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Application of game theory in wireless communication networks Huang, Wei 2012

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Application of Game Theory in Wireless Communication Networks  by Wei Huang M. Phil., Hong Kong University of Science and Technology, 2007 B. Sc., Zhejiang University, 2005  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) February 2012 c Wei Huang, 2012  Abstract The ability to model independent decision makers whose actions potentially affect other decision makers makes game theory attractive to analyze the performances of wireless communication systems. Recently, there has been growing interest in adopting game theoretic methods to wireless networks for power control, rate adaptation and channel access schemes. This thesis focuses the application of dynamic game theory and mechanism design in cognitive radio networks, Wireless Local Area Networks (WLAN) and Long Term Evolution (LTE) systems. The first part of the thesis aims to optimize the system performance through the transmission rate adaptation among wireless network users. The optimal transmission policy of each user is analyzed by formulating such a problem under general-sum Markov game framework. Structural results are obtained and provably convergent stochastic approximation algorithm that can estimate the optimal transmission policies are proposed. Especially, in the switching control Markov game theoretic rate adaptation formulation, it is proved that the optimal transmission policies are monotone in channel state and there exists a Nash equilibrium at which every user deploys a monotone transmission policy. This  ii  structural result leads to a particularly efficient stochastic-approximationbased adaptive learning algorithm and a simple distributed implementation of the transmission rate control. This thesis also considers the spectrum allocation problem in an LTE system. By incorporating cognitive capacities into femtocell base stations, the Home Node-Bs (HNBs) can be formulated as cognitive base stations competing for the spectrum resource while trying to minimize the interference introduced to the evolved Node-B (eNB) which is also referred as primary base station. Given the primary base station spectrum occupancy, the spectrum allocation problem among HNBs can be formulated under game theoretic framework. The correlated equilibrium solution of such a problem is being investigated. A distributed Resource Block (RB) access algorithm and a correlated equilibrium Q-learning algorithm are proposed to compute the spectrum allocation solutions under static and dynamic environments, respectively. The last part of the thesis uses mechanism design to design a truth revealing opportunistic scheduling system in a cognitive radio system. A mechanism learning algorithm is provided for users to learn the mechanism and to obtain the Nash equilibrium policy.  iii  Preface This thesis is completed based on the following publications. In some cases the conference papers contain materials overlapping with the journal papers.1 Book Chapters: • J. W. Huang, H. Mansour, and V. Krishnamurthy, “Transmission Rate Adaptation in Multimedia WLAN - A Dynamical Games Approach,” CRC Press, Taylor and Francis Group, 2011. • J. W. Huang, V. Krishnamurthy, “Transmission Control in Cognitive Radio as a Markovian Dynamic Game - Structural Result on Randomized Threshold Policies,” Auerbach Publications, CRC Press, Taylor and Francis Group, 2009. Journal Papers: • J. W. Huang and V. Krishnamurthy, “Transmission Control in Cognitive Radio as a Markovian Dynamic Game - Structural Result on Randomized Threshold Policies,” IEEE Transactions on Communications, vol. 58, no. 2, pp. 301-310, February 2010. 1  The author also publishes under the names of J. W. Huang and Jane Wei Huang.  iv  • J. W. Huang, H. Mansour, and V. Krishnamurthy, “A Dynamical Games Approach to Transmission Rate Adaptation in Multimedia WLAN,” IEEE Transactions on Signal Processing, vol. 58, no. 7, pp. 3635-3646, July 2010. • J. W. Huang and V. Krishnamurthy, “Cognitive Base Stations in LTE/3GPP Femtocells: A Correlated Equilibrium Game Theoretic Approach,” IEEE Transactions on Communications, December 2011. • J. W. Huang and V. Krishnamurthy, “Game Theoretical Issues in Cognitive Radio Systems,” Journal of Communications, vol. 4, no. 10, pp. 790-802, November 2009. (Invited Paper) • J. W. Huang, H. Mansour, and V. Krishnamurthy, “A Dynamical Game Approach to Transmission Rate Adaptation in Wireless Multimedia Networks,” IEEE MMTC e-letter, vol. 4, no. 8, pp. 11-13, September 2009. (Invited Paper) Conference Papers: • J. W. Huang and V. Krishnamurthy, “Transmission Control in Cognitive Radio Systems with Latency Constraints as a Switching Control Dynamic Game,” in Proceedings of IEEE CDC, pages 3823-3828, December 2008. • J. W. Huang and V. Krishnamurthy, “Rate Adaptation for Cognitive Radio Systems with Latency Constraints,” in Proceedings of IEEE Globecom, pages 1-5, November-December 2008.  v  • J. W. Huang and V. Krishnamurthy, “Truth Revealing Opportunistic Scheduling in Cognitive Radio Systems,” in Proceedings of IEEE SPAWC, June 2009. • H. Mansour, J. W. Huang, and V. Krishnamurthy, “Multi-User Scalable Video Rate Control in Cognitive Radio Networks as a Markovian Dynamic Game,” in Proceedings of IEEE CDC, December, 2009. • J. W. Huang, Q. Zhu, V. Krishnamurthy, and T. Basar, “Distributed Correlated Q-Learning for Dynamic Transmission Control of Sensor Networks,” in Proceedings of IEEE ICASSP, March 2010.  vi  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . . . . xxi  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.2  Analytical Tools . . . . . . . . . . . . . . . . . . . . . . . . .  9  1.2.1  Game Theory for Communications . . . . . . . . . . .  9  1.2.2  Stochastic Approximation Algorithms . . . . . . . . .  13  1.2.3  Mechanism Design . . . . . . . . . . . . . . . . . . . .  14  vii  1.3  Main Contributions . . . . . . . . . . . . . . . . . . . . . . . .  15  1.4  Thesis Organization . . . . . . . . . . . . . . . . . . . . . . .  21  2 Transmission Control in Cognitive Radio as a Markovian Dynamic Game . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24  2.2  Rate Adaptation Problem Formulation . . . . . . . . . . . . .  26  2.2.1  System Description and TDMA Access Rule  . . . . .  26  2.2.2  Action and Costs . . . . . . . . . . . . . . . . . . . . .  29  2.2.3  Switching Control Game and Transition Probabilities  30  2.2.4  Switching Controlled Markovian Game Formulation .  31  2.3  Randomized Threshold Nash Equilibrium for Markovian Dynamic Game . . . . . . . . . . . . . . . . . . . . . . . . . . . .  33  2.3.1  Value Iteration Algorithm . . . . . . . . . . . . . . . .  33  2.3.2  Structural Result on Randomized Threshold Policy . .  35  2.3.3  Stochastic Approximation Algorithm . . . . . . . . . .  38  2.4  Numerical Examples . . . . . . . . . . . . . . . . . . . . . . .  43  2.5  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  3 A Dynamic Game Approach to Transmission Rate Adaptation in Multimedia WLAN . . . . . . . . . . . . . . . . . . 51 3.1  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .  52  3.2  System Description and the Video Rate-Distortion Model . .  55  3.2.1  System Description . . . . . . . . . . . . . . . . . . . .  55  3.2.2  Scalable Rate-Distortion Modelling . . . . . . . . . . .  56  3.3  Uplink Rate Adaptation Problem Formulation viii  . . . . . . . .  59  3.3.1  Actions, Transmission Reward and Holding Cost . . .  61  3.3.2  Markovian Game Formulation . . . . . . . . . . . . . .  65  3.3.3  System Access Rule . . . . . . . . . . . . . . . . . . .  67  3.3.4  Transition Probabilities and Switching Control Game Formulation . . . . . . . . . . . . . . . . . . . . . . . .  3.4  67  Nash Equilibrium Solutions to the Markovian Dynamic Game 70 3.4.1  Value Iteration Algorithm . . . . . . . . . . . . . . . .  70  3.4.2  Structural Result on Randomized Threshold Policy . .  73  3.4.3  Learning Nash Equilibrium Policy via Policy Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .  75  3.5  Numerical Examples . . . . . . . . . . . . . . . . . . . . . . .  78  3.6  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  4 Cognitive Base Stations in 3GPP LTE Femtocells: A Correlated Equilibrium Game Theoretic Approach . . . . . . . 86 4.1  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .  87  4.2  Resource Allocation Among HeNBs: Problem Formulation . .  90  4.2.1  System Description . . . . . . . . . . . . . . . . . . . .  91  4.2.2  Utility Function . . . . . . . . . . . . . . . . . . . . .  95  4.3  Correlated Equilibrium Solutions with a Game-Theoretic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.1  Definition of Correlated Equilibrium . . . . . . . . . . 101  4.3.2  Decentralized RB Access Algorithm . . . . . . . . . . 102  4.3.3  Convergence of RB Access Algorithm . . . . . . . . . 105  ix  4.3.4  Correlated Equilibrium under Dynamic Environments and Curse of Dimensionality  . . . . . . . . . . . . . . 107  4.4  Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 109  4.5  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112  5 Application of Mechanism Design in Opportunistic Scheduling under Cognitive Radio Systems . . . . . . . . . . . . . . 114 5.1  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115  5.2  Opportunistic Scheduling in Cognitive Radio Systems . . . . 116  5.3  5.2.1  System States Description . . . . . . . . . . . . . . . . 117  5.2.2  Conventional Opportunistic Accessing Scheme . . . . 119  The Pricing Mechanism . . . . . . . . . . . . . . . . . . . . . 121 5.3.1  The Pricing Mechanism . . . . . . . . . . . . . . . . . 121  5.3.2  Economic Properties of the Pricing Mechanism . . . . 123  5.3.3  Mechanism Learning Algorithm . . . . . . . . . . . . . 125  5.4  Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 127  5.5  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128  6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 129 6.1  Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . 129  6.2  Discussion on Results and Algorithms . . . . . . . . . . . . . 131  6.3  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132  6.4  Future Research 6.4.1  . . . . . . . . . . . . . . . . . . . . . . . . . 133  How to Avoid the Curse of Dimensionality in Stochastic Games? . . . . . . . . . . . . . . . . . . . . . . . . 133  6.4.2  Transmission Scheduling with Partially Observable States134 x  6.4.3  More Analytical Results on Correlated Equilibrium in Stochastic Games . . . . . . . . . . . . . . . . . . . . . 135  6.4.4  Other Types of Mechanism Design . . . . . . . . . . . 136  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 A Proof of Theorem 2.3.2 . . . . . . . . . . . . . . . . . . . . . 150 A.1 Proof of Theorem 2.3.2 . . . . . . . . . . . . . . . . . . . . . . 150 A.2 Proof of Lemma A.1.1 . . . . . . . . . . . . . . . . . . . . . . 153 B Proof of The Convergence of Algorithm 3 . . . . . . . . . . 155 C Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . 157 C.1 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . 157 C.2 Proof of Lemma C.1.1 . . . . . . . . . . . . . . . . . . . . . . 160  xi  List of Tables Table 3.1  Average incoming rate and distortion (Y-PSNR) characteristics of the two video users. . . . . . . . . . . . . . . . .  xii  79  List of Figures Figure 1.1  Summary of game theoretic topics being covered in this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 2.1  5  An overlay TDMA cognitive radio system where users access the spectrum hole following a predefined access rule. Each user is equipped with a size L buffer, a decentralized scheduler and a rate adaptor for transmission control. . .  Figure 2.2  27  The transmission policy of user 1 in a switching controlled general-sum dynamic game system computed by the value iteration algorithm. The result is obtained when the channel states of user 1 and 2 are h1 = 1 and h2 = 1, respectively. 44  Figure 2.3  The transmission policy of a certain user in a switching controlled zero-sum dynamic game system obtained by value iteration algorithm. The first subfigure shows the result when the channel states of both users are h1 = 1 and h2 = 1, while the second subfigure shows the result when the channel states of user 1 and 2 are h1 = 1 and h2 = 2, respectively. . . . . . . . . . . . . . . . . . . . . .  xiii  45  Figure 2.4  The Nash equilibrium transmission control policy computed via stochastic approximation algorithm. A 2-user system is considered, and each user has a size 10 buffer. .  Figure 2.5  Performance comparison between the proposed stochastic approximation algorithm and myopic policy. . . . . . . . .  Figure 2.6  47  48  Tracking Nash equilibrium policy using the stochastic approximation algorithm (Algorithm 2). System parameters change at the 100th iteration, as specified in Section 2.3. These two figures compare the estimated randomization factor and buffer thresholds with the discrete optimal values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.1  49  A WLAN system where each user is equipped with a size B buffer, a decentralized scheduler and a rate adaptor for transmission control. The users transmit a scalable video payload in which enhancement layers provide quality refinements over the base layer bitstream.  Figure 3.2  . . . . . . . . .  54  Illustration of the modeled PSNR estimates (a) and (b), and the modeled rate estimates (c) and (d) for the base layer and two CGS enhancement layers of the sequences Foreman and Football. . . . . . . . . . . . . . . . . . . . .  xiv  59  Figure 3.3  Example of the buffer control mechanism assumed in this chapter where fin,k = 1 (k = 1, 2, . . . , K). At every timeslot, a new coded video frame enters the buffer. The buffer output depends on the scheduling algorithm involved, the buffer occupancy, and the channel quality. If a user is scheduled for transmission, then the action taken will extract a specific number l of video frame layers from up to N frames stored in the buffer.  Figure 3.4  . . . . . . . . . . . . . . .  62  The Nash equilibrium transmission policy obtained via value iteration algorithm for user 1 (a) and user 2 (b). The result (a) is obtained when h2 = 1 and b2 = 1 and (b) is obtained when h1 = 1 and b1 = 1, respectively. The transmission delay constraint is specified to be 25 ms. Each Nash equilibrium transmission policy is a randomization of two pure monotone policies. . . . . . . . . .  Figure 3.5  81  The Nash equilibrium transmission control policy obtained via policy gradient algorithm for user 1. The result is obtained with a 25 ms transmission delay constraint and the states of users 2 are h2 = 1 and b2 = 1. The transmission policy is monotone nonincreasing on its own buffer state.  xv  82  Figure 3.6  Result of the transmission of the Football sequence comparing the performance in terms of video PSNR and buffer utilization between the proposed switching control game policy and the myopic policy with 80 ms delay constraint. The result shows that the proposed switching control game policy performs better than the myopic policy. . . . . . .  Figure 3.7  84  Result of the transmission of the Foreman sequence comparing the performance in terms of video PSNR and buffer utilization between the proposed switching control game policy and the myopic policy with 80 ms delay constraint. The result shows that the proposed switching control game policy performs better than the myopic policy. . . . . . .  Figure 4.1  System schema of a single eNB macrocell which contains multiple HeNB femtocells in a 3GPP LTE network.  Figure 4.2  85  . . .  91  The effect of different values of (α2 , α3 ) defined in (4.9) on the global system performance specified in (4.3) . . . . 111  Figure 4.3  The effect of different values of (α2 , α3 ) defined in (4.9) on the global system average performance specified in (4.5).112  Figure 4.4  Performance comparison between RB access algorithm (Algorithm 5) and the “Best Response” algorithm. . . . . 113  Figure 5.1  A K secondary users cognitive radio where the central scheduler does the scheduling according to the opportunistic accessing algorithm. . . . . . . . . . . . . . . . . . 117  xvi  Figure 5.2  The MSE of the reported buffer states and throughput states using the mechanism learning algorithm. The result is of a 30 users system with L = 5, Qρ = 10 and P = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127  xvii  Glossary 3GPP 3rd Generation Partnership Project AWGN Additive White Gaussian Noise BER Bit Error Rate CCI Co-Channel Interference CGS Coarse-Grained Scalability CIF Common Intermediate Format CSMA/CA Carrier Sense Multiple Access with Collision Avoidance DCF Distributed Coordination Function DDB Drop Dependency Based DFP Dynamic Frequency Planning DPB Drop Priority Based DSL Digital Subscriber Line eNB Evolved Node-B xviii  HeNB Home Evolved Node-B HSDPA High Speed Download Package Access IEEE Institute of Electrical and Electronics Engineers IUI Inter User Interference JSVM Joint Scalable Video Model LMS Least Mean Squares LTE Long Term Evolution MAC Medium Access Control MAD Mean Absolute Difference Mbps Megabit Per Second MDP Markov Decision Process MGS Medium-Grained Scalability MSE Mean Squared Error NALU Network Abstraction Layer Unit ODE ordinary-differential-equation OFDMA Orthogonal Frequency-Division Multiple Access PER Packet Error Rate PSNR Peak Signal-to-Noise Ratio xix  QAM Quadrature Amplitude Modulation QoS Quality of Service QP Quantization Parameter RB Resource Block SNR Signal-to-Noise Ratio SON Self-Organized Network SPSA Simultaneous Perturbation Stochastic Approximation SVC Scalable Video Coding TDMA Time Division Multiple Access UMTS Universal Mobile Telecommunication System VCG Vickrey-Clark-Groves WLAN Wireless Local Area Network Y-PSNR Luminance Peak Signal-to-Noise Ratio  xx  Acknowledgments I would like to thank Dr. Krishnamurthy for his tireless efforts to provide me with the most dedicated supervision. I am indebted to Dr. Krishnamurthy for teaching me technical writing and introducing me to several exciting research areas. I am also very grateful for his encouragement, generosity with time and valuable guidance throughout my doctorate program. I also would like to thank the Enterprise Software Solutions team, Research in Motion (RIM) for the 8-months coop opportunity to work on the SharePoint project. I hugely benefited and wish to thank profusely for what I learned from the project. I also enjoyed the works I have been assigned. It is a pleasure working with the team. I am grateful to Quanyan Zhu and Dr. Tamer Basar at the University of Illinois at Urbana-Champaign, for their collaboration, as well as the discussions that helped defining some research directions for my degree. My warmest thanks are to my colleagues, whom I would miss the most when I no longer come to the lab everyday. Together we had a lot of happiness and I forever thankful for their sharing and caring, for the laughters, and the friendship that I treasure. Especially, Sahar Monfared, Raymond  xxi  Lee, Kevin Topley, Omid Namvar, Alex Wang and Mustafa Fanaswala have been extremely good friends. I also would like to thank my dearest friend from the wireless communication lab: Wing-Kwan Ng, whom I have known for more than 6 years. I will never forget the lunch time we shared together everyday. My very heartfelt thanks are to Emad Hajimiri. I appreciate him for being such a good friend during all the good times and bad times. His friendship has been a very important part of my life. My deepest thoughts are, as always, with my parents, my younger brother and my younger sister. All of life, they have always been there to take care, to teach, to support and to love. I thank my father and my mother for teaching me to be a good person, and for letting me know that their faith in me would never fade.  xxii  Dedication To my family, whom I love the most.  xxiii  Chapter 1  Introduction Wireless networks optimization is a vast area of research, in which crosslayer optimization has emerged as a new promising approach. Telecommunication systems such as cellular networks, the Internet, Wireless Local Area Network (WLAN)s, Long Term Evolution (LTE) systems and cognitive radio systems have traditionally been designed using layered architecture based models. A layered architecture allows conceptional layers to be designed independently and hence simplifies the design, implementation and deployment of communication networks. For wireless systems, however, there are other challenges which create dependencies across layers and make the traditional modular design approach inefficient. For example, the wireless channel quality, buffer state, and the resource availability are typically varying with time. This time-varying nature of the wireless communication systems immediately motivates transmission adaptation, resource allocation and opportunistic scheduling, which exploit the statics of the system parameters for transmission, resource or rate control decisions. 