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Resource management in wireless networks Pillutla, Laxminarayana S. 2008

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Resource Management in Wireless Networks by Laxminarayana S Pillutla B.E., Electrical and Electronics Engineering, University of Madras, 2000 M.S., Electrical Engineering, Wichita State University, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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) November, 2008 c Laxminarayana S Pillutla 2008  Abstract This thesis considers resource management issues in wireless sensor networks (WSNs), wireless local area networks (WLANs), and cognitive radio (CR) networks. Since energy is a critical resource in WSNs, we consider energy minimization techniques based on explicit node cooperation and distributed source coding (DSC). The explicit node cooperation based on space time block codes (STBC) improves energy efficiency of WSNs, by reducing the energy consumption per bit of each sensor node. The DSC on the other hand exploits the spatial correlation in WSNs, and thus reduces the data generated in a WSN. For the purpose of our analysis, we model the spatial correlation according to a linear Gauss-Markov model. Through our numerical results, we observe that the node cooperation combined with DSC can improve energy efficiency for many cases of interest. A unique aspect of our work is we obtain important structural results using the concepts from monotone comparative statics. These structural results provide insights into the general design of WSNs. Through our numerical results, we also demonstrate that, the cooperation based transmission can achieve better mutual information (MI)-energy tradeoff than the non-cooperation based transmission scheme. From the perspective of WLANs, we propose a price based approach to regulate the channel occupancy of lowrate users, which is known to be the primary cause for low overall throughput in WLANs. Owing to the decentralized nature of WLANs we use non-cooperative game theory as a tool for analysis. Specifically, we use supermodular game theory. Through our analysis, we show that an increase in price leads to an increase in rate of WLAN users. We also ii  Abstract prove that the best response dynamics indeed converge to the Nash equilibrium of the underlying non-cooperative game. Through our numerical results, we demonstrate that by proper tuning of the price, the proposed price based approach can lead to an improvement in overall throughput of a WLAN. Finally from the perspective of CR networks, we consider the impact of number of channels captured by a secondary user on its transmission control protocol (TCP) throughput. From our simulation results it was found that, there exists a definite optimal number of channels a secondary user needs to capture, to maximize its TCP throughput.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  Abbreviations  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiv  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xvi  Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii  1 Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.2  Wireless Local Area Networks . . . . . . . . . . . . . . . . . . . . . . . .  7  1.3  Cognitive Radio Networks . . . . . . . . . . . . . . . . . . . . . . . . . .  10  1.4  Mathematical Tools  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  13  1.4.1  Monotone Comparative Statics . . . . . . . . . . . . . . . . . . .  13  1.4.2  Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  1.4.3  Supermodular Games  18  1.5  . . . . . . . . . . . . . . . . . . . . . . . .  Summary of Results and Contributions  . . . . . . . . . . . . . . . . . .  20  iv  Table of Contents 1.6  Thesis Organization  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 Data Gathering in Correlated Wireless Sensor Networks (WSNs) 2.1  .  23  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  Introduction 2.1.1  2.2  WSN Model and Assumptions  2.3  Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks 2.3.1  . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  2.3.3  Rate Allocations of Data Gathering Stages in Correlated WSNs .  35  2.3.4  Formulation of the Minimum Energy Data Gathering (MEDG) in Correlated WSNs  . . . . . . . . . . . . . . . . . . . . . . . . . .  37  Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1  2.4.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  45  Structural result on optimal number of cooperating nodes n∗ versus correlation coefficient a  2.4.4  44  Structural result on optimal modulation rate b∗ versus correlation coefficient (a).  2.4.3  42  Expression for energy consumption for the case of simple homogeneous network  2.5  31  Expression for Total Energy Consumption per bit using MIMO Techniques  2.4  27  Expression for Total Energy Consumption per bit using SISO Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.3.2  22  . . . . . . . . . . . . . . . . . . . . . . .  46  Structural result on optimal rate b∗ versus number of cooperating nodes n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  Tradeoff Between Mutual Information and Energy in Correlated WSNs .  49  v  Table of Contents 2.5.1  Effect of observation SNR Γ on optimal sensor spacing d . . . . .  53  2.5.2  Discussion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  2.6  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  2.7  Analytical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  2.7.1  Derivation of h (yk |yk+1, yk+2, · · · , yI ) and h (yI ):  2.7.2  Proof of Theorems 2.1 and 2.2  2.7.3  . . . . . . . . .  57  . . . . . . . . . . . . . . . . . . .  58  Proof of Theorem 2.3  . . . . . . . . . . . . . . . . . . . . . . . .  61  2.7.4  Proof of Theorem 2.4  . . . . . . . . . . . . . . . . . . . . . . . .  62  2.7.5  Proof of Theorem 2.5  . . . . . . . . . . . . . . . . . . . . . . . .  64  3 A Price based Approach to Decentralized Rate Selection in IEEE 802.11 WLANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  65  3.1  65  Introduction 3.1.1  3.2  3.3  3.4  3.5  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Main Results:  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  System Model, Assumptions, and Notation  67  . . . . . . . . . . . . . . . .  68  3.2.1  System Model Description of a WLAN . . . . . . . . . . . . . . .  68  3.2.2  Notation  69  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Formulation of Non-Cooperative Game Theoretic Rate Selection with Pricing for WLANs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  71  3.3.1  Utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . .  71  3.3.2  Rules of the NRSGP . . . . . . . . . . . . . . . . . . . . . . . . .  73  Analysis of NRSGP with Supermodular Game Theory  . . . . . . . . . .  74  3.4.1  Analysis of Non-Cooperative Rate Selection with Pricing (NRSGP) 74  3.4.2  Price λ Adaptation  . . . . . . . . . . . . . . . . . . . . . . . . .  80  Extension of the Price based Approach to Multi-Channel WLANs . . . .  80  3.5.1  82  Analysis for the Two channels case . . . . . . . . . . . . . . . . .  vi  Table of Contents 3.6  Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  3.7  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  90  3.8  Analytical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  91  3.8.1  Proof of Theorem 3.1: . . . . . . . . . . . . . . . . . . . . . . . .  91  3.8.2  Proof of Theorem 3.2: . . . . . . . . . . . . . . . . . . . . . . . .  92  3.8.3  Proof of Theorem 3.3  . . . . . . . . . . . . . . . . . . . . . . . .  92  3.8.4  Proof of Theorem 3.4: . . . . . . . . . . . . . . . . . . . . . . . .  93  4 Performance of TCP in Cognitive Radio Networks . . . . . . . . . . .  94  4.1  Introduction  4.2  Cognitive Radio : Evolution, Access Techniques and Architectures  4.3  4.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  94  . . .  96  . . . . . . . .  96  . . . . . . . . . .  97  4.2.1  Evolution and Salient Features of Cognitive Radio  4.2.2  Spectrum Access Policy: Own It? or Share It?  4.2.3  Access Techniques for CR Networks  . . . . . . . . . . . . . . . .  98  4.2.4  Architectures for CR Networks . . . . . . . . . . . . . . . . . . .  98  TCP in Conventional versus Cognitive Radio (CR) Networks 4.3.1  TCP in Wired and Conventional Wireless Networks  4.3.2  Service Interruption Loss in Overlay CR Networks  . . . . . . 101 . . . . . . . 101  . . . . . . . . 102  Simulation Study of TCP Performance in an Overlay Cognitive Radio Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105  4.5  4.4.1  System Model of an Overlay CR Network  4.4.2  Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . 107  4.4.3  Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 108  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114  5 Conclusions and Future Work 5.1  . . . . . . . . . . . . . 105  . . . . . . . . . . . . . . . . . . . . . . . . 115  Summary of Work Accomplished . . . . . . . . . . . . . . . . . . . . . . 115 vii  Table of Contents 5.2  Future Work  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120  Appendices A List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134  viii  List of Tables 2.1  System parameter values for the 2.4 GHz ISM band . . . . . . . . . . . .  39  2.2  MI-Energy Tradeoff for MIMO and SISO data gathering schemes  . . . .  52  4.1  Summary of Cognitive Radio Architectures . . . . . . . . . . . . . . . . .  99  ix  List of Figures 1.1  The figure shows the research considered in this thesis under the broad framework of resource management in wireless networks. . . . . . . . . .  1.2  The figure shows a typical WSN which consists of number of sensor nodes (shown as dots) and a fusion center (FC).  1.3  . . . . . . . . . . . . . . . . .  4  The figure shows three WLANs where each WLAN consists of number of users and an access point (AP). . . . . . . . . . . . . . . . . . . . . . . .  2.1  2  8  The figure shows the scenario for data gathering in correlated random fields. Each stage of the network consists of a number of sensor nodes which cooperate during transmission of data. For each data collection period a node in each stage senses, quantizes and broadcasts to other nodes in that stage for cooperative transmission to the next stage. Note that apart from the data generated within a given stage i, the nodes in each stage also receive data from previous stages Lji that needs to be relayed to the next stage. The encoding rate of each stage is denoted as Rk , k = 1, 2, · · · , I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  28  x  List of Figures 2.2  The figure shows the plot of total energy consumption in the network versus the correlation coefficient. We fix the observation SNR (Γ) equal to 10 dB and assume the desired probability of error (pb ) equal to 10−3 . Number of stages in the network are assumed equal to 10. Distance between the various stages of the network is equal to 30 metres. . . . . . . . . . . . .  2.3  40  The figure shows the plot of total energy consumption in the network versus the correlation coefficient. We fix the observation SNR (Γ) equal to 10 dB and assume the desired probability of error (pb ) equal to 10−6 . Number of stages in the network are assumed equal to 10. Distance between the various stages of the network is equal to 30 metres. . . . . . . . . . . . .  2.4  41  The figure shows the plot of mean square error versus the stage index. We fix the observation SNR (Γ) equal to 30 dB and assume the node spacing to be 1 m, 5 m and 20 m respectively. . . . . . . . . . . . . . . . . . . . .  2.5  55  The figure shows the plot of mean square error versus the stage index. We fix the observation SNR (Γ) equal to −10 dB and assume the node spacing to be 1 m, 5 m and 20 m respectively. . . . . . . . . . . . . . . . . . . . .  3.1  56  The figure shows a snapshot of an ad hoc network with four sender and receiver pairs namely (S1 , R1 ), (S2 , R2 ), (S3 , R3 ) and (S4 , R4 ) respectively. The access point (AP) in the figure broadcasts a price λ to all the users periodically. The price is proportional to the occupancy of each user. . .  3.2  70  The figure shows the plot of equilibrium user rates on a given channel versus the average SNR. The figure shows that with an increase in the price λ the rate of each user increases as proved in Theorem 3.2.  . . . .  84  xi  List of Figures 3.3  The figure shows the total network throughput for the case of with and without pricing. The figure shows that an appropriate choice of price can lead to higher overall network throughput. . . . . . . . . . . . . . . . . .  3.4  87  The figure shows the occupancy values of the 10 dB, 15 dB and 30 dB users respectively with and without pricing. The figure shows that an increase in price reduces the occupancy of 10 dB users. This improves the occupancy of 15 dB and 30 dB users thereby leading to an increase in overall network throughput as shown in Fig. 3.3. . . . . . . . . . . . . . . . . . . . . . .  3.5  88  The figure shows the average throughput obtained for a 10 dB, 15 dB and 30 dB user. The average throughput of the 10 dB user decreases slightly due to pricing. This can be attributed to a decrease in occupancy of the 10 dB user which is shown in Fig. 3.4. However the 15 dB and 30 dB user report an increase in their throughput owing to an improvement in their occupancy.  4.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  89  System model of an overlay CR network: Np primary users and Ns secondary users access their base stations via L shared radio channels. Each radio channel has a bandwidth of br bps. The primary base station connects to the primary network, while the secondary base station connects to a server via a wired link with a bandwidth of bl bps. When needed, a primary user can force a secondary user to give up transmission, so that it can use the released channel. . . . . . . . . . . . . . . . . . . . . . . . 106  xii  List of Figures 4.2  Aggregate secondary user TCP throughput versus the maximum channels for secondary users (Ls ) when Np = 0. Due to the absence of service interruption from the primary users, this case is similar to an ad hoc network with random access. The figure shows that the aggregate TCP throughput of secondary users increases monotonically with Ls . . . . . . 109  4.3  Aggregate secondary user TCP throughput versus the maximum channels for secondary users (Ls ) in the presence of service interruption due to 30 primary users. The figure shows that the throughput is no longer monotone with respect to Ls . There exists a well defined optimal value of Ls which maximizes the aggregate TCP throughput. . . . . . . . . . . . . . . . . . 111  4.4  Aggregate secondary user UDP throughput versus the maximum channels for secondary users (Ls ) in the presence of service interruption due to 30 primary users. The figure shows that the throughput increases monotonically with respect to Ls , unlike the case of TCP. . . . . . . . . . . . . . . 111  4.5  Aggregate secondary user TCP throughput versus the maximum channels for secondary users (Ls ) for various values of load factor (A) with service interruption due to 30 primary users. The figure shows that the aggregate TCP throughput of secondary users decreases as the load factor increases. 112  4.6  Optimum number of channels (L∗s ) that a secondary user captures to maximize its TCP throughput versus load factor (A) for different values of bandwidth. The figure shows that for a given bandwidth, the optimal number of channels (L∗s ) captured is high for small and large load factor values, and low for moderate load factor values. . . . . . . . . . . . . . . 112  xiii  Abbreviations AP  Access point  BS  Base station  CR  Cognitive radio  CDMA  Code division multiple access  CSMA  Carrier sense multiple access  CTS  Clear to send  DCF  Distributed coordination function  DSA  Dynamic spectrum access  DSC  Distributed source coding  EDCF  Enhanced distributed coordination function  FC  Fusion center  FCC  Federal communications commission  FTP  File transfer protocol  IEEE  Institute of Electrical and Electronics Engineers  MAC  Medium access control xiv  Abbreviations MCS  Monotone comparative statics  MI  Mutual information  MIMO  Multiple input and multiple output  MSE  Mean square error  NE  Nash equilibrium  NS2  Network simulator version 2  OFDM  Orthogonal frequency division multiplexing  PCF  Point coordination function  QoS  Quality of Service  RTS  Request to send  SISO  Single input and single output  SNR  Signal to noise ratio  STBC  Space-time block code  TCP  Transmission control protocol  TDMA  Time division multiple access  UDP  User datagram protocol  WLAN  Wireless local area network  WSN  Wireless sensor network  xv  Acknowledgements Praise be to Lord Venkateshwara the most gracious and loving one, without whom my accomplishments would never have been possible. Most special thanks go to my supervisor Dr. Vikram Krishnamurthy for his constant support and encouragement throughout my doctoral studies at the University of British Columbia (UBC). Indeed it is my great fortune to have worked under some one as knowledgeable and learned as him. The gratitude I to owe him is truly immeasurable. He will surely be a role model to me in my professional life. Special thanks are also due to Dr. Lutz Lampe, Dr. Cyril Leung, Dr. Victor Leung, and Dr. Vincent Wong for their insightful comments and suggestions. I would also like to extend my sincere thanks to all past and present members of the statistical signal processing lab. Specially, I would like to thank Michael Maskery, Minh Ngo, Alex Wang and Irtiza Zaidi, for all the banter and helpful discussions, and Dr. Teerawat Issariyakul for his collaboration on Chapter 4. My accomplishments to date would not have been possible without the love and support of my parents, sisters, brothers-in-law, nephews and niece. My father himself being an electrical engineer, has always inspired me to achieve something really good in life. My mother has supported me in her own innocent ways, and I only wish that she could understand at least a small portion of this thesis, to realize that all her efforts for being after me during the childhood days haven’t gone wasted. Lastly, I would like to thank my adorable wife Anju for her support, patience, and unending love. To her all I xvi  Acknowledgements want to say is “Its finally over”!! I would be failing in my duty if I did not acknowledge the financial support of Natural Sciences and Engineering Research Council (NSERC) of Canada.  xvii  Dedication To Lord Venkateshwara and my parents.  xviii  Chapter 1 Introduction Wireless technology through its various forms such as cellular networks, wireless local area networks (WLANs), wireless sensor networks (WSNs) and cognitive radio (CR) networks have brought us a step closer to realizing the concept of ubiquitous communications that will allow the users to transfer and receive information on the move, anytime and anywhere. To fully realize the potential of wireless networks one needs to manage the various network resources efficiently. Thus resource management is of paramount importance in wireless networks. The term resource in a wireless network is quite general and includes such broad quantities as energy, time, channels, etc. This thesis focuses on the issue of resource management in different types of wireless networks namely, wireless sensor, wireless local area and cognitive radio networks. The research undertaken in this thesis is summarized in Fig. 1.1. From the perspective of WSNs energy is a critical resource and hence we propose resource management schemes that would minimize the overall energy consumption of a WSN. In the case of WLANs we propose a framework that would regulate the amount of time a user spends on a given channel, thus leading to overall network welfare. Finally from the perspective of CR networks, we study the need to regulate the number of channels captured by a user to maximize its transmission control protocol (TCP) throughput. We note that the tools used in this thesis are different for each of the problems considered. From the perspective of WSNs, we use non-linear optimization theory [21], [24] and monotone comparative statics [106] to prove analytical results. In the case of 1  Chapter 1. Introduction  Resource Management in Wireless Networks  Energy Management in WSNs 1. Clustering with cooperative MIMO. 2. Exploiting spatial correlation with distributed source coding (DSC). 3. Structural results using monotone comparative statics.  Access Time Management in WLANs 1. Pricing to ensure fair channel access among WLAN users. 2. Analysis using theory of supermodular games.  Channel Management in CRs 1. Impact of number of channels on TCP throughput of a user. 2. Simulation study using an NS2 simulation model.  Tools Analytical 1. Non−linear optimization. 2. Monotone comparative statics. 3. Game theory. 4. Stochastic filtering. Simulation 1. Network Simulator (ns2). 2. C/C++. 3. Matlab.  Figure 1.1: The figure shows the research considered in this thesis under the broad framework of resource management in wireless networks.  WLANs, owing to the decentralized nature of the network, we use game theory [46], [80] as a tool for analysis. Finally, we use a simulation model in NS2 [76], to study the impact of number of channels on the TCP throughput of the users. The rest of this chapter is organized as follows: we first provide an overview on wireless sensor, wireless local area and CR networks in sections 1.1, 1.2 and 1.3 respectively. We then provide an overview on mathematical tools used for analysis in Section 1.4. Next, we summarize our contributions and outline the results in Section 1.5. Finally, we describe the organization of rest of this thesis in Section 1.6. 2  1.1. Wireless Sensor Networks  1.1  Wireless Sensor Networks  A typical wireless sensor network (WSN) (shown in Fig. 1.2) consists of a fusion center (FC) and large number of spatially distributed devices called nodes which cooperate to accomplish various high level tasks [91] [7] [30]. As shown in Fig.1.2, the FC can in general be mobile. A typical sensor node can be very small, indeed some research efforts like smart dust [61] aim to reduce the size of a sensor node even further. A node in a WSN is equipped with three basic functional elements: a sensing unit, a processing unit, and a wireless transceiver unit. The sensing unit collects information from the surrounding environment; the processing unit performs some local information processing such as quantization and compression; and the wireless transceiver unit transmits the locally processed data to a fusion center where the information from different sensor nodes are aggregated and fused to generate the final intelligence. In general, the position of nodes in a WSN need not be engineered or predetermined. This allows for random deployment in inaccessible terrains or for disaster relief operations. In the following we describe some typical applications of WSNs: 1. Environmental Applications: Sensor networks can be deployed to monitor bird movements, avalanches, and for forest fire detection. The sensed data can be used to determine the migratory pattern of birds and to warn the people accordingly of any imminent danger. Please refer to [70] for an application related to habitat monitoring and to [71] for an overview on environmental sensor networks. 2. Health Applications: Some common applications for sensor networks are providing interfaces for the disabled, integrated patient monitoring, etc. Also body-area wireless sensor networks are proposed to monitor vital signs of patients [18], [19], [103]. This can be done remotely by deploying a WSN without compromising on  3  1.1. Wireless Sensor Networks  Fusion Center (FC)  Figure 1.2: The figure shows a typical WSN which consists of number of sensor nodes (shown as dots) and a fusion center (FC).  the freedom of a patient. 