodically solving optimization problem (8.46), i.e., maximizing the values of 0(t). This is illustrated in Fig. 8.4. In this figure, the trend of 0(t) for topology number 1 is shown versus the time slots. Notice that, the fading parameter A  =  10. Thus, the vector of channel states w  randomly changes every 10 time slots. This implies that the optimal solution of the maximum  219  Chapter 8.  Two-Fold Pricing  weight matching problem also changes every 10 time slots. For proper operation, TPF needs to converge to the new optimal solution accordingly. This is shown in the zoomed area in Fig. 8.4 (b). Clearly, the convergence is fast. From the results in Fig. 8.4 (a) and (b), we can also see that TFP always results in substantially higher maximum weight matching objective  e(t), compared  to SFP. The higher the maximum weight matching objective, the higher is the aggregate network utility [132]. From Lemma 7, this also implies higher network social welfare.  8.3.3  Impact of Fast-Fading  In the previous experiments, we assumed that the channels experience slow-fading. In this section, we study the impact of fast-fading. Results for all the ten simulated topologies, where the number of wireless relay nodes N  =  50 and the fading parameter A  =  2, are shown in Fig. 8.5. In this  scenario, the channel states randomly change every 2 time slots.  As a result, our proposed  distributed transmission power adjustment mechanism (see lines 22 to 29 in Algorithm 10) does not have enough time to converge to the new optimal solution of the maximum weight matching  problem after each change in the channel states. Thus, the optimal performance is not always achieved. Nevertheless, from the results in Fig. 8.5 (a) and 8.5 (c), TFP still results in 46.3% higher social welfare and 32.4% higher throughput, compared to SFP. On the other hand, from Fig. 8.5 (b), TFP is 35.2% more fair compared to SFP in this scenario. In summary, assuming the presence of slow-fading channels, our proposed TFP scheme leads to not only higher aggregate profit across the nodes and users, but also more fair revenue distri bution among the relay nodes. It also results in significantly higher network throughput. When the underlying communication channels experience fast-fading, although TFP still results in sub stantially better performance compared to SFP, optimality may not be always guaranteed. 220  Chapter 8. Two-Fold Pricing  (a)  e  2000 3000 Time Slots  (b) 3—4--—  TFP SFP  -  \  2•%  0  50  100 Time Slots  150  200  Figure 8.4: Trend of the maximum weight matching objective 9(t) versus time slots: (a) During the whole simulation time, i.e., 5000 time slots, (b) During the first 200 time slots. Notice that every T = A = 10 time slots, the channel states change randomly and the maximum weight matching objective converges to its new optimal value accordingly.  221  Chapter 8.  Two-Fold Pricing  (a)  Topology Number (b)  [1 jjj jjjjj j 1 j Topology Number (c) Cl)  40  a  .0  z a 0)  2’ I—  Topology Number  Figure 8.5: Simulation results for each of the 10 random simulated topologies when N = 50 and the communication channels experience fast-fading (i.e., A = 2): (a) Network social welfare, (b) Fairness index among the profits achieved by wireless relay nodes, and (c) Aggregate network throughput.  