1  Another motivation for cross-layer optimization and design is the fact that the spectrum resource and wireless channel is inherently a shared medium. This issue creates interdependencies among users, and also between the physical and higher layers (e.g., Medium Access Control (MAC) layer). Additionally, the wireless medium also offers some new modalities of communication that the layered architecture do not accommodate [84]. For example, the physical layer of a wireless network may be capable of receiving multiple packets at the same time, or partial collision resolution. In such scenarios, the traditional approach of designing MAC protocol independently of the physical layer will not be adequate. An additional difference between wireless and wired networks that motivates cross-layer approaches is the broadcast nature of the wireless channel allows users to collaborate [84]. The wireless networks considered in this thesis include cognitive radio networks, WLAN multimedia systems and LTE systems. Cognitive radio systems [12, 38] present the opportunity to improve spectrum utilization by detecting unoccupied spectrum holes and assigning them to secondary users. Resource management in cognitive radio networks involves user scheduling and transmission rate adaptation. Institute of Electrical and Electronics Engineers (IEEE) 802.11 WLAN has been widely accepted as the dominant technology for indoor broadband wireless networking. The most important differences between the WLAN and the MAC protocols of most wireless networks is the application of a Distributed Coordination Function (DCF) for sharing access to the medium based on the Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) protocol. In the CSMA/CA mechanism, a node 2  listens to the channel before transmission to determine whether someone else is transmitting and an acknowledgement packet is sent a short time interval after the receiving node receives the packet. Chapter 3 considers the scheduling, rate adaptation, and buffer management in a multiuser WLAN where each user transmits scalable video payload. It proposes a modified CSMA/CA channel access mechanism which takes into account the dynamic behaviours of the video variation, as well as the channel quality and transmission delay of each user. LTE is an enhancement to the Universal Mobile Telecommunication System (UMTS) which will be introduced in 3rd Generation Partnership Project (3GPP) Release 8 [1]. LTE is marketed as the 4th generation cellular technology, aiming at delivering high speed broadband connection to stationary or mobile users. One important feature of 3GPP LTE system, differentiating it from previous generations of cellular systems, is the distributed network architecture, the key issue of which is to deploy femtocells to satisfy a variety of service requirements. These femtocell access points are low-cost, low-power, plug and play cellular base stations that are well-suited to comply with the rising wireless traffic demand. These small base stations are connected to the network operator through the end-user’s broadband connection. Home Evolved Node-B (HeNB) can significantly improve the performance of current cellular networks by providing high data rates of the order of several Megabit Per Second (Mbps) and improved coverage inside home and offices [2]. The primary theoretical contributions of the thesis include important analytical results on the structure of optimal transmission policies. The 3  analytical results are original, and under the given assumptions the optimal transmission policies are monotone in channel states (see Chapters 2, 3). The monotone structural results are of utmost significance in optimization problems as they reduce the optimization dimensionality to the extreme. E.g., in Chapter 2, the transmission policy optimization problem is converted from a function optimization problem to a scalar optimization problem by utilizing the structural result on the Nash equilibrium transmission policy. The conversion allows the application of many gradient-based stochastic approximation algorithms to solve the optimal policy estimation problem [83]. Other contributions of the thesis are the derivation of equilibrium policy learning algorithms. Due to the structural results, the estimation algorithms are very efficient, the equilibrium learning algorithms derived in Chapters 2, 3, 4 can track slowly time-varying systems. The real-time learning algorithms are based on stochastic approximation and do not require a central controller, communication or exchange of information between users, or knowledge of system parameters such as the resource availability, number of active users, or the channel state probability distribution functions. The derivation of structural results and estimation algorithms in this thesis involves novel use of several powerful mathematical tools, such as game theory, stochastic approximation, mechanism design; and clever application of a number of mathematical concepts such as Nash equilibrium, correlated equilibrium, static games, stochastic Markov games, and supermodularity. The mathematical tools will be outlined later in this chapter. The remainder of this introductory chapter is organized as follows. The 4  Game Concepts  Stochastic Game (Chapter 2, 3)  Static Game (Chapter 4, 5)  Solution Concepts  Tools  Equilibrium  Mechanism Design  Nash Equilibrium (Chapters 2, 3, 5) Pricing Mechanism (Chapter 5) Correlated Equilibrium (Chapter 4)  Figure 1.1: Summary of game theoretic topics being covered in this thesis. next section contains a brief thesis overview. Section 1.2 introduces the major theoretical tools utilized in this thesis, with emphasis on their application in wireless communication systems. A summary of main contributions of this thesis is given in Section 1.3. Lastly, Section 1.4 outlines the thesis organization.  1.1  Thesis Overview  Fig. 1.1 summarizes the game theoretic topics/concepts being covered in this thesis. This thesis consists of three main parts. The first part, consisting of Chapter 2 and Chapter 3, is concerned with transmission adaptation in a cognitive radio system and a WLAN multimedia network using switching control game theoretical approach. The second part of the thesis (Chapter 4 ) investigates the correlated equilibrium resource allocation solution in an LTE system. The last part of this thesis (Chapter 5) aims to design a truth revealing pricing mechanism to improve the traditional opportunistic scheduling algorithm in a cognitive radio system. In the transmission adaptation using switching control game theoretical  5  approach problem, the aim is to analyze the Nash equilibrium transmission policy that optimizes the infinite horizon expected total discounted utility function with the infinite horizon expected total discounted latency constraint. The optimization approach taken is a switching control stochastic game theoretic approach which formulates the Markovian block fading channels as a finite state Markov chain. This stochastic game theoretic framework assumes users are selfish, rational, and aim to maximize their own expected reward. The stochastic game theoretic framework leads to a particularly useful threshold structural result and hence allows users to learn their optimal transmission policies online in a completely decentralized fashion. Compared to the centralized approach, the stochastic game theoretic formulation yields a very efficient way to optimize the transmission adaptation problem in a cognitive system or a WLAN. In particular, in the game theoretic setting, every user optimizes its transmission policy; then at equilibrium, no user will unilaterally deviate from its policy. Furthermore, instead of considering the myopic solutions, the users aim to optimize the infinite time horizon expected total discounted utility functions with latency constraints. By utilizing the system state transition probability information, it further optimizes the transmission adaptation problem. It will also be shown later in the thesis that the switching control game theoretic transmission adaptation Nash equilibrium solution has a particularly interesting and useful monotone structural result, which substantially simplifies the process of learning the optimal transmission policy. Specifically, it is shown that the Nash equilibrium transmission policy is a randomization of two pure policies, each of these two pure policies is nondecreasing (or nonincreasing) with 6  the buffer occupancy state. Therefore, to track the switching control game theoretic optimal transmission policy, each user needs to adaptively estimate only the thresholds and the randomization factor. Effectively, the function optimization problem is converted to the searching for few parameters, which can be solved easily and efficiently by many stochastic approximation algorithms. Taking into account the simplicity of the switching control game theoretic model and the implementation of switching control game theoretic solutions, the switching control game approach is practically very useful, especially for large scale wireless networks, where certain statistical properties (e.g., number of active users, channel state transition probabilities) may vary with time. The second part of this thesis focuses on the correlated equilibrium resource allocation policies in an LTE system. In particular, it considers the spectrum allocation problem in an Orthogonal Frequency-Division Multiple Access (OFDMA) LTE downlink system which consists of a macrocell base station and multiple femtocell base stations. By incorporating cognitive radio capabilities into femtocell base stations, the HeNBs are formulated as cognitive base stations seeking to maximize the spectrum utility while minimizing the interferences to the Evolved Node-B (eNB) (primary base station) in a spectrum overlay LTE system. Given the resource occupancy of the eNB, the competition for the spectrum resources among HeNBs can be formulated in a game theoretic setting. However, instead of computing the Nash equilibrium policy of the formulated game, we seek to characterize and compute the correlated equilibrium policy set [7, 8]. We formulate the resource allocation problem among HeNBs in the downlink of an LTE 7  system under static game theoretic framework, where the correlated equilibrium solutions of the formulated static game is being investigated. A distributed Resource Block (RB) access algorithm is proposed to compute the correlated equilibrium RB allocation policy. The RB access algorithm has distributed implementations that permit them to be implemented in LTE systems. The third part of the thesis uses mechanism design to improve the performance of the opportunistic scheduling in cognitive radio systems. The conventional opportunistic scheduling algorithm in cognitive radio networks does the scheduling among the secondary users based on the reported state values. However, such opportunistic scheduling algorithm can be challenged in a system where each secondary user belongs to a different independent agent and the users work in competitive way. In order to optimize his own utility, a selfish user can choose not to reveal his true information to the central scheduler. We proposed a pricing mechanism which combines the mechanism design with the opportunistic scheduling algorithm and ensures that each rational selfish user maximizes his own utility function, at the same time optimizing the overall system utility. The proposed pricing mechanism is based on the classic Vickrey-Clark-Groves (VCG) mechanism and had several desirable economic properties. A mechanism learning algorithm is then provided for users to learn the mechanism and to obtain the Nash equilibrium.  8  1.2  Analytical Tools  This thesis deploys several powerful mathematical tools that come from three research areas that have had many applications in electrical and computer engineering. In particular, we user game theory, which is a universally accepted tool in economics and social science and have been widely used in communications and networking [60]; stochastic approximation, which is a fundamental concept in stochastic control and optimization; and mechanism design, which is used to prevent malicious behaviours from game players. This section attempts to briefly overview these three areas.  1.2.1  Game Theory for Communications  Game theory is a branch of applied mathematics that provides models and tools for analyzing situations where multiple rational agents interact to achieve their goals. The classic book The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern published in 1944 [71] laid the first foundations for the research field of game theory, which now has numerous application in economics, social science, engineering and recently computer science. Recent textbooks on game theory include [13, 33, 74]. The treatment of game theory in [33] is very comprehensive and complete. Additionally, [74] and [13] are two very well-written graduate level textbooks in game theory. In communications engineering, game theory is often used for distributed resource management algorithms. They underlying motivation is that game theoretic solutions are naturally autonomous and robust. Examples of applications of game theory in wireless communications include transmission or power control [40, 65, 81], pricing [39], flow 9  and congestion control [4], and load balancing [34]. Static Game vs. Stochastic Game Static game framework has been used to compute the Nash equilibrium power allocation policies in cognitive radio networks [38, 81, 94]. Different from a dynamic game where players make sequence of decisions, a static game is one in which all players make decisions one time simultaneously, without knowledge of the strategies of other players. [81] gives an example of a cognitive radio system where each user aims to maximize its information rate subject to the transmission power constraint. A distributed asynchronous iterative water-filling algorithm is used to compute the Nash equilibrium power allocation policy of such a system using static game theoretic approach. Most games considered in wireless communication systems to date are static games. [73] and [30] are two of such examples which apply static gametheoretic analyses to address the resource allocation problem in cognitive radio networks. Stochastic dynamic game theory is an essential tool for cognitive radio systems as it is able to exploit the correlated channels in the analysis of decentralized behaviours of cognitive radios. The concept of a stochastic game, first introduced by Lloyd Shapley in early 1950s, is a dynamic game played by one or more players. The elements of a stochastic game include system state set, action sets, transition probabilities and utility functions. It is an extension of the single player Markov Decision Process (MDP) to include the multiple players whose actions all impact the resulting payoffs and next state. A switching control 10  game [31, 68, 90] is a special type of stochastic dynamic game where the transition probability in any given state depends on only one player. It is known that the Nash equilibrium for such a game can be computed by solving a sequence of Markov decision processes. In the first part of Chapter 3, we formulate the resource allocation problem among secondary base stations under static environment using static game theoretic framework. In the second part of Chapter 3, such resource allocation problem is considered under dynamic environment and stochastic game theoretic tools are being applied. Chapter 1 and Chapter 2 use switching control game to solve the transmission control problem in a cognitive radio system and a WLAN system, respectively. Equilibrium and Learning A major breakthrough in game theory was due to John Nash when he introduced the solution concept of Nash equilibrium in the early 1950’s. A strategy combination constitutes a Nash equilibrium if each agent’s strategy is optimal against other agent’s strategies [74]. As a result, at a Nash equilibrium agents have no motivation to unilaterally deviate from their strategies. Nash equilibrium plays an essential role in analysis of conflict and cooperation in economics and social sciences. However, Nash equilibrium suffers from limitations, such as non-uniqueness, loss of efficiency, non-guarantee of existence. In game theory, a correlated equilibrium is a solution concept which is more general than the Nash equilibrium [7, 8]. A correlated equilibrium is defined as follows. Each player in a game chooses his action according to his observation of the value of a 11  signal. A strategy assigns an action to every possible observation a player can make. If no player would deviate from the recommended strategy, the distribution is called a correlated equilibrium. Compared to Nash equilibria, correlated equilibria offer a number of conceptual and computational advantages, including the facts that new and sometimes more “fair” payoffs can be achieved, that correlated equilibria can be computed efficiently for games in standard normal form, and that correlated equilibria are the convergence notion for several natural learning algorithms. Furthermore, it has been argued that the correlated equilibria are the natural equilibrium concept consistent with the Bayesian perspective [8]. In this thesis, Chapter 2 and Chapter 3 look in to the Nash equilibrium solutions of the transmission policy, and Chapter 4 is seeking for the correlated equilibrium resource allocation policies. While game theory and the equilibrium solution concepts are extremely useful and have numerous applications, the theory of learning in games is yet relatively less complete in the sense that convergence results are hard to establish and may require either restrictive, or hard to verify conditions. [32] contains essential existing results in the field of learning in games together with original work by the authors. The most studied methods of learning in games are fictitious play and best response dynamic [32]. These two methods have similar asymptotic behaviours and convergence to Nash equilibrium, in general, cannot be proved for either of the methods. The exceptions are potential games, where users have a common utility function; then both fictitious play and best response dynamic converge to a Nash equilibrium. Emergence of a Nash equilibrium can be proved for some 2-person games. 12  Other methods of learning in games include regret-minimizing algorithm. It has been shown in literature that if a game is repeated infinitely many times such that every player plays according to a regret-minimizing strategy, then the empirical frequencies of play converge to a set of correlated equilibrium [36, 52].  1.2.2  Stochastic Approximation Algorithms  Stochastic approximation/optimization algorithms are widely used in electrical computer engineering to recursively estimate the optimum of a function or its root, see for example [54, 83] for excellent expositions of this area. The first papers on the stochastic approximation methods are those by Robbins and Monro [79] and Kiefer and Wolfowitz [49] in the early 1950’s. The well-known Least Mean Squares (LMS) adaptive filtering algorithm is a simple example of a stochastic approximation algorithm with a quadratic objective function. Stochastic approximation algorithm has been applied to reinforcement learning for stochastic control problems in [10], learning equilibria games [43, 44], as well as optimization and parametric identification problems (e.g., recursive maximum likelihood and recursive expectation maximization algorithms [50]). In tracking applications, the step size of a stochastic approximation algorithm is chosen as a small constant. For such constant step size algorithm, one typically proves weak convergence of the iterates generated by the stochastic approximation algorithm. Weak convergence is a generalization of convergence in distribution to a function space. The weak convergence analysis of stochastic approximation algorithms with Markovian noise has 13  been pioneered by Kushner and co-workers, see [54] and references therein. It was demonstrated in the 1970’s that the limiting behaviour of a stochastic approximation algorithm can be modeled as a deterministic ordinarydifferential-equation (ODE). This is the basis of the so-called ODE method for convergence analysis of stochastic approximation algorithms. In wireless networks, as there are usually underlying dynamics (e.g., a correlated fading channel) or factors that can only be measured in noise such as expected system throughput, stochastic approximation algorithms play an important role in optimization. In this thesis, gradient-based stochastic approximation algorithms are used for adaptive and distributed estimation of the optimal transmission policies in Chapters 2, 3.  1.2.3  Mechanism Design  The theory of mechanism design in economics/game theory deals with the design of multi-agent interactions of rational agents [63]. It is fundamental to Economics, as well as to the attempt to design protocols for non-cooperative computational wireless communication environment [15]. The importance of its application in the wireless communication world has increased dramatically due to the desire to design economic interactions in the wireless network setup. In a typical mechanism design setup, a center attempts to perform some task, or to arrive at some decision, based on information available to a set of agents. The major problem is that the agents might not supply the desired information to the center, and might cheat the center in order to increase their individual payoffs. The central problem is to design a mechanism that when the agents interact rationally through that 14  mechanism, the center will be able to obtain its objective. In order to do so, a mechanism has to be designed; typically, this mechanism determines actions to be taken by the center as a function of messages sent to its by the agents; the agents’ strategies correspond to the messages they send to the center as a function of the private information they have. Such interaction is modeled by a Bayesian game. The aim is to design a mechanism that when the agents behave rationally (i.e., according to an equilibrium of the corresponding game), a desired behaviour is obtained. A milestone in mechanism design is the VCG mechanism, which is a generalization of Vickrey’s second price auction [89] proposed by Clark [20] and Groves [35]. The particular pricing policy of the VCG mechanism makes reporting true values the dominant strategy for all the players. Chapter 5 is an example where we model each user in a cognitive radio as a selfish player aiming to optimize his own utility and we try to find a mechanism which ensures efficient resource allocation within the network [42].  