3. Military and Homeland Security Applications: WSNs can be an integral part of military command, control, communications, computing, intelligence, surveillance, reconnaissance and targeting (C4ISRT) systems. Further unattended ground sensor networks (UGSN) can be deployed in inaccessible areas to monitor terrorist movements. Also bio-sensors can be deployed along the national borders to detect the smuggling of bio-weapons by terrorists. In the above list we summarize some important applications of WSNs. Indeed, poten4  1.1. Wireless Sensor Networks tial applications of sensor networks are limited only by the imagination. Please refer to [7] for a more detailed discussion on applications of sensor networks. It is envisioned that, in future, WSNs will have a far bigger impact than the present day personal computers. Typically the sensor nodes are equipped with small batteries and hence are subject to severe energy constraints. Consequently, the main objective of sensor network research is to design energy-efficient devices and algorithms to support various aspects of network operation. The µAMP [29] and the PicoRadio [90] projects at MIT and UC Berkeley respectively are aimed in this direction. Owing to the severe energy constraint on the sensor nodes the authors in [47] call for a cross-layer design based approach for the design of general energy constrained networks. Cross-layer design represents a new protocol design paradigm, where a network is designed by optimizing across multiple layers of the protocol stack. There are numerous works in literature that study the interaction between various layers of the protocol stack for throughput maximization in wireless networks. For example, joint scheduling and power control was considered in [43], while cross-layer design based on computation of optimal power control, link schedule and routing flow was described in [35]. Joint routing, power control and scheduling for a Time Division Multiple Access (TDMA)- code division multiple access (CDMA) network was considered in [22] and [113]. However, from the perspective of WSNs since energy is a critical resource. Therefore most of the works in the literature aim to jointly design the physical, medium access contrl (MAC) and routing layers so that the total energy is minimized. Note that most WSNs are short range, therefore the circuit energy is comparable to or greater than that of the transmission energy [38]. Consequently ignoring circuit energy consumption can lead to sub-optimal performance. Accordingly in [37], the authors propose an optimization framework to determine optimal modulation schemes that take into account the circuit  5  1.1. Wireless Sensor Networks energy consumption. In [39], the authors consider joint design of physical, routing and MAC layer to minimize the overall consumption in interference-free WSNs, that includes both the circuit and transmission energy consumption. Other works that consider crosslayer design in WSNs include [69], where the authors consider joint physical, MAC and routing layers from the perspective of interference limited WSNs. Other techniques to improve energy efficiency of WSNs include clustering and cooperation of sensor nodes. In [52], the authors propose an application specific protocol architecture based on clustering called low-energy adaptive clustering hierarchy (LEACH). In [37], the authors identify explicit cooperation among sensor nodes based on the wellknown multiple input and multiple output (MIMO) antenna techniques called space-time block codes (STBC) [82]. Note that the distributed STBC proposed in [37] is just one example of node cooperation. For example, in [60], the author considers node cooperation based on the popular V-BLAST architecture [82]. In [36], the authors consider joint routing and scheduling in WSNs based on STBC based node cooperation. The authors also demonstrate that in the presence of circuit energy multi-hop routing may not be optimal. The WSNs, apart from being energy constrained, offer new avenues to improve its energy efficiency and performance. For example, in WSNs the sensor nodes observe a common physical phenomenon of interest, therefore the data sensed by them is spatially correlated. The spatial correlation can be exploited by sensor nodes to reduce the amount of data generated, and therefore the total energy of the WSN as shown in [32]. Alternatively, spatial correlation can be exploited to improve application level performance of a WSN. For example, in [96], the authors present a new approach to routing that maximizes application level detection performance. In Chapter 2 of this thesis, we consider a combination of cooperation and distributed  6  1.2. Wireless Local Area Networks source coding (DSC) to improve the energy efficiency of WSNs. We model the spatial correlation using a linear Gauss-Markov model, through which we determine, the sensor encoding rates and the route assignment. A unique aspect of our work is structural results, through which one can study as how the modulation rate and number of cooperating nodes vary with the correlation. We also study the mutual information (MI)-energy tradeoff for cooperative and non-cooperative transmission schemes.  1.2  Wireless Local Area Networks  A WLAN is the wireless equivalent of local area networks (LAN) commonly used in universities, offices, etc. [34]. A typical WLAN consists of an access point (AP) and a number of users as shown in Fig.1.3. The purpose of an AP is to connect a given WLAN to other networks thereby increasing the range of it. Typical range of WLANs is of the order of a few hundred metres, which is quite small compared to the cellular networks. However, the loss in range comes at the benefit of increased data rates for a WLAN. Another important difference between WLANs and cellular networks is WLANs use orthogonal frequency division multiplexing (OFDM), while the 2G and 3G cellular networks use either TDMA or CDMA or both [92]. The Institute of Electrical and Electronics Engineers (IEEE) has set up the IEEE 802.11 task group to develop set of standards for WLANs [1]. The IEEE 802.11 group is divided into several sub-groups such as IEEE 802.11a, IEEE 802.11b, IEEE 802.11g, IEEE 802.11n, IEEE 802.11e, etc., to address various operational aspects of a WLAN. The IEEE 802.11a sub-group defined the standard for operation in the 5 Giga Hertz (GHz) range of the spectrum. The IEEE 802.11g defines the operational standards for the 2.4 GHz range. The IEEE 802.11g uses the same OFDM scheme as IEEE 802.11a and is designed to be backward compatible with the IEEE 802.11b standard [107]. The most recent IEEE 802.11n standard provides 7  1.2. Wireless Local Area Networks WLAN 1 AP 1  AP 2 AP 3  WLAN 2  WLAN 3  Figure 1.3: The figure shows three WLANs where each WLAN consists of number of users and an access point (AP).  higher throughput enhancements using the popular multiple input and multiple output (MIMO) antenna technology [2]. For the MAC the IEEE 802.11 specifies two different mechanisms namely, the contention-based distributed coordination function (DCF) and the polling-based point coordination function (PCF). At present only the DCF is implemented in most IEEE 802.11 compliant products, hence we focus on it for the rest of this subsection. In the rest of this sub-section, we describe briefly about the operation of DCF, mention about the enhancements to DCF, and finally discuss an anomaly of DCF and its enhancements that tend to decrease the overall throughput of a WLAN. The DCF is the fundamental access method used to support asynchronous data transfer in WLANs. All the users in a WLAN must support DCF. The DCF is based on carrier sense multiple access mechanism used commonly in (wired) LANs. However in WLANs the users operate with collision avoidance (CA) feature instead of collision detection (CD) feature used in LANs because the users are unable to listen to the collisions while transmitting [34]. In the DCF mechanism, each WLAN user checks whether the channel is  8  1.2. Wireless Local Area Networks idle before attempting to transmit. If the channel has been sensed idle for a distributed inter frame space (DIFS) period, the transmission can begin immediately. If the medium is determined to be busy, the user shall defer until the end of the current transmission. After deferral, the user will select a random backoff interval and shall decrement the backoff interval counter while the channel is idle. Once the backoff interval has expired, the user begins transmission. More specifically, the user selects a random number called backoff time, in the range of 0 to contention window (CW). The backoff timer decrements the backoff time each time the channel is detected to be idle for an interval of one slot time. As soon as the backoff timer becomes zero, the user can begin to transmit. If the transmission is not successful, a collision is considered to have occurred. In this case, the contention window is doubled, and a new backoff procedure starts. The process will continue until the transmission is successful or discarded [34]. Note that the DCF mechanism does not differentiate users or traffic classes. All users and traffic classes have the same priority to access the channel. Thus, different delay and bandwidth requirements of applications are not supported by DCF. However certain high-layer applications such as data, video and audio would require priority access to the channel due to their strict quality of service (QoS) requirements in terms of bandwidth, delay, jitter and packet loss. To meet these QoS requirements of various traffic types the IEEE 802.11e standard proposed a new channel access method called the enhanced DCF (EDCF) [3]. In EDCF QoS support is realized with the introduction of traffic categories. EDCF defines the access category (AC) mechanism that provides support for the priorities of the users. Each user may have up to four ACs to support eight user priorities (UPs). One or more UPs are assigned to one AC. Each AC is assigned a different value of CW such that a high priority AC will be able to transmit before low priority ones [48]. To achieve further differentiation different inter frame spacing  9  1.3. Cognitive Radio Networks (IFS) is used for different ACs. Each AC within a given user behaves like a virtual user. It contends for access to the wireless channel similar to DCF, by sensing the channel for a sufficiently long time. Collision between ACs within a single user are resolved such that the data frame of higher-valued AC receives the transmission opportunity [48]. The EDCF in general works well for differentiated data services [48]. Both DCF and its enhanced cousin EDCF suffer from a performance anomaly that was first reported in [53] and further analyzed in [100]. The anomaly is due to the so called capture effect of the low rate users. Essentially, the capture effect refers to the phenomenon, where a low rate user spends inordinately more time on the channel, thereby depriving other users of their fair share to transmit, leading to a reduction in the overall throughput of a WLAN. To overcome this capture effect of the low rate users, in Chapter 3 of this thesis, we consider a price based approach to regulate the time spent by a user on a given channel. Pricing is a technique commonly used in economics to generate revenue for the system, and to encourage users to use the system resources efficiently [72]. In Chapter 3, we use pricing for the latter purpose.  1.3  Cognitive Radio Networks  Cognitive radio (CR) represents a new wireless design paradigm where a communication device (or radio) adapts its transmission parameters such as power, rate, operating frequency etc., in response to the changing wireless medium. The term cognitive radio was coined by Mitola III in [58]. Note that CR represents a design paradigm rather than a wireless application like WSNs, WLANs, etc. We provide a more in depth discussion on CR networks in Chapter 4. In this section, we provide a brief introduction to CR networks and outline some design challenges in CR networks. A specific instance of CR design paradigm called dynamic spectrum access (DSA) 10  1.3. Cognitive Radio Networks [114], which corresponds to adaptation of operating frequency of the radio has received a lot of attention recently. DSA has the potential to alleviate the problem of spectrum underutilization, by allowing the unlicensed users to access portions of the licensed spectrum. The current interest in DSA is due to a report published by the Federal Communications Commission (FCC) pointing out that vast portions of the spectrum are underutilized [44]. The term DSA stands opposite to the current static spectrum management policy, where a given frequency band is allocated exclusively for a licensee [114]. The DSA strategies can be broadly classified under three models: dynamic exclusive model, open sharing model, and hierarchical access model [114]. A detailed description about dynamic exclusive model and open sharing model can be found in [114]. In this thesis, we focus on the hierarchical access model of DSA, since it is perhaps the most compatible with the current spectrum management policies and legacy wireless systems. The basic idea in hierarchical model is to open the spectrum designated for primary (licensed ) users to secondary (unlicensed) users, by regulating the interference perceived by primary users. Two approaches to spectrum sharing between primary and secondary users between primary and secondary users are spectrum-underlay and spectrum-overlay [114]. In the spectrum-underlay approach, the secondary users regulate their total power so that the total interference perceived by primary users is kept at a minimum [114]. In the spectrum-overlay approach, the secondary users access the spectrum opportunistically (i.e., when primary users are not active). Therefore the spectrum-overlay approach is also referred to as opportunistic spectrum access [114]. Owing to the difference in the two approaches namely, spectrum-underlay and spectrumoverlay, the design challenges between the two are accordingly different. For example, in spectrum-underlay approach, since the power of secondary users need to be regulated, therefore the issue of secondary user power control is important. In [56], the authors  11  1.3. Cognitive Radio Networks propose power control based on auction mechanisms. Specifically, the authors model the secondary user power control problem under an interference temperature constraint as a non-cooperative game, where the secondary users submit bids in response to the price announced by the spectrum manager. Further, the authors propose two algorithms that converge globally to the Nash equilibrium of the proposed game. In [110], the authors consider the problem of power allocation for secondary users based on pricing. Specifically, the authors formulate the problem as a non-cooperative game, where the aim of each secondary user is to maximize the sum rate on all the links. The authors in [110] further propose two provably convergent algorithms namely, Sequential Iterative Water Filling (SIWF) and Parallel Iterative Water Filling (PIWF) that converge to a unique Nash equilibrium. On the other hand for the spectrum-overlay approach to be feasible, efficient detection of primary receivers is important. To address this in [112], the authors consider detection of primary users, where it was proposed to exploit the use of local oscillator leakage current for the detection of primary receivers. This approach requires low cost sensors to be deployed close to the primary receivers for efficient detection. Another approach to primary receiver detection is to transform to that of primary transmitter detection. This helps in transforming the primary receiver detection problem into a classical signal processing problem which can be solved using traditional signal detection techniques such as matched filter (or) radiometer as noted in [27]. The spectrum sensing in the spectrum overlay networks induces an overhead on the secondary user in terms of energy which can be quite considerable in energy constrained networks such as WSNs. To address this, the authors in [115] adopt a cross-layer approach to the design of protocols for the spectrum-overlay approach. Most existing works in literature focus on the physical, MAC and routing layer aspects  12  1.4. Mathematical Tools of CR networks. In Chapter 4 of this thesis, we focus on the performance of TCP in a CR network based on spectrum-overlay approach. TCP operates at the transport layer of the protocol stack and is used ubiquitously for reliable transfer of data across the internet. The performance of TCP was analyzed in conventional (i.e., without cognition) wired and wireless networks extensively (see for example [64] and [14]) under congestion and other losses. In CR networks based on spectrum-overlay approach, the secondary users are subject to a new type of loss called service interruption loss. We analyze the impact of this service interruption loss on the secondary user TCP throughput on an NS2 [76] simulation model. The simulation results suggest that there exists an optimal number of channels, which the secondary user needs to capture to maximize its TCP throughput.  1.4 1.4.1  Mathematical Tools Monotone Comparative Statics  In this subsection we summarize some key concepts and results that are important for our analysis in Chapter 2. Monotone comparative statics (MCS) are used extensively in economics literature to study as how the optimal solution varies with respect to the underlying parameters of the problem. It makes the assumption of differentiablity unnecessary which are required in standard comparative statics [11]. The most popular result in MCS is due to Topkis [104]. We refer the interested readers to [106] for an in depth coverage on MCS and its associated results. Also refer to [11] for a comprehensive survey on MCS, supermodularity and supermodular games. Here we present a brief summary on important results in MCS. We start with the following definition of lattice. Definition 1.1. Let Ω ⊆ Rn , x = (x1 , x2 , · · · , xm ) and y = (y1 , y2 , · · · , ym ) such that x, y ∈ Ω. Also denote x ∨ y  (max[x1 , y1 ], max[x2 , y2 ], · · · , max[xm , ym]) as the meet of 13  1.4. Mathematical Tools x, y, and x ∧ y  (min[x1 , y1 ], min[x2 , y2 ], · · · , min[xm , ym ]) as the join of x, y. Then the  set Ω is a lattice if for any x, y ∈ Ω, the meet x ∨ y and join x ∧ y of x and y are also in Ω. Next we define the concept of increasing differences. The concept of increasing and decreasing differences are central to the Topkis theorem [104] presented later in this subsection. Definition 1.2. A function F : S × A → R is said to satisfy (strictly) increasing differences in (s, a) if F (s′, a′ ) − F (s′ , a)(>) ≥ F (s, a′ ) − F (s, a) ∀a′ > a, s′ > s i.e., if F (:, a′ ) − F (:, a) is an increasing function for all a′ > a. On the other hand if the difference F (:, a′ ) − F (:, a) is a decreasing function for all a′ > a, then the function F is said to satisfy decreasing differences. Remark 1.1. If a function F : S × A → R is continuous and differentiable then proving that F satisfies (strict) increasing differences in (s, a) is equivalent to showing that ∂ 2 F (s,a) (>) ∂a∂s  ≥ 0 [11].  Definition 1.3. A function F (x1 , x2 , · · · , xn ) is said to be supermodular if it satisfies the definition of increasing differences for every xi and xj such that i = j. We now state the following simplified version of Topkis’s monotonicity theorem in [104]. This theorem is useful in proving structural results in Chapter 2 of the thesis. Theorem 1.1. Consider the following optimization problem: minimize F (s, a). Suppose s∈S  that S forms a lattice and has at least one solution for each a ∈ A. Suppose also that F satisfies (strictly) increasing differences in (s, a). Then optimal s is always decreasing in a. Proof. Please refer [104] for a detailed proof. 14  1.4. Mathematical Tools Remark 1.2. In Theorem 1.1 if the function F satisfies decreasing differences, instead of increasing differences with all the other assumptions in tact then optimal s is increasing in a. Remark 1.3. In Theorem 1.1, if we replace the objective function by a monotone transformation (such as logarithm) of the objective function, still the parametric monotonicity of Theorem 1.1 holds. The above remark might be useful in cases, when it is not possible to say conclusively if the function is super (sub) modular without any additional assumptions. Please refer [11] for an interesting example.  1.4.2  Game Theory  Game theory is a mathematical tool used to study interactive decision problems among intelligent rational decision makers. It has numerous applications in economics, social sciences, engineering and recently computer science. In this section, we will briefly introduce the necessary definitions and solution concepts that are relevant to this thesis. Please refer to [46], [80] for greater details on game theory. There are two types of game theory namely, cooperative and non-cooperative game theory. For the purpose of this thesis we focus only on the non-cooperative game theory. Any game is characterized by three quantities namely, the players, the actions, the utilities (or payoffs), known collectively as the rules of the game, which are described next. Players are the individuals who make decisions, denoted by a set N = {1, 2, · · · , N}. An action ai is the choice player i can make. Player i’s action set Ai = {ai } is the set of all choices he can make. An action profile a = (ai )N i=1 is a sequence of all player’s actions, one for each player. Player i’s utility ui (a) is a function of the action profile a, and describes how much the players gain from the game for each possible action profile. 15  1.4. Mathematical Tools Henceforth in this thesis we use the notation G  [N , {Ai}, {ui}] to denote a general  non-cooperative game. An important assumption in game theory is that all players are rational, i.e., they want to choose actions to maximize their utilities. In order to maximize their utilities, the players would design contingent plans known as strategies, which map a player’s information to his actions. A strategy could be pure in which case it only contains one deterministic action for each information set; or it could be mixed, in which case it specifies a set of actions each chosen according to a probability vector for each information set. In this thesis, we are mainly concerned about pure strategies. A strategy profile is a sequence of the players strategies, one from each player. The concept of equilibrium is central to most game theoretic analysis. The most popular equilibrium concept in game theory is Nash equilibrium (NE), which was named in honour of the economics Nobel laureate John F. Nash. In simple words, NE represents a strategy profile at which there is no incentive for any player to change its strategy. A more mathematically rigorous definition is as follows: Definition 1.4. A strategy profile a∗ = (a∗1 , a∗2 , · · · , a∗N ) is a NE of a game if for every i ∈ N , ui (a∗i , a∗−i ) ≥ ui (a′i , a∗−i ) for all a′i ∈ Ai . Here we use the standard game theoretic notation a = (ai , a−i ), where a−i denotes the actions of all players except i. One concept closely related to NE is the player i’s best response correspondence  1  BRi (a−i ), which is a set of strategies such that  ui (ai , a−i ≥ ui (a′i , a−i ) ∀ai ∈ BRi (a−i ) and a′i = ai . From the definition of best response correspondence one can define NE a∗ as a point such that a∗i ∈ BRi (a∗−i ). Besides Nash equilibrium another popular equilibrium concept 1  The correspondence concept is more general than that of a function. If the correspondence is a singleton set then it reduces to a function.  16  1.4. Mathematical Tools in game theory is that of correlated equilibrium [15], [16], which was first introduced by another Economics Nobel Laureate Robert J. Aumann in [15]. The correlated equilibrium is a generalization of Nash equilibrium, where it is assumed that the players adapt their strategies according to a common distribution. In this thesis, we use the notion of Nash equilibrium for decentralized rate selection in WLANs. Game theory and the associated equilibrium concepts are useful for analysis in most decentralized decision making problems. However the theory of learning in games, which involves proving the convergence of various adaptive procedures to equilibrium, is relatively incomplete. The book [45] contains some existing results along with some original works of the authors. The two popular adaptive procedures used for learning in games are fictitious play and best response dynamics [45]. In general proving the convergence with these two popular adaptive procedures to an equilibrium requires restrictive conditions that are hard to verify in practice with the exception of supermodular [77] and potential games [78]. We use the theory of supermodular games for our game theoretic analysis in Chapter 3. Other popular adaptive learning procedures in games include the regret-minimizing procedure. It has been shown in literature that if a game is played infinitely many times such that every player plays to minimize his regret, then the empirical frequencies of play converge to a set of correlated equilibria [49]. Game theory is used extensively in wireless network design to characterize the impact of interaction between various users of the network on their individual utility (such as throughput) and on a global system utility. For example it has been used extensively for power control in CDMA cellular and ad hoc networks [93], [51], [57], and in cognitive radio networks [56]. It has also been used for medium access control in wireless networks [68] [79], and for flow, routing control in wireless networks [9]. Please see [10], for a detailed survey on applications of game theory to telecommunication networks. Other  17  1.4. Mathematical Tools interesting related applications of game theory include decentralized activation of sensor nodes in a wireless sensor network [73], and design of missile deflection systems for military applications [74], [75].  