222  Chapter 8. Two-Fold Pricing  8.4  Summary  In this chapter, we proposed a market-based wireless access network model with two-fold pricing (TFP), where several self-interested wireless relay nodes provide the connectivity service for wire less users. The relay-prices are used as incentives to encourage nodes to collaborate and forward each other’s packets. The interference-prices are also used to leverage optimal resource alloca tion. Together, the relay and interference prices incorporate both cooperative and competitive interactions among the nodes. The positive profit for each individual wireless relay node is guar anteed for a wide range of pricing functions. Assuming that CDMA technology is being used for transmissions, the relay and interference pricing functions are then determined to also maximize the network social welfare and aggregate network utility. Compared with the single-fold price (SFP) scheme in [126], where only the relay prices are taken into account, simulation results show that TFP increases the network social welfare and aggregate throughput by 24.6% and 14.7%, respectively. TFP also leads to significantly more fair revenue sharing and profit distribution among the wireless relay nodes with 18.3% higher fairness index. For future work, we plan to extend our model to other multiple access technologies, such as the contention-based medium access control scheme in the IEEE 802.11 distributed coordination function. We shall also consider the effect of user mobility and roaming.  8.5 8.5.1  Analytical Proofs Proof of Theorem 28  Knowing that all the constraints are linear and the objective function is strictly concave, problem (8.14) is a convex optimization problem. Therefore, it has a unique local (thus global) optimal 223  Two-Fold Pricing  Chapter 8.  solution. In a high SINR regime, the optimal solution should satisfy the following necessary and sufficient Karush-Kuhn-Tucker (KKT) optimality conditions [15, Proposition 3.3.1]: (Cm(t))  nm 0  / *(Cm(t))(t) (r),pnm  $‘(t)  .  N (Zmemn  out  *(cnm(t)) Prim (  )  Vm é  =  —  =  o,  mP*(m(t nm ‘ ) = 0  (8.27) JfOut  Vm E  ,  t:7m> 0,  0,  (8.26)  Nout  (8.28) (8.29)  , t VméAf,’  where ? denotes the Lagrange niultiplier corresponding to constraint  *(Cm(t))  ZmEN0Ut  Pnm  *(Cm(t)) (t) and CTm denotes the Lagrange multiplier corresponding to constraint Pnm  e  each wireless link (n, m)  m(t(t)  <  prnax  £ot  *(c) Pnm  (t)  •(Cm(t))  Onm  ZmEJqout  )  —  (Cm(t))(t) 0 n m (ZrnE ,tout n  —  *(c)  > 0, we have Prim (t)  =  =  prnax (Zae\{n}  Notice that, in this case, for each link (n, m) $m(t))(t)  Om  C  =  =  0 for all links  Cm(t)  such that  (8.31)  aEK\{ri} t)),  —  then the KKT conditions are satisfied by setting mn —  0 and  ‘(t)). On the other hand, if  /(I_daEN  önm(t))ft\  nm  =  and any commodity  In this case, for each link (n, m) e  S(t) > 0, we h ave  (8.30)  a\{n} )(t)),  then the KKT conditions (8.26)-(8.29) are satisfied by setting A  e  0 for  We can show that if  Zmat  (n. m)  (t <  e  0 for all links (n, m) t))) /prnax  (ca(t))  (aE./,ut  Sria  and  by (8.31)  0.  >  Lt and any commodity c  P,”/  e Lt  (),j.  =  Cm(t)  (8.32) such that  Thus, since p(t)  is  the only point that satisfies the KKT conditions in (8.26)-(8.29), it is indeed the unique global optimal solution for the local transmission power control problem in (8.14).  224  Two-Fold Pricing  Chapter 8.  8.5.2  Proof of Theorem 29  From (8.13) and (8.18), for any node n e 1 A and at any time t E T, we have:  xn(t)  =  X(t) +  (c  nn(p(t),(t))(t)  ZcjV\{n} ZmN  ZcEjV\{n} ZmEut (p(t), c (t)) ((t)  =  [  Zcein ZmEut (n) ZaEK\{n}  +  [  —  +  zCwfl  (t)  -  t)  -  max)  (p(t), w(t)) S(t) ZcE\{n} ZmEut  (8.33)  p(t)]  (c) (k)(t)+ mE1mfl(P(t), (ZEDZkEDr  (c) ZcN\{n} ZmEut nrn  (t)  ZaEJf\{n}bn  (p(t), (t))  ZdEN\{a}ZbEut  ( ()  w(t)))  kt)  max)]  p (t).  Since the optimal objective function in (8.14) is non-negative, (p(t),w(t))S(t)— cEN\{n} mt  >  ij(t)  aEf\{n}  >  p(t)  0.  (8.34)  cE\{n} mN t  Following the proof of [126, Theorem 2b], we can also show that: (k) ZcEN\{n} (ZievnZkevcni (t)  +ZmEN  (c)  —  (p(t), w (t))) t)  (8.35)  ZcE\{m} ZmEt nm(p(t),  (t)) ((t)  —  max)  0.  By replacing (8.34) and (8.35) in (21) we have:  xn(t)  (m) ZaeAr\{n}  Adding up both sides for t= 1,  ...  ,  (t)  ZdEAr\{a} ZbEJ%fut  p(t).  T, the inequality in (8.19) results.  225  (8.36)  Chapter 8.  