1.3  Main Contributions  This section presents a summary of methodologies, main structural results and learning algorithms developed in this thesis. As mentioned previously, the thesis consists of three parts. The first part of the thesis focuses on the Nash equilibrium solutions under switching control stochastic game setting. The second part of the thesis looks into the correlated equilibrium solutions of resource allocation policies in a static game environment. The last part of the thesis studies the mechanism design and its application in opportunistic scheduling. In each of the considered optimization problem, analytical 15  results and then numerical algorithms that exploit the analytical results are derived. An outline of the progression of ideas in the thesis is as follows. Chapter 2 and Chapter 3 constitute the first main part of the thesis, where the transmission adaptation problem under a constrained switching control stochastic game framework is analyzed. Switching control game is a special type of dynamic game where the transition probability in any given state depends on only one player. It is known that the Nash equilibrium for such a game can be computed by solving a sequence of Markov decision processes. Chapter 2 considers the rate adaptation problem in a Time Division Multiple Access (TDMA) cognitive radio system model. The interaction among secondary users is characterized as a competition for the spectrum hole and can naturally be formulated as a dynamic game. By modeling transmission channels as correlated Markovian sources, the transmission rate adaptation problem for each user can be formulated as a general-sum switching control Markovian dynamic game with a latency constraint. The structure of the Nash equilibrium transmission policy is studied and a stochastic algorithm is proposed for numerically solving the optimization problem. The main results in the chapter include the followings: • We formulate the secondary user rate adaptation problem in a cognitive radio network as a constrained general-sum switching control Markovian dynamic game. The TDMA cognitive radio system has a pre-specified channel access rule that is typically a function of secondary user channel qualities and buffer occupancies. The Markovian  16  block fading channels are formulated as a finite state Markov chain, and each secondary user aims to optimize its own utility with a transmission delay constraint. • It is shown that under reasonable assumptions, the Nash equilibrium policy of the transmission control game (formulated as a general-sum switching control game) is a randomized mixture of two pure policies. Each of the policies is monotone nondecreasing in the buffer occupancy state. We combine the latency constraint with the optimization objective using a Lagrange multiplier. The original constrained problem is then transformed into an unconstrained Markovian game. It is shown that the Nash equilibrium policy of such a game can be computed by using a value iteration algorithm. • Based on the structural result on the Nash equilibrium policy, a stochastic approximation algorithm is proposed to compute the policy. The algorithm provides insight into the nature of the solution without brute force computation and is able to adapt the Nash equilibrium policy in real time to the non-stationary channel and user statistics. Numerical results of the stochastic approximation algorithm are shown to verify the performance of the algorithm. Chapter 3 considers the transmission adaptation problem in an IEEE 802.11 WLAN which deploys the CSMA/CA. By exploiting the correlated time varying sample paths of the channels and buffers, the transmission rate adaptation problem can be formulated as a Markovian dynamic game. We also describe the details of the WLAN system under study as well as the 17  dynamic behaviour of video sources and the scalable rate-distortion models used to represent this behaviour. The main results presented in Chapter 3 are listed as follows. • We consider a multiuser WLAN system where each user is equipped with a scalable video quality encoder delivering video bitstream that conforms with scalable video coding (SVC) which is the scalable extension of H.264/AVC. We address the scheduling, rate-control, and buffer management of such system, and formulate the problem as a constrained switching control stochastic dynamic game combined with rate-distortion modeling of the source. The video states and the block fading channel qualities of each user are formulated as a finite state Markovian chain, and each user aims to optimize its own utility under a transmission latency constraint when it is scheduled for transmission. We then formulate the rate adaptation in the WLAN multimedia system as a constrained switching control dynamic Markovian game. • We propose a value iteration algorithm that obtains the Nash equilibrium transmission policy of the constrained general-sum switching control game. The Nash equilibrium policy of such constrained game is a randomization of two deterministic policies. The value iteration algorithm involves a sequence of dynamic programming problems with Lagrange multipliers. In each iteration of the value iteration algorithm, a Lagrange multiplier is used to combine the latency cost with the transmission reward for each user, the Lagrange multiplier is then updated based on the delay constraint. The algorithm converges to 18  the Nash equilibrium policy of the constrained general-sum switching control game. • The main result of the chapter is the structural result on the Nash equilibrium policy. It is shown that under reasonable assumptions, the Nash equilibrium policy is monotone nonincreasing on the buffer state occupancy. This structural result on the transmission policy enables us to search for the Nash equilibrium policy in the policy space via a policy gradient algorithm. The structural result is especially useful in dealing with multiuser systems where each user has a large buffer size. Finally, numerical results of the proposed switching control Markovian dynamic game policy in transmitting scalable video are presented. The second part of the thesis is presented in Chapter 4. The chapter considers the downlink spectrum allocation problem in a macrocell area containing multiple femtocells within an LTE system. By incorporating cognitive capabilities into femtocell base stations, the HeNBs can be formulated as cognitive base stations seeking to maximize the spectrum utility while minimizing the interference to primary base station (eNB). Given the resource occupancy of the eNB, the competition for the spectrum resources among HeNBs can be formulated in a game theoretic setting. However, instead of computing the Nash equilibrium policy of the formulated game, we seek to characterize and compute the correlated equilibrium. The main results in Chapter 4 include the following: • We formulate the RB allocation among HeNBs in the downlink LTE system as a game. Given the RB usage of the eNB in an LTE macro19  cell, the cognitive HeNBs are modelled as selfish players competing to acquire the common frequency resources. This framework borrows the idea of cognitive radio systems [38, 67], where we formulate the cognitive HeNBs as secondary users and the eNB as the primary user in the shared spectrum system. A global utility function is defined to evaluate the overall LTE network performance. To achieve the optimal global utility value in a distributed set-up, we also define a local utility function for each cognitive HeNB. This local utility comprises of components that incorporate self-interest, fairness and power consumption of each HeNB. • We define the correlated equilibrium of the formulated game. An RB access algorithm (Algorithm 5) is proposed which converges to the correlated equilibrium solution. This RB access algorithm is based on the regret matching algorithm [36, 51, 64] and it has a distributed nature as each cognitive HeNB does not require the information of other HeNBs. We also prove that this proposed algorithm will converge to the correlated equilibrium set of the formulated game. Numerical examples are shown to verify the results. The third part of this thesis is presented in Chapter 5 where we applies the mechanism design to eliminate the malicious behaviour in cognitive radio networks with conventional opportunistic scheduling algorithm. In the conventional opportunistic scheduling algorithm, the central scheduler adopts opportunistic scheduling to schedule the users under an overall transmission power constraint. The scheduling is based on the reported states of 20  each user. When the secondary users belong to different independent selfish agents, the users may lie about their true state values in order to optimize their own utilities, sometimes in the price of reducing the overall system performance. The main results of Chapter 5 can be concluded as follows. • We combine the mechanism with the opportunistic scheduling to eliminate the users from lying to ensure the optimality of the overall system performance. The pricing mechanism we use for each user is based on VCG mechanism and it maintains the same desirable economic properties as that of the VCG mechanism. • A mechanism learning algorithm is proposed to implement the pricing mechanism.The convergence of this mechanism is shown numerically in the chapter.  1.4  Thesis Organization  The remainder of the thesis consists of five chapters. Chapter 2 and Chapter 3 constitute the first part of the thesis, which concerns transmission adaptation. Chapter 2 studies the transmission control among cognitive radios using switching control Markov game theoretical approach. Structural result on the Nash equilibrium transmission policy and stochastic approximation algorithm are presented. Chapter 3 considers scheduling, rate adaptation, and buffer management in a multiuser WLAN where each user transmits scalable video payload. Algorithms are proposed to compute the optimal transmission policy and simulation highlight the performance gain compared to the myopic policies. Chapter 4 is the second part of the thesis, it con21  cerns the correlated equilibrium solution is a game formulation. Chapter 4 investigates the resource allocation problem in an OFDMA LTE downlink system which consists of a macrocell base station and multiple femtocell base stations. The resource allocation problem is considered under static environment. The third part of the thesis (Chapter 5) presents the application of mechanism design to enforce the truth revealing feature in a conventional opportunistic scheduling algorithm under a cognitive radio network. Lastly, Chapter 6 contains the discussion of main results, conclusions and proposal of future research directions. A review of the work related to the specific optimization context is also given in each chapter.  22  Chapter 2  Transmission Control in Cognitive Radio as a Markovian Dynamic Game 1 This chapter considers an uplink overlay Time Division Multiple Access (TDMA) cognitive radio network where multiple cognitive radios (secondary users) attempt to access a spectrum hole. We assume that each secondary user can access the channel according to a decentralized predefined access rule based on the channel quality and the transmission delay of each secondary user. By modeling secondary user block fading channel qualities as a finite state Markov chain, we formulate the transmission rate adaptation problem of each secondary user as a general-sum Markovian dynamic game 1 This chapter is based on the following publication. J. W. Huang and V. Krishnamurthy, “Transmission Control in Cognitive Radio as a Markovian Dynamic Game Structural Result on Randomized Threshold Policies,” IEEE Transactions on Communication, vol. 58, no. 2, pp. 301-310, February 2010.  23  with a delay constraint. Conditions are given so that the Nash equilibrium transmission policy of each secondary user is a randomized mixture of pure threshold policies. Such threshold policies can be easily implemented. We then present a stochastic approximation algorithm that can adaptively estimate the Nash equilibrium policies and track such policies for non-stationary problems where the statistics of the channel and user parameters evolve with time.  2.1  Background  Resource management in cognitive radio networks involves user scheduling and transmission rate adaptation. Among the non-game theoretic approaches to such a problem, we discuss [62] and [95]. [62] deals with dynamic resource management based on quantized channel state information for multi-carrier cognitive radio networks. Using a stochastic dual approach, optimum dual prices are found to optimally allocate resources across users per channel realization without requiring knowledge of the channel distribution. Alternatively, [95] applies decentralized cognitive MAC protocols to address dynamic spectrum management for single-hop networks. Most games considered in wireless communication systems to date are static games. [73] and [30] are two of such examples which apply static gametheoretic analyses to address the resource allocation problem in cognitive radio networks. Stochastic dynamic game theory is an essential tool for cognitive radio systems as it is able to exploit the correlated channels in the analysis of decentralized behaviours of cognitive radios. In this chapter, we are going to formulate the rate adaptation problem in a cognitive radio 24  network as a general-sum constrained switching control Markovian dynamic game. This chapter uses a Time Division Multiple Access (TDMA) cognitive radio system model (as specified in the IEEE 802.16 standard [46]) that schedules one user per spectrum hole at each time slot according to a predefined decentralized scheduling policy. Therefore, the interaction among secondary users is characterized as a competition for the spectrum hole and can naturally be formulated as a dynamic game. In a TDMA cognitive radio system, the system state transition probability at each time slot only depends on the active user action. This feature fulfills the property of a special type of dynamic game which is called a switching control game [31, 68, 90], where the transition probability in any given state depends on only one player. It is known that the Nash equilibrium for such a game can be computed by solving a sequence of Markov decision processes. In this chapter, the transmission rate control problem is formulated as a constrained switching control Markovian game [3]. Lagrangian dynamic programming [48] has recently been applied to Markov decision processes in transmission scheduling. Here, we extend its application to dynamic games. In the problem formulation, we combine the latency constraint with the optimization objective using a Lagrange multiplier. The original constrained problem is then transformed into an unconstrained Markovian game. It is shown that the Nash equilibrium policy of such a game can be computed by using a value iteration algorithm. One of the main goals of this thesis is to use mathematical tools (supermodularity, etc.) to characterize the nature of the optimal solution rather 25  than do brute force computation. To the best of our knowledge, the structural results proved in the thesis are some of the most general available in the literature. They hold under reasonable assumptions as described in Section 2.3.2.  2.2  Rate Adaptation Problem Formulation  This section describes the system model (Fig. 2.1). We consider an overlay TDMA cognitive radio system with K secondary users, each user tries to transmit delay-sensitive data with Quality of Service (QoS) requirement. At each time slot, only one user can access the channel according to a predefined decentralized access rule. The access rule will be described later in this section. By modelling the correlated block fading channel of each user as a Markov chain, the rate control problem can then be formulated as a constrained Markovian dynamic game. More specifically, under the predefined decentralized access rule, the problem presented is a special type of game, namely a switching control Markovian dynamic game. Note that the problem formulation of this chapter can be extended to a scenario with multiple spectrum holes. In which case, the rate adaptation problem at each spectrum hole follows the formulation of this chapter.  2.2.1  System Description and TDMA Access Rule  The channel quality state of user k at time n is denoted as hnk and it is assumed to belong to a finite set {0, 1, . . . , Qh }. The channel state can be obtained by quantizing a continuous valued channel model comprising of circularly symmetric complex Gaussian random variables that depend only  26  Spectrum Hole  Buffer  Access Control  1  2  Incoming Traffic  Secondary User 1  Secondary User 2  L  Rate Adaptor  Transmit Data  Decentralized Scheduler  Secondary User K  Figure 2.1: An overlay TDMA cognitive radio system where users access the spectrum hole following a predefined access rule. Each user is equipped with a size L buffer, a decentralized scheduler and a rate adaptor for transmission control. on the previous time slot. The composition of channel states of all the K users can be written as hn = {hn1 , . . . , hnK }. Assuming that the channel state hn ∈ H, n = 1, 2, . . . , N is block fading and each block length equals to one time slot, the channel state can be modeled using a finite states Markov chain model. The transition probability of the channel states from time n to (n + 1) can be denoted as P(hn+1 |hn ). Let bnk denote the buffer occupancy state of user k at time n and it belongs to a finite set bnk ∈ {0, 1, . . . , L}. The composition of the buffer states of all the K users can be denoted as bn = {bn1 , . . . , bnK } and bn is an element of the secondary user buffer state space B. New packets arrive at the buffer at each time slot and we denote the number of new incoming packets of the kth user at time n as fkn , fkn ∈ {0, 1, 2, . . . , ∞}. The composition of the incoming traffic of all the K users 27  n }, it is an element of the incoming traffic can be denoted as f n = {f1n , . . . , fK  space F. For simplicity, the incoming traffic is assumed to be independent and identically distributed (i.i.d.) in terms of time index n and user index k. The incoming traffic is not a part of the system state but it affects the buffer state evolution. We use snk = [hnk , bnk ] to denote the state of user k at time n. The system state at time n can then be denoted as sn = {sn1 , . . . , snK }. The finite system state space is denoted as S, which comprises channel state H and secondary user buffer state B. That is, S = H × B. Here × denotes a Cartesian product. Furthermore, Sk is used to indicate the state space where user k is scheduled for transmission. S1 , S2 , . . . , SK are disjoint subsets of S with the property of S = S1 ∪ S2 ∪ · · · ∪ SK . This chapter adopts a TDMA cognitive radio system model (IEEE 802.16 [46]). A decentralized channel access algorithm can be constructed as follows: At the beginning of a time slot, user k attempts to access the channel after a certain time delay tnk . The time delay of user k can be specified via an opportunistic scheduling algorithm [28], such as tnk =  γk . bnk hnk  (2.1)  Here γk is a user specified QoS parameter and γk ∈ {γp , γs }. If user k is a primary user, then, γk = γp , otherwise, γk = γs . By setting γp << γs , the network does not allow the transmission of secondary users with the presence of primary users. As soon as a user successfully access a channel, the remaining users detect the channel occupancy and stop their attempt to access. We use k∗(n) to denote the index of the first user which successfully 28  accesses the spectrum hole. If there are multiple users with the same minimum waiting time, k∗(n) is randomly chosen from these users with equal probability.  2.2.2  Action and Costs  If the kth user is scheduled for transmission at the nth time slot, its action ank ∈ {0, 1, . . . , A} represents the bits/symbol rate of the transmission. Assuming the system uses an uncoded M-ary Quadrature Amplitude Modulation (QAM), bits/symbol rate determines the modulation scheme, n  that is, M = 2ak . Transmission cost: When user k is scheduled for transmission at time n, that is, sn ∈ Sk , the cost function of user k depends only on ank , as all the other users are inactive. In this chapter, we choose the transmission cost of the kth user ck (sn , ank ) to be its transmission Bit Error Rate (BER). The costs of all the users in the system can be specified as: ck (sn , ank ) ≥ 0, ci,i=k (sn , ank ) = 0.  (2.2)  For notational convenience, in the following sections we will drop the subscript k, by defining c(sn , ank ) := ck (sn , ank ).  Holding cost: Each user has an instantaneous QoS constraint denoted as di (sn , ank ), i = 1, . . . , K. In this chapter, we choose the QoS constraint to be the delay (latency constraint) and di (sn , ank ) is a function of the buffer state 29  bni .  2.2.3  Switching Control Game and Transition Probabilities  With the above setup, the decentralized transmission control problem in a Markovian block fading channel cognitive radio system can now be formulated as a switching control game. In such a game [31], the transition probabilities depend only on the action of the kth user when sn ∈ Sk (2.3). This feature enables us to solve such a game by a finite sequence of Markov decision processes.    P(sn+1 |sn , a1 ), if       P(sn+1 |sn , a2 ), if n+1 n P(s |s , a1 , a2 , . . . , aK ) =   ...       P(sn+1 |sn , aK ), if  sn ∈ S1 sn ∈ S2  .  (2.3)  sn ∈ SK  If sn ∈ Sk , the transition probability between the current system state  sn = [hn , bn ] and the next state sn+1 = [hn+1 , bn+1 ] can be specified as: K  P(sn+1 |sn , ank ) =  i=1  K  P(hn+1 |hni ) · i  i=1,i=k  P(bn+1 |bni ) · P(bn+1 |bnk , ank ). i k  In state sn ∈ Sk , the buffer occupancy of user k evolves according to Lindley’s equation [24], bn+1 = min [bnk − ank ]+ + fkn , L . k  (2.4)  Here, [·]+ is defined as: [x]+ = x if x ≥ 0, otherwise, [x]+ = 0. The buffer state transition probability of user k is a function of its action and its  30  incoming traffic distribution, which can be specified as   P(f n = bn+1 − [bn − an ]+ ), k k k k n n P(bn+1 |b , a ) = k k k ∞  n P(f = x), −an ]+ x=L−[bn k k  k  if bn+1 <L k if bn+1 =L k  .  (2.5)  The buffer state of user i (i = k) evolves according to the following rule: bn+1 = min(bni + fin , L). i  The buffer state transition probability of user i (i = k) is a function of its incoming traffic, which is   P(f n = bn+1 − bn ), i i i P(bn+1 |bni ) = i ∞  P(f n = x), x=L−bn i  2.2.4  i  if bn+1 <L i if bn+1 =L i  .  (2.6)  Switching Controlled Markovian Game Formulation  We use πi (i = 1, 2, . . . , K) to denote the transmission policy vector of the ith user. With a slight abuse of notation, πi (s) is used to denote the transmission policy of user i in state s and it is a component of πi . πi (s) lives in the same space as the action ai of the ith user. Assume at time instant n user k is scheduled for transmission according to the system access rule (2.1), the infinite horizon expected total discounted cost2 of any ith (i = 1, 2, . . . , K) 2 There are two cost criteria commonly used for MDPs: expected average cost criterion and expected total discounted reward criterion. We choose the discounted cost criterion because it is mathematically simpler. The average cost criterion has several technicalities involved with the existence of stationary optimal policies. In addition, if the stationary policy exists, as the discount factor β → 1, the policy obtained under the discounted cost criterion is the same as that under the average cost criterion.  31  user under transmission policy πi can be written as: Ci (πi ) =  Eπi  ∞ n=1  β n−1 · ci (sn , ank ) ,  (2.7)  with 0 ≤ β < 1 is a discount factor which models the fact that the future rewards worth less than the current reward. This discount factor can be justified as an inflation rate, as an otherwise unmodeled probability that the simulation ends at each time-step. The expectation of the above function is taken over the system state sn which evolves over time. Denoting the holding cost of user i at the nth time slot as di (sn , ank ), the infinite horizon expected total discounted latency constraint can be written as Di (πi ) = Eπi  ∞ n=1  β n−1 · di (sn , ank ) ≤ Di ,  (2.8)  where Di is a system specified parameter. In order to ensure the validity of the latency constraint, Di is chosen so that the set of policies that satisfy such a constraint is non-empty. This is the feasibility assumption A 2.3.1 which will be discussed more specifically in Section 2.3.2. Equations (2.3, 2.7, 2.8) define a constrained switching control Markovian game. It is assumed that all the users are transmitting delay sensitive traffic with QoS requirement. Our goal is to compute a Nash equilibrium3 policy πi∗ , i = 1, . . . , K (which is not necessarily unique) that minimizes the discounted transmission cost (2.7) subject to the latency constraint (2.8). The following result shows that a Markovian switching control game can be solved using a sequence of Markov decision processes. For the proof, see 3  A Nash equilibrium [31] is a set of policies, one for each player, such that no player has incentive to unilaterally change its action.  32  [31, 68]. Result: [31, Chapter 3.2] The constrained switching control Markovian game (2.7, 2.8) can be solved by a finite sequence of MDPs (as described in Algorithm 1). At each step, the algorithm iteratively updates the transmission policy πin of user i given the transmission policies of the remaining users. The optimization problem at each iteration is: ∗(n)  πi  2.3  = {πin : min Cin (πi ) s.t. Din (πi ) ≤ Di , i = 1, . . . , K}. πi  (2.9)  Randomized Threshold Nash Equilibrium for Markovian Dynamic Game  This section presents a value iteration algorithm to compute the Nash equilibrium policy of the general-sum Markovian dynamic switching control game. We then present a structural result on the Nash equilibrium policy. Finally, we propose a computationally efficient stochastic approximation algorithm to compute the Nash equilibrium.  2.3.1  Value Iteration Algorithm  A value iteration algorithm is presented in [31] to compute the Nash equilibrium of an unconstrained general-sum Markovian dynamic switching control game. By using the Lagrangian dynamic programming, we combine the optimization objective (2.7) with the latency constraint (2.8) via Lagrange multipliers λi , i = 1, 2, . . . , K. We then compute the Nash equilibrium of the resulting unconstrained switching control Markovian dynamic game by  33  the value iteration algorithm described in Algorithm 1. In Algorithm 1, Vi denotes the value vector of user i, each of its components vi (s) denotes the discounted weighted sum of the transmission cost and the latency cost of user i in state s (the weighting factor is the Lagrange multiplier λi ). Algorithm 1 Value Iteration Algorithm 0 Step 1: Set n = 0; Initialize V10 , . . . , VK , λ1 , λ2 , . . . , λK . Step 2: Update the transmission policy and value vector of each user: for k = 1 : K do for each s ∈ Sk ,  πkn (s) = arg minak  c(s, ak ) + λk · dk (s, ak ) + β  |S| s =1  P(s |s, ak )vkn (s ) ;  |S|  vkn+1 (s) = c(s, πkn (s)) + λk · dk (s, πkn (s)) + β s =1 P(s |s, πkn (s))vkn (s ); |S| vin+1 (s) = λi · di (s, ak ) + β s =1 P(s |s, πkn (s))vin (s ), (where i = 1, 2, . . . , K, i = k); end for Step 3: If Vkn+1 < Vkn , k = 1, . . . , K, set n = n + 1, and return to Step 2; otherwise, πkn for k = 1, 2, . . . , K is the optimal policy for user k.  With slight abuse of notation, we denote the iteration index as n. In Step 1 of Algorithm 1, we first set n = 0, then initialize the value vector and the Lagrange multiplier of each user Vk0 , λk for k = 1, 2, . . . , K. The Lagrange multipliers λ1 , λ2 , . . . , λK are chosen to be reasonable positive constants. Algorithm 1 is specified for fixed values of Lagrange multipliers. The optimal values of these Lagrange multipliers can be computed by applying the stochastic approximation algorithm which will be introduced in Section 2.3.3. Step 2 solves a kth user controlled game. It computes the optimal strategy for user k (if s ∈ Sk ), with the strategies of the remaining users fixed. The value vectors of all the users are then computed. The algorithm terminates when the value vector of each user converges as Vkn+1 = Vkn (k = 1, 2, . . . , K); otherwise, it returns to Step 2.  34  Theorem 2.3.1 Under the feasibility assumption A 2.3.1 (specified in Section 2.3.2) and given the Lagrange multipliers λk , k = 1, 2, . . . , K, the value iteration algorithm (Algorithm 1) converges to the Nash equilibrium of the constrained switching control general-sum Markovian game. Please refer to [31, Chapter 6.3] for the proof of Theorem 2.3.1. The intuition behind the proof is as follows: The value vector Vkn (k = 1, 2, . . . , K) is nonincreasing on the iteration index n in the value iteration algorithm. There are only a finite number of strategies available for the optimal policy πk∗ for k = 1, 2, . . . , K. It can be concluded that the algorithm converges in a finite number of iterations. Thus, the Nash equilibrium of the switching control dynamic game under the discounted reward criterion can be obtained by the value iteration algorithm. Remark: The primary purpose of the value iteration algorithm is to prove the structural results on the Nash equilibrium policy. The value iteration algorithm (Algorithm 1) is not designed to be implement in a practical system because at each iteration of the algorithm, a user k is required to know the channel states of all the other users and the state transition probability matrix (2.3).  2.3.2  Structural Result on Randomized Threshold Policy  We aim to characterize the structure of the Nash equilibrium policy in this subsection. First, we list three assumptions. Based on these three assumptions, our main structural result is introduced in Theorem 2.3.2. • A 2.3.1: The set of policies that satisfy the system delay constraint (2.8) is non-empty. 35  First, we define the holding cost of any user i (i = 1, 2, . . . , K) when the state s ∈ Sk : di (s, ak ) =  bi . f  (2.10)  Here f is the average number of incoming packets, and is a system specified parameter. Assumption A 2.3.1 holds if there exists an action such that ak > f . • A 2.3.2: Transmission cost c(s, ak ) and holding cost dk (s, ak ) are submodular4 functions of bk , ak given channel quality hk of the current user, and are independent of the incoming traffic fk . c(s, ak ) and dk (s, ak ) are also nondecreasing functions of bk for any hk and ak . The system transmission cost (2.2) is chosen to be the transmission BER. Assuming the channel states are quantized by system parameterized quantization thresholds Γ(h)1 , Γ(h)2 , . . . , the system transmission cost BER(γ, ak ) is a function of the random channel gain γ ∈ [Γ(h)i , Γ(h)i ). The transmission cost is [19], BERk (γ, ak ) = BERki (hk , ak )  =  0.2 × exp Γ(h)i Γ(h)i−1  −1.6γ , (2ak − 1)  BER(γ, ak )g(γ)dγ  Γ(h)i Γ(h)i−1  g(γ)dγ  (2.11) .  (2.12)  We assume an uncoded M-ary quadrature modulation (QAM). ak is the number of bits per symbol and different values of ak correspond to different modulation scheme. g(γ) denotes the probability distri4  A function f : A × B × C → R is said to be submodular in (a, b) for any fixed c ∈ C if f (a , b ; c) − f (a, b ; c) ≤ f (a , b; c) − f (a, b; c) holds for all a ≥ a and b ≥ b.  36  bution of the Signal-to-Noise Ratio (SNR) γ. For a channel state hk belongs to quantization region [Γ(h)i−1 , Γ(h)i ), the expectation of BERki (hk , ak ) is taken over γ for γ ∈ [Γ(h)i−1 , Γ(h)i ). From (2.11) and (2.12), it can be seen that the average transmission cost is independent of the buffer occupancy bk . Furthermore, it is clear from (2.10) that the holding cost function dk (s, ak ) is nondecreasing on bk and independent of ak . Thus, assumption A 2.3.2 holds. • A 2.3.3:  L bk =l  P(bk |bk , ak ) is a submodular function of bk , ak and is  nondecreasing on bk for any l and ak . As stated in (2.4), the buffer occupancy state evolves according to Lindley’s recursion equation. Given the current state buffer occupancy and action, the transition probability depends on the incoming traffic distribution, which is shown in (2.5). Thus, the buffer state transition probability can be rewritten as P(bk |bk , ak ) = P(bk |(bk − ak )). Assume the incoming traffic is evenly distributed in 0, 1, . . . , L, which can be mathematically written as: P(fk < 0 or fk > L) = 0 and P(0 ≤ fk ≤ L) =  1 L+1 .  Then the buffer state transition probability is  P(bk |bk , ak ) =      1 L+1 , 1+[bk −ak ] L+1  +  if  bk < L  , if  bk = L  .  (2.13)  Therefore, the buffer occupancy state transition probability is independent of bk and ak when bk < L, and is first order stochastically nondecreasing on bk −ak when bk = L. This result verifies  L bk =l  P(bk |bk , ak )  is nondecreasing on bk in A 2.3.3. According to (2.13) we can see that P(bk = L|bk , ak ) is submodular in bk , ak . This verifies A 2.3.3. 37  The following theorem is our main result. It states that the Nash equilibrium policy of a constrained dynamic switching control game is a randomized mixture of two pure monotone policies.  Theorem 2.3.2 Consider the rate adaptation problem in a cognitive radio system as described in Section 2.2. Assume assumption A 2.3.1-A 2.3.3 hold. Then the Nash equilibrium policy of any user k (k = 1, 2, . . . , K) πk∗ is a randomized mixture of two pure policies: πk1 and πk2 . Each of these two pure policies is nondecreasing on the buffer occupancy state bk . The proof of Theorem 2.3.2 is given in Appendix A.  2.3.3  Stochastic Approximation Algorithm  The structural result stated in Theorem 2.3.2 can be exploited to compute the Nash equilibrium policy. We now present a stochastic approximation algorithm that exploits this structure. Suppose the action set of each user contains two actions, that is ak ∈ {0, 1}. The two actions can be chosen as No Transmission (ak = 0) and Transmission (ak = 1). Using the structural result on the Nash equilibrium policy as shown in Theorem 2.3.2, the Nash equilibrium πk∗ (s) is a randomized policy parameterized by three parameters:  πk∗ (s) =    0, if 0 ≤ bk < bl (s)         p, if bl (s) ≤ bk < bh (s) . 1, if bh (s) ≤ bk  38  (2.14)  The parameter p ∈ [0, 1] is a randomization probability. The parameters bl (s) and bh (s) are the lower and higher buffer state thresholds, respectively. The search for the Nash equilibrium policy over the intractably large buffer space reduces to estimating the three parameters, bl (s), bh (s) and p. The above structural result can be extended to cognitive radio systems with larger action sets. For example, when the action set of each secondary user is ak = {0, 1, 2}, the Nash equilibrium policy can be parameterized by 6 parameters (4 buffer state thresholds and 2 randomization factors). In the stochastic approximation algorithm, we compute the continuous optimal values of bl (s) ∈ [0, L] and bh (s) ∈ [bl (s), L] (bh (s) ≥ bl (s)), then round them off to the nearest discrete values. This is a relaxation of the original discrete stochastic optimization problem as the buffer states in the problem setup are discrete. For convenience, let θk denote the vector of all the parameters that will be estimated for user k (k = 1, 2, . . . , K). The composition of the parameter vectors of all the K users can be denoted as Θ = {θ1 , θ2 , . . . , θK }. A primal and dual gradient projection method requires the Lagrangian to be locally convex at the optimum in order to ensure the convergence. In our problem, the optimization objective Ck (2.7) and latency constraint Dk (2.8) (k = 1, 2, . . . , K) are not convex over Θ since Θ = {θ1 , θ2 , . . . , θK } and θk is defined on the estimation parameters of the transmission policy of user k (e.g., bl (s), bh (s) and p). However, the augmented Lagrangian method can “convexify” the problem by adding a penalty term to the objective [53]. Based on which, the stochastic approximation algorithm is introduced in Algorithm 2. The gradients of Ck (2.7) and Dk (2.8) are estimated by using the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm [83]. The essential feature of SPSA is the underlying gradient ap39  proximation, which requires only two objective function measurements per iteration, regardless of the dimension of the optimization problem. These two measurements are made by simultaneously varying, in a properly random fashion, all of the variables in the problem.  Algorithm 2 Stochastic Approximation Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9:  Initialization: Θ(0) , Λ0 ; n = 0; ρ = 4; Initialize constant perturbation step size µ and gradient step size α; Main Iteration if sn ∈ Sk then mk = |θkn |; Generate ∆n = [∆n1 , ∆n2 , . . . , ∆nmk ]T ; ∆ni are Bernoulli random variables with p = 12 . n θk+ = θkn + µ × ∆n ; n θk− = θkn − µ × ∆n ; ∆Ckn  n n c(sn ,θk+ )−c(sn ,θk− ) [(∆n1 )−1 , (∆n2 )−1 , . . . , (∆nmk )−1 ]T ; 2µ  10: 11:  = ∆Dkn  12:  =  13:  θkn+1 = θkn − α × ∆Ckn + ∆Dkn · max 0, λnk + ρ · D(sn , θkn ) − Dk  13:  λn+1 = max (1 − k  n n d(sn ,θk+ )−d(sn ,θk− ) [(∆n1 )−1 , (∆n2 )−1 , . . . , (∆nmk )−1 ]T ; 2µ  α ρ  ;  · λnk ), λnk + α · D(sn , θkn ) − Dk ;  end if The parameters of other users remain unchanged; 16: n = n + 1; 17: The iteration terminates when the values of the parameters Θn converge; else return back to Step 3. 14: 15:  In Algorithm 2, µ and α are used to denote the constant perturbation step size and constant gradient step size, respectively. The composition of the Lagrange multipliers of all the users can be written as Λn = {λn1 , λn2 , . . . , λnK }. In the main body of the algorithm, the SPSA algorithm is used to estimate the gradients of Ck and Dk , then the parameters are updated iteratively. Specifically, when user k is scheduled for transmission at time slot n, pa40  rameters θkn and the Lagrange multiplier λnk can be updated after introducing a random perturbation vector ∆n . Meanwhile, the parameters of the other users remain unchanged. ∆Ckn and ∆Dkn are used to denote the gradient estimators of Ckn and Dkn , respectively. More specifically, they can be written as: ∆Ckn  n = gC (θkn ) + onCk , k  ∆Dkn  n = gD (θkn ) + onDk , k  ∂Ckn (θkn ) , ∂θkn ∂Dkn (θkn ) n gD (θkn ) = , k ∂θkn  n gC (θkn ) = k  (2.15) (2.16)  with onCk and onDk denoting the noise of the gradient estimations. Theorem 2.3.3 Consider the sequence of estimates {Θn (α, µ), Λn (α, µ)} generated by the stochastic approximation algorithm. Define associated piecewise constant interpolated continuous-time processes as follows: Θt (α, µ) = Θn (α, µ),  for t ∈ [nα, (n + 1)α),  (2.17)  Λt (α, µ) = Λn (α, µ)  for t ∈ [nα, (n + 1)α).  (2.18)  Assume onCk and onDk are uncorrelated noise processes with zero means and fin (θ n ) and g n (θ n ) are uniformly bounded nite variances and the gradients gC Dk k k k  for sufficiently large ρ. Then, as α → 0,  α µ2  → 0 and t → ∞, {Θt (α), Λt (α)}  converge in distribution to the Kuhn Tucker (KT) pair of (2.9) [54, Theorem 8.3.1], which is specified in (2.19). According to [54, Theorem 8.3.1], Algorithm 2 converges in distribution under the following conditions. •  α µ2  n (θ n ) and g n (θ n ) are uniformly bounded. → 0, α → 0. gC k Dk k k  • The uncorrelated noise processes onCk and onDk have zero means. 41  • ∆Ckn and ∆Dkn have finite variances. This condition holds because onCk and onDk have finite variances. n (·) and g n (·) that are continuous uniformly • There are functions gC Dk k  in n and random variables onCk and onDk such that n n En [Ckn (θk+ ) − Ckn (θk− )]/2µ =  n gC (θkn ) + onCk , k  n n ) − Dkn (θk− )]/2µ = En [Dkn (θk+  n gD (θkn ) + onDk . k  According to the definition of ∆Ckn (2.15) and ∆Dkn (2.16), this condition holds. The optimal policies of (2.9) satisfying the KT condition can be defined as follows. θi∗ belongs to the KT set when KT = {θi∗ : ∃λi > 0, such that ∇θi Ci + ∇θi λi (Di − Di ) = 0, i = 1, . . . , K},(2.19)  where Ci and Di are the optimization objective (2.7) and discounted time delay (2.8), respectively. Moreover, θi∗ satisfies the second order sufficiency conditions: ∇2θi Ci + ∇2θi (Di − Di ) ≥ 0 is positive definite for all the i, and (Di − Di ) = 0, λi > 0, i = 1, . . . , K. Note that in the stochastic approximation algorithm, we first compute the continuous values of bl (s) and bh (s), then round them off to the nearest discrete values. This relaxation leads to the continuous value of θi during calculation, thus, (2.19) is differentiable on θi . Remark: At each iteration of the stochastic approximation algorithm, user k is only required to know its own transmission cost and holding cost. The transmission cost and holding cost of user k are functions of its own 42  channel state and buffer state (Section 2.3.2). Thus, the stochastic approximation algorithm is distributed and implementable in a practical system.  2.4  Numerical Examples  This section presents numerical results on the Nash equilibrium transmission policy. For convenience, all the secondary users in the simulation setup are assumed to be identical. The simulation results shown here are of the transmission policies of the first user. In the system model, each user has a size 10 buffer, and the channel quality measurements are quantized into two different states, namely {1, 2}. In the system configuration, the transmission costs, the holding costs and buffer transition probability matrices are chosen to ensure assumption A 2.3.2-A 2.3.3 (specified in Section 2.3.2). The channel transition probability matrices are generated randomly. In the system models used in Fig. 2.2 and Fig. 2.3, each user has three different action choices. The system uses an uncoded M-QAM modulation scheme, and different actions are followed by different modulation modes, leading to different transmission rates. The three different actions are ak = 1, 2, 3 and consequently, the modulation schemes are 2-QAM, 4-QAM and 8-QAM. When the Nash equilibrium policy of a user equals to 0 in a state, it means that user is not scheduled for transmission in that state. The transmission policies shown in Fig. 2.2 and Fig. 2.3 are obtained by fixing the Lagrange multiplier to be λ1 = λ2 = 1.3 and then apply the value iteration algorithm (Algorithm 1). a) General-Sum Constrained Game: Structure of the Nash: Fig. 2.2 shows the structured optimal transmission policy of a two-user general-sum 43  General−sum Deterministic Monotone Policy with h1=1, h2=1  3  Action  2  1  0 1  2  3  4  5 6 Buffer State of the Other User 7  7  8  9  10  1  2  3  4  5  8  9  10  6 Buffer State  Figure 2.2: The transmission policy of user 1 in a switching controlled general-sum dynamic game system computed by the value iteration algorithm. The result is obtained when the channel states of user 1 and 2 are h1 = 1 and h2 = 1, respectively. system model with h1 = 1 and h2 = 1. The figure is the Nash equilibrium policy of a switching control game with general-sum transmission costs and zero-sum holding costs. The transmission policy is monotone nondecreasing on the buffer occupancy state. b) Zero-Sum Constrained Game: Shapley’s Theorem says a discounted, zero-sum, two-person stochastic game possesses a value vector that is the unique solution of the game [31]. Fig. 2.3 considers a two-user zero-sum game model and shows the structural result on the optimal transmission policy. The first subfigure is of the result with h1 = 1 and h2 = 1, and the second subfigure is of that with h1 = 1 and h2 = 2. It can be seen from the figures that the policies are deterministic, and under a certain 44  Zero−sum Deterministic Monotone Policy with h1=1, h2=1  3  Action  2  1  0 1  2  3  4  5  6 Buffer State of the Other User 7  7  8  9  10  1  2  3  4  5  8  9  10  6 Buffer State  Zero−sum Deterministic Monotone Policy with h =1, h =2 1  2  3  Action  2  1  0 1  2  3  4  5 6 Buffer State of the Other User 7  7  8  9  10  1  2  3  4  5  8  9  10  6 Buffer State  Figure 2.3: The transmission policy of a certain user in a switching controlled zero-sum dynamic game system obtained by value iteration algorithm. The first subfigure shows the result when the channel states of both users are h1 = 1 and h2 = 1, while the second subfigure shows the result when the channel states of user 1 and 2 are h1 = 1 and h2 = 2, respectively. 