1.4.3  Supermodular Games  Supermodular games are those in which the marginal returns to increasing a player’s strategy rise with an increase in the competitor’s strategies. Supermodular games were introduced by Topkis in [105] and were further analyzed by Vives in [108], [109]. The class of supermodular games encompass many of the important applications of non-cooperative game theory. Important applications of supermodular games in wireless networks are power control in CDMA networks [93], [57]. We use supermodular games in Chapter 3 for decentralized rate selection in WLANs. In the following we provide a definition of supermodular games and state some of its important properties. First we provide the following general definition for a supermodular game. Definition 1.5. A non-cooperative game G  [N , {Ai}, {ui}] is a supermodular game  if for each i ∈ N the following conditions are satisfied (i) Ai is a non-empty sublattice, (ii) ui is continuous in all players’ strategies and supermodular in its own strategy and (iii) ui has increasing differences between any component of player i’s strategy and any component of other player’s strategy [77], [106]. As evident from the definition above supermodular games do not rely on the differentiability of the player utility functions. This is a very important observation since in most practical problems the strategy set of players are discrete. For example in the decentralized rate selection problem considered in Chapter 3 of this thesis, the actions of the players are rates which take only integer values. Further supermodular games do not require the action sets Ai of players to be convex. It only requires the action 18  1.4. Mathematical Tools spaces of players to be lattices. However for the case when player utility functions are differentiable one can use a simplified version for the definition of supermodular games (please see Theorem 4 in [77]). For a supermodular game the best response correspondence of every player is nondecreasing. Consequently, the overall best response correspondence is also nondecreasing. The choice of any point in this overall best response correspondence has a fixed point, which in fact is the pure strategy NE. This property of supermodular games is summarized in the following theorem. Theorem 1.2. For any supermodular game there exists a pure strategy Nash equilibrium. Moreover, the Nash equilibrium is contained in a set that is a complete lattice i.e., it contains a smallest and largest element in the given order. More specifically, the NE a∗ is contained in a set defined as follows: NEsupermodular = {a∗ ∈ A : a∗S ≤ a∗ ≤ a∗L } , where A  (1.1)  ×N i=1 denotes the joint strategy space of the game, the operator × denotes the  Cartesian product of sets, and a∗S , a∗L denote the smallest and largest elements of the set NEsupermodular . Proof. The existence part can be established using Tarski’s fixed point theorem [102]. For the latter part refer [77]. Note that the Theorem 1.2 only says that the NE of a supermodular game lies in a set defined as in Eq. (1.1). However it does not mean that all the elements of such a set are Nash equilibria. On the other hand the smallest and largest elements in the set NEsupermodular of Eq. (1.1) are indeed pure strategy NE. We now state the following theorem. The theorem studies the variation of largest and 19  1.5. Summary of Results and Contributions smallest Nash equilibrium of the set NEsupermodular defined in Eq. (1.1), with respect to an exogenous parameter µ. An exogenous parameter is one which does not depend on any player’s actions. Theorem 1.3. Let Gµ = [N , {Ai}, {uµi}] be a parameterized supermodular game such that the utility function uµi is in general a function of parameter µ, then the smallest and largest NE in the set NEsupermodular (defined in Eq. (1.1)) denoted respectively as a∗S and a∗L are non-decreasing in µ if the player utilities ui satisfy the notion of increasing differences between players actions and the exogenous parameter µ. Proof. Refer [77] for details. An important aspect of supermodular games as noted in Section 1.4.2 is, it can be proved that the learning procedures such as best response dynamics and fictitious play, would converge to a NE. In this thesis, we focus on the best response dynamics. In the case of best response dynamics, at any update instant, a player or group of players choose their best reply based on the previous actions of the other players. For a supermodular game it can be proved that the best response dynamics converge to the smallest (largest) NE of the set defined in Eq. (1.1) starting from the smallest (largest) point of the joint strategy space [106], [109]. However starting from an arbitrary feasible strategy, the best response dynamics will eventually lie in a set defined in Eq. (1.1) [77]. An important practical aspect of interest regarding best response dynamics in supermodular games is, the update between players can be asynchronous.  1.5  Summary of Results and Contributions  This thesis covers several resource management problems in various types of wireless networks. The results are divided into three chapters. The contributions in each chapter 20  1.5. Summary of Results and Contributions are as follows. • In Chapter 2 of the thesis, we consider the issue of resource management from the perspective of WSNs. Specifically, we consider the problem of data gathering in correlated WSNs with the objective of minimizing the total energy consumption of a WSN. To this end, we consider a combination of node cooperation and DSC. We initially formulate a minimum energy data gathering optimization problem, for the case of non-cooperation based single input and single output (SISO), and cooperation based MIMO transmission scheme. Next, using the concepts of super and sub modularity on a lattice, we analytically quantify as how the optimal modulation rate and optimal number of cooperating nodes vary with respect to the spatial correlation coefficient. We also study numerically the mutual information (MI)-energy tradeoff between the SISO and MIMO transmission scheme. Finally, from our analysis and simulation results we study the impact of sensor spacing (and hence the spatial correlation) on the performance achieved at destination. • In Chapter 3, we consider the problem of decentralized rate selection in IEEE 802.11 WLANs. Owing to the decentralized nature of WLANs we formulate the problem as a non-cooperative game where individual users of a WLAN can pick their actions from a finite set of physical layer modulation rates. The utility of each user is the difference of throughput and a cost incurred due to the price imposed by the access point. We prove the resulting non-cooperative game to be supermodular, and hence has at least one pure strategy Nash equilibrium, that is contained in a set bounded by the smallest and largest Nash equilibria. We next analyze the equilibrium properties of the non-cooperative game. We also present an algorithm to compute the best response of each user asynchronously, that converges almost surely to the smallest Nash equilibrium of the game. Next we extend our 21  1.6. Thesis Organization price based approach to the multi-channel case and prove the resulting game to be supermodular in the special case of two channels. Finally, through our simulation results we show the improvement in overall network throughput with appropriate tuning of the price. • In Chapter 4, we illustrate the performance of TCP in an overlay CR network under DSA. We show that the performance of TCP in overlay CR networks that implement DSA to be quite different from its performance in conventional networks, which do not allow DSA. The key difference is that secondary users in an overlay CR network have to cope with a new type of loss called service interruption loss, due to the existence of primary users. We demonstrate on an NS2 simulation model the surprising result: Excessive radio resource usage leads to a decrease in aggregate TCP throughput. This behavior is in contrast to the behavior of TCP in conventional networks, where throughput increases monotonically with the available radio resource.  1.6  Thesis Organization  The rest of this thesis is organized as follows. The issue of data gathering in correlated WSNs is considered in Chapter 2. The issue of decentralized rate selection in IEEE 802.11 WLANs is presented in Chapter 3. The performance of TCP in spectrum-overlay CR networks is considered in Chapter 4. Finally, the conclusions and possible future extensions are presented in Chapter 5. All the main chapters of the thesis are selfcontained. A review of the related work for each chapter appears in the corresponding introduction section. The notations are also defined separately for each chapter, but is kept consistent throughout the thesis for the ease of understanding.  22  Chapter 2 Data Gathering in Correlated Wireless Sensor Networks (WSNs)2 2.1  Introduction  The design of a wireless sensor network (WSN) differs significantly from the design of conventional wireless networks, owing to the severe energy constraint of sensor nodes, as noted in Section 1.1 of Chapter 1. Further in a WSN the data sensed by nodes is spatially correlated because of the proximity of sensor nodes and them having to observe the same underlying physical activity. Two representative examples where data sensed by nodes can be spatially correlated are: (i) acoustic sensor networks, where the neighboring sensor nodes record the time delayed version of the same signal, and (ii) in sensor networks deployed to detect gas leaks. In the following, we summarize a few important works in the literature that aim to improve the energy efficiency of a WSN by various techniques. Towards the end of this section, we present some key results of this chapter and provide a brief outline of what is to follow. To reduce energy consumption of each node, and hence prolong the lifetime of network, a cooperation based architecture was proposed in [37]. Specifically, the authors assume that the sensor nodes exchange information, and transmit cooperatively, using the well 2  The work in this chapter will appear as a book chapter [86]. The work also appeared in parts as conference papers [85] and [87] respectively.  23  2.1. Introduction known multiple input and multiple output (MIMO) encoding technique due to Alamouti [8]. Such cooperation based transmission schemes are attractive for applications such as, environmental, habitat and avalanche monitoring. In such applications, the nodes are expected to last for long time without battery replacement. Deploying cooperative MIMO based scheme for such applications can prolong lifetime of the network, since each node would spend a fraction of total energy required in non-cooperative based transmission scheme. Unlike single-hop transmission considered in [37], the paper [36] considers cooperative MIMO techniques for multi-hop routing. However, the issue of correlation in the data sensed by nodes was not considered in [36]. But in practice owing to the close proximity of nodes, the data sensed by neighboring nodes can be correlated, hence we consider the issue of data correlation in this work. Architecturally our system requires local information circulation to exploit the correlation, which is not the case in [36]. An important aspect of our work is, we obtain structural results on modulation rate and number of cooperating nodes using concepts in monotone comparative statics (MCS) [106], which to the best of our knowledge are not found elsewhere. These structural results provide important design guidelines for data gathering in correlated WSNs. There exist other works in literature which consider correlated data gathering in WSNs. For example in [32], the authors consider distributed source coding (DSC) and routing in correlated sensor networks. In this approach, the joint problem of optimizing the transmission structure and source coding becomes decoupled. The optimal transmission structure can be determined in a simple manner, however the coding becomes complex, owing to the need for global network knowledge and correlation structure [32]. Contrary to the approach in [32], in [33], the authors consider the case where correlation structure is exploited through explicit communication. In this case, the rate  24  2.1. Introduction allocation for source coding becomes simple, however determining the optimal transmission structure becomes difficult. In fact the authors prove the problem of determining optimal transmission structure to be NP-complete, and hence propose heuristic approximation algorithms that provide good solutions for this difficult problem. Similar works where the interplay of data compression and routing was studied can be found in [94] and [81]. In [81], using an empirical correlation model, the authors evaluate three qualitatively different schemes namely: distributed source coding, routing driven compression and compression driven routing. Further, the authors prove cluster-based tree structures to have good performance, depending on the correlation level. In all of the works mentioned above, the effect of data correlation on physical layer design variables such as, modulation rate and number of cooperating nodes was not considered. In general, the energy expended by a sensor node is a function of modulation rate. Modulation rate represents the number of bits that can be transmitted in a given symbol. Further in cases where nodes cooperate to transmit data, the energy expended is also a function of number of cooperating nodes. In WSNs, since the data sensed by nodes is correlated, it is of interest to know as “how the physical layer design variables such as modulation rate and number of cooperating nodes vary with the correlation in data”, which we address in this chapter. Another unique feature of our work is, we study the tradeoff between Energy-Mutual Information (MI) and Energy-Mean Square Error (MSE) for Single Input and Single Output (SISO) and MIMO transmission schemes in correlated WSNs. Further, we also consider the issue of node placement in correlated WSNs, so that MI at the destination is maximized. Again here we use concepts in monotone comparative statics, to obtain a structural result on variation of node spacing with respect to the observation signal to noise ratio (SNR). The implication of this structural result on MSE achieved at the sink  25  2.1. Introduction (or destination) is investigated. Using this analysis we obtain important guidelines for placement of sensors in correlated WSNs. We now briefly comment on monotone comparative statics that are used extensively in economics literature to study as how the optimal solution of a parameterized optimization problem varies monotonically with respect to the parameter. Please refer to Section 1.4.1 in Chapter 1 for details. The following are the important results of this chapter.  2.1.1  Main Results  • We prove that the optimal modulation rate adopted by a sensor node to minimize total energy consumption in the network is an increasing function of the correlation coefficient. This implies that when data is correlated the sensor node can adopt higher modulation rates than when data is uncorrelated. • For the case of cooperative transmission based on MIMO techniques we prove that the optimal number of cooperating nodes to minimize total energy consumption in the network is a decreasing function of the correlation coefficient. This implies that when data is correlated the cooperation among nodes can be less than the case when data is uncorrelated. • In a cooperative MIMO transmission scheme we prove that the optimal modulation rate adopted by each cooperating node to minimize energy consumption decreases with increase in number of cooperating nodes. The reverse implication is also true i.e., the optimal number of cooperating nodes to minimize energy consumption also decreases with an increase in modulation rate. • In a correlated WSN the spacing between nodes to maximize MI at the sink (or destination) is an increasing function of observation SNR. Using this structural 26  2.2. WSN Model and Assumptions result and MSE as a performance measure we conclude that at high observation SNR correlation does not have significant impact on performance. However at low observation SNR, we conclude that the performance depends critically on the correlation. • Finally, using our numerical results we observe that for large values of intercluster spacing in a network, MIMO achieves better Energy-MI (or MSE) tradeoff than the SISO transmission. However for small values of intercluster spacing SISO achieves better Energy-MI (or MSE) tradeoff. The rest of this chapter is organized as follows: Section 2.2 contains system model description and assumptions. Section 2.3 contains the formulation of minimum energy data gathering in correlated WSNs. Section 2.4 contains structural results on optimal modulation rate and number of cooperating nodes. Section 2.5 studies the MI-Energy tradeoff for MIMO and SISO data gathering schemes and also studies the variation of optimal node spacing with observation SNR. Finally, Section 2.6 contains concluding remarks.  2.2  WSN Model and Assumptions  We consider data gathering using WSN. For the purpose of our analysis we assume the signal along the route to the destination forms a one dimensional stationary GaussMarkov process. Such a model was used in [98] and [97] for routing in correlated random fields. This is valid in the case of acoustic sensor networks where multiple acoustic sensor nodes are deployed to record sound field created by an acoustic source. In such a case each of the sensor nodes deployed receives time delayed version of the same signal, and hence it is reasonable to assume that the signal sensed by successive stages of the 27  2.2. WSN Model and Assumptions  Physical activity modelled as correlated random field  LiI  L1i 11d12 00 00 11 00 11  Stage 1  d  11 ij 00 00 11 00 11  11 00 00 11 00 11  L2i Stage 2 L  2j  R1  LII+1  Lij  L12  R2  11 00 00 11 00 11  LjI Stage I Stage j LjI+1 Rj RI  Stage i Ri  d  11 II+1 00 00 11 00 11  X  Sink Stage (I+1) RI+1  − Transmission node Sensing node  11 00 00 11 − 00 11 00 11  Figure 2.1: The figure shows the scenario for data gathering in correlated random fields. Each stage of the network consists of a number of sensor nodes which cooperate during transmission of data. For each data collection period a node in each stage senses, quantizes and broadcasts to other nodes in that stage for cooperative transmission to the next stage. Note that apart from the data generated within a given stage i, the nodes in each stage also receive data from previous stages Lji that needs to be relayed to the next stage. The encoding rate of each stage is denoted as Rk , k = 1, 2, · · · , I. network is correlated. We use k = 1, 2, · · · , I to denote discrete time. Equivalently k also represents stage number, since stage (k + 1) receives observation at time (k + 1) due to delay. The Gauss-Markov assumption on the underlying signal process leads us to the following state-space representation of the signal process (See [98] and [97]):  xk = ak−1 xk−1 + uk−1 , for k = 1, 2, · · · , I, uk−1 ∼ N 0, P0 (1 − a2k−1 ) ,  (2.1)  28  2.2. WSN Model and Assumptions where ak−1 denotes the correlation between (k − 1)th and k th stages of the network. In general the models used for correlation are classified into four standard groups: Spherical, Power Exponential, Rational Quadratic and Mat´ern [20]. The key properties such as non-negativity of correlation and decrease in correlation with distance are satisfied by all the four models. However for our analysis in Section 2.5 we model correlation ak−1 using the popular Power Exponential model as ak−1 = e−Adk−1 , where A is the field diffusion coefficient and dk−1 denotes the distance between (k − 1)th and k th stage of the network. We believe the analysis in Section 2.5 can be extended to other correlation models as well. We also remark that the structural results on modulation rate and number of cooperating nodes derived in Section 2.4 are independent of the correlation model. Further each stage of the network on the way to the destination observes a noisy version of the process i.e., yk = xk + nk , nk ∼ N 0, σ 2 ,  (2.2)  where σ 2 denotes the variance of observation noise. Due to stationarity of the random field, the observation SNR at each stage of the network is same and is defined as  Γ  P0 . σ2  (2.3)  The observations of each sensor are transmitted to the destination through multihop routing, which in turn aims at extracting relevant features of the original correlated random field (which in our case is generated by an acoustic source) using judicious signal processing.  29  2.2. WSN Model and Assumptions We abstract the WSN at the network layer as a directed graph G = (V, E) where V denotes the set of stages and E denotes the set of directed links. We consider the case of single sink data gathering only. We assume there are (I + 1) stages in the network, where 1, 2, · · · , I denote the data gathering stages and the (I + 1)th stage denotes the sink node (or destination node). Every directed link (i, j) ∈ E that connects two stages i, j ∈ V has a cost (Eij ) associated with it. In general the link cost Eij depends on modulation rate bij adopted on the link, square of length of the link dij and circuit power consumption. An (I + 1) stage3 network depicting the scenario considered in this work is shown in Figure 2.1. Each stage can contain one node (for SISO transmission case) or more than one node (for MIMO transmission case). In the SISO case, the only node present at a given stage senses, quantizes and transmits along with the data received from previous stages to the next stage. Note that a given stage can divide the number of bits among more than one subsequent stage. The number of bits received by each of the subsequent stages is determined by solution to the energy optimization problem formulated in Section 2.3. In MIMO case a node called the sensing node makes an observation of the underlying process, quantizes and broadcasts locally to other nodes in that stage for cooperative transmission. Upon local information circulation within each stage, the nodes transmit cooperatively the data originated at its own stage along with the data received from previous stages to the next stage. As in the SISO case, the next stage is determined by solution to the energy optimization problem formulated in Section 2.3. The cooperative transmission between various stages of the network is accomplished based on the wellknown space-time block codes (STBC) [101]. 3  A stage in a network represents single node for the case of SISO transmission or group of nodes for the case of MIMO transmission. Henceforth, we use the word stage to denote either a single node (or) group of nodes.  30  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks Since for short range communications circuit energy can be significant, we consider circuit energy consumption along with the transmission energy consumption as in [37], [38] and [36]. To model the power consumption blocks of a sensor node we use the model in [38]. The antenna of a sensor node is typically at the ground level, hence we model the channel between sensors to be of Rayleigh fading type. We assume each sensor node employs M-ary quadrature amplitude modulation (M-QAM) for digital modulation. We assume the cooperating nodes in stage k are distributed in a circle of radius dmax,k . Since the cluster radius of stage k is much less than the inter cluster distance, it is reasonable to assume as in paper [37] that each node of a given stage is equidistant from the neighboring stage.  2.3  Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks  In this section, we formulate the problem of minimum energy data gathering in sensor networks, for the linear Gauss-Markov state-space model assumed in Eqs. (2.1) and (2.2). The modulation rate b and number of cooperating nodes n can be determined so that the total energy consumption in the network is minimized.  2.3.1  Expression for Total Energy Consumption per bit using SISO Techniques  In this subsection, we determine the total energy consumption per bit in a sensor node employing SISO transmission (i.e., without node cooperation) in a Rayleigh fading environment. In SISO transmission since there is no local information circulation, the single node 31  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks present in a given stage encodes at a rate Rk as determined in Section 2.3.3. The total power consumed in a sensor node employing QAM while transmitting over link (i, j) is given as [37] ij PSISO =  1 (1 + α) 6  1 pb  2bij − 1 N0 G0 Ml d2ij B + (PT + PR ) (for bij ≥ 2) ,  (2.4)  Circuit power  Transmission power  where α is a constant defined by the power amplifier efficiency, pb is the desired probability of error, bij denotes the modulation rate adopted on link (i, j), N0 is the noise spectral density, G0 =  (4π)2 Gt Gr λ2  is the attenuation factor, Ml denotes the link margin compensating  for the hardware process variations, background noise and fading, dij is length of the link (i, j), B is the symbol rate and PT , PR denote the transmitter and receiver circuit power consumption. Thus the energy consumption per bit in transmitting over link (i, j) using SISO transmission can be written as ij ESISO =  ij PSISO B  1 bij  1 1 (1 + α) = 6 pb  2.3.2  2bij − 1 bij  N0 G0 Ml d2ij +  (PT + PR ) 1 (for bij ≥ 2) (2.5) . B bij  Expression for Total Energy Consumption per bit using MIMO Techniques  In this subsection, we determine the total energy consumption per bit for cooperative transmission based on MIMO antenna techniques in Rayleigh fading environment. As mentioned in Section 2.2, for cooperative transmission to be possible there needs to be local information circulation. To accomplish this, we assume a node in each stage  32  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks of the network makes an observation, encodes at rate Rk (as determined in Section 2.3.3) and broadcasts to the other nk − 1 nodes in that stage for cooperative transmission. The energy consumption for this local broadcast in k th stage can be written as in Eq. (2.5) to be k Elocal =  2blocal − 1 blocal  1 1 (1 + α) 6 pb  G0 N0 Ml d2max,k + (nk − 1)  (PT + PR ) (2.6) B blocal  where blocal denotes the modulation rate adopted for local transmission and dmax,k denotes the cluster radius of stage k. The factor of (nk − 1) in the second term of Eq. (2.6) is due to the fact that during local broadcast there are always (nk − 1) circuits listening. For the purpose of our analysis and simulation in this chapter we assume blocal as k fixed. Consequently, the energy consumption for local broadcast at the k th stage (Elocal )  is a function of only the number of cooperating nodes nk present in the k th stage. If we assume ni nodes in a given stage i cooperate and transmit, then the energy consumption per bit ETijx−LH for cooperative transmission over the link (i, j) of length dij can be written as [37]  ETijx−LH =  2 (1 + α)ni 3  4 pb  1 ni    bij     2 1 − 1  N0 G0 Ml +1 n biji  1 r  d2ij (for bij ≥ 2) , (2.7)  where as before α is a constant defined by the power amplifier efficiency, bij is the modulation rate adopted for cooperative transmission on link (i, j), pb is the desired probability of error, N0 is the noise spectral density, G0 is the attenuation factor, Ml is the link margin and r is the spatial rate of STBC used for cooperative transmission. The factor r in Eq. (2.7) accounts for the bandwidth expansion due to employing an STBC with spatial rate r. For the case of Alamouti code employed in [36] (where number of cooperating nodes ni are equal to two) r is equal to one. For the general case of ni 33  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks cooperating nodes as observed in [101], it may not be possible to design r = 1 STBC for complex modulation schemes such as QAM or phase shift keying (PSK) (this was formally proved in [65]). Consequently, r in our formulation depends on the number of cooperating nodes ni . The total circuit power consumed for cooperative transmission over the link (i, j) is ij equal to ni (PT + PR ). Therefore the circuit energy consumption ECircuit−LH per bit for  long haul transmission can be written as ij ECircuit−LH =  ni (PT + PR ) , Brbij  (2.8)  where in Eq. (2.8) B denotes the symbol rate and PT , PR denote the power consumption in transmitter and receiver circuits. Thus, the total energy consumption per bit for transmission over the link (i, j) with MIMO transmission can be written as sum of Eqs. (2.7) and (2.8)  ij EM IM O =  1 ni    bij     2 1 − 1  N0 G0 Ml d2ij +1 n biji (PT + PR ) +ET x−local + (ni − 1) , B blocal 2 (1 + α) ni 3  4 pb  1 r  + ni  (PT + PR ) Br  1 bij (2.9)  where ET x−local is given by first term in Eq. (2.6). A close look at Eq. (2.9) reveals that when a r ≤ 1 STBC is used then, it can lead to increased energy consumption compared to the case when r = 1.  