Two-Fold Pricing  Proof of Lemma 7  8.5.3  Replacing ?9(t) and  xn(t)  in (8.20) by (8.12) and (8.18), respectively, we have:  1 ZiED 9(t) x(t) + Z = z  (z ZiED  Z kEDC r (t) (t)  (c) w(t)) (t) iimm(P(t),  ±  ZmjV  +  (a) aEfif\{n} (t)dEp\{a}ZbENout  ±  Zl ZnE ziv (Zc\{n} ZkeD g  ZcEf\{n}  (c)  Zci\{n} ltnm(p(t),  —  (c)  (t)_ZaEN\{nøi  (ZCE\{} ZiED ZkED r  (  +  —  (t) (t)  (Znv cE1\{n} ZmEt nm(p(t), (a)  ((zfl  —  +  Z  ZaEN\{n}  (c)  (t) (t))  (Zne ZaE\{n}  w(t)) (t)))  (t) ZdEN\{a} bEut p (t))  (m)  (t) ZcN\{n} ZmEut  p(t)))  g(k) (r(k) (t))  jED cN\{n} ZkED  (k)  =  (c)  (t) (t))  (c)  (c)  1 Z  r  r  tmn(P(t),1’(t))(t))  (ZnEJrZCeiv\{n}  —  +  nE\{n} ZkED (k)  —  ‘\  (t) cI\{n} Zme1t Pnm(t)) (k)  (k)((k)(t))  = zi  2 (t) w(t))q$  Zt=i Zneiv ZiED cf\{n} ZkED g  (k)  (r  (t)).  (8.37) The last line in (8.37) is indeed the same as (8.21). Notice that we have: (p(t),w(t)))(t) nEJV cE.Af\{n}  =  flEJ\1 c)’f\{n}  mEJ\I  p(t),w(t))(t),  (8.38)  )(t)p(t).  (8.39)  mE.fV1t  and ,(a)(d) =  nEJ\1 aE.A[\{n} dE.Af\{a}  t bE.N”  nEJ.f  t aiV\{n} cEiV\{n} mEN,9’  In (8.38), the left hand side denotes the aggregate relay price that all wireless relay nodes  receive  while the right hand side denotes the aggregate relay price all wireless relay nodes pay. Similarly, 226  Chapter 8. Two-Fold Pricing in (8.39), the left hand side is the aggregate interference price that all relay nodes receive while the right hand side is the aggregate interference price all relay nodes pay.  8.5.4  Proof of Theorem 30  Given t’ e T, for each time slot t e {t’,.  ..  , t’ + T },  consider two arbitrary non-negative valued  3(t) and 7 5(t) such that: transmission power vectors 1  (t)  -<  (8.40)  3(t), 1  where the inequality is interpreted coordinate-wise. That is, for any wireless link (n, m) e £ and each commodity c E N\{n}, we have (t)  l/(inrn(n(t)) + m) < > 1/  ZaE)1’\{n}  thin)  ((t))  From (8.3), we can show that:  1nm(_n(t)),  Vn e  N, m e  (8.41)  3_n(t)) + im), 1 l/(Inm(  Vu E  N, m e N0t  (8.42)  (t)),  Vn eN, a c- N\{n},  (8.43)  Vu C N.  (8.44)  1/ ZaEV\{n}  /4) ((t)),  Replacing (8.44) in (8.17), we have: (t +1)  3(t + 1).  (8.45)  From (8.40) and (8.45), the update formulation in (8.17) forms a monotone mapping [18]. Mono tone mappings satisfy both synchronous convergence and box conditions [18, pp. 431]. Thus, from the asynchronous convergence theorem [18], the transmission powers will converge to a fixed point, assuming that T  =  A is large enough. By definition, p should denote the optimal solution  of the local optimization problem in (8.14) for all relay nodes. Next, we show that p also denotes  227  Two-Fold Pricing  Chapter 8.  the unique optimal solution of the following maximum weight matching problem [131, 132]:  max  t(p(t), (t)) s$(t’)  ZnEAI ZcEr\{n}ZmEvout  ZcEj\{n}ZmEft p(t) <Pt,  s.t.  p(t)  =  (8.46)  Vn eN  VneN, ceN\{n}, m74 or c  0,  Cm(t’)  or S(t’)  0.  Notice that the objective function in (8.46) is different from the objective function in (8.14) as it is the weighted summation of the data rates over all links. Problem (8.46) is indeed the key resource allocation problem to be solved by the back-pressure algorithms [131, 132]. Using the  logarithmic change of variables (cf. [30, Theorem 1]), we can transform problem (8.46) to an equivalent convex problem. Thus, problem (8.46) has a unique optimal solution (cf. [15]). Let p* denote the unique optimal solution of problem (8.46). From KKT conditions, we have: (db(t’))(/) (t’)) Pnrr  (  *(Cnm(t’))  rnetv’t Pnrn  (i)  —  pmax)  * QnmPnmm(t(t)  P where for each node n  *(cm(t’)) (i) p  V  e N, m e N0ut  (8.47)  =  0,  Vn  eN  (8.48)  =  0,  V n E N, m  0,  Vm  0,  Nout  (8.49)  e N. m e Nout  (8.50)  C  P.m  denotes the Lagrange multiplier corresponding to con  0. Comparing with (8.27)-(8.29), the KKT conditions (8.48)-(8.50) hold if  Pnm  we set p  Pnm,  p denotes the Lagrange multiplier corresponding to constraint  (t) <P’ and  mEJvtPflm  straint  e N,  =  *  Z\{m} ZbEut(t) hnb/*(t))+  =  Az,, and  m  =  m’  for all nodes n  e N and all links (n, m)  case, since  aJ’.f\{n} bEJVa0ut  hb I(*) +  by (8.24)  1b 1  228  —  aEfi/\{n}  E0ut.  