45  channel state value, the optimal action policy is monotone nondecreasing on the buffer occupancy state. By comparing subfigures 1 and 2 we see that as the channel state of the other user improves, the current user has lower probability to transmit and its transmission policy becomes more aggressive. c) Stochastic Approximation Algorithm for Learning Nash Equilibrium Policy: Fig. 2.4 and Fig. 2.6 consider a two-user system with each user having two different action choices denoted as {1, 2}. As it is a constrained switching controlled Markovian game, the Nash equilibrium policy is a randomized mixture of two pure policies. The optimal transmit policy can be parameterized by lower threshold bl (s), upper threshold bh (s) and randomization factor p, as described in (2.14). The stochastic approximation algorithm is applied to estimate the Nash equilibrium policy. The simulation results of user 1 with h2 = 1, b2 = 1 are shown in Fig. 2.4. The figure shows that the optimal transmission policy is a randomized mixture of two pure policies, each of these policies is monotone nondecreasing on the buffer state. In order to demonstrate the effectiveness of the proposed stochastic approximation algorithm, we compare its performance with that of the myopic policy. The myopic policy does not consider the effect of current actions on future states, it can be computed by choosing the discount factor β = 0 in equations (2.7,2.8). It is shown in Fig. 2.5 that under the same channel conditions, the proposed algorithm has better BER performance and buffer management compared to the myopic policy. The stochastic approximation algorithm proposed in Section 2.3.3 is able to adapt its estimation of the Nash equilibrium policy as the system 46  Randomized Optimal Policy 2  Policy  1.5  1  0.5 Channel State  2  1 1  0  2  3  4  5  Buffer State  6  7  8  9  10  Figure 2.4: The Nash equilibrium transmission control policy computed via stochastic approximation algorithm. A 2-user system is considered, and each user has a size 10 buffer. parameters evolve with time. Fig. 2.6 shows the time evolution of lower threshold bl (s), upper threshold bh (s) and randomization factor p for h1 = 1, h2 = 1 and b2 = 1. The system configuration used from iteration 101 to 200 is different from that used from iteration 1 to 100. We can see from the simulation result that when the system parameters evolve with time, the stochastic approximation algorithm can adapt the transmission policy accordingly. The estimated three parameters of the policy at iteration 100 are bl (s) = 2.4, bh (s) = 7.6 and p = 0.5. After detecting the change of the system parameters, the new estimated parameters are updated to be bl (s) = 3.8, bh (s) = 9.7 and p = 0.29. In Fig. 2.6, the estimated upper and lower thresholds of the system computed by Algorithm 2 are shown  47  Performance Comparison between Proposed Algorithm with Myopic Policy  Discounted Latency  200  150  100 Proposed Algorithm Myopic Policy  50  0  0  10  20  30  40  50 Time Slot  60  70  80  90  100  2  Discounted Cost  1.5  1  Proposed Algorithm Myopic Policy  0.5  0  0  10  20  30  40  50 Time Slot  60  70  80  90  100  Figure 2.5: Performance comparison between the proposed stochastic approximation algorithm and myopic policy. by solid lines. By rounding the estimated parameters off to the nearest discrete values, the discrete optimal parameters are obtained (shown by dashed lines).  48  Tracking of Randomization Factor 1 Optimal Randomization Factor Estimated Randomization Factor  0.9 0.8  Probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  50  100 Number of Iterations  150  200  Tracking of Lower and Upper Thresholds 10 9 8 Optimal Lower Threshold Estimated Lower Threshold Optimal Upper Threshold Estimated Upper Threshold  Buffer State  7 6 5 4 3 2 1 0  0  50  100 Number of Iterations  150  200  Figure 2.6: Tracking Nash equilibrium policy using the stochastic approximation algorithm (Algorithm 2). System parameters change at the 100th iteration, as specified in Section 2.3. These two figures compare the estimated randomization factor and buffer thresholds with the discrete optimal values. 49  2.5  Summary  By extending the structural results in [72] and [29],we formulated the rate adaptation problem of secondary users in cognitive radio systems as a constrained general-sum switching control Markovian game, where the block fading channels were modeled as a finite state Markov chain. Assuming perfect information on the primary user behaviour, each secondary user accessed the channel according to a decentralized access rule. The mechanism design ideas can be incorporated in the access rule as in [41]. We showed the Nash equilibria of the game can be calculated by the value iteration algorithm. Under reasonable assumptions, we proved that the Nash equilibrium policy was a randomized mixture of two pure policies, and each of them was monotone nondecreasing on the buffer occupancy state. We then exploited this structural result to devise a computationally efficient stochastic approximation algorithm. Numerical examples were provided to verify these results.  50  Chapter 3  A Dynamic Games Approach to Transmission Rate Adaptation in Multimedia WLAN 1 This chapter considers the scheduling, rate adaptation, and buffer management in a multiuser WLAN where each user transmits scalable video payload. Based on opportunistic scheduling, users access the available medium (channel) in a decentralized manner. The rate adaptation problem of the WLAN multimedia networks is then formulated as a general-sum switching control dynamic Markovian game by modelling the video states and block 1  This chapter is based on the following publication. J. W. Huang, H. Mansour, and V. Krishnamurthy, “A Dynamical Games Approach to Transmission Rate Adaptation in Multimedia WLAN,” IEEE Transactions on Signal Processing, vol. 58, no. 7, pp. 3635-3646, July 2010.  51  fading channel qualities of each user as a finite states Markovian chain. A value iteration algorithm is proposed to compute the Nash equilibrium policy of such a game and the convergence of the algorithm is also proved. We also give assumptions on the system so that the Nash equilibrium transmission policy of each user is a randomization of two pure policies with each policy nonincreasing on the buffer state occupancy. Based on this structural result, we use policy gradient algorithm to compute the Nash equilibrium policy.  3.1  Background  The time-varying nature of wireless channels and video content motivate the need for scalable media which can adapt the video bit-rate without significantly sacrificing the decoded video quality. The Scalable Video Coding (SVC) project provides spatial, temporal, and signal-to-noise ratio (SNR) scalability which ensures a graceful degradation in video quality when faced with channel fluctuations [77]. In this chapter, we consider multiple users transmitting scalable video payload in a WLAN system. The aim is to adapt the video layers of each user to the fluctuations in channel quality and the transmission control polices, such that, the transmission buffer of each user does not saturate and the transmission delay constraint is not violated. Rate allocation for multimedia transmission in wireless networks has been studied extensively. Previous works that are of interest include [55, 56, 96]. In [56], joint radio link buffer management and scheduling strategies are compared for wireless video streaming over High Speed Download 52  Package Access (HSDPA) networks. We use the results of [56] to choose an appropriate buffer management strategy for our proposed switching control stochastic dynamic game solution. In [96] and [55] video rate and distortion models are used to improve the transmitted video quality for low latency multimedia traffic. Rate and distortion modelling has become a keystone in model-based video rate and transmission control algorithms. The models are used to predict the coded video packet size and the distortion of the corresponding decoded picture prior to the actual encoding process in order to be used in a constrained optimization framework. However, these models cannot be used for characterization of scalable video data. In this chapter, we consider an IEEE 802.11 [45] wireless local area network (WLAN) which deploys the CSMA/CA. We propose a modified CSMA/CA channel access mechanism (Section 3.3.3) which takes into account the dynamic behaviours of the video variation, as well as the channel quality and transmission delay of each user. Due to the fact that there is no central controller for resource allocation in a WLAN system, there is strong motivation to study the case where individual users seek to selfishly maximize their transmission rate while taking into account the access rule. This is akin to individuals minimizing their tax (throughput subject to latency constraint in our case) without violating the tax-law (the modified CSMA/CA channel access rule in our case). The interactions among system users can be naturally formulated using game theory. Furthermore, by exploiting the correlated time varying sample paths of the channels and buffers, the transmission rate adaptation problem can be formulated as a Markovian dynamic game. Section 3.2 describes the details of the WLAN system under study 53  Figure 3.1: A WLAN system where each user is equipped with a size B buffer, a decentralized scheduler and a rate adaptor for transmission control. The users transmit a scalable video payload in which enhancement layers provide quality refinements over the base layer bitstream. as well as the dynamic behaviour of video sources and the scalable ratedistortion models used to represent this behaviour. One of the main goals of this thesis is to use mathematical tools to characterize the nature of the optimal solution rather than do brute force computation. To the best of our knowledge, the structural results proved in the thesis are some of the most general available in the literature. They hold under reasonable assumptions as described in Section 3.4.2.  54  3.2  System Description and the Video Rate-Distortion Model  This section introduces the time-slotted WLAN system model (Fig. 3.1) which consists of multiple users attempting to transmit their data over available channels (medium) [45]. Each user is equipped with a buffer, a decentralized scheduler and a rate adaptor to transmit a scalable video payload which consists of a base layer and multiple enhancement layers. We use game theory to formulate the decentralized behaviours of all the users access the common spectrum resource.  3.2.1  System Description  We consider a K user WLAN system where only one user can access the channel at each time slot according to a modified CSMA/CA mechanism (Section 3.3). The system model also assumes there is no hidden nodes in the network. Thus, we do not need to consider packet collision in our formulation. The rate control problem of each user can be formulated as a constrained dynamic Markovian game by modelling the correlated block fading channel as a Markov chain. Under the predefined decentralized access rule (Section 3.3.3), the problem presented is a special type of game namely a switching control Markovian dynamic game. We assume that K users (indexed by k = 1, . . . , K) are equipped with scalable video encoders delivering video bitstreams that conform with SVC [80]. In SVC, quality scalability is achieved using Medium-Grained Scalability (MGS) or Coarse-Grained Scalability (CGS) where scalable enhancement packets deliver quality refinements to a preceding layer representation by 55  re-quantizing the residual signal using a smaller quantization step size and encoding only the quantization refinements in the enhancement layer packets [80]. Moreover, MGS/CGS enhancement layer packets are coded using the key-picture concept where the coding loop is closed at the highest scalability layer for key-pictures only, while the loop is closed at the base-layer quality for the remaining pictures. This approach achieves a tradeoff between the coding efficiency of enhancement layers and the drift at the decoder [80]. Assume that each user k encodes its video stream at fixed base and enhancement layer Quantization Parameter (QP) values ql,k , where l = 0 corresponds to the base layer, l ≥ 1 to every subsequent enhancement layer. This results in a fluctuation of the video bitrate and distortion as the scene complexity and level of motion varies. The video variation can be captured using the Mean Absolute Difference (MAD) of the prediction residual since it quantifies the mismatch between the original un-coded picture and the Intra/Inter prediction. Therefore, we resort to video rate and distortion models to predict the video distortion of the corresponding decoded picture prior to the actual encoding process. In [61], new rate and distortion models are proposed that capture the variation of MGS/CGS scalable coded video content as a function of two encoding parameters, namely the quantization parameter (QP) and the MAD of the residual signal. We will summarize the results in the next subsection.  3.2.2  Scalable Rate-Distortion Modelling  In this section, we present new rate and distortion estimation models for individual frame representation of MGS/CGS scalable coded video content. 56  Two real-time encoding parameters are commonly used in rate-distortion estimation models, namely the QP and the MAD of the residual signal. Dropping the user subscript k, let ql , ql ∈ {0, 1, ...51} [47] be the QP values of two distinct video layers l and l , respectively. We have found a simple relationship that estimates the residual MAD m ˜ l of a specific frame given the residual MAD of the same frame at an initial ql as shown below: m ˜l = m ˜ l 2ς(ql −ql  )  (3.1)  where ς is a model parameter typically valued around 0.07 for most sequences [47, 61]. In MGS/CGS scalability, enhancement layer packets contain refinements on the quantization of residual texture information [80]. Therefore, we use the expression in (3.1) to estimate the perceived MAD of each of the MGS/CGS enhancement layers. Rate model Here we present the rate model for base and CGS enhancement layer SVC coded video frames. The MGS/CGS [55] base and enhancement layer bitrate in SVC can be expressed as follows: ˜ l 2ς(ql −ql r(l, m ˜ l ) = cl (m ˜ l )u 2−ql /6 = cl m  ) u 2−ql /6  (3.2)  where cl is a model parameter, and u is a power factor that depends on the frame type, such that, u = 1 for Inter-coded frames and u ≈ Intra-coded frames [47, 61].  57  5 6  for  Distortion model The distortion model estimates the decoded picture Luminance Peak Signalto-Noise Ratio (Y-PSNR). Let ql be the QP value of layer l, and let m ˜ l be the prediction MAD estimate. The expression of the video Peak Signal-toNoise Ratio (PSNR) of achieved by decoding all layers up to layer l is then expressed as follows: δ(l, m ˜l )  = ν1 log10 ((m ˜ l )u + 1) · ql + ν2 = ν1 log10  m ˜ l 2a(ql −ql  ) u  (3.3)  + 1 · ql + ν2 ,  where u is the same power factor described in (3.2), ν1 and ν2 are sequencedependent model parameters typically valued at 0.52 and 47, respectively [61]. The parameters ν1 and ν2 can be refined for each sequence during encoding. Fig. 3.2 illustrates the performance of the proposed distortion model compared to actual encoding of two reference video sequences: Foreman and Football. Fig. 3.2 (a) and (b) measure the Y-PSNR estimation results, while (c) and (d) measure the rate estimation results. The video streams used in the simulation have one base layer and two CGS enhancement layers. The accuracy of this rate distortion model (3.3) can be easily seen in Fig. 3.2. Note that the parameters ς, ν1 , ν2 , and cl are only calculated once and remain constant for the entire video sequence.  58  Figure 3.2: Illustration of the modeled PSNR estimates (a) and (b), and the modeled rate estimates (c) and (d) for the base layer and two CGS enhancement layers of the sequences Foreman and Football.  3.3  Uplink Rate Adaptation Problem Formulation  We assume every user has an uplink buffer which holds the incoming coded video frames. When a user is scheduled for transmission, the buffer output rate is chosen depending on the channel quality and buffer occupancy. Therefore, the rate adaptation problem is reformulated as a buffer control problem for every user. In this section, we give a Markovian dynamic game description of this problem.  59  We assume that the time slot coincides with the video frame duration (30-40 ms [80]). Each user has a fixed packet arrival rate and the the incoming packets are SVC bitstreams with both base and enhancement layers. We denote the incoming number of video frames of user k as fin,k (k = 1, 2, . . . , K). Let bnk represent the buffer occupancy state of user k at time n and bnk ∈ {0, 1, . . . , B}, where B is the maximum number of coded video frames that can be stored in the buffer. The composition of buffer states of all the K users is bn = {bn1 , . . . , bnK } and bn ∈ B, where B is used to denote the space of system buffer states. Furthermore, we use bn−k to denote the buffer state composition of all the users excluding user k. The channel quality of user k at time n is denoted as hnk . The channel is characterized using circularly symmetric complex Gaussian random variables which depend only on the previous time slot. By quantizing the channel quality metric, we can denote the resulting discrete channel state space by hnk ∈ {1, 2, . . . }. Let hn = {hn1 , . . . , hnK } be the composition of channel states of all the K users. Assuming that hn ∈ H, n = 1, 2, . . . , N is block fading, with block length equal to each time period, the channel states constitute a Markov process with transition probability from time n to (n + 1) given by P(h(n+1) |hn ). Let mnk be the MAD of user k at time n. In [96], the MAD between two consecutive video frames is modeled as a stationary first-order Gauss Markov process. We quantize the range of mnk to achieve a video variability state space denoted by mnk ∈ M = {0, 1, 2 . . . , M }. The video states constitute (n+1)  a Markov process with transition probabilities given by P(mk 60  |mnk ). The  system video state at time n is a composition of the video states of all the users and mn = {mn1 , mn2 , . . . , mnK }. The system state at time n is composed of the channel state, video state and buffer state, which is denoted as sn = {hn , mn , bn }. Let S denote the finite system state space, which comprises channel state space H, video state space M, and user buffer state space B. That is, S = H × M × B. Here × denotes a Cartesian product. Sk is used to indicate the states where user k is scheduled for transmission. S1 , S2 , . . . , SK are disjoint subsets of S with the property of S = S1 ∪ S2 ∪ · · · ∪ SK .  3.3.1  Actions, Transmission Reward and Holding Cost  The action ank by user k at time slot n is defined to be the number of coded video frames taken from the buffer and transmitted. The rate control algorithm considered in this chapter adjusts the output frame-rate ank according to the channel quality and buffer occupancy. Fig. 3.3 illustrates the buffer control mechanism adopted in this chapter. Actions: Let ank denote the action of the kth user at the nth time slot. We express ank as the number of coded video frames included in the outgoing traffic payload. If sn ∈ Sk , then ank ∈ A = {amin , . . . , amax }, otherwise, ank = 0. amin and amax denote the minimum and maximum video frames that user k can output at a time n, respectively. We assume that the number of video layers lank at the output window at time slot n is dictated by a channel dependent rule. Therefore, we express  61  Figure 3.3: Example of the buffer control mechanism assumed in this chapter where fin,k = 1 (k = 1, 2, . . . , K). At every time-slot, a new coded video frame enters the buffer. The buffer output depends on the scheduling algorithm involved, the buffer occupancy, and the channel quality. If a user is scheduled for transmission, then the action taken will extract a specific number l of video frame layers from up to N frames stored in the buffer. the output number of layers of the video frames at time slot n as follows: lank = g(hnk ),  (3.4)  where g(.) is the channel dependent rule and it is an increasing function on the channel state. The remaining video layers l > lank are dropped. This assumption is to reduce the action space and allows us to derive the structural results as illustrated later in Section 3.4.2. Consequently, the action space constitutes the number of video frames to be transmitted and is given by A = {1, 2, · · · , W }. Let Lk be the number of video layers admitted into the buffer of user k, each input packet of user k has Lk layers. The video layer index is denoted as l ∈ L = {0, 1, · · · Lk − 1}, where l = 0 corresponds to the base layer, l ≥ 1 62  to every subsequent enhancement layer. The buffer output packet ank of user k at time n is determined by the modified CSMA/CA decentralized channel access algorithm. If user k is scheduled for transmission at time slot n, ank > 0, otherwise, ank = 0. However, the system selectively transmits some packet layers according to the channel quality. E.g., the system transmits all of the Lk layers of each packet if the channel quality hnk is good, while it only transmits the base layers of each packet if the channel quality hnk is poor. Transmission reward: Let ck (sn , an1 , . . . , anK ) denote the transmission reward of user k at time n. Specifically, ck (sn , an1 , . . . , anK ) is chosen to be the expected video PSNR resulting from transmitting lank video layers from ank video frames stored in the buffer of user k. Let δ(l, mnk ) be the video PSNR achieved by decoding all packets up to layer l of user k at time n achieved with video state mnk , where mnk is the base-layer MAD. Let pl (hnk ) be the Packet Error Rate (PER) encountered by user k at the lth layer during the transmission. The transmission reward can be expressed as [61]: ck (sn , an1 , . . . , anK ) = ank · (1 − p0 (hnk )) · δ(0, mnk ) + lan −1 k  l  l=1 j=0  (1 − pj (hnk )) · [δ(l, mnk ) − δ(l − 1, mnk )] .  (3.5)  The bit errors are assumed to be independent identically distributed (i.i.d.), and δ(l, mnk ) is given in (3.3). Note here that the video distortion measure δ(l, mnk ) can either be considered as the picture mean-square-error (MSE) or the PSNR. In the case of PSNR, the objective will be to maximize ck (sn , ank ), otherwise it is minimized. In this chapter, we consider the PSNR as the video 63  quality metric. Note in equation (3.5), we assume a target Packet Error Rate (PER) which allows the flexibility of modulation schemes and channel coding selections. It simplifies both the expression of the transmission reward function (3.5) and the derivation of the Nash equilibrium policy. It can be seen from (3.5) that at time slot n if the kth user is scheduled for transmission the performance of user k depends solely on its own channel state hnk and not on the actions. Thus, for sn ∈ Sk the rewards of all the users in the system are ck (sn , an1 , . . . , anK )  = ck (sn , ank ) ≥ 0  (3.6)  ci (sn , an1 , . . . , anK )  = 0, (i = k).  (3.7)  For notational convenience, in the following sections we will drop the subscript k by defining c(sn , ank ) := ck (sn , ank ).  Holding cost: Each user has an instantaneous Quality of Service (QoS) constraint denoted as dk (sn , an1 , . . . , anK ) where k = 1, . . . , K. If the QoS is chosen to be the delay (latency) then dk (sn , an1 , . . . , anK ) is a function of the buffer state bnk of the current user k. The instantaneous holding costs will be subsequently included in an infinite horizon latency constraint. We express the holding cost of user k with sn ∈ Sk as follows: dk (sn , ank ) =  1 · ([bnk − ank + fin,k ]+ )τ , τ ≥ 1 κ 64  (3.8)  where κ is the average output frame rate which is determined by the system. τ is a constant factor which is specified to be τ ≥ 1, this is due to the fact that the data overflow probability becomes higher with more data in the buffer. The latency cost at time slot n is evaluated by the buffer state after taking action ank . Since the transmission latency is independent of the actions of all the remaining users, it can be simplified as dk (sn , an1 , . . . , anK ) = dk (sn , ank ) ≥ 0, where sn ∈ Sk .  (3.9)  For the remaining K − 1 users who are not scheduled for transmission, their holding costs can be expressed as di,i=k (sn , ank ) =  3.3.2  1 · min(bni + fin,i , B)τ . κ  (3.10)  Markovian Game Formulation  The intuition behind the Markovian game formulation for the rate adaptation problem in a WLAN system is as follows: A WLAN system does not have a central authority for the resource allocation and the system access rule is typically a decentralized opportunistic scheme. Due to the selfish nature of the system users, each user aims to maximize its own payoff. The interaction among users is characterized as the competition for the common system resources, which can best be formulated using game theory. By modelling the transmission channel as correlated Markov process, the rate adaptation problem can further be formulated as a Markovian game. 65  At time instant n, assume user k is scheduled for transmission according to the system access rule specified in Section 3.3.3. We define c(sn , ank ) ≥ 0  to be the instantaneous reward of user k when the system is in state sn .  We use Φ1 , Φ2 , . . . , ΦK to represent the set of all the deterministic policies of each user, respectively. The infinite horizon expected total discounted reward of any user i given its transmission policy πi (πi ∈ Φi ), can be written as Ci (πi ) =  Esn  ∞ n=1  β n−1 · ci (sn , ank ) πi  (3.11)  where 0 ≤ β < 1 is the discount factor, the state sn ∈ Sk and the expectation  is taken over action ank as well as system state sn evolution for n = 1, 2, . . . . The holding cost of user i at that time is di (sn , ank ) and the infinite horizon expected total discounted latency constraint can be written as Di (πi ) = Esn  ∞ n=1  β n−1 · di (sn , ank ) πi ≤ Di ,  (3.12)  with Di being a system parameter depending on the system requirement on user i. Optimization Problem: Given the policies of the other users, the optimal transmission policy of user i, πi∗ , is chosen so as to maximize the overall expected discounted reward subject to its constraint, this can be mathematically written as: πi∗ = {πi : max Ci (πi ) s.t. Di (πi ) ≤ Di , i = 1, . . . , K}. πi ∈Φi  (3.13)  Not that the choice of the optimal rate transmission policy of ith user  66  is a function of the policies of other users. Each user aims to optimize his own discounted reward (3.11) under the latency constraint (3.12).  3.3.3  System Access Rule  This chapter adopts a WLAN system model (IEEE 802.11 [45]) with a modified CSMA/CA mechanism which will be implemented by a decentralized channel access rule. The decentralized channel access algorithm can be constructed as follows: At the beginning of a time slot, each user k attempts to access the channel after a certain time delay tnk . The time delay of user k can be specified via an opportunistic scheduling algorithm [28], such as tnk =  γk . bnk hnk  (3.14)  Here γk is a user specified quality of service (QoS) parameter and γk ∈ {γp , γs }. As soon as a user successfully access the channel, the remaining users detect the channel occupancy and stop their attempt to access. Let k∗n denote the index of the first user which successfully accesses the channel. If there are multiple users with the same minimum waiting time, k∗n is randomly chosen from these users with equal probability.  3.3.4  Transition Probabilities and Switching Control Game Formulation  With the above setup, the system feature satisfies the assumptions of a special type of dynamic game called a switching control game [31, 68, 90], where the transition probabilities depend on only one player in each state. It is known that the Nash equilibrium for such a game can be computed by solving a sequence of Markov decision processes [31]. The decentralized 67  transmission control problem in a Makovian block fading channel WLAN system can now be formulated as a switching control game. In such a game [31], the transition probabilities depend only on the action of the kth user when the state s ∈ Sk , this enables us to solve such type of game by a finite sequence of Markov decision processes. According to the property of the switching control game, when the kth user is scheduled for transmission, the transition probability between the current composite state s = {h, m, b} and the next state s = {h , m , b } depends only on the action of the kth user ak . The transition probability function of our problem can now be mathematically expressed by the following equation. P(s |s, a1 , a2 , . . . , aK ) =  P(s |s, ak ) K  = i=1  K  P(hi |hi ) ·  i=1  K  P(mi |mi ) ·  i=1,i=k  P(bi |bi ) · P(bk |bk , ak ).  (3.15)  As each user is equipped with a size B buffer, the buffer occupancy of user k evolves according to Lindley’s equation [24, 25] bk = min([bk − ak + fin,k ]+ , B).  (3.16)  The evolution of the buffer state of user i = 1, 2, . . . , K when i = k follows the following rule: bi = min(bi + fin,i , B).  For user k, the buffer state transition probability depends on input and  68  output date rate. This can be expressed as follows: P(bk |bk , ak ) = I{bk =min([bk −ak +fin,k ]+ ,B)} ,  (3.17)  where I{·} is the indicator function. For those users that are not scheduled for transmission, the buffer state transition probability depends only on the distribution of incoming traffic, which can be written as P(bi |bi ) = I{bi =min(bi +fin,i ,B)} .  (3.18)  Equations (3.11, 3.12, 3.15) define the constrained switching control Markovian game we consider. Our goal is to solve such games. That is, we seek to compute a Nash equilibrium2 policy πi∗ , i = 1, . . . , K (which is not necessarily unique). However, if both the reward (3.11) and constraint (3.12) are zero-sum among all the users, then all the Nash equilibria have a unique value vector and are globally optimal [31]. The Markovian switching control game can be solved by constructing a sequence of MDP as described in Algorithm 3. We refer to [31, Chapter 3.2], [68] for the proof. So, the constrained switching controlled Markovian game (3.11, 3.12) can be solved by iteratively updating the transmission policy πi∗n of user i, i = 1, . . . , K with the policy of remaining users fixed. Here n denotes the iteration index as mentioned in Algorithm 3. At each step, ith 2  A Nash equilibrium [31] is a set of policies, one for each player, such that no player has incentive to unilaterally change their action. Players are in equilibrium if a change in policies by any one of them would lead that player to earn less than if it remained with its current policy.  69  user aims to maximize the overall expected discounted reward subject to its constraint as specified in (3.13).  3.4  Nash Equilibrium Solutions to the Markovian Dynamic Game  This section first presents a value iteration algorithm to compute the Nash equilibrium solution to the formulated general-sum dynamic Markovian switching control game and proves the convergence of such algorithm. Then, we introduce four assumptions on the transmission reward, holding cost and state transition probability functions which lead us to the structural result on the Nash equilibrium policy. This structural result enables us to search for the Nash equilibrium policy in the policy space using a policy gradient algorithm and it has reduced computational cost.  3.4.1  Value Iteration Algorithm  In [31], a value iteration algorithm was designed to calculate the Nash equilibrium for an unconstrained general-sum dynamic Markovian switching control game. Therefore, we convert the problem in (3.13) to an unconstrained one using Lagrangian relaxation and use the value iteration algorithm specified in Algorithm 3 to find a Nash equilibrium solution. n and The algorithm can be summarized as follows. We use Vk=1,2,...,K  λm k=1,2,...,K to represent the value vector and Lagrange multiplier, respectively, of a user k (k = 1, . . . , K) at the nth and mth time slot. The algorithm mainly consists of two parts: the outer loop and the inner loop. The outer loop updates the Lagrange multipliers of each user and the inner loop  70  Algorithm 3 Value Iteration Algorithm Step 1: Set m = 0; Initialize l. 0 }, {λ0 , λ0 , . . . , λ0 }. Initialize {V10 , V20 , . . . , VK 1 2 K Step 2: Inner Loop: Set n=0; Step 3: Inner Loop: Update Transmission Policies; for k = 1 : K do for each s ∈ Sk , πkn (s)  β  =  |S| s =1 P(s  − c(s, ak ) + λm · dk (s, ak ) + k  arg minπkn (s)∈A |s, ak )vkn (s ) ;  |S|  n n n vkn+1 (s) = −c(s, πkn (s)) + λm k · dk (s, πk (s)) + β s =1 P(s |s, πk (s))vk (s ); |S| n+1 n n n vi=1:K,i=k (s) = λm i · di (s, πk (s)) + β s =1 P(s |s, πk (s))vi (s ); end for Step 4: If Vkn+1 − Vkn ≤ ε, k = 1, . . . , K, set n = n + 1, and return to Step 3; Otherwise, go to Step 5. Step 5: Update Lagrange Multipliers for k = 1 : K do  λm+1 = λm k + k  1 l  n)−D Dk (π1n , π2n , . . . , πK k  end for Step 6: The algorithm stops when λm k , k = 1, 2, . . . , K converge, otherwise, set m = m + 1 and return to Step 2. optimize the transmission policy of each user under fixed Lagrange multipliers. The outer loop index and inner loop index are m and n, respectively. Note from Algorithm 3 the interaction between users is through the update n+1 (s) is a function of πkn (s). of value vectors since vi=1:K,i=k  In Step 1, we set the outer loop index m to be 0 and initialize the step 0 size l, the value vectors Vk=1,2,...,K and Lagrange multipliers λ0k=1,2,...,K .  Step 3 is the inner loop where at each step we solve kth user controlled game and obtain the new optimal strategy for that user with the strategy of the remaining players fixed. Note here that the objective of the system is to 71  maximize the transmission reward and minimize the holding cost as shown in (3.13). Since −c(s, ak ) is used for each step the optimal transmission policy πkn (s) is obtained by doing the minimization. Variable ε in step 4 is a small number chosen to ensure the convergence of Vkn . Step 5 updates the Lagrange multipliers based on the discounted delay value of each user given n }. the transmission policies {π1n , π1n , . . . , πK  1 l  is the step size which satisfies  the conditions for convergence of the policy gradient algorithm. This sem quence of Lagrange multipliers {λm 1 , . . . , λK } with m = 0, 1, 2, . . . converges  in probability to {λ∗1 , . . . , λ∗K } which satisfies the constrained problem defined in (3.13) [50, 83]. The algorithm terminates when λm k=1,2,...,K converge, otherwise, go to Step 2. Since this is a constrained optimization problem, the optimal transmission policy is a randomization of two deterministic polices [24, 25]. We use λ∗k=1,2,...,K to represent the Lagrange multipliers obtained with the above algorithm. The randomization policy of each user can be written as: πk∗ (s) = qk πk∗ (s, λk,1 ) + (1 − qk )πk∗ (s, λk,2 ),  (3.19)  where 0 ≤ qk ≤ 1 is the randomization factor and πk∗ (s, λk,1 ), πk∗ (s, λk,2 ) are the unconstrained optimal policies with Lagrange multipliers λk,1 and λk,2 . Specifically, λk,1 = λ∗k − ∆ and λk,2 = λ∗k + ∆ for a perturbation parameter ∆. The randomization factor of the kth user qk is calculated by:  qk =  Dk − Dk (λ1,2 , . . . , λK,2 ) Dk (λ1,1 , . . . , λK,1 ) − Dk (λ1,2 , . . . , λK,2 )  72  (3.20)  The convergence proof of the inner loop of Algorithm 3 is shown in Appendix B and this value iteration algorithm obtains a Nash equilibrium solution to the constrained switching control Markovian game with general sum reward and general sum constraint [31]. Remark: The primary purpose of the value iteration algorithm is to prove the structural results on the Nash equilibrium policy. The value iteration algorithm (Algorithm 3.4.1) is not designed to be implemented in a practical system because at each iteration of the algorithm, a user k is required to know the channel states of all the other users and the system state transition probability matrix.  3.4.2  Structural Result on Randomized Threshold Policy  In this subsection, we characterize the structure of the Nash equilibrium achieved by Algorithm 3. First, we list four assumptions. Based on these four assumptions, Theorem 3.4.1 is introduced. • A 3.4.1: The set of policies that satisfy constraint (3.12) is non-empty, to ensure the delay constraint of the system is valid. A 3.4.1 is the feasible assumption on the system constraint, and it is assumed to be satisfied.  • A 3.4.2: Transmission reward c(s, ak ) is a supermodular3 function of 3  If a function f : A × B × C → R is supermodular in (a, b) for any c ∈ C. Then for all a ≥ a and b ≥ b, f (a , b ; c) − f (a, b ; c) ≥ f (a , b; c) − f (a, b; c) holds. If f is a supermodular function in (a, b), then −f is a submodular in (a, b).  73  bk , ak for any channel quality hk and MAD mk of the current user. c(s, ak ) is also a integer concave function of bk − ak for any hk and mk . It can be seen from Section 3.3.1 that the transmission reward c(s, ak ) is linear in ak (3.5) and independent of the buffer state bk . Thus the assumption A 3.4.2 holds.  • A 3.4.3: Holding cost dk (s, ak ) is a submodular function of bk , ak for any channel quality hk and MAD mk of the current user. dk (s, ak ) is also integer convex in bk − ak for any hk and mk . The holding cost of user k with s ∈ Sk as follows: dk (s, ak ) = κ · (bk − ak + fin,k )τ .  (3.21)  The submodularity property of the holding cost function dk (s, ak ) can be easily verified. It is also straightforward to show dk (s, ak ) is integer convex in bk − ak .  • A 3.4.4: P(bk |bk , ak ) is second order stochastically increasing on (bk − ak ) for any bk and ak . P(bk |bk , ak ) expression given in (3.17) shows it is first order stochastically increasing in (bk − ak ). First order stochastic dominance implies second order stochastic dominance [24, 25], thus assumption A 3.4.4 holds. 74  The following theorem is one of our main results. It shows that the Nash equilibrium of a constrained dynamic switching control game is a randomization of two pure monotone policies. Theorem 3.4.1 For each user k and a given channel state, the Nash equilibrium policy πk∗ (s) is a randomization of two pure policies πk1 (s) and πk2 (s). Each of these two pure policies is a nonincreasing function with respect to buffer occupancy state bk . The detailed proof of Theorem 3.4.1 is given in Appendix C.  3.4.3  Learning Nash Equilibrium Policy via Policy Gradient Algorithm  The value iteration algorithm in Algorithm 3 is applicable to a general type of constrained switching control game. In the case that the system parameters satisfy assumptions A 3.4.1-A 3.4.4 as stated above, we can exploit the structural result (Theorem 3.4.1) to search for the Nash equilibrium policy in the policy space. This results in a significant reduction on the complexity of computing the Nash equilibrium policies. The main idea of searching the Nash equilibrium policy in the policy space is as follows. If there are three actions available with action set ak =  75  {1, 2, 3}, the Nash equilibrium policy πk∗ (s) is a randomized mixture:    1,       p1 ,    ∗ πk (s) = 2,      p2 ,       3,  if  0 ≤ bk < b1 (s)  if  b1 (s) ≤ bk < b2 (s)  if  b2 (s) ≤ bk < b3 (s) .  if  b3 (s) ≤ bk < b4 (s)  if  b4 (s) ≤ bk  (3.22)  Here, {p1 , p2 } ∈ [0, 1] is the randomization factor, b1 (s), b2 (s), b3 (s) and b4 (s) are the buffer thresholds. The search for each Nash equilibrium policy problem is now converted to the estimation of these 6 parameters. Note here that the policy search applies to a system with any number of actions, here we consider a 3 action system as an example. The simultaneous perturbation stochastic approximation (SPSA) method [83] is adopted to estimate the parameters. SPSA is especially efficient in high-dimensional problems in terms of providing a good solution for a relatively small number of measurements for the objective function. The essential feature of SPSA is the underlying gradient approximation, which requires only two objective function measurements per iteration regardless of the dimension of the optimization problem. These two measurements are made by simultaneously varying, in a properly random fashion all of the variables in the problem. The detailed algorithm is described in Algorithm 4. The first part of the algorithm initializes system variables. θ n represents the union of all the parameters we search for at the nth time slot, and θ n = {θ n1 , θ n2 , . . . , θ nK }. θ nk indicates parameter vector of the kth user. The Lagrange multipliers of all the K users are defined to be λn = {λn1 , λn2 , . . . , λnK }. 76  Algorithm 4 Policy Gradient Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9:  Initialization: θ (0) , λ0 ; n = 0; ρ = 4; Initialize constant perturbation step size β and gradient step size α; Main Iteration for k = 1 : K do dimk = 6 × |hk | × |hi | × |bi |; Generate ∆n = [∆n1 , ∆n2 , . . . , ∆ndimk ]T ; ∆ni are Bernoulli random variables with p = 12 . θnk+ = θnk + β × ∆n ; θnk− = θ nk − β × ∆n ; ∆Ckn  = 10:  =  n n ck (sn ,θn k+ )−ck (s ,θ k− ) [(∆n1 )−1 , (∆n2 )−1 , . . . , (∆nmk )−1 ]T ; 2β n ∆Dk n n dk (sn ,θn k+ )−dk (s ,θ k− ) [(∆n1 )−1 , (∆n2 )−1 , . . . , (∆nmk )−1 ]T ; 2β  (n+1)  = θnk − α × ∆Ckn + ∆Dkn · max 0, λnk + ρ · D(sn , θnk ) − Dk  (n+1)  = max (1 −  11:  θk  11:  λk  α ρ  ;  · λnk ), λnk + α · D(sn , θ nk ) − Dk ;  end for 13: The parameters of other users remain unchanged; 14: n = n + 1; 15: The iteration terminates when the values of the parameters θ n converge; else return back to Step 3. 12:  β and α denote the constant perturbation step size and constant gradient step size, respectively. When α → 0,  α β2  → 0 and n → ∞, the policy gradi-  ent algorithm converges weakly [54, Theorem 8.3.1]. In the main part of the algorithm, SPSA algorithm is applied to iteratively update system parameters. When the kth user is scheduled to transmit at time slot n, parameters θ nk and the Lagrange multiplier λnk can be updated after introducing a random perturbation vector ∆n . Meanwhile, the parameters of the other users remain unchanged. The algorithm terminates when θ n converge. Remark: At each iteration of the policy gradient algorithm, user k is only required to know its own transmission reward and holding cost. The 77  transmission reward and holding cost of user k are functions of its own channel state and buffer state (Section 3.4.2). Thus, the policy gradient algorithm is distributed and implementable in a practical system.  3.5  Numerical Examples  In this section, we illustrate the performance improvement of the proposed dynamic switching control game policy over myopic policies. We also present numerical examples of the Nash equilibrium transmission policy. For our comparisons, we used the Joint Scalable Video Model (JSVM)-9-8 reference software [87] to encode two reference video sequences in Common Intermediate Format (CIF) resolution: Foreman and Football. Each sequence is composed of 100 frames encoded in SVC, with an H.264/AVC compatible base layer, and two CGS enhancement layers. The coded video streams are composed of I and P frames only, with only the first frame encoded as an I frame, a frame rate of 30 fps, and minimum QoS guarantee corresponding to the average base-layer quantization parameter value of q0 = 38. The first and second CGS enhancement layers have average QP values of 32 and 26, respectively. The video MAD mk is quantized into three states mk ∈ M = {0, 1, 2}, such that,  mk =    0, if M ADk < µk −         1, if µk −  σk 2  σk 2 ,  ≤ M ADk ≥ µk +  2, if M ADk > µk +  σk 2 ,  σk 2  where µk and σk are the mean and standard deviation of the MAD of user k. Table 3.1 lists the bit-rate in bits per second and average distortion in terms of the luminance peak signal-to-noise ratio (Y-PSNR) of each video 78  layer of the two sequences. Table 3.1: Average incoming rate and distortion (Y-PSNR) characteristics of the two video users. Sequence: Video Layer Base l = 0 First CGS l = 1 Second CGS l = 2  Foreman r¯l,1 (kbps) Y-PSNR (dB) 40.9 30.75 103.36 34.59 242.64 38.87  Football r¯l,2 (kbps) Y-PSNR (dB) 55.82 28.17 139.03 32.47 252.97 37.29  In all the cases considered here, the channel quality measurements hk of user k are quantized into two different states, hk ∈ {1, 2}. In the models used, each user has a size 10 buffer with an incoming rate equivalent to increasing the buffer size by one during a transmission time slot (fin,1 = fin,2 = · · · = fin,K = 1). The transmission time slot duration is set equal to 10 ms thus allowing a maximum transmission delay of 100 ms. This system configuration ensures that the transmission rewards, holding costs and buffer transition probability matrices satisfy assumptions A 3.4.2-A 3.4.3 specified in Section 3.4.2. The channel is assumed to be Markovian and the transition probability matrices are generated randomly. In the system models used, each user has 3 different action choices when it is scheduled for transmission. The 3 different actions are ak = 1, 2, 3 and correspond to the user buffer output of one, two, and three frames, respectively. The action ak = 0 corresponds to the No-Transmit state. Therefore, when the Nash equilibrium policy is 0, the current user is not scheduled for transmission and ak = 0. However, the outgoing traffic is still one frame fin,k = 1. 