34  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks  2.3.3  Rate Allocations of Data Gathering Stages in Correlated WSNs  The rate allocations Rk of data gathering stages k = 1, 2, · · · , I are determined so that the rate admissibility constraint is satisfied. For a network with single sink the rate admissibility constraint can be written as below with yk∆ denoting the quantized value of the continuous observation yk at stage k I  k=1  Rk ≥ H y1∆ , y2∆, · · · , yI∆ ,  (2.10)  where H(z1 , z2 , · · · , zK ) denotes joint entropy of z1 , z2 , · · · , zK . In the presence of knowledge about correlation between various stages of the network, we can choose individual Rk as [32] ∆ ∆ Rk = H yk∆ |yk+1 , yk+2 , · · · , yI∆ for k = 1, 2, · · · , I − 1,  (2.11)  RI = H yI∆ ,  (2.12)  where H (zk |zk+1 , zk+2 , · · · , zI ) denotes the conditional entropy of zk given zk+1 , zk+2 , · · · , zI and H (zI ) denotes the entropy of zI . To compute Rk for k = 1, 2, · · · , I in Eqs. (2.11) and (2.12) for the process and observation model in Eqs. (2.1) and (2.2) we approximate Eqs. (2.11) and (2.12) for sufficiently small quantization stepsize ∆ as [31]  Rk ≈ h (yk |yk+1, yk+2, · · · , yI ) − log2 ∆ for k = 1, 2, · · · , I − 1,  (2.13)  RI ≈ h (yI ) − log2 ∆ ,  (2.14)  where h (zk |zk+1 , zk+2 , · · · , zI ) denotes the conditional differential entropy of zk given 35  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks zk+1 , zk+2 , · · · , zI and h (zI ) denotes the differential entropy of zI . For the linear Gauss-Markov state-space model assumed in equations (2.1) and (2.2), we can express h (yk |yk+1, yk+2, · · · , yI ) in close form as equal to 12 log2 (2πeσ 2 ) 1 + while h(yI ) can be written as  1 2  Σk|k−1 σ2  log2 [(2πeσ 2 ) (Γ + 1)] (see Section 2.7 for detailed deriva-  tion). Thus Eqs. (2.13) and (2.14) can be re-written as 1 log2 2 1 log2 = 2  Rk = RI  2πeσ 2 ∆2 2πeσ 2 ∆2  1+  Σk|k−1 σ2  (Γ + 1)  , for k = 1, 2, · · · , I − 1 ,  ,  (2.15) (2.16)  where Σk|k−1 is the predictor covariance, ∆ is the quantization stepsize, Γ is the observation SNR defined as in Eq. (2.3) and σ 2 is the observation noise variance. For the case of stage 1, we choose Σ1|0 = P0 , hence the encoding rate of stage 1 is equal to  R1 =  1 log2 2  2πeσ 2 ∆2  (Γ + 1)  .  (2.17)  To simplify Rk for k = 2, 3, · · · , I − 1 we can write the predictor covariance Σk|k−1 using Kalman recursion [12] as Σk|k−1 = a2k−1 γk−1 + Γ 1 − a2k−1 , σ2  (2.18)  where γk−1 =  Σk−1|k−2 . Σk−1|k−2 + σ 2  (2.19)  Here Γ is the observation SNR defined as in Eq. (2.3) and σ 2 is the observation noise.  36  ,  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks By noting γk−1 ≤ 1 we can upper bound Rk for k = 2, 3, · · · , I − 1 in Eq. (2.15) as Rk ≤  1 log2 2  2πeσ 2 ∆2  (Γ + 1) − a2k−1 (Γ − 1)  for k = 2, 3, · · · , I − 1. (2.20)  The advantage of approximation in Eq. (2.20) over Eq. (2.15) is that to compute rate (Rk ), the k th stage requires only the position of (k − 1)th stage, which can be provided at the time of initial network setup. For the rest of our analysis in this chapter we assume that the sensing node in stage 1 encodes at rate R1 given by Eq. (2.17), stages k = 2, 3, · · · , I − 1 encode at a rate equal to the upper bound in Eq. (2.20) and the sensing node in stage I encodes at rate RI given by Eq. (2.16).  2.3.4  Formulation of the Minimum Energy Data Gathering (MEDG) in Correlated WSNs  In this subsection we formulate the problem of minimum energy data gathering (MEDG) in correlated WSNs. In a WSN since energy consumption needs to be minimized to enhance the lifetime of the network, the optimization problem to minimize energy consumption can be stated  37  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks as follows: ij ESISO (bij ) Lij (for SISO case)  minimize [bij Lij ]  (i,j)∈E ij EM IM O (bij , n) Lij (for MIMO case)  (OR) minimize [bij n Lij ]  (2.21)  (i,j)∈E  Subject to  j:(i,j)∈E  Lij −  2 ≤ bij ≤  j:(j,i)∈E  Lji = Ri for i = 1, 2, 3, · · · , I + 1  bij max  (2.22) (2.23)  1≤n≤N  (2.24)  ij ij where ESISO (.) and EM IM O (., .) denote the cost in terms of energy associated with the  link (i, j) for the SISO and MIMO cases which are given by Eq. (2.5) and Eq. (2.9) respectively, Lij denotes the flow (in terms of bits) associated with the link (i, j). The constraint in Eq. (2.22) is the flow conservation constraint which ensures at any stage the difference between outgoing flow and the incoming flow equals the data generated at the corresponding stage. In Eq. (2.23) bij max is determined so that the sensor node does not transmit beyond its peak power. The rate allocations Ri for i = 1, 2, · · · , I in Eq. (2.22) are given by Eqs. (2.17), (2.16) and (2.20) respectively. Since the sink node (corresponding to the stage with index (I + 1)) absorbs the data from the data gathering stages 1, 2, · · · , I, we can write I  RI+1 = −  Rk .  (2.25)  k=1  For a fixed modulation rate bij on a given link (i, j) and fixed number of cooperating nodes n the optimization problem in Eqs. (2.21) - (2.24) is a Minimum Cost Network Flow (MCNF) problem [5] and can be efficiently solved using the Matlog toolbox [55]. 38  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks Table 2.1: System parameter values for the 2.4 GHz ISM band fc = 2.45 GHz =⇒ λ(wavelength of radiation) = 0.12 m η = 0.35 N0 = −174 dBm/Hz B = 10 KHz Gt (transmit antenna gain) = Gr (receive antenna gain) = 0 dB PADC = 6.7 mW PDAC = 16.3 mW Pfiltx = Pfilrx = 2.5 mW PIFA = 3 mW PLNA = 20 mW Pmix = 30.3 mW PT PDAC + Pmix + Pfiltx 2 PR PLNA + Pmix + PIFA + Pfilrx + PADC G0 G(4π) 2 ≈ 40 dB t Gr λ α = 7.6 Ml = 40 dB By solving the optimization problem in Eqs. (2.21) - (2.24) we can determine the number of bits Lij to be transmitted on a link (i, j). To give a numerical example we consider a ten stage network (i.e., (I + 1) = 10), with each stage separated by a distance of thirty metres (i.e., d = 30 m). The first 9 stages of the network denote data gathering stages, while the 10th stage denotes the sink node. The system parameter details of 2.4 GHz radio in the ISM band are shown in Table 2.1 [38]. We choose the probability of error pb to be 10−3 . For the MIMO transmission case we assume the number of cooperating nodes in each stage as equal to two (i.e., ni = 2 for i = 1, 2, · · · , I), hence the spatial rate is equal to one (i.e., r = 1). We assume the cluster radius of stage k as equal to one metre i.e., dmax,k = 1 m. We assume the observation SNR is equal to ten dB (i.e., Γ = 10 dB). We assume the observation noise variance σ 2 = 0.01 and the quantization stepsize ∆ = 0.001. The plot of optimal energy consumption values versus the correlation coefficient is shown in Figure 2.2. As expected, the energy consumption value decreases with increase in value of correlation coefficient. It can be observed from the figure that when circuit energy is taken into account, then MIMO transmission may not be efficient. However, when transmission energy alone is taken into account then MIMO transmission is efficient. Also, when transmission energy alone is taken into account then the optimal energy consumption value obtained by solving MCNF problem coincides with the energy 39  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks  −3  3  x 10  Plot of total energy consumption in the network versus correlation coefficient  Energy Consumption in Joules  2.5  2  1.5  1  Energy consumption with MIMO transmission (circuit + transmission energy) Energy consumption with SISO transmission (circuit + transmission energy) Energy Consumption with MIMO transmission (transmission energy alone) Energy consumption with SISO transmission (transmission energy alone) Energy consumption SISO (circuit + transmission energy) along the shortest path Energy consumption with SISO transmission (transmission energy alone) along the shortest path  0.5  0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  Correlation coefficient  Figure 2.2: The figure shows the plot of total energy consumption in the network versus the correlation coefficient. We fix the observation SNR (Γ) equal to 10 dB and assume the desired probability of error (pb ) equal to 10−3 . Number of stages in the network are assumed equal to 10. Distance between the various stages of the network is equal to 30 metres.  consumption value obtained via shortest path routing. However, when circuit energy consumption is taken into account then the energy consumption value obtained via shortest path routing is higher. From this, we can conclude that when transmission energy alone is considered then shortest path routing is optimal. The plot of optimal energy consumption values versus the correlation coefficient for the case when probability of error pb = 10−6 is shown in Figure 2.3. For the case when number of cooperating nodes is equal to four, spatial rate (r) is equal to 3/4 as it is possible to construct an orthogonal STBC with this spatial rate as shown in [101], while for the case when the number of cooperating nodes is equal to eight, spatial rate (r) is equal to 5/8 as shown in [66]. From Figure 2.3 it can be observed that when the desired probability of error is small 40  2.3. Minimum Energy Data Gathering (MEDG) In Correlated Wireless Sensor Networks Plot of total energy consumption in the network versus correlation coefficient 0.4  Energy Consumption in Joules  0.35  0.3  0.25  0.2  0.15  0.1  Energy consumption with SISO transmission Energy consumption with MIMO transmission, 2 cooperating nodes Energy consumption with MIMO transmission, 4 cooperating nodes Energy consumption with MIMO transmission, 8 cooperating nodes  0.05  0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  Correlation coefficient  Figure 2.3: The figure shows the plot of total energy consumption in the network versus the correlation coefficient. We fix the observation SNR (Γ) equal to 10 dB and assume the desired probability of error (pb ) equal to 10−6 . Number of stages in the network are assumed equal to 10. Distance between the various stages of the network is equal to 30 metres.  MIMO transmission outperforms SISO transmission in terms of energy efficiency. For the given system parameter values, it has been found that a cluster with 4 cooperating nodes can be more energy efficient than compared to the case when the cluster has 2 and 8 cooperating nodes. For the case when number of cooperating nodes are equal to 4, the diversity advantage provided by additional cooperating nodes is more than the increase in energy consumption due to local communication and use of STBC with spatial rate less than one. While for the case of 8 cooperating nodes, the increase in energy consumption due to local communication and use of STBC with spatial rate less than one can be more than the diversity advantage provided by additional cooperating nodes. Hence, there exists an optimal number of cooperating nodes that can lead to high energy efficiency, which in the current case is equal to 4. 41  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes In this section we have derived the encoding rates adopted by sensor nodes for data gathering in correlated WSNs for the linear Gauss-Markov state-space model assumed in Eqs. (2.1) and (2.2) respectively. We also formulated the problem of MEDG in correlated WSNs and demonstrated the efficacy of exploiting correlation through a numerical example for both MIMO and SISO data gathering schemes. However as mentioned in Section 2.1 our aim in this chapter is to obtain structural results on physical layer design variables such as modulation rate and number of cooperating nodes. These structural results are proved in Section 2.4.  2.4  Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes  In this section, we present structural results on variation of optimal modulation rate and optimal number of cooperating nodes with respect to the correlation coefficient. To simplify analysis in this section we assume for both SISO and MIMO transmission, each stage adopts minimum distance routing i.e., every stage of the network transmits to the immediate next stage (ith stage to (i + 1)th stage, (i + 1)th stage to (i + 2)th stage and so on). To simplify notation, we denote the modulation rate adopted by ith stage to transmit to (i + 1)th stage as bi . Note that the distance between ith stage and (i + 1)th stage is denoted as di . Since each stage transmits the data generated locally along with the data received from previous stages, the amount of data transmitted by k th stage is given as k  Qk = i=1  Ri for k = 1, 2, · · · , I,  (2.26)  where R1 is given by Eq. (2.17), Ri for i = 2, 3, · · · , I − 1 is given by Eq. (2.20) and RI 42  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes is given by Eq. (2.16). As described in Section 2.2, in SISO transmission protocol there exists only one node per stage that senses, quantizes and transmits to the next stage. Thus, the total energy consumption in this case using the expression for energy consumption per bit in (2.5) is given by I  ǫSISO  ESISO = k=1  1 6  where we define ǫSISO  2bk − 1 bk  d2k +  PT + PR Bbk  Qk ,  (2.27)  (1 + α) N0 G0 Ml , bk is modulation rate adopted by stage k, dk  denotes the distance between k th and (k + 1)th stage and Qk is defined as in (2.26). For the MIMO transmission protocol, in a given stage k one node would sense, encode at a rate Rk and transmit to the remaining (nk − 1) nodes for cooperative transmission to the next stage of network. Using the expression for energy consumption per bit for the cooperative MIMO techniques in Eq. (2.9), we can write the expression for total energy consumption using MIMO techniques as I  EM IM O =      bk −1      ǫM IM O nk  2 1  d2k + nk (PT + PR ) + Elocal  Qk , (2.28) +1 Bbk r n k=1 bk k  where we define ǫM IM O  2 3  (1 + α) N0 G0 Ml ( 1r ), bk is modulation rate adopted for co-  operative transmission by stage k, nk is number of cooperating nodes in stage k and Elocal EC−local  ET x−local + (nk − 1)EC−local , where ET x−local is defined as under Eq. (2.9) and (PT +PR ) . Bblocal  43  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes  2.4.1  Expression for energy consumption for the case of simple homogeneous network  For the purpose of proving structural results in this section we assume a homogeneous network, where in each stage of the network is separated by a distance of d as in [36] i.e., dk = d. Further, we assume each stage of the network adopts the same modulation rate b for transmission i.e., bk = b. Also for MIMO transmission case, we assume the number of nodes participating in cooperative transmission in each stage to be same and equal to n i.e., nk = n. Since we assume the spacing between two successive stages of the network to be same, the correlation between signal samples sensed by successive stages of the network is equal i.e., ak−1 = a. Thus the encoding rate Rk adopted by stages k = 2, 3, · · · , I − 1 given in Eq. (2.20) can be expressed in terms of common correlation coefficient a as  R =  2πeσ 2 1 log2 (Γ + 1) − a2 (Γ − 1) 2 ∆2  .  (2.29)  With these assumptions Q1 = R1 , Qk = (k − 1)R + R1 for k = 2, 3, · · · , I − 1, while QI = (I − 2)R + R1 + RI , where R1 , R and RI are given by Eqs. (2.17), (2.29) and (2.16) respectively. Hence the total energy consumption for SISO transmission can be written as (1)  ET otal =  ǫSISO  2b − 1 b  d2 +  (PT + PR ) Bb  (I − 2) (I + 1) R + IR1 + RI (2.30) . 2  44  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes Similarly, the total energy consumption for MIMO transmission can be written as (2)  ET otal =  2b − 1  1  ǫM IM O n (K) n  b  1 +1 n  d2 +  n (PT + PR ) + Elocal Bb  (I − 2) (I + 1) R + IR1 + RI 2  2.4.2  × (2.31)  Structural result on optimal modulation rate b∗ versus correlation coefficient (a).  For the purpose of analysis in this subsection consider the following optimization problem: (1)  minimize ET otal (b, a) b  subject to 2 ≤ b ≤ bmax  (2.32)  (1)  where Etotal is given by Eq. (2.30) and bmax is determined by peak power of the node. We now state our first structural result as the following theorem. The theorem states that the optimal modulation rate b∗ is an increasing function of correlation coefficient a. Theorem 2.1. For a given number of stages I in the network, spacing d, observation SNR Γ the optimal modulation rate b∗ of the optimization problem in Eq. (2.32) is always an increasing function of the correlation coefficient a. i.e., b∗ (a)  (1)  arg min ET otal (b, a) | 2 ≤ b ≤ bmax b  is an increasing function of a. Proof. See Section 2.7.  Interpretation of Theorem 2.1: In general a = 0 corresponds to the case when 45  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes data is uncorrelated, while a > 0 corresponds to the case when data is correlated, hence from Theorem 2.1 it can be concluded that while transmitting correlated data the sensor nodes can employ higher modulation rates. (2)  Remark 2.1. Theorem 2.1 can be proved with ET otal as objective function in Eq. (2.32).  2.4.3  Structural result on optimal number of cooperating nodes n∗ versus correlation coefficient a  For the purpose of analysis in this subsection consider the following optimization problem (2)  minimize ET otal (n, a) n  subject to 1 ≤ n ≤ N  (2.33)  (2)  where ET otal is given by Eq. (2.31). We now state our second structural result as the following theorem. The theorem proves that for arbitrarily low probability of error pb , the optimal number of cooperating nodes (n∗ ) is a decreasing function of the correlation coefficient (a). Theorem 2.2. For a given number of stages I in the network, spacing d, modulation rate b, observation SNR Γ, the optimal number of cooperating nodes n∗ of the optimization problem in Eq. (2.33) is a decreasing function of the correlation coefficient a for arbitrarily low probability of error pb . i.e., as pb → 0+ n∗ (a)  (2)  arg min ET otal (n, a) | 1 ≤ n ≤ N n  is a decreasing function of correlation coefficient (a). Proof. See Section 2.7. 46  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes Interpretation of Theorem 2.2: The conclusion of Theorem 2.2 is interesting. The theorem suggests that for arbitrarily low probability of error (pb ), when the data is correlated, the number of nodes participating in cooperative transmission can be less than the case when data is uncorrelated. Remark 2.2. Note that the Theorem 2.1 and Theorem 2.2 hold even when the different stages of the network, employ different modulation rate and number of cooperating nodes. This can be proved straightforwardly by showing each term of the summation of Eq. (2.27) and Eq. (2.28) to be submodular and supermodular respectively.  Structural result on optimal rate b∗ versus number of  2.4.4  cooperating nodes n In this subsection we study as how the optimal rate b∗ varies with respect to the number of cooperating nodes n. For analysis in this subsection we ignore circuit power consumption in the transmitter and receiver of the sensor node. By ignoring the circuit energy consumption of the node the expression for total energy consumption using MIMO transmission in Eq. (2.31) can be re-written as 1  (3)  ET otal = ǫM IM O n (K) n  2b − 1 b  1 +1 n  d2 + ET x−local  (I − 2) (I + 1) R + IR1 + RI 2 (2.34)  where ǫM IM O  2 3  (1 + α) N0 G0 Ml  1 r  and K  4 pb  .  47  2.4. Structural Results on Optimal Modulation Rate and Number of Cooperating Nodes For the analysis in this subsection consider the following optimization problem: (3)  minimize ET otal (b, n) subject to 2 ≤ b ≤ bmax ,  (2.35)  (3)  where ET otal is given by Eq. (2.34). We now state our third structural result as the following theorem. In this theorem we study as how the optimal modulation rate b∗ varies with the number of cooperating nodes. Theorem 2.3. For a given number of stages (I) in the network, observation SNR (Γ), correlation coefficient (a) and spacing (d), the optimal modulation rate (b∗ ) at which each sensor transmits to minimize the transmission energy consumption is a decreasing function of the number of cooperating nodes (n). i.e., b∗ (n)  (3)  arg min ET otal (b, n) | 2 ≤ b ≤ bmax b  (2.36)  is a decreasing function of n. Proof. See Section 2.7. Remark 2.3. Since supermodularity doesn’t differentiate between the two variables, from Theorem 2.3 it can be inferred that the optimal number of sensors n∗ participating in cooperative transmission to the sink, is a decreasing function of the rate b.  Interpretation of Theorem 2.3 and Remark 2.3: Theorem 2.3 and Remark 2.3 are intuitively appealing. Theorem 2.3 implies that owing to the cooperation among sensor nodes, the modulation rate adopted by each sensor participating in cooperative 48  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs transmission to minimize energy consumption decreases as the number of participating nodes increase. While the Remark 2.3 implies that as the modulation rate adopted by each cooperating node increases, the number of cooperating nodes to minimize the energy consumption decreases. Remark 2.4. Theorem 2.3 is valid only when transmission energy alone is considered. In presence of circuit energy the objective function in (2.35) is not supermodular. However Theorem 2.3 is interesting in its own right since transmission energy dominates circuit energy when probability of error (pb ) is arbitrarily small.  2.5  Tradeoff Between Mutual Information and Energy in Correlated WSNs  In this section, we study the tradeoff between Mutual Information and energy consumption in correlated WSNs employing SISO and MIMO data gathering techniques. For the purpose of exposition assume stage 1 of the network is fixed. From our notation in Section 2.2 stage (I + 1) is the sink which receives data from data gathering stages 1, 2, · · · , I. Also, d1 , d2 , · · · , dI−1 denote the distance between data gathering stages 1 and 2, 2 and 3, · · · , I − 1 and I respectively. The following theorem computes the Mutual Information (MI) between the signal and observation process of data gathering stages 2, 3, · · · , I for the model assumed in Eq. (2.1) and Eq. (2.2). Theorem 2.4. Define X  (x2 , x3 , · · · , xI ) and Y  (y2 , y3 , · · · , yI ) where xi and yi  denote the instance of signal and observation process respectively, then the mutual information I (X, Y ) between the signal sequence (X) and the observation sequence (Y ) for 49  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs the linear Gauss-Markov model in Eq. (2.1) and Eq. (2.2) is given by  I (X; Y ) =  1 2  I  log2 1 + k=2  Σk|k−1 σ2  ,  (2.37)  Proof. See Section 2.7. Using the expression for predictor covariance in Eq. (2.18), we can further express RHS of Eq. (2.37) as  I(X; Y )  = high SNR  ≈  1 2 1 2  I  k=2 I  k=2  log2 1 + Γ − (Γ − γk−1) a2k−1 log2 1 + Γ − (Γ − 1) a2k−1  ,  ,  (2.38)  where Γ denotes the observation SNR, γk−1 is defined as in Eq. (2.19) and ak−1 denotes the correlation between (k − 1)th and k th stage respectively. With the Power Exponential model assumed to model the correlation i.e., ak−1 = e−Adk−1 , we can express the MI in Eq. (2.38) in terms of interstage spacing d1 , d2 , · · · , dI−1 and observation SNR Γ as 1 G(d1 , d2 , · · · , dI−1 ; Γ) = 2  I  k=2  log2 1 + Γ − (Γ − 1) e−2Adk−1  ,  (2.39)  where dk−1 is the distance between (k − 1)th and k th stages of the network. It is well known that traditional MIMO transmission achieves better MI and energy tradeoff than SISO transmission. However in a WSN employing MIMO data gathering scheme it is not clear if MIMO transmissions achieve better MI and energy tradeoff compared to the SISO transmission because of the additional overhead in terms of local 50  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs communication involved in MIMO transmission (even if MIMO transmissions are energy efficient it is of interest to know over what distance ranges they achieve better tradeoff compared to the SISO transmission). To study this tradeoff between MI and total energy consumption in the network for MIMO and SISO data gathering schemes, we formulate the optimization problem as follows:  I  Minimize  [d1 ,d2 ,··· ,dI−1 ]  ESISO (dk−1)  λ k=2  − G(d1 , d2 , · · · , dI−1 ; Γ)  I  (OR) Minimize  [d1 ,d2 ,··· ,dI−1 ]  λ  EM IM O (dk−1 ) k=2  − G(d1 , d2 , · · · , dI−1; Γ)  Subject to: 0 ≤ dk−1 ≤ D for k = 2, 3, · · · , I  (2.40)  where G(.) is given by Eq. (2.39), ESISO and EM IM O are energy consumption per bit values for SISO and MIMO data gathering schemes given by Eqs. (2.5) and (2.9) respectively, λ (> 0) is the scanning parameter (expressed in bits/joule) and D is the maximum distance range that the sensors can transmit and is typically determined by peak power of the sensor nodes. By varying the scanning parameter λ, we obtain the tradeoff curve between total energy consumption and MI. For a given λ, the optimization problem in Eq. (2.40) is convex  4  and hence can be solved efficiently [24].  