In this  Chapter 8. Two-Fold Pricing the KKT condition in (8.47) is also resulted from (8.26). Thus, p  =  p is indeed the unique  optimal solution of the maximum weight matching problem in (8.46). In other words, given the interference pricing model in (8.24), optimization problem (8.46) is solved every T  =  A time  slots. This, together with (8.23), results in achieving maximum aggregate network utility (cf. [132] and [134, Theorem  41).  From Lemma 7, obtaining the maximum aggregate network utility  also implies achieving maximum network social welfare.  229  Chapter 9  Conclusions and Future Work 9.1  Summary of Work Accomplished  In this thesis we have considered various optimal resource management problems in wireless access networks More specifically, we have considered three problems under the general framework of network utility maximization:  Cross-Layer Interface Assignment and Channel Allocation for MC-WMNs: In Chapters 2, 3, 4, and 7 of the thesis we developed practical cross-layer channel and interface assignment algorithms to optimize the network performance in MC-WMNs. It was a chal lenging task because of the lack of accurate and tractable capacity models and the inherent discrete properties of the channel and interface assignment problems. We first developed a linear link capacity model using the concepts of multi-channel contention graph and binary linearization (see Chapters 2, 3). We also proposed a novel continuous capacity model in the context of multi-interface multi-channel random access (see Chapter 7). Unlike the prior work, we took into account not only the orthogonal (i.e., non-overlapped) frequency channels, but also the partially overlapped channels (see Chapter 4). This specially en abled us to fully utilize the available frequency spectrum. We then formulated a family of NUM problems which could incorporate the features of the multi-channel WMNs. Using  230  Chapter 9. Conclusions and Future Work various techniques from the optimization theory and the theory of parallel and distributed computation, we solved the formulated problems both in centralized and distributed fash ions. Compared to the previous channel and interface assignment algorithms, our proposed algorithms achieve better efficiency-fairness tradeoff and have reduced computational com plexity. They also closely collaborate with other network resource management algorithms such as MAC, routing, and congestion control. Simple, Robust, and Optimal Random Access: In Chapters 5 and 6, we considered the NUM problem at MAC layer when the optimization variables are the persistent prob abilities of the wireless nodes. Most of the previous NUM-based random MAC algorithms have one or more of the following performance bottlenecks: (1) extensive signaling overhead, (2) requiring synchronization among wireless nodes, (3) slow convergence speed, and (4) supporting a limited range of utility functions under which the NUM is shown to be a con vex problem. Thus, most of the previously proposed NUM-based random MAC algorithms are not adequate for practical implementation. Our proposed random MAC algorithm in Chapter 5 overcomes these bottlenecks in all four aspects. First, it only requires a limited message passing among the nodes. The signaling overhead is reduced by a factor of 10. Sec ond, our algorithm is fully asynchronous. This minimizes the coordination overhead and allows maximum heterogeneity among wireless nodes in terms of computational complexity and software implementation. Furthermore, our