79  General-Sum Constrained Game: Randomized Monotone Transmission Policy. Fig. 3.4 considers the same two user system of the previous example with a 25 ms transmission delay constraint. As it is a constrained switching controlled Markovian game, the Nash equilibrium policy is a randomization of two pure policies. The figure shows that the optimal transmission policies are no longer deterministic but are a randomization of two pure policies, and that each Nash equilibrium policy is monotone nonincreasing on the buffer state. The results of Fig. 3.4 are obtained by applying the value iteration algorithm. Fig. 3.4 (a) is the optimal policy of user 1 when h2 = 1 and b2 = 1, while, Fig. 3.4 (b) shows the optimal policy of user 2 when h1 = 1 and b1 = 1. Result by Policy Gradient Algorithm: The policy gradient algorithm (Algorithm 4) we propose uses the structural result on the optimal transmit policy and each policy can be determined by three parameters, namely, lower threshold bl (s), upper threshold bh (s) and randomization factor p as described in (3.22) (assume there are 2 actions). The simulation results of the policy of user 1 with h2 = 1, b2 = 1 shown in Fig. 3.5. In the system setup, each user has a buffer size of 10 and 3 actions and the system transmission delay constraint is 25ms. By comparing Fig. 3.5 to Fig. 3.4 (a) we can see that the results obtained by the value iteration algorithm and policy gradient algorithm are very close. Transmission Buffer Management: We adopt a priority based packet drop similar to the Drop Priority Based (DPB) buffer management policy proposed in [56]. With this policy, low priority packets that have resided longest in the transmission buffer are dropped when the buffering delay ap80  Monotone Transmission Policy 3  Optimal Policy  2.5 2 1.5 1 0.5 Channel State  0  1  2 10  9  8  7  6 5 Buffer State  4  3  2  1  Monotone Transmission Policy 3  Optimal Policy  2.5 2 1.5 1 0.5 0  Channel State 1 2 10  9  8  7  6 5 Buffer State  4  3  2  1  Figure 3.4: The Nash equilibrium transmission policy obtained via value iteration algorithm for user 1 (a) and user 2 (b). The result (a) is obtained when h2 = 1 and b2 = 1 and (b) is obtained when h1 = 1 and b1 = 1, respectively. The transmission delay constraint is specified to be 25 ms. Each Nash equilibrium transmission policy is a randomization of two pure monotone policies. 81  Monotone transmission policy  3  Optimal Policy  2.5 2 1.5 1 0.5 Channel State 1  0  2 10  9  8  7  6 5 Buffer State  4  3  2  1  Figure 3.5: The Nash equilibrium transmission control policy obtained via policy gradient algorithm for user 1. The result is obtained with a 25 ms transmission delay constraint and the states of users 2 are h2 = 1 and b2 = 1. The transmission policy is monotone nonincreasing on its own buffer state.  proaches the delay constraint. In [56], the DPB policy was found to be second best to the Drop Dependency Based (DDB) which requires video dependency information that cannot be collected in real time. The DPB policy is attractive since it requires minimal priority information about the video payload which is readily available in the video Network Abstraction Layer Unit (NALU) header of a MGS/CGS coded video stream. We implement this policy by shifting the user action ak to a lower state that satisfies the buffer delay constraint. In order to demonstrate the effectiveness of the proposed dynamic switching control game algorithm, we compare its performance to that of a myopic 82  policy that selects at each time slot the user action that maximizes the video PSNR while satisfying the transmission buffer delay constraint. The time slot duration equals to the video frame capturing duration which is around 30ms. The myopic policy does not consider the effect of current actions on future states. We assume that the delay constraint corresponds to the packet play-out deadline, therefore, all buffered packets that exceed the delay constraint are assumed to be lost and are discarded form the user buffer. Figs. 3.6 and 3.7 show the results of simulating the transmission of two users Foreman and Football for both the proposed switching control game policy and the myopic policy. The figures show that under the same channel conditions, the proposed policy has better Y-PSNR performance and transmission buffer utilization for both users.  3.6  Summary  This chapter considers a time-slotted multiuser WLAN system where each user delivers SVC video bitstream. The modified CSMA/CA mechanism of such system is implemented through an opportunistic scheduling system access rule. By modelling video states and block fading channels as a finite state Markov chain, the rate adaptation problem among users can be formulated as a constrained general-sum switching control Markovian game. The Nash equilibrium transmission policy of such game can be computed through a value iteration algorithm. Given four assumptions on the transmission reward, holding cost and transition probability functions (Section 3.4.2), the Nash equilibrium policy is a randomized mixture of two pure policies with each policy monotone nonincreasing on the buffer state occupancy. This 83  Y−PSNR  Football 40 30 20  0  20  40  60  80  100  0  20  40  60  80  100  0  20  40  60  80  100  0  20  40  60  80  100  Action  2 1  buffer state  channel state  0 2 1.5 1 10 5 0  Time Slot  Figure 3.6: Result of the transmission of the Football sequence comparing the performance in terms of video PSNR and buffer utilization between the proposed switching control game policy and the myopic policy with 80 ms delay constraint. The result shows that the proposed switching control game policy performs better than the myopic policy.  structural result enables us to search for the Nash equilibrium policy in the policy space via a policy gradient algorithm.  84  Y−PSNR  Foreman 40 30 20  0  20  40  60  80  100  0  20  40  60  80  100  0  20  40  60  80  100  0  20  40  60  80  100  Action  4 2  buffer state  Channel state  0 2 1.5 1 10 5 0  Time Slot  Figure 3.7: Result of the transmission of the Foreman sequence comparing the performance in terms of video PSNR and buffer utilization between the proposed switching control game policy and the myopic policy with 80 ms delay constraint. The result shows that the proposed switching control game policy performs better than the myopic policy.  85  Chapter 4  Cognitive Base Stations in 3GPP LTE Femtocells: A Correlated Equilibrium Game Theoretic Approach 1 This chapter considers downlink spectrum allocation in an LTE system macrocell which contains multiple femtocells. By incorporating cognitive capabilities into femtocell base stations, the Home evolved Node Bs (HeNBs) can be formulated as secondary base stations seeking to maximize the spectrum utility while minimizing interference to primary base stations (evolved Node-Bs). The competition amongst cognitive HeNBs for spectrum re1  This chapter is based on the following manuscript. J. W. Huang and V. Krishnamurthy, “Cognitive Base Stations in LTE/3GPP Femtocells: A Correlated Equilibrium Game Theoretic Approach,” IEEE Transactions on Communications, September 2011.  86  sources is formulated as a non-cooperative game-theoretic learning problem where each agent (HeNB) seeks to adapt its strategy in real time. We formulate the resource block (RB) allocation among HeNBs in the downlink of an LTE system using a game-theoretic framework, where the correlated equilibrium solutions of the formulated game are being investigated. A distributed RB access algorithm is proposed to compute the correlated equilibrium RB allocation policy.  4.1  Background  An important feature of 3GPP LTE systems [1] is that it allows distributed implementation of femtocells to meet a variety of service requirements. The femtocell access points, denoted as or Home evolved Node-B (HeNB) in 3GPP, are low-cost, low-power, plug-and-play cellular base stations. In order to provide broadband connectivity, these HeNBs will need to possess adaptive/cognitive facilities. In the October 2010 release of 3GPP, HeNBs are described as self-optimized nodes in a Self-Organized Network (SON) which need to maintain quality of service (QoS) with minimal intervention from the service operator [86]. HeNBs are equipped with cognitive functionalities for load balancing, interference management, random access channel optimization, capacity and coverage optimization, and handover parameter optimization. With the above motivation, this chapter considers spectrum resource allocation in an orthogonal frequency division multiple access (OFDMA) LTE downlink system which consists of a macrocell base station (eNB) and multiple femtocell base stations (HeNBs). By incorporating cognitive capabilities 87  into these self-optimized femtocell base stations, the cognitive HeNBs aim to maximize the spectrum utility by utilizing the unoccupied frequencies while minimizing interference to the eNB (primary base station) in a spectrum overlay LTE system. The unit of spectrum resource to be allocated in an LTE system is called a resource block (RB) and it is comprised of 12 subcarriers at a 15 kHz spacing. Given the RB occupancy of the eNB, the competition for the spectrum resources among HeNBs can be formulated in a game-theoretic setting [91]. Instead of computing the Nash equilibrium policy of the formulated game, we seek to characterize and compute the correlated equilibrium policy set [7, 8]. The set of correlated equilibria is a convex polytope. It includes the set of Nash equilibria – indeed the Nash equilibria are isolated points at the extrema of this set [69, 70]. The set of correlated equilibria [8] is arguably the most natural attractive set for a decentralized adaptive algorithm such as the one considered here, and describes a condition of competitive optimality between agents (cognitive femtocell base stations). It is more preferable than Nash equilibria since it directly considers the ability of agents to coordinate their actions. This coordination can lead to higher performance than if each agent was required to act in isolation. Indeed, Hart and Mas-Colell observed in [36] that, for most simple adaptive procedures, “... there is a natural coordination device: the common history, observed by all players. It is thus not reasonable to expect that, at the end, independence among players will obtain.” Since the set of correlated equilibria is convex, fairness between players can be addressed in this domain. Finally, decentralized, online adaptive procedures naturally converge to the correlated equilibria, 88  whereas the same is not true for Nash equilibria (the so-called law of conservation of coordination [37]). There are several important issues being addressed in recent literature regarding the deployment of HeNB femtocells in an LTE system. One area of interest is how to mitigate the interferences among HeNBs so as to improve the system performance. In [59] and [58], Lopez-Perez et al. used Dynamic Frequency Planning (DFP), an interference avoidance technique, to decrease the inter-cell interference and increase the network capacity by dynamically adapting the radio frequency parameters to the specific scenario. They also verified the performance of the DFP technique in an OFDMA WiMAX macrocells and femtocells system. Choi et al. investigated in [18] how to minimize the interference caused by femtocells in an open access (spectrum underlay) network. They showed adaptive open access will maximize the value of a femtocell both to its owner and to the network using numerical results. Attar et al. studied the benefits of developing cognitive base-stations in a UMTS LTE network [6]. Radio resource management protocols are not specified by standards, such as 3GPP’s UMTS LTE. Thus, there is considerable flexibility in their design. The insufficiency of traditional coexistence solutions in LTE context is shown in [6]. It is argued that cognitive base-stations are crucial to an efficient and distributed radio resource management of LTE given its distributed architecture. One motivation for such argument is the lessons learnt from wide-spread deployment of wireless local area network (WLAN) access points. The simple plug-and-play nature of WLAN routers, along with the unlicensed nature of WLAN spectrum access, alleviated the 89  need for time and cost-intensive network plannings. This in turn helped a rapid proliferation of WLAN hotspots. However, as the number of coexisting WLAN networks increases, so does their mutual interfering effect, rendering such simple, selfish coexistence strategies problematic. By incorporating the main three cognitive radio capabilities into the LTE base-stations, which are 1. radio scene analysis; 2. online learning based on the feedback from RF environment; 3. and agile/dynamic resource access schemes; a successful coexistence strategy among eNBs and HeNBs can be achieved.  4.2  Resource Allocation Among HeNBs: Problem Formulation  We consider a macrocell area with a number of femtocells randomly deployed by home and offices within an OFDMA LTE network (Fig. 4.1). By incorporating the cognitive capacities into femtocell base stations, the spectrum occupancy behaviour of the macrocell base station (eNB) can be formulated as a primary base station and that of the femtocell base stations (HeNBs) can be formulated as secondary base stations (cognitive base stations) in a spectrum overlay network2 . Due to the selfish nature of each base station, the competition for the common spectrum resource among cognitive base stations can be formulated using a game-theoretic framework. 2  As described in [6], it is also possible to study an underlay cognitive femtocell strategy in which RBs accessed by eNB and HeNBs are not orthogonal. However, as the objective of our analysis is modelling the competition among selfish HeNBs, the proposed solutions can be readily extended to the aforementioned spectrum underlay LTE systems.  90  Figure 4.1: System schema of a single eNB macrocell which contains multiple HeNB femtocells in a 3GPP LTE network.  4.2.1  System Description  The resource allocation process in LTE networks follows a time slotted system model where each time slot length equals to that of an RB (0.5 ms), with RB being the smallest unit of resource that can be allocated. Let t ∈ {1, 2, . . . , T } denote the time slot index. T defines the time horizon of the formulated game. Let N denote the total number of available RBs in t denote the number of RBs occupied by HeNBs at time t the system, Nhenb t and NeN B denotes the number of RBs occupied by eNB at time t. As we t t consider a spectrum overlay system, Nhenb ≤ N − NeN B .  91  Let K denote the total number of HeNBs coexisting in the network, t } denote k ∈ K = {1, 2, . . . , K} denote the user index and f ∈ {1, 2, . . . Nhenb  the RB index occupied by cognitive base stations (HeNBs). We use ptk (f ) ∈ {0, 1} to denote the action of the kth HeNB on the f th RB at time t, where 0 represents not transmit and 1 represents transmit. t )} ∈P k denote the action of the kth HeNB Let ptk = {ptk (1), . . . , ptk (Nhenb  over all the available RBs to HeNBs at time t, with Pk denoting the action space of the kth HeNB. pt = {pt1 , . . . , ptK } ∈P is used to denote the composition of the actions from all the HeNBs at time t. P is the joint action space of all HeNBs. We use stk (f ) ∈ {1, 2, . . . , Qs } to denote channel quality state of the kth HeNB on the f th RB at time t after quantization. For example, the channel quality can be obtained by quantizing a continuous valued channel model comprising of circularly symmetric complex Gaussian random variables into t )} denote the channel Qs different states. Let stk = {stk (1), . . . , stk (Nhenb  state composition of the kth HeNB over all the available RBs and st = {st1 , . . . , stK } denote the channel state composition over all the HeNBs. Let Ikt (f ) denote the interference introduced to the kth HeNB at time t on the f th RB. The interference comprises two parts, namely, noise and Co-Channel Interference (CCI). The noise ntk (f ) is assumed to be Additive White Gaussian Noise (AWGN) with a noise covariance of σk2 (f ) and the CCI is introduced by having different HeNBs sharing the same RB. An interference matrix wt (f ) is introduced to model the CCI among all HeNBs on the f th RB at time t. The elements of this interference matrix, i.e., t (f ) where i, j ∈ K, denote the cross channel quality between the ith and wi,j  92  jth HeNBs. Assuming channel reciprocity, wt (f ) is a symmetric matrix, t (f ) = w t (f ). It is clear that the interference matrix has zeroi.e., wi,j j,i t (f ) = 0 for ∀ i ∈ K. The value of wt (f ) valued diagonal elements, i.e., wi,i  depends on the location of all the HeNBs. We assume the kth (k ∈ K) HeNB has the following information at the beginning of a time slot t. t : the number of available RBs for HeNBs, 1. Nhenb  2. stk , i.e., the channel state vector of the kth HeNB, 3. Ikt (f ) which is the received interference at HeNB k on RB f at time t, 4. the current demand level of the kth HeNB which is denoted by dtk . t Nhenb can be obtained through a fixed broadband access network (e.g., Dig-  ital Subscriber Line (DSL), Cable) as described in [85]. stk and Ikt (f ) can be obtained by channel sensing mechanism during the guard interval at the beginning of each time slot. dtk is used to denote the system resource demand level of HeNB k at time t, it is of the same unit as that of system capacity. dtk is determined by the user requirement with in a HeNB cell at time t. In the case that there are more devices transmit data using frequency bandwidth within HeNB k at time t, dtk is of higher value. An important characteristic of this model is that radio-specific quantity dtk needs only to be known to the kth HeNB cell, thus allowing decentralized resource allocation algorithms. Based on the above information, HeNB k chooses its action vector ptk , selfishly, to maximize its local utility function. The definition of utility will be presented in Section 4.2.2. 93  Note that in the case that HeNBs are owned by different agents and they are so sophisticated to behave maliciously, they can opt not to reveal their true demand levels dtk (k = 1, . . . , K) to optimize their own utilities at the cost of reducing the overall system performance. It requires mechanism design theory in order to prevent this from happening. Similar problems has been studied in [42] where the authors applied pricing mechanism to ensure each rational selfish user maximize its own utility function, at the same time optimizing the overall system utility. Reputation based mechanism design is another area of research, where the system uses reputation as a tool to motivate cooperation between users and indicate a good behaviour within the network. If a user does not pay attention to its reputation and keep acting maliciously, it will be isolated and discarded. Such reputation based mechanism has been applied in ad hoc networks and sensor networks [16, 78]. However, this chapter assumes all the malicious behaviours have been eliminated in the system and each HeNB uses its true demand level dtk to compute its utility function. The distributed decision making process amongst HeNBs defines the action set pt , which in turn leads to a different realization of interference t level Ikt (f ), ∀ k ∈ K and f = {1, . . . , Nhenb }. Therefore, the action of one  femtocell base station (HeNB) affects the utilities of other femtocells, which motivates the use of game-theoretic approaches to analyze and compute the RB allocation policies among all the HeNBs. In the following subsection, we define the global system utility function and the local utility functions for femtocell base stations.  