To give a numerical example we consider a ten stage network i.e., I + 1 = 10, wherein stages 1, 2, · · · , 9 correspond to data gathering stages and 10th stage corresponds to the sink. We choose the system parameter values as in Table 2.1. We assume the modulation rate for SISO and MIMO data gathering scheme as equal to two i.e., b = 2. We set the 4  Since the energy consumption is quadratic in dk−1 and the negative of MI can be verified to be convex.  51  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs Table 2.2: MI-Energy Tradeoff for MIMO and SISO MI in Optimal MSE ESISO in bits Spacing Joules (d∗ ) in m 11.90 3.20 0.0756 6.7730 × 10−5 14.00 6.70 0.0874 1.2670 × 10−4 14.35 7.85 0.0876 1.5486 × 10−4 14.77 9.85 0.0883 2.1509 × 10−4 14.94 10.96 0.0889 2.5466 × 10−4 15.07 12.10 0.0893 2.9904 × 10−4 15.31 15.30 0.0901 4.4759 × 10−4 15.55 30 0.0909 16 × 10−4  data gathering schemes EM IM O in Joules 1.5239 10−4 1.5249 10−4 1.5254 10−4 1.5264 10−4 1.5271 10−4 1.5278 10−4 1.5304 10−4 1.5496 10−4  × × × × × × × ×  probability of error pb = 10−6. For the MIMO data gathering scheme we assume the number of cooperating nodes as equal to two i.e., ni = 2. Our optimization variables in this case are d1 , d2 , · · · , d8 , which correspond to the distance between successive data gathering stages of the network. We set the maximum distance that the nodes can transmit D to be 30 metres. Finally, we set observation SNR Γ = 10 dB and the field diffusion coefficient A = 0.1. The optimal energy-MI tradeoff values and the optimal distance values for MIMO and SISO data gathering schemes are shown in Table 2.2. From the values in the table it can be concluded that for small distance ranges i.e., upto 6.5 metres, SISO data gathering scheme requires less energy compared to the MIMO data gathering scheme to achieve the same amount of MI. But for large distances (roughly greater than 7.5 metres) MIMO data gathering scheme achieves better MI and energy tradeoff.  52  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs Table 2.2 also shows the value of MSE achieved at the destination for the corresponding optimal distance values. The MSE values are obtained by using the well-known Kalman filter recursion. From Table 2.2 it can be observed that in general as the distance increases, the MSE increases. This is to be expected since an increase in distance reduces the correlation in the data sensed by nodes, which leads to performance degradation. Further it can be observed that for small distances, SISO data gathering consumes less energy than the MIMO data gathering scheme. However for large distances, MIMO data gathering scheme achieves the same MSE at a lower energy compared to the SISO data gathering scheme. Thus from this simple numerical example we can conclude that for small distances SISO based data gathering achieves better energy-MI (or MSE) tradeoff than the MIMO based scheme. However for moderate and large distances MIMO based scheme achieves better energy-MI (or MSE) tradeoff.  2.5.1  Effect of observation SNR Γ on optimal sensor spacing d  In this subsection we study the impact of observation SNR Γ on sensor placement so that the mutual information is maximized. Further, we also study the effect of observation SNR on sensor spacing so that the error achieved at the destination is minimized. To put the problem into perspective consider the following optimization problem, wherein we consider optimal placement of sensors to maximize mutual information:  Maximize G([d1 , d2 , · · · , dI−1 ]; Γ)  [d1 d2 ··· ,dI−1 ]  s.t.: 0 ≤ dk−1 ≤ D for k = 2, 3, · · · , I.  (2.41)  where G(.) is the MI from source to destination given by Eq. (2.39).  53  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs We now state the structural result on the variation of optimal spacing dk−1 w.r.t the observation SNR Γ as the following theorem. Theorem 2.5. The optimal spacing [d1 , d2 , · · · , dI−1]∗ between sensors to maximize the MI from source to destination is in general an increasing function of observation SNR Γ. i.e., [d1 , d2 , · · · , dI−1]∗  arg  max  [d1 ,d2 ,··· ,dI−1 ]  [G ([d1 , d2 , · · · , dI−1]; Γ)]  s.t. 0 ≤ dk−1 ≤ D for k = 2, 3, · · · , I  (2.42)  is an increasing function of Γ. Proof. See Section 2.7.  Interpretation of Theorem 2.5: At high observation SNR since the signal component is strong, correlation in data does not contribute additional information. Therefore in this case the sensor nodes can be separated by large distance. However at low observation SNR since the signal is weak, the correlation in data contributes additional information. Therefore in this case the sensor nodes need to be placed closely so that the correlation in data can be exploited.  2.5.2  Discussion  The increase in optimal spacing with observation SNR implies correlation does not have much impact on performance at high observation SNR, since correlation decreases with increase in distance. While at low observation SNR correlation can affect the performance significantly, since at low observation SNR sensors need to be more closer than at high observation SNR as proved in Theorem 2.5. 54  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs −3  1.03  x 10  ObservationSNR = 30 dB  Mean Square Error  1.02 1.01 1 0.99 0.98 0.97 1  Node spacing (d) = 1 m. Node spacing (d) = 5 m. Node spacing (d) = 20 m. 2  3  4  5 6 Stage index  7  8  9  10  Figure 2.4: The figure shows the plot of mean square error versus the stage index. We fix the observation SNR (Γ) equal to 30 dB and assume the node spacing to be 1 m, 5 m and 20 m respectively.  To verify this claim we present simulation case study with Mean Square Error (MSE) as a performance measure at high and low observation SNR. For the high SNR case we assume observation SNR (Γ) = 30 dB, while for the low SNR case we set the observation SNR (Γ) = −10 dB. For both cases we set the distance between nodes as equal to 1, 5 and 20 metres respectively. We set the value field diffusion coefficient (A) to 0.1. For the network model we assume the same ten stage network. Figure 2.4 shows the plot of MSE versus stage index of the network for high observation SNR. Here we assume Γ = 30 dB. The figure shows that the MSE achieved when node spacing is equal to 1, 5 and 20 metres is almost same. This is because at high observation SNR, the signal component in the observation is strong and has greater influence on MSE than the noise averaging effect of correlation. Consequently, the performance is independent of correlation and hence spacing between nodes can be large when observation SNR is high (since correlation does not convey any new information), as proved in  55  2.5. Tradeoff Between Mutual Information and Energy in Correlated WSNs  0.95  Mean Square Error  0.9  Observation SNR =−10 dB  0.85  0.8  0.75  0.7 1  Node spacing (d) = 1 m. Node spacing (d) = 5 m. Node spacing (d) = 20 m. 2  3  4  Decreasing Node Spacing (d) 5 6 7 Stage index  8  9  10  Figure 2.5: The figure shows the plot of mean square error versus the stage index. We fix the observation SNR (Γ) equal to −10 dB and assume the node spacing to be 1 m, 5 m and 20 m respectively.  Theorem 2.5. Figure 2.5 shows the plot of MSE versus stage index of the network for low observation SNR. Here we assume Γ = −10 dB. The figure shows that node spacing corresponding to 1 metre leads to lower value of MSE compared to the case when node spacing is equal to 5 and 10 metres respectively. This is because at low SNR, the signal component in the observation is weak and hence the MSE achieved at the destination depends significantly on the noise averaging effect of the correlation. Thus at low observation SNR performance is dependent on correlation, and hence spacing between nodes has to be small at low observation SNR to exploit the correlation as proved in Theorem 2.5. This analysis provides valuable insight into the placement of sensors for a given application. For example if the observation SNR is high, then the sensors can afford greater spacing between them without loss in performance. On the other hand when the observation SNR is low, the sensors need to be placed closely to exploit the correlation.  56  2.6. Summary  2.6  Summary  We considered minimum energy gathering of correlated data in sensor networks. Under Gauss-Markov assumption on the signal process we formulated minimum energy correlated data gathering in sensor networks. Using the concepts in monotone comparative statics, we proved the optimal modulation rate adopted for transmission to be an increasing function of correlation coefficient. For cooperative transmission case, we proved that the optimal number of cooperating nodes to be a decreasing function of correlation coefficient. Further, the optimal modulation rate adopted for cooperative transmission decreases with an increase in number of cooperating nodes. We also proved, the optimal number of cooperating nodes decrease with an increase in modulation rate adopted for cooperative transmission. From our numerical results, we observed that SISO data gathering scheme achieves better Energy-MI (or MSE) tradeoff for short distances than MIMO data gathering scheme. However for long distances MIMO achieves better EnergyMI (or MSE) tradeoff compared to the SISO data gathering scheme. Finally, we proved the spacing between nodes in a correlated WSN to maximize MI to be an increasing function of observation SNR. Through our simulation and analysis, we observed that the performance at high observation SNR is independent of correlation; however at low observation SNR the correlation has a significant impact on performance.  2.7 2.7.1  Analytical Proofs Derivation of h (yk |yk+1, yk+2, · · · , yI ) and h (yI ):  Using the definition of conditional differential entropy we can write  h (yk |yk+1, yk+2, · · · , yI ) = h (yk , yk+1, · · · , yI ) − h (yk+1, yk+2, · · · , yI ) . (2.43) 57  2.7. Analytical Proofs For the linear Gauss-Markov state-space model assumed in Eq. (2.1) and Eq. (2.2), the joint probability density function (pdf) of observations (yk , yk+1 , · · · , yI ) is given as [95]  f (yk , yk+1, · · · , yI ) ∼ N µk|k−1 , diag(Sk ) , where µk|k−1  xˆk|k−1, xˆk+1|k · · · , xˆI|I−1  notes a diagonal matrix and Sk  T  , xˆi|i−1  (2.44)  E {xi |y1, y2 , · · · , yi−1 }, “diag” de-  Σk|k−1 + σ 2 denotes the innovation variance.  In general, the differential entropy of a Gaussian random vector z ∼ N (µ, Υn×n ) is given by [31] 1 log2 (2πe)n det Υ , 2  h (z) =  (2.45)  where “det” denotes determinant of the matrix. Using Eqs. (2.44) and (2.45), we can write Eq. (2.43) as 1 log2 (2πeSk ) 2 Σk|k−1 1 = log2 2πeσ 2 1 + 2 σ2  h (yk |yk+1, · · · , yI ) =  (2.46)  hence Eq. (2.15) follows. Using the definition of differential entropy we also obtain h (yI ) = 12 log2 (2πeσ 2 (Γ + 1)).  2.7.2  Proof of Theorems 2.1 and 2.2  It is straightforward to show using Definition 1.1 of Chapter 1 that constraints in optimization problem Eq. (2.32) and Eq. (2.33) form a lattice. Thus the condition of constraints having to form a lattice to invoke Theorem 1.1 and Remark 1.2 in Chapter 1  58  2.7. Analytical Proofs are satisfied, therefore, the desired result of Theorem 2.1 (Theorem 2.2) follows by proving the objective function of optimization problem in Eq. (2.32) (Eq. (2.33) for Theorem 2.2 respectively) as submodular (supermodular for Theorem 2.2) and invoking Remark 1.2 (Theorem 1.1) in Chapter 1. (1)  Using the expression for R in Eq. (2.29), we can re-write the expression for ET otal in Eq. (2.30) as (1)  ET otal =  (PT + PR ) 2b − 1 d2 + × b Bb 2πeσ 2 (I − 2) (I + 1) log2 (Γ + 1) − a2 (Γ − 1) 4 ∆2 ǫSISO  + IR1 + RI  , (2.47)  where R1 and RI are defined as in Eqs. (2.17) and (2.16) respectively. From Remark 1.1 in Section 1.4.1 of Chapter 1 a function is submodular (supermodular) if its mixed derivative is less (greater) than or equal to zero. (1)  Thus, differentiating ET otal in Eq. (2.47) w.r.t b and a successively we obtain (1) 1 ǫSISO d2 2b (b ln 2 − 1) + 1 − (PT + PR )/B ∂ 2 ET otal × = ∂b∂a ln 2 b2       (I − 2) (I + 1)  ×   4       −2a   .  Γ+1 2 −a   Γ−1  (2.48)  >1  The term (b ln 2 − 1) is positive for any b ≥ 2, also PT + PR is usually small and B is of the order of KHz, hence the term ǫSISO d2 2b (b ln 2 − 1) + 1 − (PT + PR )/B is greater than zero. Thus, the mixed derivative in Eq. (2.48) is less than or equal to zero, therefore (1)  the function ET otal is submodular in b and a. Now invoking Remark 1.2 in Section 1.4.1 59  2.7. Analytical Proofs of Chapter 1, Theorem 2.1 follows. (2)  Using the expression for R in Eq. (2.29), we can re-write the expression for ET otal in Eq. (2.31) as (2)  ET otal =  1  ǫM IM O n (K) n  2b − 1 b  d2 +  1 +1 n  n (PT + PR ) + Elocal Bb  2πeσ 2 ∆2  (I − 2) (I + 1) log2 4  ×  (Γ + 1) − a2 (Γ − 1)  + IR1 + RI  , (2.49)  where R1 and RI are defined as in Eqs. (2.17) and (2.16) respectively. (2)  Upon differentiating ET otal in Eq. (2.31) w.r.t n and a, we obtain (2)  ∂ 2 ET otal = ∂n∂a  1 (I − 2) (I + 1) ln 2 4  2a × Γ+1 2 −a Γ−1 >1    As pb → 0+ , K  4 pb  term1  ǫM IM O d2  2b −1 b  ln  K b  −n  n term 2  −    (PT + PR ) − EC−local  . (2.50) Bb  → ∞, hence term 2 → ∞, but term 1 is finite and positive, (2)  hence the mixed derivative is greater than or equal to zero. Thus the function ET otal is supermodular in n and a. Now invoking Theorem 1.1 in Section 1.4.1 in Chapter 1, Theorem 2.2 follows.  60  2.7. Analytical Proofs  2.7.3  Proof of Theorem 2.3 (3)  To prove Theorem 2.3 we need to prove ET otal as supermodular in b and n. For conve(3)  nience we re-write ET otal in Eq. (2.34) as follows: 1  (3)  ET otal = ǫM IM O n(K) n ET x−local  2b − 1 1 +1 n  d2  (I − 2)(I + 1)R + IR1 + RI + 2  b (I − 2)(I + 1)R + IR1 + RI 2  (2.51)  Since the optimal b i.e., b∗ obtained does not depend on constant, we can ignore the second term in Eq. (2.51). (3)  Thus, proving ET otal as supermodular reduces to proving the first term in Eq. (2.51) as supermodular. The general technique of taking the mixed derivative w.r.t b and n and checking its sign for supermodularity as stated in Remark 1.1 of Section 1.4.1 in Chapter 1 may not be possible because of the unruly nature of objective function. Fortunately as observed in Remark 1.3 of Section 1.4.1 in Chapter 1, the result of Theorem 1.1 is invariant under monotonic transformation of the objective function. Thus by proving logarithm of first term in Eq. (2.51) as supermodular we can still invoke Theorem 1.1 to prove Theorem 2.3. Proceeding as stated above we obtain (3)  1 ∂ 2 ln ET otal = , ∂b∂n Kn2 b  (2.52)  which is positive for any b > 0. The result of Theorem 2.3 follows now by invoking Theorem 1.1 of Section 1.4.1 in Chapter 1.  61  2.7. Analytical Proofs  2.7.4 Define X  Proof of Theorem 2.4 (x2 , x3 , · · · , xI ) and Y  (y2 , y3 , · · · , yI ), then the Mutual information  I(X; Y ) denotes the reduction in the uncertainty of X when Y is observed and viceversa [31]. i.e.,  I(X; Y ) = H(X) − H(X|Y ) = H(Y ) − H(Y |X) .  (2.53)  From the definition in (2.53), we have  I(X; Y ) = Ep(Y |X) (ln p(Y |X)) − Ep(Y ) (ln p(Y )) .  (2.54)  The conditional density of Y given X, can be written as I  p(Y |X) =  k=2  p(yk |xk )  ,  (2.55)  distributed as N (xk ,σ2 )  σ 2 denotes the variance of observation noise. Eq. (2.55) follows from the fact that given the state xk , the conditional densities p(.|.) on the RHS are independent of each other. Taking logarithm of (2.55), we obtain I  ln p(Y |X) =  I 2 − 21  ln(2πσ ) k=2  −  k=2  (yk − xk )2 . 2σ 2  (2.56)  62  2.7. Analytical Proofs The density p(Y ) is given as [95] I  p(Y )  1 √ 2πSk k=2  =  n  ln p(Y )  =⇒  −  k=1  −  e  (yk −ˆxk|k−1 )  Pn  k=1  1 ln (2πSk ) − 2  n  k=1  E (xk |y2, · · · , yk−1) and Sk  where we define xˆk|k−1  2  2Sk  yk − xˆk|k−1 2Sk  2  ,  yk − xˆk|k−1  E  (2.57)  2  |y2 , · · · , yk−1  is called the innovations variance. Upon substituting (2.56) and (2.57) into the definition of I(X; Y ) in (2.54) we obtain I  I(X; Y ) = k=2  I  (yk − xk )2 2σ 2  −1 ln 2πσ 2 − E 2 k=2  I  + k=2  1 ln (2πSk ) 2  =(I−1)/2 I  +  E k=2  yk − xˆk|k−1 2Sk  2  .  (2.58)  =(I−1)/2  Thus, (2.58) can be simplified and written as 1 I(X; Y ) = 2  n  ln k=1  Sk σ2  .  (2.59)  Using Kalman recursion equation, the innovations variance Sk can be expressed as Sk = Σk|k−1 + σ 2 ,  (2.60)  where Σk|k−1 denotes the predictor covariance [12]. Using (2.60) in (2.58) leads to (2.37), which concludes the proof.  63  2.7. Analytical Proofs  2.7.5  Proof of Theorem 2.5  The constraints of optimization problem in Eq. (2.41) form a lattice, hence Theorem 2.5 follows by verifying the objective function given by Eq. (2.39) as supermodular. Using the definition of supermodular function in Remark 1.1 of Section 1.4.1 in Chapter 1, we can verify the quantity inside the summation of MI expression in Eq. (2.39) to be supermodular. i.e., ∂ 2 log2 1 + Γ − (Γ − 1)e−2Adk−1 ∂Γ∂dk−1  1 4Adk−1e−2Adk−1 ln 2 (1 + Γ − (Γ − 1)e−2Adk−1 )2 > 0.  =  , (2.61)  Now since each term of MI expression in Eq. (2.39) is symmetric and sum of supermodular functions is supermodular, the objective in Eq. (2.41) is supermodular. Now invoking the Theorem 1.1 due to Topkis in Section 1.4.1 of Chapter 1, the result of Theorem 2.5 follows.  64  Chapter 3 A Price based Approach to Decentralized Rate Selection in IEEE 802.11 WLANs5 3.1  Introduction  The IEEE 802.11 [1] is the first international standard for wireless local-area networks (WLANs) and it has been used widely in most commercial WLAN products available in the market. The IEEE 802.11 specifies two different medium access control (MAC) mechanisms in WLANs: the contention-based Distributed Coordination Function (DCF) and the polling-based Point Coordination Function (PCF). At present, only the mandatory DCF is implemented in the 802.11-compliant products. A detailed performance analysis of DCF can be found in [23]. The DCF however has an adverse effect on the overall throughput of the network due to the presence of few low rate data stations as demonstrated in [53]. The reason for this anomaly can be explained as follows: if a user transmits at a lower rate then it tends to occupy the channel for a long time, thereby depriving other users from their fair share to transmit on the channel. This leads to a reduction in the overall network throughput as noted in [99] and [100]. In the follow5  A preliminary version of the work in this chapter will appear as a conference paper [84]. A more detailed version has been submitted as a journal article [88].  65  3.1. Introduction ing, we will sample a few important works that have focused on overcoming the above mentioned anomaly of DCF. Towards the end of this section, we will outline important results of this chapter, and outline what is to follow. The authors in [99] call for time-based fairness notion to improve the performance in multi-rate WLANs, instead of the commonly used throughput-based fairness notion. Further, the paper also presents an algorithm called Time-based Regulator, which ensures time-based fairness by regulating the packets sent by various users in the network. In [54], the authors present a new method called Idle Sense where each host adjusts its contention window size dynamically. Idle Sense overcomes the performance anomaly of IEEE 802.11 by proper scaling of contention window size of various users. In [67], the authors propose a new MAC protocol called CoopMAC. In CoopMAC the high rate users help the low rate users, thereby helping to mitigate the adverse effect low rate users have on the overall network throughput. The cooperation between high rate and low rate users is possible because the transmission in a wireless channel is typically overheard by the surrounding neighboring stations. Different from the works mentioned above, in this chapter, we consider a price-based approach to ensure fair channel access among various users of the network, thus improving the overall network throughput. Pricing is a common technique used extensively in economics to achieve a socially desirable result. Pricing based techniques have been previously used for power control in code division multiple access (CDMA) based wireless networks to improve the overall network performance [93]. At this point it is worthwhile to make an important distinction between networks based on CDMA and carrier sense multiple access (CSMA) mechanisms. The WLANs based on IEEE 802.11 standard fall under the latter category. In CDMA based networks a given user’s throughput is directly affected by the transmission power of other users. However in CSMA based networks the  66  3.1. Introduction other user rates have an indirect effect on the throughput of a given user as noted in [53] and [100]. This can lead to a reduction in the overall network throughput. Owing to the decentralized rate selection of each user in a WLAN, we use noncooperative game theory [46] and [80] for our analysis in this chapter. The objective of each user (who are indeed players of the game) is to maximize a utility function, which is equal to the difference between the throughput a user would obtain for transmitting at a specific rate, and a cost that is proportional to the fraction of time a user occupies the channel. The cost is incurred due to the price imposed by access point (AP) on each user. The actions of the users in the game are their respective rates. The proposed price based approach does not require any changes to the existing IEEE 802.11 standard. It only requires periodic broadcast of the price by the AP. The key results of this chapter are summarized below.  3.1.1  Main Results:  1. We initially formulate a non-cooperative game where the actions of players are choosing rates on s given channel to maximize their utilities (cf. Section 3.3). The resulting game is a supermodular game because of which the game always has at least one pure strategy Nash equilibrium that is contained in a set bounded by the smallest and largest Nash equilibria [77]. 2. Also, owing to the property of non-decreasing differences of player utility with respect to its action variable and the price imposed by access point, we prove the smallest and largest Nash equilibria to be non-decreasing functions of the price. Furthermore, since the player utilities exhibit the property of Negative Externalities [41], therefore we prove the smallest Nash equilibrium to be Pareto-dominant. 3. We propose a totally decentralized algorithm to compute the best response of each 67  3.2. System Model, Assumptions, and Notation user, that converges to the smallest Nash equilibrium almost surely (a.s.). 4. We next consider extension of our price based approach to multi-channel WLANs and formulate a game where each user tries to maximize its aggregate utility and the actions of the users would be to choose a rate vector for transmission on a given channel. The resulting game can again proved to be supermodular for the special case of two channels. 5. Finally through our numerical simulation results, we observe that by tuning the price judiciously the access point can reduce the occupancy of low rate users. This improves the occupancy of moderate and high rate users thereby leading to an increase in overall network throughput. The rest of this chapter is organized as follows: Section 3.2 contains system model description, assumptions, and notation. Section 3.3 contains the non-cooperative game theory formulation of the problem considered in this chapter. Section 3.4 contains analysis of the game formulated in Section 3.3. Section 3.5 contains extension of the proposed price based approach to multi-channel WLANs. Section 3.6 contains simulation results and discussion. Section 3.7 contains conclusions.  3.2 3.2.1  System Model, Assumptions, and Notation System Model Description of a WLAN  We consider a snapshot of wireless local area network (WLAN) shown in Fig. 3.1 with a set N = {1, 2, · · · , N} of distinct sender and receiver pairs which we refer to as “users” in a given WLAN . We use the terms “user” and “player” interchangeably in the ensuing description and analysis. Before the transmission of a given data frame the transmitter of 68  3.2. System Model, Assumptions, and Notation a given user broadcasts a request to send (RTS) frame. The intended receiver for its part would estimate the channel SNR and convey the modulation rate to the sender via the clear to send (CTS) frame by solving an optimization problem proposed in Section 3.3. The access point (AP) in Fig. 3.1 broadcasts a price apart from its usual functionality of connecting the users in a given network to other networks. Note that for the transmission of a frame the sender adopts the distributed coordination function (DCF) mechanism. We assume that each bit in a frame experiences fading which is independent and identically distributed (i.i.d). We also assume the background noise to be additive white Gaussian. We assume the sender uses Quadrature Amplitude Modulation (QAM) for baseband modulation. The conditional probability of bit error with uncoded QAM (with a square constellation approximation) can be written as [89]  pb (b|x) = Q =  √  2bx  1 4 1− b b 22  for (b = 1) Q  3bx −1  2b  for (b ≥ 2),  (3.1)  where b denotes the modulation rate i.e., number of bits transmitted per symbol, henceforth we refer to it as rate for convenience, Q(y)  √1 2π  ∞ − z2 e 2 dz, y  and x denotes the per  bit received SNR. The error probability expression in Eq. (3.1) will be useful in the next section where we formulate the utility expression for each user.  3.2.2  Notation  Throughout this chapter, we use Rn to denote an n-dimensional Euclidean space, and use x (i.e., lower case bold faced letters) to denote a vector that belongs to the set X ⊆ Rn . We use the standard game-theoretic notation ui (bi , b−i ) to indicate the dependence of the utility function on the ith player’s action bi , and on other player’s actions denoted as  69  3.2. System Model, Assumptions, and Notation  AP  λ  λ  λ S1 R4  S2  S3 R3  R2  S4  R1 Figure 3.1: The figure shows a snapshot of an ad hoc network with four sender and receiver pairs namely (S1 , R1 ), (S2 , R2 ), (S3 , R3 ) and (S4 , R4 ) respectively. The access point (AP) in the figure broadcasts a price λ to all the users periodically. The price is proportional to the occupancy of each user.  70  3.3. Formulation of Non-Cooperative Game Theoretic Rate Selection with Pricing for WLANs b−i i.e., b−i  [b1 , b2 , · · · , bi−1 , bi+1 , · · · , bN ]. Occasionally we will use the notation ui(b)  to denote the outcome of the game in terms of the selected actions of all players i.e., b  [b1 , b2 , · · · , bi , · · · , bN ]. We use “×” to denote the Cartesian product of sets, and Ex to  denote the mathematical expectation with respect to the random variable x. We assume the partial ordering in this chapter to be “≥”. Also if we define x y  [y1 , y2, · · · , yN ], then x  3.3  [x1 , x2 , · · · , xN ] and  y implies xi ≥ yi for all xi , yi respectively.  