algorithm is robust to arbitrary large but finite message passing delays as well as message loss. Third, our algorithm can achieve much faster convergence. Finally, our proposed algorithm has provable convergence, optimality, and robustness properties under a wide range of utility functions, even if the corresponding  231  Chapter 9. Conclusions and Future Work NUM problems are non-convex. We then modified the algorithm to eliminate the need for frequent message passing in Chapter 6. In the proposed algorithm, each wireless node simply observes the contention history and accordingly assigns its persistent probabilities to maximize the NUM problem. This is of critical importance in practice, since any explicit message passing among wireless users will lead to further contentions in the network and reduce the network performance. • Two-Fold Pricing for Multi-Hop Wireless Access Networks: In Chapter 8, we proposed a market-based model with two-fold pricing for wireless access networks. In our model, the relay-pricing is used to encourage nodes to forward each other’s packets. That is, each node receives a payment for the relay service it provides.  We also considered  interference-pricing to leverage optimal resource allocation. Together, the relay and the interference prices incorporate both cooperative and competitive interactions among the nodes. We proved that two-fold pricing guarantees positive profit for each individual wireless node under a wide range of pricing functions. The profit increases as the node forwards more packets. Thus, the cooperative nodes are well rewarded. We then determined the relay and the interference pricing functions such that the network social welfare and the aggregate network utility are maximized. Simulation results show that, compared to a recently proposed single-fold pricing model where only the relay prices are considered, our proposed two-fold pricing scheme significantly increases the total network profit and the network throughput.  It also leads to more fair revenue sharing and profit distribution  among the wireless nodes.  232  Chapter 9. Conclusions and Future Work  9.2  Future Work  This thesis can be extended in several aspects. Here, we present overview some possible directions for future work.  Tackling Non-Convex Optimization Problems: The analysis techniques we proposed in Chapter 5 are quite general and can be used to tackle various other non-convex opti mization problems in communications and networking. The key idea is to break down the original problem into several local and myopic problems from each node’s point of view. In many cases (as in the case for random access as well as multi-interface multi-channel random access), the local problem turns out to be convex. This implies that each node can easily solve its local resource management problem. In some cases, the closed-form solu tion can also be obtained as in Theorems 10 and 11. Then, we can apply several mapping analysis techniques (see Section 1.2.3) to derive the behavior of the system when each node only solves its own local and myopic optimization problem. • Algorithms with No Explicit Frequent Message Passing: The novel distributed learning strategies in Chapter 6 can also be applied to other resource management problems to achieve signalling-free protocols. In particular, it is easy to extend the multi-interface multi-channel random access algorithm in Chapter 7. • Partially Overlapped Channel Assignment: The mathematical formulations in Chap ter 4 can be used to extend a wide range of frequency channel assignment algorithms in the literature that only use orthogonal channels. In fact, different from the general theme of this thesis which uses optimization as the main solution tool, the proposed models in  233  Chapter 9. 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