Formulation of Non-Cooperative Game Theoretic Rate Selection with Pricing for WLANs  In this section we motivate the choice of utility function for the rate selection considered in this chapter. We also formulate a non-cooperative game for rate selection in WLANs.  3.3.1  Utility function  Let Mi , Wi , and bi and δi denote the MAC layer frame size, symbol rate (expressed in symbols per second (sps)), rate and bit overhead of the user i respectively. For our analysis we assume Wi to be fixed. A user i by varying its rate bi can vary its transmission rate Ri = Wi bi . Also denote f (bi ) as the average throughput user i obtains for transmitting a frame of size Mi at a rate of Wi bi bits per second (bps). With this notation in place the throughput f (.) for best effort traffic can be written as  f (bi ) =  1−  δi Mi  Wi bi Eγ (1 − pb (bi |γ))Mi ,  (3.2)  71  3.3. Formulation of Non-Cooperative Game Theoretic Rate Selection with Pricing for WLANs where Eγ denotes the mathematical expectation with respect to the random variable γ where γ represents the per bit received SNR in a given MAC frame of size Mi . The quantity pb (.) is the probability of bit error expression defined in Eq. (3.1) with b equal to bi and x equal to γ. Note that in Eq. (3.2) we assumed each bit of the frame is corrupted independently and the fading distribution is identical. The throughput expression in Eq. (3.2) is same as the expression used in [93] for power control. However in [93], the bits in a frame are assumed to be corrupted only by the background noise. Now consider the following utility function for each user i for i = 1, 2, · · · , N. In the equation below λ (expressed in bps) represents the price imposed by an access point (AP) on the users.  ui (bi , b−i ) = f (bi ) − λ Reward  Mi Wi bi Mi Wi bi  +  Mj j=i Wj bj  .  (3.3)  cost  Note that the cost paid by user i increases monotonically with an increase in its occupancy equal to  Mi Wi bi  P M Mi + j=i W jb Wi bi j j  . The cost in Eq. (3.3) could actually deter the users from  deploying a small rate on a given channel. Note that by employing a smaller rate a user occupies the channel for a long time, thereby preventing other users from accessing the channel. This is known to be the primary cause for a lower overall system throughput in WLANs as remarked in [99] and [100]. An appropriate choice of price λ can lead to better occupancy levels for all the users, and therefore an increase in overall system throughput. Note that we impose the same price on all the users, thus it is enough for the AP to broadcast a single price to all the users. Otherwise the AP has to transmit different price values to each user. Further to compute these price values, the AP would require the channel SNRs of each user leading to an increase in communication between the AP and the users. The adaptation of λ  72  3.3. Formulation of Non-Cooperative Game Theoretic Rate Selection with Pricing for WLANs is governed by the system designer’s policy. We will elaborate on this aspect in Section 3.4.2. We remark that due to the lack of dependence of the throughput function f (.) on the actions of other users, the analysis in this chapter hold for any reasonable choice of the throughput function f (.).  3.3.2  Rules of the NRSGP  In this subsection, we formulate the Non-Cooperative Rate Selection Game with Pricing (NRSGP). Let GNRSGP = [N , {Bi}, {ui}] denote the NRSGP where N = {1, 2, · · · , N} denote the set of users in a WLAN, Bi is the strategy set of user i, and ui defined in Eq. (3.3) is their utility function. In an NRSGP each user i selects a rate bi on a channel such that bi ∈ Bi , where Bi is the user strategy set defined as in Eq. (3.4) below. The strategy set of user i namely Bi in an NRSGP is a compact subset of R i.e., Bi  {bi,min , · · · · · · , bi,max } ,  (3.4)  where bi,min denotes the minimum transmission rate that a user i can use to transmit on a given channel. Unlike a transmission control problem where the actions of the users are either to transmit at a fixed rate or not to transmit, here we consider a rate selection problem where the aim of each user is to select an appropriate rate for transmission so that the utility ui (.) in Eq. (3.3) is maximized. Therefore we assume that bi,min > 0. The term bi,max in Eq. (3.4) denotes the maximum rate that a user can transmit on a given channel. It is typically determined by factors such as the peak power of a user battery source. In an NRSGP every user maximizes its utility in a distributed fashion. Formally, the 73  3.4. Analysis of NRSGP with Supermodular Game Theory distributed optimization problem can be stated as follows:  (NRSGP) max ui (bi , b−i ) , bi  s.t. bi ∈ Bi for all i ∈ N , (3.5)  where ui (., .) is given by Eq. (3.3), Bi is defined as in Eq. (3.4) and N denotes the set of players.  3.4  Analysis of NRSGP with Supermodular Game Theory  In this section, we prove NRSGP to be a supermodular game (cf. [77] for definition), analyze the equilibrium properties of the game, provide an algorithm that would attain the Nash equilibrium of the game, and finally present a method to tune the price λ of the users.  3.4.1  Analysis of Non-Cooperative Rate Selection with Pricing (NRSGP) [N , {Bi }, {ui}] is a supermodular game if bi,min in  Theorem 3.1. The game GNRSGP Eq. (3.4) is chosen so that bi,min ≥  P  Mi Wi  Mj j=i Wj bj  .  Proof. See Section 3.8. Remarks on Theorem 3.1:  74  3.4. Analysis of NRSGP with Supermodular Game Theory 1. The user i to compute its best response requires the quantity  Mj j=i Wj bj .  This  can be obtained by user i from the RTS frame broadcasted by the user k during its most recent data transmission, by subtracting out the time required to receive CTS frame, one acknowledgement frame, and 3 times the value of short inter frame space (SIFS) interval (cf. [1] for the RTS frame format used in IEEE 802.11 standard). This avoids the need for separate transmission of rate adopted by each user to all other users in the network. Updating the best response of a given user based on the previous actions of other users is referred to as myopic best response (MBR) update. In a supermodular game the MBR updates converge to the NE [77]. Therefore a given user need not know the current actions of other users to compute bi,min in Theorem 3.1. 2. Note that proving GNRSGP as supermodular is only one of the ways to proving the existence of pure strategy NE (cf. Theorem 3.2 below). Consequently, the condition imposed in Theorem 3.1 to prove NRSGP as supermodular ensures that a pure strategy NE always exists. However it does not imply that if the condition in Theorem 3.1 is not met then there is no equilibrium. An upshot of a game being supermodular is that the best response correspondence of every player is non-decreasing in other players’ actions. Consequently, the overall best response correspondence is also non-decreasing. The choice of any point in this overall best response correspondence has a fixed point, which in fact is the pure strategy NE (this can be established using Tarski’s fixed point theorem [102]). This property of supermodular games is summarized in the following theorem for NRSGP. Theorem 3.2. The game GNRSGP  [N , {Bi }, {ui}] has at least one pure strategy Nash  equilibrium. Furthermore, the set of Nash equilibria are contained within the set NENRSGP λ  75  3.4. Analysis of NRSGP with Supermodular Game Theory defined as follows:  NENRSGP = {b(λ) ∈ B : bS (λ) ≤ b(λ) ≤ bL (λ)} , λ where B  (3.6)  ×N i=1 Bi (with Bi defined as in Eq. (3.4)), bs (λ), and bL (λ) denote the smallest  and largest Nash equilibria. Furthermore bS (λ) and bL (λ) are non-decreasing functions of λ, the price. Proof. See Section 3.8. Interpretation of Theorem 3.3: The second part of Theorem 3.2 implies that with an increase in price λ the rates of various users increase, since NE in our case corresponds to the rates of various users. The increase in NE with respect to price λ has another practical significance in the computation of user best response. To see this define b∗i (λ1 ) as the rate of user i at NE, when the price λ = λ1 . Now for any λ2 > λ1 it is enough for a given user i to search over Bi such that Bi  {b∗i (λ1 ), · · · , bi,max }, owing to the fact that  NE increases with an increase in price λ. This could lead to a reduction in computational complexity especially if the strategy space of users is large. The following theorem proves that the smallest Nash equilibrium in NEλ namely bS (λ) is a Pareto dominant equilibrium in the set NENRSGP . λ Theorem 3.3. The smallest NE namely bS (λ) ∈ NENRSGP is the Pareto dominant λ ) of the game equilibrium (i.e., uj (bS (λ)) ≥ uj (b(λ)) for all j and b(λ) ∈ NENRSGP λ GNRSGP . Proof. See Section 3.8. Remark 3.1. The Theorem 3.3 is not true for any general supermodular game. However in our case, the Theorem 3.3 is true since the utilities of all the players satisfy a property 76  3.4. Analysis of NRSGP with Supermodular Game Theory referred to in the Economics literature as Negative Externalities (i.e., the utilities of all the players are decreasing in other players actions) [41]. If the utilities of all the players satisfy Positive Externalities ( i.e., the utilities of all the players are increasing in other players actions ) then the largest NE is Pareto dominant [41]. Interpretation of Theorem 3.3: From Theorem 3.2, we can conclude that the Nash equilibria of NRSGP are contained within the set NENRSGP . The Theorem 3.3 on the other hand says that, among all the λ equilibria within the set NEλ , no equilibrium other than the smallest Nash equilibrium bS (λ) will yield higher utilities for all the players. Another important consequence of a supermodular game apart from Theorem 3.2 is that, if each player in the game adopts a myopic best response update (i.e., each player chooses his best action based on the previous actions of other players), then it converges to a Nash equilibrium in the set NEλ given by Eq. (3.6) [77]. Based on the above mentioned property of supermodular games, we can devise an asynchronous decentralized algorithm (cf. Algorithm 1) for each user to compute its best strategy based on the immediate previous actions of other users. Note that the convergence of myopic best response updates to a Nash equilibrium in supermodular games can be established even if, more than one user adapts it strategy simultaneously [106]. However, we deliberately restrict ourselves to the case where only a single user adapts its strategy at a given time instant, because in the carrier sense multiple access (CSMA) based ad hoc networks at any given time instant, only a single user transmits on a given channel. The following theorem proves the convergence of Algorithm 1 to the smallest Nash equilibrium bS (λ). Theorem 3.4. Let Ti be the set of unbounded positive time instances at which a given 77  3.4. Analysis of NRSGP with Supermodular Game Theory  Algorithm 1 Algorithm for computation of best response using stochastic approximation (SA). 1: At any update instant k ∈ T the ith user computes its best response correspondence (k) (k−1) BRi (b−i ) as below: a. Compute the bi,min of ith user strategy space Bi as follows: bi,min =  Mi Wi  (3.7)  Mj j=i (k−1) Wj b j  where  Mj (k−1)  Wj bj  for all j = i such that j ∈ N in practice can be obtained via  the RTS frame broadcasted previously by all j as noted before (cf. Remark 1 below Theorem 3.1). b. Using the independent SNR estimates γq (q = 1, 2, · · · , Q) obtain an estimate (k−1) of the utility function uˆi (bi , b−i ) for each bi ∈ Bi as below:   Q Mi δi 1 (k−1) Wi bi , (1 − )Wi bi (1 − pb (bi |γq ))Mi −λ  M uˆi(bi , b−i ) = Mj i Q q=1 Mi + j=i (k−1) Wi bi  Wj bj  (3.8)  where λ is broadcasted by the access point. (k)  c. Assign BRi (.) = arg max uˆi(.), where uˆi is defined as in Eq. (3.8) above. bi ∈Bi  2:  (k)  Assign the rate bi (k)  (k)  = min BRi (.) , in case if the best response correspondence  BRi (.) is not unique.  78  3.4. Analysis of NRSGP with Supermodular Game Theory user i updates its rate bi . Also define T as the set of update instances of T1 ∪ T2 ∪ · · · ∪ TN sorted in ascending order. Also assume the initial rate vector b(0) = b, where b denotes the smallest point in the joint strategy space ×N i=1 Bi . With this notation in place if a (k)  given user i updates its rate bi  at time instant k ∈ T according to Algorithm 1 then the  rate vector b(k) would converge to the smallest Nash equilibrium bS (λ) in the set of Nash equilibria NENRSGP defined in Eq. (3.6) almost surely (a.s.). λ Proof. See Section 3.8. Remarks on Algorithm 1: 1. The convergence of Algorithm 1 to the smallest Nash equilibrium bS relies critically on the fact that, the algorithm starts at the smallest point b of the joint strategy space. On the other hand, if the algorithm starts at the largest point b of the joint strategy space, and if “min” is replaced by “max” in the step 4 of the algorithm, then the Algorithm 1 converges to the largest Nash equilibrium bL . In general, convergence to the smallest or largest Nash equilibrium starting from an arbitrary point is not ensured. 2. Note that at each update instant k, a given user i uses an estimate uˆi of its utility (k)  function to compute its best response correspondence BRi (.). The estimate is computed by using the independent channel SNR estimates γq (for q = 1, 2, · · · , Q). This is necessary because the utility function of each user namely ui cannot be computed analytically. Also since the utility function is defined over a discrete set Bi , hend the notion of gradient used in continuous optimization cannot be (k)  used. Therefore to compute the best response correspondence BRi  we use an  exhaustive search since the search space is relatively small. If the search space is relatively large one can use algorithms (cf.[13] and [42] ) proposed recently in the 79  3.5. Extension of the Price based Approach to Multi-Channel WLANs discrete stochastic optimization literature. These algorithms were used recently for spreading code optimization in CDMA networks [62].  3.4.2  Price λ Adaptation  For example, the system designer may determine the price of each user so that all the users have the same occupancy level. However such a policy could be detrimental to users with bad channel conditions, since if the channel is bad they may be forced to transmit at a lower rate. A more rational policy for price determination could be that the system increase the price until one or more of the users report a reduction in their individual throughput. This is similar to the well-known Max-Min fairness concept used in networks to ensure fairness among various users of the network. Algorithm 2 Algorithm to tune the price λ of users (executed by the Access Point (AP)). 1: At the start all the nodes set λ = 0. 2: The nodes determine their rates according to Algorithm 1. 3: The AP increment λ as λ = λ + ∆. 4: If any node reports decrease in utility the AP stops price increment, else go to step 3 and continue. The algorithm to tune the price λ of the network is shown in Algorithm 2. As mentioned above, the access point (AP) stops the price increment when one or more of the users report reduction in their throughput.  3.5  Extension of the Price based Approach to Multi-Channel WLANs  In this section, we provide an extension of the proposed price based approach to the multi-channel case. Since in a multi-channel case each user pursues transmission on 80  3.5. Extension of the Price based Approach to Multi-Channel WLANs more than one channel, the actions of each user in this case are rates across various channels. Therefore as per our notation we denote the action of player i as bi We also denote the rates across users for a given channel l as bl  bli  N . i=1  bli  L . l=1  The actions  of other users is denoted as b−i as before. The users in the multi-channel case are subject to a delay constraint of the form L Mi l=1 Wi bli  ≤ Ti , where Ti denotes the acceptable delay of user i. Therefore the strategy  set of user i for the multi-channel case can be written as follows: L  BiMC  =  bi : l=1  Mi ≤ Ti , and bli,max ≤ bli ≤ bli,min , ∀ l = 1, 2, · · · , L Wi bli  ,  (3.9)  where Mi and Wi denote the frame size and symbol rate of user i respectively and bli,min , bli,max denote the minimum and maximum transmission rates on channel l. The total throughput obtained by a given user in the multi-channel case is equal to the sum throughput obtained on each channel. The users also pay a cost that is proportional to their occupancy on a given channel. The cost is incurred due to the price imposed by the access point (AP) as noted before. Therefore the (net) utility uMC of a given user i i in the multi-channel case can be written as follows: L  uMC i (bi , b−i ) =  l=1    f (bli ) − λl   Mi Wi bli Mi Wi bli  +  Mj j=i Wj blj    ,  (3.10)  where f (.) is defined as in Eq. (3.2) and λl denotes the price imposed by AP on a given channel l. With the utility function as above one can define a non-cooperative game called MultiChannel Non-Cooperative Rate Selection Game with Pricing (MC-NRSGP), where the aim of each user is to maximize the utility defined in Eq. (3.10). However in the presence of delay constraint i.e.,  L Mi l=1 Wi bli  ≤ T , the strategy sets BiMC of users are not sublattices. 81  3.5. Extension of the Price based Approach to Multi-Channel WLANs Therefore we apply a strategy transformation which is described for the special case of L = 2 channels. The transformation cannot be extended for more than two channels.  3.5.1  Analysis for the Two channels case  In this subsection we analyze the game MC-NRSGP for the case of L = 2 channels under a strategy transformation. For L = 2 channels case, the strategy sets are sublattices under the transformation 1./b1 −1./b2 , where the operator “./” denotes the component wise division of a given vector. The transformed strategy space BiMC can equivalently be expressed as BiMC = (bli )2l=1 , b1i = values that  bli  1 b1i  2 l+1 l bi l=1 (−1)  bi :  ≤  Ti Wi Mi  and bli,min ≤ bli ≤ bli,min  where bi  and b2i = − b12 , and bli,min , bli,max denote the minimum and maximum i  can take. With this transformation we formally define the distributed  optimization problem for the case of L = 2 channels as below. (MC-NRSGP) max uMC bi , b−i i ei b  ,  s.t. bi ∈ BiMC for all i ∈ N ,  (3.11)  MC where uMC is defined as above and N denotes the i (.) is defined as in Eq. (3.10), Bi  set of players as before. The MC-NRSGP can be proved as supermodular for L = 2 channels, this is summarized in the following theorem. Theorem 3.5. For the L = 2 channels case, the game GMC-NRSGP  [N , {BiMC }, {uMC i }]  is supermodular if the maximum transmission rate bli,max on channel l in BiMC is chosen so that bli,max ≤  P  Mj e blj  j=i Mi Wi  Wj  .  Proof. The proof of this theorem is similar to the proof of Theorem 3.1 and follows straightforwardly from the Definition 1.5.  82  3.6. Simulation Results Since the game GMC-NRSGP is supermodular over the transformed strategy space BiMC therefore the set of Nash equilibria is a complete lattice and therefore the NE of MCNRSGP lie in a set bounded by the smallest and largest element similar to the NENRSGP λ defined in Eq. (3.6). Also, since the user utilities uMC increase with an increase in other i users actions, therefore in the case of MC-NRSGP the largest NE is Pareto-dominant. Finally, similar to Algorithm 1 each user i can compute its best response asynchronously over the transformed strategy space BiMC . The convergence of such an algorithm to the largest NE happens almost surely provided the algorithm starts at the largest point in the joint strategy space and each user selects maximum of its best response correspondence as noted in Remark 1 below Theorem 3.4.  3.6  Simulation Results  In this section, we present simulation results for the pricing based rate selection game considered in Section 3.4 and corroborate some of the analytical results obtained. We set the frame size Mi of user i equal to 2000 bytes. We assume the combined bit overhead δi = 100 bytes. This implies the actual payload is equal to 1900 bytes. The symbol rate Wi of all the users is set equal to 106 symbols per second. The maximum rate bi,max of all the users is set equal to 20. We set the mean channel SNR values for users (1-5) as 10, 15, 20, 25, 30 respectively. We assume the fading in the channel to be Rayleigh distributed. Therefore, the received per bit SNR γ has an exponential distribution. For the case of Fig. 3.2 we first get the equilibrium rates for the case when the price factor λ = 0. This corresponds to the case where there is no price imposed on a user for channel occupancy. Once the equilibrium is attained in this case (which is guaranteed by our analysis in Section 3.4), all the nodes in the network will increment their price factor as noted in Algorithm 2, and converge to the corresponding equilibrium. The price 83  Equilibrium User Rates in bits per modulation symbol  3.6. Simulation Results  2  10  1  10  Price factor (λ)=0 Price factor (λ)=λB  0  10 1 10  2  10 Average Signal to Noise Ratio (SNR) of users  3  10  Figure 3.2: The figure shows the plot of equilibrium user rates on a given channel versus the average SNR. The figure shows that with an increase in the price λ the rate of each user increases as proved in Theorem 3.2.  84  3.6. Simulation Results increment is continued until one of the user (which is user 1 in our case since it has the lowest mean channel SNR) reports reduction in utility. The corresponding price λB is indeed the best for the given network scenario. Figure 3.2 shows the plot of equilibrium rates of various users on a given channel versus the SNR. The figure shows that an increase in price λ forces the users to adopt higher rates as proved in Theorem 3.2, where we proved the smallest and largest equilibrium to be non-decreasing with respect to the price λ (note that the Algorithm 1 attains the smallest equilibrium almost surely as proved in Theorem 3.4). Also notice that the increase in rate with respect to the price is significant at lower SNR compared to that at high SNR. This is to be expected since the price may not be too large for the users to increase their rates drastically. We next present simulation results on the impact of pricing on the overall network throughput. To this end we consider a worst case scenario in WLANs. A typical worst case scenario contains large number of low-rate users and a smaller number of moderate and high rate users. The aforementioned scenario is worst from the overall network throughput perspective, since the low rate users occupy the channel for a long time and deny the moderate and high rate users of their fair share to transmit on the channel. Accordingly we assume there are 5 users with average SNR equal to 10 dB, 3 users with average SNR equal to 15 dB and 2 users with average SNR equal to 30 dB. To mimic the DCF mechanism used in WLANs we assume each user before the transmission of a frame accesses the channel with a probability equal to 0.1. This choice of access probability ensures all the users obtain a fair share to transmit on the channel. Indeed the DCF mechanism ensures this in the long run as noted in [100]. In case if more than one user decides to transmit a collision occurs and we drop the frames of both the users. In a wireless environment since the users are subject to channel errors, therefore  85  3.6. Simulation Results the frame transmitted by a given user i is dropped with a probability equal to 1 − pfi , where pfi  [1 − pb (bi |γ)]M denotes the frame success probability of user i. In our case  we compute pfi for user i empirically as follows: S  pfi = j=1  [1 − pb (bi |γj )]M ,  (3.12)  where pb denotes the bit error expression for transmitting at a rate bi and γj denote the SNR values generated according to a known distribution and S denotes the SNR sample size. We assume each user transmits at a rate bi . The rate bi is chosen equal to the equilibrium rates. The equilibrium rates for the current simulation scenario are obtained by using the same procedure described for Fig. 1 above (i.e., we first assume there is no price (λ = 0), allow the system to converge to the equilibrium point, then increase the price and allow it to converge to the corresponding equilibrium point. The procedure is continued until at least one user reports a reduction in its equilibrium utility). Figure 3.3 shows the plot of total network throughput with and without pricing. The figure shows that the total network throughput increases significantly due to pricing. This can be attributed to an increase in occupancy level for the 15 dB and 30 dB users respectively which is shown in Fig. 3.4. Fig. 3.4 also shows that an increase in rate due to pricing of the 10 dB user reduces its occupancy. Finally the Fig. 3.5 shows the average throughput of 10 dB, 15 dB and 30 dB users respectively with and without pricing. As the Figure 3.5 shows the average throughput of the 10 dB user decreases slightly due to pricing. This can be attributed to a decrease in occupancy of the 10 dB user which is shown in Fig. 3.4. However the 15 dB and 30 dB user report an increase in their throughput owing to an improvement in their occupancy which is again shown in Fig. 3.4. 86  3.6. Simulation Results  Figure 3.3: The figure shows the total network throughput for the case of with and without pricing. The figure shows that an appropriate choice of price can lead to higher overall network throughput.  87  3.6. Simulation Results  Figure 3.4: The figure shows the occupancy values of the 10 dB, 15 dB and 30 dB users respectively with and without pricing. The figure shows that an increase in price reduces the occupancy of 10 dB users. This improves the occupancy of 15 dB and 30 dB users thereby leading to an increase in overall network throughput as shown in Fig. 3.3.  88  3.6. Simulation Results  Figure 3.5: The figure shows the average throughput obtained for a 10 dB, 15 dB and 30 dB user. The average throughput of the 10 dB user decreases slightly due to pricing. This can be attributed to a decrease in occupancy of the 10 dB user which is shown in Fig. 3.4. However the 15 dB and 30 dB user report an increase in their throughput owing to an improvement in their occupancy.  89  3.7. Summary From this simple simulation example we can conclude that the proposed pricing based approach can lead to an increase in overall network throughput due to better occupancy levels for moderate and high rate users.  3.7  Summary  We considered price based decentralized rate selection in IEEE 802.11 WLANs. Owing to the decentralized nature of WLANs we adopt a non-cooperative game theoretic approach for our analysis. The aim of each user is to maximize a utility function which is equal to the throughput a user obtains for transmitting at a specific rate and a cost that is proportional to its occupancy. The actions of the users are their respective rates. We proved the resulting game to be supermodular, and therefore has at least one pure strategy Nash equilibrium (NE). We also showed the smallest and largest NE to be non-decreasing functions of the price. Further we proved the smallest Nash equilibrium to be Pareto-dominant. We also presented an asynchronous decentralized algorithm for computation of best response of each user which converges to the smallest Nash equilibrium almost surely. We also proposed an extension of the price based approach to the multi-channel case and proved the resulting game to be supermodular for the special case of two channels. Through our simulation results, we observed that by tuning the price it is possible to contain the occupancy of low rate users and improve the occupancy of high rate users thereby leading to an increase.in total network throughput.  90  3.8. Analytical Proofs  3.8 3.8.1  Analytical Proofs Proof of Theorem 3.1:  We use the Definition 1.5 in Section 1.4.3 of Chapter 1 to prove NRSGP as supermodular. The condition (i) of Definition 1.5 requires the strategy space of all users Bi , i ∈ N to be lattices. This can be easily verified to be true by using the Definition 1.1 of lattices in Section 1.4.3 of Chapter 1. To prove the utility functions ui (i = 1, 2, · · · , N) as continuous in all players strategies we note that the joint strategy space B  ×N i=1  is discrete and finite (since Bi (i = 1, 2, · · · , N) is discrete and finite). This discrete nature of B ensures that the utility function ui (i = 1, 2, · · · , N) is (strongly) continuous by Theorem 3 of [40]. Further a function is always supermodular in a single variable. Therefore condition (ii) of Definition 1.5 is also (trivially) satisfied . Finally to verify condition (iii) of Definition 1.5 which requires the utility of all the users ui, i ∈ N satisfy the property of increasing differences between bi and bk (∀k ∈ N such that k = i). Therefore using the Definition of increasing differences in Definition 1.2 in Section 1.4.3 of Chapter 1 we can write for (bk > b′k ) as ui (bi , bk ) − ui (bi , b′k ) = λ    Mk Wk bk  −  Mk Wk b′k  ×  term 1 Mi Wi bi Mi Wi bi  +  Mj j=i,k Wj bj  +  Mk Wk b′k  Mi Wi bi  term 2  +  Mj j=i,k Wj bj  +  Mk W ke bk     ,  (3.13)  Note that the term 1 in Eq. (3.13) is always negative. Therefore the property of increasing differences of ui with respect to (w.r.t) bi and bk is equivalent to showing that the term 2 is decreasing w.r.t bi . To accomplish this we apply the following simple trick: we first prove  91  3.8. Analytical Proofs that the gradient of term 2 is negative over [bi,min , bi,max ]. Now since Bi ⊂ [bi,min , bi,max ] it can be concluded that term 2 is decreasing w.r.t bi , therefore ui satisfies increasing differences w.r.t. bi and bk . Applying this trick it can be shown that the gradient of term 2 is decreasing if bi,min ≥  3.8.2  P  Mi Wi Mj j=i Wj bj  . This completes the proof.  Proof of Theorem 3.2:  Since GNRSGP is a supermodular game, therefore the proof of the first part of Theorem 3.2 can be established via Theorem 5 and its corollaries in [77] which is summarized in Theorem 1.2 of Section 1.4.3 in Chapter 1, To prove that the smallest and largest Nash equilibria viz., bS and bL are nondecreasing with respect to the price λ we need to verify that ui satisfies the property of increasing differences w.r.t bi and λ as noted in Theorem 1.3 of Section 1.4.3 in Chapter 1. Applying the definition of increasing differences as in the proof  of Theorem3.1 we can write ui (bi , λ) − ui (bi , λ′ ) for λ > λ′ as equal to  λ′ − λ   1  1+  bi Wi  Mj P j=i Wj bj Mi  . The  term with in square brackets in the previous expression is decreasing w.r.t bi , therefore the overall expression is increasing since (λ′ − λ) < 0. Therefore ui satisfies increasing differences w.r.t bi and λ, thus by Theorem 1.3 of Section 1.4.3 in Chapter 1, the smallest and largest Nash equilibria namely bS and bL are non-decreasing functions of λ the price. This completes the proof.  3.8.3 Define b  Proof of Theorem 3.3 [b1 , b2 , · · · , bi · · · , bN ], b′  [b′1 , b′2 · · · , b′i · · · , b′N ] such that b, b′ ∈ NENRSGP λ  and b ≥ b′ . Also recall from our notation b−i  [b1 , b2 , · · · , bi−1 , bi+1 , · · · , bN ] and b′−i  represents b′1 , b′2 , · · · , b′i−1 , b′i+1 , · · · , b′N . From the utility expression in Eq. (3.3) it can be concluded that the ith user utility decreases with an increase in other users rates. 92  3.8. Analytical Proofs Therefore, ui (bi , b−i ) ≤ ui bi , b′−i in Eq. (3.3). Further, the definition of Nash equilibrium implies ui bi , b′−i ≤ ui b′i , b′−i . Therefore ui (b) ≤ ui(b′ ). Since bS represents the ˜ ≤ ui(bS ) where b ˜ ∈ NENRSGP such smallest NE in the set NENRSGP . Therefore ui(b) λ λ ˜ ≥ bS , therefore the smallest NE is Pareto dominant in the set NENRSGP . that b λ  3.8.4  Proof of Theorem 3.4:  In Algorithm 1 for any fixed bi ∈ Bi , (1 −  δi )Wi bi (1 Mi  − pb (bi |γq ))Mi  is an i.i.d. sequence  of random variables. Therefore, by virtue of the strong law of large numbers δi )Wi bi (1 Mi  − pb (bi |γq ))Mi → (1 −  δi )Wi bi Eγ Mi  (1 − pb (bi |γq ))Mi  1 Q  Q q=1 (1 −  almost surely (a.s.) as  Q → ∞. This and the finiteness of Bi imply that as Q → ∞ arg max uˆi(bi , b−i ) → arg max ui(bi , b−i ) a.s. bi ∈Bi  (3.14)  bi ∈Bi  (k)  Therefore the best response correspondence BRi (.) of user i in step 1c of Algorithm 1 at any update instant k would converge to the set of global optimizers. Furthermore in a supermodular game, if the players start at the smallest point b in the joint strategy space ×N i=1 Bi , and take “minimum” of the best response correspondence, then the rate vector b(k) converges to the smallest Nash equilibria bS (λ) in the set NENRSGP defined in Eq. (3.6) (cf. Theorem 8 in [77]). However in our case, this conλ vergence happens almost surely because of the almost sure convergence of the estimate uˆi to ui in step 1b of Algorithm 1. This completes the proof.  93  Chapter 4 Performance of TCP in Cognitive Radio Networks6 4.1  Introduction  Cognitive Radio (CR) is an emerging communications paradigm, wherein a wireless transceiver unit can sense the surrounding environment and adapt itself accordingly. Cognitive radio technology, along with Dynamic Spectrum Access, has the potential to alleviate the shortage of radio resource. For many years, it was believed that the spectrum shortage was due to an increasing number of wireless applications, and their substantial bandwidth usage. However, spectrum measurements in Washington, D.C., New Orleans, San Diego, Atlanta, Chicago, and other metropolitan areas show that vast portions of licensed spectrum are not in use (i.e., they are underutilized) [44]. This finding suggests that an improvement in spectrum utilization can alleviate the spectrum shortage problem. A transceiver equipped with the cognitive radio capability can access and release the spectrum with more agility. Such agility can allow secondary users (who are unlicensed) to access the spectrum when primary users (i.e., spectrum licensees) are inactive. Referred to in the literature as Dynamic Spectrum Access (DSA), this approach clearly reduces spectrum underutilization, and hence alleviating the spectrum shortage problem. Most of the CR research focused at the lower layers (i.e., physical and MAC layers) 6  The work in this chapter will appear as an article in IEEE Communications Magazine [59].  94  4.1. Introduction of the network protocol stack. Despite sporadic works that study routing issues in CR networks, no work to date has considered transport layer issues in CR networks [6]. In this chapter, we study the performance of Transmission Control Protocol (TCP) in a CR network. TCP has become a ubiquitous transport layer protocol for reliable non-real time data transmission over the internet. TCP was originally designed to control network congestion in wired networks, and was later enhanced to cope with channel errors in wireless networks [17]. The advent of DSA introduces a new type of packet loss called the service interruption loss (see the definition in Section 4.3.2), in addition to the existing packet losses resulting from network congestion and channel errors. Consequently, there is a strong motivation to study the performance of TCP under service interruption loss in CR networks that employ DSA. This chapter provides a detailed survey of CR networks. We present their salient features, access techniques, and architectures. We also discuss a surprising result regarding TCP performance in overlay CR networks (see the definition in Section 4.2.3): the TCP throughput of a secondary user does not always increase with the available radio resource. Due to service interruption from the primary user, TCP performance of a secondary user could degrade when it tries to acquire too much radio resource. This finding poses the need to optimally control how secondary users obtain the radio resource, in order to fully utilize the available radio resource. The rest of this chapter is organized as follows. Section 4.2 discusses CR evolution and provides a survey of various regulatory issues, access techniques, and architectures proposed in the literature. Section 4.3 describes the basic operation of TCP in conventional networks, and emphasizes the need to study TCP performance over a CR network. Section 4.4 shows our simulation study of TCP performance in CR networks allowing  95  4.2. Cognitive Radio : Evolution, Access Techniques and Architectures DSA. Finally, conclusions are given in Section 4.5.  4.2  Cognitive Radio : Evolution, Access Techniques and Architectures  4.2.1  Evolution and Salient Features of Cognitive Radio  Cognitive radio is a key enabling technology to achieve DSA of licensed bands of the spectrum. The current interest in CR technologies is due to a report published by the Federal Communications Commission (FCC) pointing out that vast portions of the spectrum are underutilized [44]. This finding suggested that granting access to unlicensed users could lead to significant improvement in spectrum utilization. The term Cognitive Radio was coined by Mitola III in [58], although its genesis can be traced to the concept of Software Defined Radio (SDR). According to the FCC, “an SDR is one in which operating parameters such as frequency range, modulation type or output power can be altered by software without making any changes to the hardware components”. Taking the definition of SDR a step further, Mitola III defined CR as “an SDR that senses its environment, tracks changes and reacts accordingly” [58]. Cognitive radio is characterized by two main features: • Cognitive capability: This refers to the ability of the radio to identify spectrum opportunities and adapt to the situation accordingly. The typical steps involved in this adaptive operation are spectrum sensing, spectrum analysis and spectrum decision. Spectrum sensing involves detection of white spaces (i.e., the spectrum portion that is not in use). Spectrum analysis involves analyzing the characteristics of the detected white space (e.g., characterizing the white space based on its time-varying radio characteristics and the primary user activity). The spectrum 96  4.2. Cognitive Radio : Evolution, Access Techniques and Architectures decision specifies the action (e.g., adjusting transmission rate, power levels, and/or bandwidth) which should be taken after spectrum sensing and analysis [6] [50]. • Reconfigurability: This refers to the ability of the radio to be configured dynamically according to the environment. The various parameters that can be reconfigured are, for example, operating frequency, modulation type and transmission power [50]. A general CR network consists of two sub-networks: a primary network and a secondary network. The primary network consists of primary users and the primary base station, whose function is to control and regulate the users of the network, similar to a cellular system. The secondary network does not own a license and tries to utilize the spectrum in an opportunistic manner. A secondary network, may or may not have a base station (BS), depending on the architecture (see Section 4.2.4). We note that CRs can be conceived for even the unlicensed bands of the spectrum, as noted in [6]. However, in this chapter, we focus our attention on CR networks for access to the licensed bands only.  4.2.2  Spectrum Access Policy: Own It? or Share It?  The inception of cognitive radio has raised spectrum regulatory issues. Two schools of thought–namely Property Rights and Spectrum Commons–are being debated extensively [83]. Advocates of Property Rights argue that spectrum owners should have an absolute right over their spectrum. Proponents of Spectrum Commons, on the other hand, believe that sharing spectrum would be more efficient, and will greatly alleviate the problem of spectrum under utilization. Hence, there are constant regulatory and legal policy discussions on these two conflicting ideas, the result of which is still inconclusive.  97  4.2. Cognitive Radio : Evolution, Access Techniques and Architectures  4.2.3  Access Techniques for CR Networks  Access techniques for CR networks can be classified into two types: • Overlay Approach (or Interference-Free Approach): In this approach, the secondary users access the portion of the spectrum that is not used by primary users. As a result, there is virtually no interference to the primary users. We refer CR networks that employ overlay access techniques as Overlay CR networks. • Underlay Approach (or Interference-Tolerant Approach): In this approach, the secondary users access the network by spreading their signals over a wide frequency band. The underlay approach imposes severe constraints on the transmission power of secondary users. Operating below the noise floor of primary users, the secondary users are allowed to interfere with primary users up to a certain tolerable level. We refer CR networks that employ underlay access techniques as Underlay CR networks. The current spectrum management policy adopted by the FCC, based on the Property Rights model, does not allow secondary user operation in the licensed spectrum, irrespective of the access techniques. Hence, there may be regulatory and legal discussions on how secondary users should operate in the licensed spectrum.  4.2.4  Architectures for CR Networks  The architecture for CR networks can be either centralized or distributed. Table 4.1 summarizes various CR architectures proposed in the literature.  98  4.2. Cognitive Radio : Evolution, Access Techniques and Architectures Table 4.1: Summary of Cognitive Radio Architectures Centralized Sensing entity Secondary users / Spectrum broker Spectrum broker Separate Entity / Primary BS / Secondary BS Access granted to Primary and Secondary users Examples Spectrum pooling [111] DIMSUMnet [26]  Distributed Secondary users None Secondary users CORVUS [25] Nautilus [116] [28]  Centralized CR Network Architectures A general centralized network architecture consists of two main entities. One is a base station, which schedules the data transmission of users in the network. Another entity is the spectrum broker, which is responsible for allocating the radio resource to users (primary, secondary, or both). The spectrum broker can be a primary base station, or a secondary base station [6], or a dedicated entity dealing with spectrum allocation [111] [26]. The sensing functionality in a centralized CR network can be performed by secondary users or the spectrum broker. The sensed spectrum information is used by the spectrum broker to create a spectrum allocation map for radio resource allocation. The followings are examples of the centralized CR architectures proposed in the literature. A more comprehensive survey of CR architectures is given in [6]. • Spectrum pooling: In this architecture, spectrum from different owners is put together in a common pool [111]. The architecture is based on the well-known Orthogonal Frequency Division Multiplexing (OFDM). The advantage of an architecture based on OFDM is that interference to the primary users can be minimized by allocating zero power to the subcarriers occupied by them. • DIMSUMnet (Dynamic Intelligent Management of Spectrum for Ubiquitous Mobile network) [26]: In DIMSUMnet, a spectrum broker owns a contiguous chunk of  99  4.2. Cognitive Radio : Evolution, Access Techniques and Architectures spectrum called the Coordinated Access Band (CAB)7 . The CAB is then leased according to requests from different users, for a specified time. Distributed CR Network Architectures This type of architecture has no centralized agent like a spectrum broker or a base station to coordinate spectrum access of secondary users. It is similar to that used in wireless ad hoc networks. The main difference from wireless ad hoc networks is the presence of primary users and service interruption loss, which will be defined in Section 4.3.2. Distributed CR networks can be classified into cooperative or non-cooperative networks. In a cooperative network, the users share the interference information and determine spectrum allocation based on this shared information. In a non-cooperative network, the users access the spectrum based on local policies, since there is no communication regarding interference information among users of the network. In general, the cooperative approach achieves better throughput than the non-cooperative approach. However, a cooperative approach may incur extra overhead owing to the communication among users. We now briefly summarize the existing distributed CR network architectures: • CORVUS (COgnitive Radio approach for usage of Virtual Unlicensed Spectrum) [25] : In CORVUS, secondary users form a communication group called a Secondary User Group (SUG). The secondary users in an SUG help each other in sensing the channel for primary user activity. If primary users are inactive on a channel, then SUG takes that channel for communication within the group. We note that CORVUS can be used even in the centralized mode. In each SUG, a secondary user could be selected to control spectrum access of other secondary 7  CABs can be designated frequency bands in cellular, PCS, or TV broadcasting systems.  100  4.3. TCP in Conventional versus Cognitive Radio (CR) Networks users. • Nautilus [116],[28] : This project devises distributed, efficient, and scalable algorithms for spectrum sharing in CR ad hoc networks. The project proposes spectrum access algorithms based on two approaches: one based on graph coloring [116], and another based on local bargaining [28]. In the former approach, the spectrum allocation is transformed into a graph-coloring problem, where a secondary user corresponds to a vertex, while a radio channel corresponds to a color. Since a graph-coloring problem is NP-hard, the authors propose heuristic approximation algorithms to solve the problem [116]. In the latter approach, the secondary users affected by topological changes form local bargaining groups and adapt their spectrum assignment to a new optimal conflict free assignment [28].  4.3  TCP in Conventional versus Cognitive Radio (CR) Networks  4.3.1  TCP in Wired and Conventional Wireless Networks  TCP was originally conceived for wired networks as a means to avoid and control network congestion, and to provide reliable end-to-end delivery of user data. Whenever the sender identifies loss of data (either in the form of several duplicated acknowledgments or an absence of acknowledgment for a timeout interval), TCP reacts to the loss by reducing its transmission window size before retransmitting packets. This reduction in window size eases the load on intermediate links, hence controlling congestion in the network. The subsequent increase in window size depends on whether TCP is in slow start phase or congestion avoidance phase. In the slow start phase, the window size grows linearly,  101  4.3. TCP in Conventional versus Cognitive Radio (CR) Networks in response to every acknowledged packet, while the window size grows sub-linearly in the congestion avoidance phase. In general, the essential functionality of TCP remains the same, despite the modifications to improve its performance. For example, TCP variants, namely, TCP-Tahoe, TCP-Reno, implement the same window based adaptation mechanism, but have minor differences. TCP was originally developed for wired networks, and later enhanced for wireless networks. Unlike in a wired network where packet loss is mainly caused by congestion, the majority of packet loss in a wireless network is due to channel errors (caused by fading, interference and shadowing) or packet collision (i.e., when more than one mobile accesses the channel simultaneously). While two consecutive packet losses due to network congestion are highly correlated, those due to channel errors or collision may not be related at all. In general, the suggested solutions to improve performance of TCP in wireless networks are classified into three basic groups based on their fundamental philosophies: end-to-end solutions, split-connection solutions and link-layer solutions [17]. Among these three solution approaches, the link-layer solution is the most popular, since it does not require any modification of the network architecture or TCP operation.  4.3.2  Service Interruption Loss in Overlay CR Networks  The concept of DSA introduces a new type of loss called the service interruption loss for the secondary users of an overlay CR network. Service interruption loss refers to the loss experienced by secondary users due to the intervention of primary users while transmitting the data. The service interruption loss is absent in the underlay CR networks, since in an underlay approach, the secondary users operate much below the power level of the primary users, hence ruling out the possibility of collisions with the primary users of the  102  4.3. TCP in Conventional versus Cognitive Radio (CR) Networks spectrum. The amount of service interruption loss for the secondary users of the overlay CR network is determined by the amount of primary user activity in the network. To capture the amount of primary user activity, we define load factor (A) as the amount of traffic all the primary users generate on a single channel. Mathematically, it is stated as  A=  Np TON × , TON + TOF F L  (4.1)  where TON and TOF F denote the mean ON time and OFF time of the primary users, Np denotes the number of primary users, and L denotes the total number of channels in the spectrum. The rationale behind Equation (4.1) is as follows. First, the average traffic generated by each primary user is TON /(TON + TOF F ). Hence, the total traffic generated by all the primary users is Np · TON /(TON + TOF F ). Since this traffic is generated over L radio channels, the total traffic (or load) generated by the primary users on a single channel is given by (A) defined as in Equation (4.1). In words, if the traffic generated by all the primary users is distributed equally among all the radio channels, then A in Equation (4.1) denotes the amount of primary user traffic on one radio channel. For example, A = 0.5 implies that every radio channel is used by all primary users, on an average for 50% of the time. In general, an higher value of the load factor (A) implies lower aggregate TCP throughput of the secondary users. The service interruption loss is different from the following three major losses that occur in a conventional network. First, the losses due to network congestion are usually caused by buffer overflow, and are assumed to be correlated. Second, the losses due to channel errors depend on channel characteristics. Depending on whether the user is moving slowly or rapidly the losses due to channel errors can be assumed to be correlated 103  4.3. TCP in Conventional versus Cognitive Radio (CR) Networks or independent. The third and final type of losses are due to the collision between various users of the network, that depend on the underlying Medium Access Control (MAC) mechanism. The service interruption loss due to DSA on the other hand, does not depend on network conditions, channel characteristics, nor on the underlying MAC mechanism. It depends on the amount of primary user activity in the spectrum, which in turn depends on extraneous factors such as geographical location, and time of the day. The report in [44] notes that, in a given geographical area for most part, the spectrum is devoid of primary user activity. In such cases, the duration of service interruption loss may be small, since it is possible for the secondary user to find an alternate channel for transmission almost immediately. However, if a given geographical region has heavy primary user activity, then it may not be possible for the secondary user to find an alternate channel immediately, and hence leading to a longer duration of the service interruption loss. TCP performance under the congestion losses, channel error losses, and/or data collision losses has been studied extensively. However, due to the service interruption loss, TCP performance in an overlay CR network could be significantly different from that in a conventional network. This poses a clear need to study TCP performance in an overlay CR network. As we shall see from our simulation study presented in the next section, a definite number of channels must be carefully chosen in order to optimize TCP performance in an overlay CR network.  104  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network  4.4  Simulation Study of TCP Performance in an Overlay Cognitive Radio Network  4.4.1  System Model of an Overlay CR Network  In this chapter, we consider an overlay CR network with Np primary users and Ns secondary users (see Fig. 4.1). Primary users send their data to a primary network via its base station. On the other hand, each secondary user transfers a file with infinite size to the server connecting to its base station. Both primary and secondary users share L radio channels to communicate with their base stations. We assume that the data rate and propagation delay of each radio channel are br bps and dr seconds, respectively. Also, the link which connects the base station and the server has a data rate of bl bps and a propagation delay of dl seconds. Each secondary user is allowed to access the free channels only. While using a channel, a secondary user may be asked (by a primary user) to give up its ongoing transmission, and return the channel to the primary user. In this case, the packet transmitted by the secondary user will be lost due to service interruption. We model the primary user activity according to a continuous-time Markov chain with two states namely, ON and OFF respectively. This modeling assumption implies each primary user generates a data packet that holds the channel for an exponentially distributed time with mean TON . After transmitting a packet, a primary user turns OFF for an exponentially distributed time with mean TOF F . When switching to an ON state, a primary user will first look for a free channel to transmit data. If there is no free channel, a primary user will force one of the secondary users (if any) to give up transmission on the radio channel. If all the channels are occupied by primary users, the new primary user will not be able to transmit its data. 105  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network  Figure 4.1: System model of an overlay CR network: Np primary users and Ns secondary users access their base stations via L shared radio channels. Each radio channel has a bandwidth of br bps. The primary base station connects to the primary network, while the secondary base station connects to a server via a wired link with a bandwidth of bl bps. When needed, a primary user can force a secondary user to give up transmission, so that it can use the released channel.  106  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network Each secondary user transfers a file by using TCP as its transport layer protocol. Again, based on a perceived packet loss pattern, TCP feeds and stops feeding packets to the underlying link layer. A secondary user attempts to transmit each link layer packet according to the following spectrum access mechanism. At any time, a secondary user can access a maximum of Ls radio channels which implies there are always (L − Ls ) channels reserved for the primary users. Among the Ls number of channels, a secondary user chooses to transmit (or) not to transmit on a free channel with probability p and 1 − p, respectively8 . If more than one secondary user chooses to transmit on the same channel simultaneously, all the transmitted packets will be lost. The above access mechanism is invoked repeatedly until all the link layer packets are transmitted or the number of occupied channels is greater than Ls .  4.4.2  Simulation Parameters  We run the simulation using an open source network simulation tool NS2. Unless specified otherwise, we assume Np = 30, L = 20, TOF F = 0.1 seconds, br = 100 kbps, dr = 2 milliseconds, bl = 100 Mbps, and dl = 2 milliseconds. The value of TON is determined by the choice of the load factor value (A). The value of br chosen above is typical for wireless data applications. For example, the CDMA 2000 1xEV-DO standard (which is one of the six radio interfaces under IMT-2000) supports a maximum data rate of 154 kbps [4]. We run the simulation for a duration of 500 seconds, which we found to be sufficient for TCP to reach its steady state. Each secondary user employs TCP new Reno9 to upload the file, and chooses transmission probability value (p) equal to 1/Ns , where Ns denotes 8  This type of random access is widely used for ad hoc wireless networks, where there is no base station to control and coordinate the transmission of secondary users. 9 TCP New Reno is an improved version of Reno that avoids multiple reductions of congestion window size when several TCP packets from the same window of data are lost [63].  107  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network the number of secondary users10 . Finally, we assume there are no losses due to channel errors (or) buffer overflow on all the TCP flows of secondary users.  4.4.3  Simulation Results  Fig. 4.2 shows the plot of aggregate TCP throughput of secondary users versus the maximum channels for secondary users (Ls ), for the case when there are no primary users i.e., Np = 0. With no primary users, this case corresponds to a conventional random access ad hoc network. As can be seen from Fig. 4.2, without the service interruption from the primary users, the aggregate TCP throughput of the secondary users increases monotonically as Ls increases. In a random access, there is no collision for Ns = 1. Therefore this case has higher aggregate TCP throughput than the multi-user case (i.e., Ns = 5 and Ns = 10). Note that the aggregate TCP throughput for Ns = 1 saturates at Ls = 18, since the TCP window size has reached its maximum value. However for the multi-user case (i.e., Ns = 5 and 10), the aggregate TCP throughput increases continuously, since in these cases the TCP window size has not reached its maximum. Fig. 4.3 shows the impact of service interruption on the aggregate TCP throughput of secondary users. Here, we set the number of primary users (Np ) equal to 30. In this case, the throughput gradually increases as the maximum channels for secondary users (Ls ) increase up to a certain point, beyond which the throughput decreases. Hence, there exists a well defined optimal value of Ls (denoted by L∗s ) which maximizes the aggregate TCP throughput. This behavior is in contrast to that of user datagram protocol (UDP) behavior shown in Fig. 4.4. As can be seen from Fig. 4.4, the UDP throughput of secondary users increases monotonically with the number of channels Ls . This non-monotonic behavior of TCP in Overlay CR networks that implement DSA 10  This choice of p = 1/Ns minimizes the collision probability among Ns secondary users.  108  Aggregate secondary user TCP Throughput in bps  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network  5  18  x 10  Number of secondary users (Ns)=1  16  Number of secondary users (Ns)=5  14  Number of secondary users (N )=10 s  12 10 8 6 4  Number of primary users (N )=0 p  2 0 0  5 10 15 Maximum number of channels for secondary users (L )  20  s  Figure 4.2: Aggregate secondary user TCP throughput versus the maximum channels for secondary users (Ls ) when Np = 0. Due to the absence of service interruption from the primary users, this case is similar to an ad hoc network with random access. The figure shows that the aggregate TCP throughput of secondary users increases monotonically with Ls .  109  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network can be explained as follows. Intuitively, as the maximum channels for secondary users (Ls ) increase, we expect an increase in aggregate TCP throughput. However, an increase in Ls also increases the probability of service interruption from a primary user, leading to increased packet loss. This packet loss (detected via timeout or duplicated acknowledgement) leads to a decrease in TCP window size, and hence significant degradation of aggregate TCP throughput of secondary users. When adopting DSA in overlay CR networks, the secondary users need to be judicious in choosing the number of channels for data transmission. We also observe from Fig. 4.3 that the optimal value of Ls (denoted by L∗s ) also increases as the number of secondary users increases. Since every secondary user accesses the channel with the same probability p = 1/Ns , on an average, all the free channels are distributed equally among the secondary users. For example, suppose the number of free channels are 15. In a single-user case, all the 15 free channels are used by only one user. When Ns = 5, each secondary user employs, on the average, 15/5 = 3 channels to transmit TCP packets. With a small number of channels per user, a success in single transmission could gain a significant increase in TCP throughput when compared to the packet loss due to service interruption. Therefore, the peaks (i.e., L∗s ) in Fig. 4.3 shift to the right as the number of secondary users increases. Fig. 4.5 shows the variation of aggregate TCP throughput with respect to the maximum number of channels Ls for different values of the load factor (A). In this case, we set Np = 30 and Ns = 10, and the load factor A is chosen to be 10%, 30%, and 50% respectively. A higher value of the load factor implies an increase in the probability of service interruption from the primary user, and also a decrease in the transmission opportunities for a secondary user. Hence, the aggregate TCP throughput of the secondary users decreases as the load factor increases.  110  Aggregate secondary user TCP Throughput in bps  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network 5  14  x 10  Number of secondary users (Ns)=1 Number of secondary users (Ns)=5  12  Number of secondary users (N )=10 s  10 8 6 4  Number of primary users (Np)=30  2 0 0  5 10 15 Maximum number of channels for secondary users (L )  20  s  Aggregate secondary user UDP Throughput in bps  Figure 4.3: Aggregate secondary user TCP throughput versus the maximum channels for secondary users (Ls ) in the presence of service interruption due to 30 primary users. The figure shows that the throughput is no longer monotone with respect to Ls . There exists a well defined optimal value of Ls which maximizes the aggregate TCP throughput. 5  16  x 10  14 12  Number of secondary users (Ns)=1 Number of secondary users (Ns)=5 Number of secondary users (N )=10 s  10 8 6 4 2 0 0  Number of primary users (Np)=30  5 10 15 Maximum number of channels for secondary users (L )  20  s  Figure 4.4: Aggregate secondary user UDP throughput versus the maximum channels for secondary users (Ls ) in the presence of service interruption due to 30 primary users. The figure shows that the throughput increases monotonically with respect to Ls , unlike the case of TCP.  111  Aggregate secondary user TCP Throughput in bps  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network 5  14  x 10  12  Load factor (A)=10 % Load factor (A)=30 % Load factor (A)=50 %  10 8 6  Number of primary users (Np)=30 Number of secondary users (Ns)=10  4 2 Increasing load factor (A) 0 0  5 10 15 Maximum number of channels for secondary users (L )  20  s  Figure 4.5: Aggregate secondary user TCP throughput versus the maximum channels for secondary users (Ls ) for various values of load factor (A) with service interruption due to 30 primary users. The figure shows that the aggregate TCP throughput of secondary users decreases as the load factor increases.  Increasing logical channel bandwidth 18  *  Optimal number of channels (Ls) captured  20  16 Number of primary users (N )=30 14  12  10  p  Number of secondary users (Ns)=1 Logical channel bandwidth = 0.1Mbps Logical channel bandwidth = 0.3Mbps Logical channel bandwidth = 0.5Mbps −6  −4  −2  10 10 10 Primary user network load factor (A)  Figure 4.6: Optimum number of channels (L∗s ) that a secondary user captures to maximize its TCP throughput versus load factor (A) for different values of bandwidth. The figure shows that for a given bandwidth, the optimal number of channels (L∗s ) captured is high for small and large load factor values, and low for moderate load factor values.  112  4.4. Simulation Study of TCP Performance in an Overlay Cognitive Radio Network Fig. 4.6 plots the optimal value of Ls (i.e., L∗s ) versus the load factor (A). We set the number of primary and secondary users to be 30 and 1, respectively. The non-monotone behavior of L∗s with respect to the load factor A in Fig. 4.6 can be explained as follows. First, a small load factor implies that primary users rarely need the channel; service interruption is rare in this case. With small service interruption probability, a secondary user should take a chance and transmit on as many channels as possible to increase the TCP throughput. Secondly, as the load factor increases, the service interruption probability increases. In this case, we should decrease L∗s to avoid service interruption. The decrease of L∗s (with respect to A) stops, when the traffic load is extremely large. In this case, primary users tend to occupy all the channels for most of the time, and the secondary users may not be able to acquire a channel. Therefore, the secondary users should take a chance and transmit packets on as many channels as possible, even though they are prone to increased service interruption from the primary users. Due to the above non-monotone behavior, a secondary user needs to be judicious in setting the maximum number of channels for the secondary users viz., Ls . Fig. 4.6 also shows the effect of radio channel bandwidth on L∗s . An increase in radio channel bandwidth decreases packet transmission time. In this case, packet transmission could be complete before a primary user service interruption. Increasing radio channel bandwidth therefore decreases the service interruption probability. In a high bandwidth case, we can allow a secondary user to acquire more radio channels without increasing the risk of service interruption. Hence, we observe in Fig. 4.6 that increasing the radio channel bandwidth can lead to a higher optimal value of Ls (i.e., L∗s ).  113  4.5. Summary  4.5  Summary  In this chapter, we reviewed the evolution of cognitive radio (CR), and the various architectures envisaged for its realization. The dynamic spectrum access (DSA) technique introduces a new type of loss called the service interruption loss for the secondary users in an overlay CR network. We showed via simulations that the TCP performance under service interruption loss in an overlay CR network is significantly different to its performance in a conventional network. In particular, we observed that in an overlay cognitive radio network that adopts DSA, there exists an optimal number of channels that the secondary users need to capture, to maximize their aggregate TCP throughput. After all, a secondary user cannot blindly utilize all the available channels and expect the best result.  114  Chapter 5 Conclusions and Future Work 5.1  Summary of Work Accomplished  In this thesis we considered resource management problems in different types of wireless networks. More specifically, we considered resource management in wireless sensor networks (WSNs), wireless local area networks (WLANs) and cognitive radio (CR) networks. The contributions with respect to each of the problems considered in individual chapters are summarized below: • In Chapter 2, we considered the issue of energy minimization in WSNs, since energy is a critical resource in WSNs. We specifically considered a combination of explicit node cooperation and distributed source coding (DSC) to improve energy efficiency of WSNs. The explicit node cooperation among nodes improves the energy efficiency of WSNs by reducing the per bit energy consumption for transmission. The DSC is used to exploit the spatial correlation in data sensed by nodes, which is unique to WSNs. We model the spatial correlation in WSNs according to a linear Gauss-Markov model. With this correlation model, we determined the encoding rates of each sensor cluster in a multi-hop environment. Using these encoding rates we formulated the problem of minimum energy data gathering in correlated WSNs. From our numerical results, we concluded that the cooperation among nodes is beneficial, especially for very low probability of error values. Next using the con-  115  5.1. Summary of Work Accomplished cepts in monotone comparative statics we analyzed as how the optimal modulation rate and number of cooperating nodes vary with the correlation coefficient. Indeed we proved the optimal modulation rate and number of cooperating nodes to be increasing and decreasing functions of the correlation coefficient. We also proved the optimal modulation rate to be a decreasing function of the number of cooperating nodes. We then considered mutual information (MI)-energy tradeoff for cooperation and non-cooperation transmission schemes in WSNs. Through our numerical results, we observed that there exists a distance threshold, beyond which the cooperation based transmission scheme achieves better MI-energy tradeoff than the non-cooperation based transmission scheme. Also, it was observed that the correlation improves performance significantly at low observation SNR. • In Chapter 3 of the thesis, we considered decentralized rate selection in WLANs. Rate selection is unique to WLANs and was studied extensively in literature. However here, we considered a price based approach for rate selection. Pricing as noted in Chapter 1 is used extensively in Economics literature to motivate social behavior among the system users. In Chapter 3, we considered pricing to prevent low-rate users from occupying the channel for a long time, thereby preventing other users from their fair share to transmit on the channel. Thus leading to a reduction in the overall throughput of a WLAN. The price is imposed by the access point (AP) on each user and is proportional to the channel occupancy of the corresponding user. Owing to the decentralized rate selection in WLANs, we formulated the problem as a non-cooperative game, where the players of the game are WLAN users, their actions are (modulation) rates, and their utilities are equal to the difference of the throughput obtained for transmitting at a specific rate and a cost incurred due to the price imposed by AP. The resulting non-cooperative game was analyzed under  116  5.1. Summary of Work Accomplished the framework of supermodular games. Supermodular games as noted in Section 1.4.3 of Chapter 1 have rich structure, and further allow the player strategy sets to be discrete. Also the best response dynamics converge to a NE of the game. Through our simulation results, we observed that by appropriately tuning the price one can obtain higher overall throughput. This can be attributed to the reduction in occupancy of low-rate users and a corresponding increase in occupancy of moderate and high rate users. An important practical aspect of the proposed pricing based approach is, it does not require too many changes to the existing IEEE 802.11 standard. It only requires the AP to broadcast a price periodically. • In Chapter 4 of the thesis, we considered the performance of transmission control protocol (TCP) in spectrum- overlay cognitive radio (CR) networks under dynamic spectrum access (DSA). As noted before, a spectrum-underlay CR networks contains two types of users namely, primary users and secondary users. The secondary users access the channels designated for exclusive use of primary users opportunistically. Most existing research on CR networks concentrate on lower layer (such as physical, MAC, and routing layers) design aspects. In contrast here, we investigated performance of higher layer protocols, specifically TCP in spectrum-overlay CR networks. Specifically, through a simulation testbed in NS2 [76], we investigated the impact of number of channels on secondary user TCP throughput. This is important because, multi-channel transmission is a unique aspect of CR networks. Our simulation results suggest that, there exists an optimal number of channels that each secondary user needs to capture to maximize its TCP throughput. We also presented a brief survey, on CR networks, and on the various architectures proposed in literature for its realization.  117  5.2. Future Work  5.2  Future Work  The work in this thesis can be extended in several ways. Here, we present a few possible extensions for the future. • Node Selection in WSNs: In Chapter 2 of the thesis, we considered cooperative MIMO as one of the means to improve energy efficiency of WSNs. Another interesting area of research in this direction would be, selecting group of sensor nodes so that the total energy of WSN is minimized, but at the same time obtaining desired performance (in terms of estimation error) at the fusion center (FC). In general, activating too many sensors is not desirable owing to the increase in total energy consumption of WSN. Also one do not want to activate too few sensors, since it may affect the performance at the FC. Note that here we are interested not only to determine how many sensors to activate, but also decide on which sensors to activate. Further it would be of interest to track the time varying set of active sensor nodes. This is important to account for the target dynamics or sensor failure. • Socially Optimal Decentralized Rate Selection in IEEE 802.11 WLANs: In Chapter 3 of the thesis, we considered a price based rate selection to improve the performance of IEEE 802.11 WLANs. Another interesting solution approach would be to consider rate selection, so that the overall throughput of the WLAN is maximized. This is similar to the socially optimal power control problem considered in [57]. However the difference here is that the optimization variable is rate instead of power. The utility of each user is also different. Here the utility would be sum of product of throughput of each user and its channel occupancy. A more important difference is that the optimization variable (rate) takes only discrete values. • Modeling of TCP and other transport protocols with ARQ in CR Net118  5.2. Future Work works: In Chapter 4 of the thesis, we analyzed performance of TCP in CR networks via an NS2 simulation testbed. As part of future work it would be of interest to model its behavior in the multi-channel CR setting. 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Pillutla and V. Krishnamurthy, “Data Gathering in Correlated Wireless Sensor Networks with Cooperative Transmission,” Book chapter in Cooperative Communications for Improved Wireless Network Transmission, IGI-Global Publications, Editors: M. Uysal, 2008 (invited). • L. S. Pillutla and V. Krishnamurthy, “A Price based Approach to Decentralized Rate Selection in IEEE 802.11 WLANs,” submitted to IEEE Trans. on Wireless Communications, 2008. Conference Papers • L. S. Pillutla and V. Krishnamurthy, “Game Theoretic Rate Adaptation in SpectrumOverlay Cognitive Radio Networks,” to appear in the Proceedings of IEEE Global Communications Conference (Globecom) 2008, New Orleans, Louisiana, Nov. 2008. 134  Appendix A. List of Publications • L. S. Pillutla and V. Krishnamurthy, “Minimum Energy Data Gathering in Correlated Sensor Networks with Cooperative Transmission,” in Proceedings of IEEE ICC 2007, Glasgow, Scotland, June 2007. • L. S. Pillutla and V. Krishnamurthy, “Mutual Information and Energy Tradeoff in Correlated Wireless Sensor Networks,”, in Proceedings of IEEE ICC 2008, Beijing, China, May 2008. • L. S. Pillutla and V. Krishnamurthy, “Structural Results on Optimal Rate and Number of Clusters in Cluster based Cooperative MIMO Sensor Networks,” in proceedings of Asilomar Conference on Computers, Signals and Systems 2005, Monterey, CA, Nov. 2005 • L. S. Pillutla and V. Krishnamurthy, “Joint Rate and Cluster Optimization in Cooperative MIMO Sensor Networks”, in proceedings of IEEE SPAWC 2005, New York, NY, June, 2005.  135  

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