94  4.2.2  Utility Function  The goal of this chapter is to optimize the global resource allocation problem using a decentralized approach. We should demonstrate a connection between global utility function and the local utility function that will guide the allocation decision of each HeNB. This connection is presented through the derivation of global and local performance measures. This subsection defines a global utility function to evaluate the overall system performance, based on which a local utility function is defined for each cognitive HeNB. Each HeNB selfishly maximizes its local utility function which does not guarantee the global system performance. We aim to design local utility functions which ensure global system performance quality at the correlated equilibrium of the formulated game. Let Ckt denote the capacity of HeNB k at time t. If a capacity-achieving code such as turbo code or low-density parity-check code (LDPC) code is applied for error correction, Ckt can be expressed as follows using ShannonHartley’s theorem [93]. t Nhenb  Ckt = f =1  ω · log2 1 +  ptk (f ) × stk (f ) Ikt (f )  (4.1)  K  Ikt (f )  =  σk2 (f )  + i=1  t wk,i (f ) × pti (f ),  (4.2)  where ω denotes the bandwidth of an RB. We assume that all the HeNBs t = 0 for k = treat the interferences as Gaussian noises. Note that wk,k  1, . . . , K. (4.2) models the Inter User Interference (IUI) as a function of actions of all the other users in the system. However, in a practical system, 95  Ikt (f ) is obtained by channel estimation through downlink common pilots. Thus, a user k is not required to have full knowledge of pt in order to compute Ckt . Global Utility Function Since all cognitive HeNBs have equal priority in accessing system resources, a global objective is chosen to maximize the performance of the worst-off HeNB such that the available resources are fairly allocated among HeNBs. Therefore, given the action vector pt of all the HeNBs, the global utility function at time t is defined as follows. UG (pt ) = min min( k∈K  Ckt , 1) . dtk  (4.3)  Ct  Here, the term min( dtk , 1) represents the satisfaction level of the kth HeNB k  and it is a function of the instantaneous capacity of the kth HeNB at time t divided by its current demand level. Note that mathematically (4.3) is equivalent to UG (pt ) = mink∈K  Ckt , dtk  Ct  operation min( dtk , 1) is applied because k  the satisfaction level is within the range of [0, 1]. The action vector among all the HeNBs at time t pt is chosen to maximize the minimum satisfaction level among all the K HeNBs. That is, pt = arg max UG (p) = max min min( p∈P  p∈P k∈K  Ckt , 1) . dtk  (4.4)  This global utility optimization problem (4.4) aims to maximize the minimum satisfaction level among all the K HeNBs. Note the global utility function can be of other forms, e.g., aiming to maximize the average sat96  isfaction level among all the K HeNBs, in which case the global utility function can be specified as follows (4.5). However, this chapter focuses on the scenario where the global utility is chosen to maximize the worst-off user (4.3). UG (pt ) =  1 K  min( k∈K  Ckt , 1) . dtk  (4.5)  The system objective is to find the action vector pt so as to maximize the satisfaction level of the worst-off HeNB UG (pt ). The proposed approach to achieve the correlated equilibrium policy in a decentralized way, is to allow each cognitive HeNB choosing its action ptk (k ∈ K) based on the optimization of its local utility function. The question that remains to be answered is how to select a proper local utility function which also ensures a good overall system performance. In the next subsection, we will derive such a utility function and determine its relation to the global utility objective. Local HeNB Utility Function If each cognitive HeNB has a reasonable estimate of the global utility function, UG , then the decentralized resource allocation policy can be directly realized by each cognitive HeNB choosing an action which maximizes its estimate of UG . However, as the global utility function also depends on the private information of other players, i.e., the demand levels dti and actions pti (i ∈ K and i = k) of other HeNBs, such an assumption is not practical. Below, we construct a local utility function Uk for HeNB k (k ∈ K) consisting of three parts, where each part addresses a certain aspect of the  97  problem at hand. The first part of the local utility function reflects the self interest component of a cognitive HeNB, given by, Uk [1](ptk ) = min  Ckt ,1 . dtk  (4.6)  Maximizing Uk [1](ptk ) is equivalent to maximizing HeNB k’s portion of the global utility UG (4.3). However, a game with (4.6) as the only component of the local utility function would resemble a congestion game, which may not be solvable in closed form. Moreover, (4.6) only shows the self interested part of the global objective (4.3) and it neglects the inter-relation of decisions of each player on the achieved utility of other players. Since each HeNB known only its own demand leven dtk and action ptk , we induce such interaction through the following two principles. 1. At each time t, the capacity of the kth HeNB Ckt should not exceed its demand level dtk , as it leaves less resource to other HeNBs. 2. Each HeNB should minimize its transmission power, as higher transmission power decreases the performance of other HeNBs by introducing higher interferences. The first principle is satisfied by introducing the second component of the local utility function where a penalty is introduced if the choice of resources of a HeNB exceeds its demand level. The details of the second component  98  of the local utility function are shown as follows: Uk [2](ptk ) = −  1 C t − dtk dtk k  +  ,  (4.7)  where (x)+ denotes the operation max(x, 0). The second local utility component suppresses greedy HeNB behaviour, and it brings Ckk closer to its demand dtk . This local utility component also helps to maintain the fairness among all the HeNBs. The third component of the local utility function is used to implement the second principle by considering power as part of the cost of a HeNB, thus, encouraging each HeNB to minimize its power consumption. The details of the third component are shown as follows: t Nhenb  Uk [3](ptk )  =−  ptk (f ).  (4.8)  f =1  Note that we assume unit power transmission of all the HeNBs on each of the RB in our system model. In this system model, each HeNB is assumed to be a selfish user aiming to maximize its own utility function with the minimum cost. By including the power consumption cost as part of the local utility function, it helps the local utility to represent the global utility. Based on the above definitions, the local utility function of a HeNB k can be defined as follows. Uk (ptk ) = Uk [1](ptk ) + α2 · Uk [2](ptk ) + α3 · Uk [3](ptk ),  99  (4.9)  where α2 and α3 are the weighting factors introduced to combine the three utility components assuming Uk [1] has a unit weighting factor, i.e., α1 = 1. These weighting factors are necessary because the actual effect of each component is unknown. By carefully adjusting the values of (α2 , α3 ), we can change the effect of each of the three utility components, which then enable the local utility to mimic the behaviour of the global utility (4.3) in the best way. Thus, the question remains to be how to choose (α2 , α3 ) which lead to the best overall system performance (4.3). This chapter does not provide a closed form solution to this question, instead, we choose the weight factors according to the numerical studies in (4.4). The selected weighting factors (α2 , α3 ) does not ensure the maximization of the instantaneous global utility function at time t, instead, it maximizes the expected system performance under different channel realizations, i.e., Est1 ,st2 ,...st UG (Pt ) . K  Recall that a cognitive HeNB k, ∀k ∈ K, tries to maximize its utility function Uk (ptk ) selfishly by choosing the action vector ptk at the beginning of each time slot. In the following sections, we will show the existence of the correlated equilibrium solution, given the distributed decision making process of HeNBs in a static environment.  4.3  Correlated Equilibrium Solutions with a Game-Theoretic Approach  This section uses game-theoretic approach to formulate the resource allocation among cognitive femtocell base stations (HeNBs) in a static environment, each of HeNB is formulated as a selfish game player aiming to maximize its local utility function (4.9). A static environment is where the 100  system parameters (e.g., the channel statistics stk (k ∈ K), the primary base station behaviour and the number of HeNBs K) are constants or slowly evolve with time. We investigate the correlated equilibrium solution of this static game, which can be obtained via a distributed RB access algorithm. The algorithm is an adaptive variant of the regret matching procedure of [36]. It dynamically adapts the behaviour of HeNBs to time varying environment conditions. We also prove the RB access algorithm converges to the correlated equilibrium set of the defined game.  4.3.1  Definition of Correlated Equilibrium  In a K-player (HeNB) game set-up, each HeNB k (k ∈ K) is a selfish game player aiming to devise a rule for selecting an action vector ptk at each time slot to maximize (the expected value of) its utility function Uk (ptk ). Since each player only has control over its own action ptk , the optimal action policy depends on the rational consideration of the action policies from other users. We focus on the correlated equilibrium solution of the considered game [7, 8], this solution is an important generalization of the Nash equilibrium and is defined as follows. Definition 4.3.1 Define a joint policy π to be a probability distribution on the joint action space P = P1 × P2 . . . PK . Given actions of other players pt−k , the policy π is a correlated equilibrium, if for any alternative policy ptk ∈ Pk , it holds that,  pt−k ∈P−k  π(pt−k , ptk )Uk (pt−k , ptk ) ≥  pt−k ∈P−k  101  π(pt−k , ptk )Uk (pt−k , ptk ). (4.10)  Correlated equilibrium can be intuitively interpreted as if π provides the K players a strategy recommendation from the trusted third-party. The implicit assumption is that the K − 1 other players follow this recommendation, and player k ask itself whether it is of its best interest to follow the recommendation as well. The equilibrium condition states that there is no deviation rule that could award player k a better expected utility than π. Any Nash equilibrium can be represented as a correlated equilibrium when users can generate their recommendations independently. One of the advantages of using correlated equilibrium is that it permits coordination among users, generally through observation of a common signal, which leads to an improved performance over a Nash equilibrium [8].  4.3.2  Decentralized RB Access Algorithm  The RB access algorithm is an adaptive extension of the regret matching procedure [36], it enables HeNBs to adapt their policies to time varying system environment. Let Hk (pt ) denote the regret matrix of the kth cognitive HeNB at time t, it is of size |Pk | × |Pk | with its (|i|, |j|)th entry (i, j ∈ Pk ) specified as, (|i|,|j|)  Hk  (pt ) = 1(pt =i) × [Uk (j, pt−k ) − Uk (i, pt−k )], k  (4.11)  where 1(·) is an indicator function. Furthermore, we define θ tk to be the overall regret matrix of HeNB k and it is also of size |Pk | × |Pk |. The regret value θ tk (|j|, |i|) measures the average gain of user k at time t if k had chosen action i in the past (from time 0 to t) instead of j. If the gain is positive, k is  102  more likely to switch to action i in the future, otherwise, k is more likely to stay with j. Specifically, policy transition matrix is a function of the current regret matrix θ kt−1 as specified in (4.12). Note this regret based scheme requires users to know the reward for each action, even if that action is not taken. Users updates its policy based on the calculated regret matrix (4.12). An RB access algorithm is proposed to compute the correlated equilibrium resource allocation policy, the details of which are listed in Algorithm 5. Algorithm 5 LTE Cognitive HeNB RB Access Algorithm Step 1 Set t = 0; Initialize p0 and set θ0k = Hk (p0 ) for ∀k ∈ K. Step 2 for t = 1, 2, 3, . . . do Action Update: For ∀k ∈ K, choose ptk = i with probability  t−1 max(θk (|j|,|i|),0)  if i = j t−1 t−1 t µ P(pk = i | pk = j, θ k ) = t−1 (|j|,|i|),0) max(θ k  1− if i = j i=j µ  (4.12)  Regret Matrices Update: Based on the new action, the overall regret matrices are updated according to the following stochastic approximation algorithm with step size εt . θtk = θt−1 + εt · (Hk (pt ) − θt−1 k k ), ∀k ∈ K.  (4.13)  end for  Algorithm 5 can be summarized as follows. Step 1 initializes the system by setting time index t = 0 and the initial values of p0 and θ 0k . Step 2 is the main iteration of the algorithm which is composed of two parts: actions update and regret matrices update. A HeNB k chooses its action for time slot t according to the current action pkt−1 and the regret matrix θ kt−1 . In (4.12), µ is a constant parameter which is chosen to be µ ≥  t−1 i=j max(θk (|j|, |i|), 0)  to ensure the probabilities are non-negative. The choice of step size εt used in  103  (4.13), can be either a decreasing step size εt = 1/t or a small constant step size εt = ε (0 < ε  1). If system parameters do not evolve with time, using  decreasing step size convergences the algorithm to the correlated equilibrium set with probability one. Using a constant step size enables the algorithm to track of the correlated equilibrium set if system parameters slowly evolve with time. Depending on the selection of the step size εt , the regret matrix θ tk can be rewritten as follows. θtk (|j|, |i|)  =  θtk (|j|, |i|)  =  1 t  l≤t,plk =j  l≤t,plk =j  Uk (i, pl−k ) − Uk (j, pl−k ) , if εt =  1 ; t  ε(1 − ε)t−l Uk (i, pl−k ) − Uk (j, pl−k ) , if εt = ε. (4.14)  The above RB access algorithm is a modification of the regret matching procedure in [36]. In the regret matching approach, the decisions of each HeNB is based on the average history of all past observed performance results. This choice, however, is not desirable in our scenario since the system parameters may vary over time. Instead, our algorithm adapts the regret matrices θ tk (k ∈ K) according to the updated system parameters which captures the time-varying nature of the system. The decentralized feature of this RB access algorithm permits its implementation among the distributed femtocells in a 3GPP LTE network. Regret matrix θ tk is one of the key parameters in Algorithm 1, based on which a cognitive HeNB adjusts its future action. The interpretation of the (|i|, |j|)th entry of θ tk is that measures the average gain that a cognitive 104  HeNB k would have, had it chosen action j in the past (i.e, in the (t − 1)th time slot) instead of i. Remark: RB access algorithm (Algorithm 5) is a distributed algorithm, the k user is only required to know its own action and interference at each iteration. IUI Ikt (f ) is obtained by channel estimation through downlink common pilots. Ikt (f ) in (4.2) provides the mathematical formulation of IUI.  4.3.3  Convergence of RB Access Algorithm  By using the result from [9], we introduce Theorem 4.3.2 which proves that the RB access algorithm (Algorithm 5) converges to the correlated equilibrium under certain conditions. Let θ k denote the regret matrix of the kth HeNB when t → ∞. We use ΓΩ (θ k ) to denote the projection of parameter θ k on Ω, where Ω is the closed negative orthant of R|pk |×|pk | . Let < x, y > denote the inner product of x and y. Theorem 4.3.2 The RB access algorithm (Algorithm 5) is ensured to converge to the correlated equilibrium set of the formulated game. Proof: By using Proposition 3.8 and Corollary 5.7 in [9], we know if every HeNB follows the strategy in Algorithm 5, it is enough to prove that the following inequality, given by (4.15), holds in order to prove that Algorithm 5 converges to the set of correlated equilibria of the defined game.  < θ k − ΓΩ (θ k ), θ k − ΓΩ (θ k ) >≤ 0.  105  (4.15)  Condition (4.15) is originated from the Blackwell’s sufficient condition for approachability [14]. Therefore, we only need to demonstrate that (4.15) holds in order to prove the convergence of RB access algorithm. Note that the negative orthant Ω is a convex set. The left hand side of (4.15) can be expressed as,  < θ k − ΓΩ (θ k ), θ k − ΓΩ (θ k ) > = < θ k − ΓΩ (θ k ), θ k > − < θ k − ΓΩ (θ k ), ΓΩ (θ k ) >,  (4.16)  where < θ k −ΓΩ (θ k ), ΓΩ (θ k ) >= 0 due to the definition of projection. Thus, in order to establish (4.15) we need to prove < θ k − ΓΩ (θ k ), θ k >≤ 0. Let us construct a Markov chain with the transition probability specified in (4.12) and use τ|i| (i ∈ Pk ) to denote the stationary distribution of such a Markov chain. τ|i| can be specified as follows. τ|i| =  |j|=|i|  τ|j|  θk+ (|j|, |i|) + τ|i| · 1 − µ  where µ is a constant chosen to be µ >  |i|=|j|  θk+ (|i|, |j|) , µ  + |j|=|i| θk (|j|, |i|).  (4.17)  Then, (4.17) is  equivalent to the following equation.  |j|=|i|  τ|j| θk+ (|j|, |i|) = τ|i|  106  |j|=|i|  θk+ (|i|, |j|).  (4.18)  Since the projection is on the negative orthant Ω, θ k − ΓΩ (θ k ) = θ + k. < θ k − ΓΩ (θ k ), θ k > can then be written as < θ k − ΓΩ (θ k ), θ k >=< θ + k , θk > = j  i=j  = i=j  j  θk+ (|j|, |i|) Uk (i, p−k ) − Uk (j, p−k ) τ|j| θk+ (|i|, |j|)Uk (j, p−k )τ|i| −  = j  i=j  θk+ (|i|, |j|)τ|i| −  i=j  j  i=j  θk+ (|j|, |i|)Uk (j, p−k )τ|j|  θk+ (|j|, |i|)τ|j| Uk (j, p−k )  = 0.  (4.19)  Therefore, the condition stated in (4.15) is proved to hold. This concludes the proof of Theorem 4.3.2.  4.3.4  Correlated Equilibrium under Dynamic Environments and Curse of Dimensionality  In the case that system contains large number of active mobile users, it causes high dynamic macro base station behaviour. In which scenario, the problem can be described as: resource allocation among femtocells (HeNBs) in an OFDMA LTE downlink system under a dynamic environment where the resource occupancy behaviour of the primary base station (system state) is varying quickly while other system parameters (e.g., number of HeNBs) are constants or evolving slowing. By formulating the dynamic of system state as a Markov chain, it requires Markov game-theoretic approach to formulate the resource allocation problem among femtocell base stations (HeNBs) as a Markov game. Dif-  107  ferent from static game, system state and state transition probabilities are important elements in dynamic games as they abstract the time-varying nature of a dynamic environment. A reasonable choice of the system state t , st1 , . . . , stK ] which is composed of the number of available RBs for is [Nhenb  HeNBs and the channel states of HeNBs. By defining the correlated equilibrium or Nash equilibrium of such a dynamic stochastic game, different optimization algorithms can be used to compute the equilibrium transmission polices. E.g., [43] proposed iterative value optimization algorithm and stochastic approximation algorithm to compute the Nash equilibrium policies in the formulated Markovian game. Potential applications notwithstanding, there remains substantial hurdles in the application of dynamic stochastic games as a modelling tools in practice. Discrete-time stochastic games with finite number of states are central to the analysis of strategic interactions among selfish HeNBs in dynamic environment. The usefulness of discrete-time games, however, is limited by their computational burden; in particular, there is “curse of dimensionality”. In a discrete-time dynamic stochastic game, each game player (HeNB) is distinguished by an individual state at each time slot. The system state is a vector encoding the number of players with each possible t , st1 , . . . , stK ]. The system state is value of the individual state variable [Nhenb  exponential to the number of the HeNBs in the system. How to efficiently reduce the state space is yet an issue to solve before the implementation of stochastic games in LTE systems. One direction of research is to consider continuous-time stochastic game models. E.g., [27] aims to reduce the dimensionality by exploring the alternative continuous-time stochastic 108  games with a finite number of states and show that the continuous time has substantial advantages.  4.4  Numerical Examples  Algorithm 5 is designed to compute the correlated equilibrium policies for cognitive HeNBs in the downlink of an OFDMA LTE system. This section illustrates the performance of the RB access algorithm (Algorithm 5) in a game set-up. The RB access algorithm is highly scalable and is able to handle large number of users. We consider K = 6 HeNBs in this section just for demonstration purpose. For the kth HeNB (k ∈ K), its channel quality at any RB f (f ∈ F) is sk (f ) ∈ {1, 2, 3}, its demand level belongs to the set dk ∈ {10, 20, 30, 40}. The action of the kth HeNB at the f th RB is specified as pk (f ) ∈ {0, 1}, where 0 represents no transmission and 1 represents transmit. In our simulation set-up, we specify the noise covariance to be σk2 (f ) = 0.1 (∀k ∈ K and ∀f ∈ F). In this simulation model, the number of available RB for HeNBs is fixed to be Nhenb = 10. In first example, we are going to investigate the impact of the pricing parameters (α2 , α3 ) in the local utility function (4.9) on different global utilities (4.3) and (4.5). The pricing parameters is chosen to ensure the global system performance. This is an off-line calculation procedure. The simulation results in Fig. 4.2 and Fig. 4.3 are averaged over 1000 iterations, where 20 different scenarios with different noise nk (f ), channel states sk are being considered in each iteration. In both figures, x-axis and y-axis denote α2 and α3 , respectively. Z-axis denotes the global system performance specified in (4.3) in Fig. 4.2, while it denotes the system average performance 109  specified in (4.5) in Fig. 4.3. Based on Fig. 4.2, we notice the completely selfish behaviours from HeNBs (α2 = 0, α3 = 0) do not ensure the optimum of global system performance; while α2 = 5, α3 = 0.25 lead the least satisfaction level among HeNBs to 0.9, i.e., UG (pt ) = 0.9. We will specify α2 = 5 and α3 = 0.25 in the simulation for Fig. 4.4. Fig. 4.3 shows the system average performance (4.5) is less sensitive to the change of pricing parameters (α2 , α3 ). System average performance is of similar level when the pricing parameters are of the range 0 ≤ α2 ≤ 7, 0.05 ≤ α3 ≤ 0.5. Thus, in the case we choose (4.5) as the global utility function, the selection of (α2 , α3 ) is not unique, they can be of any values within the above range. It can also be noticed from Fig. 4.3 that the system average performance decrease dramatically when α2 > 7. This can be explained as follows: if the power consumption cost is greater than a certain threshold, a HeNB will choose not to transmit as the payoff is less than the cost. Thus, a very high power consumption cost weighting factor (α3 ) can have a negative effect on HeNB performances. The next example (Fig. 4.4) compares the performance of the proposed RB access algorithm with the existing “Best Response” algorithm with a global utility function specified in (4.3). In the simulation, we use small constant step size and it is specified as εt = ε = 0.05. The results are averaged over 50 scenarios and there are 2000 iterations. Best Response is a simple case where each HeNB chooses its action ptk at each time slot solely maximizing its local utility (4.9) and each HeNB assumes that the actions of other HeNBs are fixed. Thus, Best Response is 110  The Effect of Different Selections of (! , ! ) on the System Performance  System Performance (Worst Player)  2  3  1 0.9 0.8 0.7 0.6 0.5 0.4 0.3  6 0.2  4 0.1  !  3  2 0  0  !  2  Figure 4.2: The effect of different values of (α2 , α3 ) defined in (4.9) on the global system performance specified in (4.3) . a special case of the proposed RB access algorithm with the step size chosen to be εt = 1. The action update in the Best Response is not a function of the previous regret θ kt−1 but only a function of the current instantaneous regret matrix Hk (pt ). From Fig. 4.4 we can see that the system performance using the RB access algorithm reaches 0.9 after 1400 iterations and the result from the “Best Response” algorithm stays at around 0.6. It can be seen that the RB access algorithm (Algorithm 5) improves the system performance greatly compared to the “Best Response” algorithm. We can also observe from Fig. 4.4 that both RB access algorithm and “Best Response” algorithm  111  The Effect of (! ,! ) on System Average Performance 2  3  System Average Performance  1 0.95 0.9 0.85 0.8 0.75 0.5 0.4 0.3 0.2 0.1 !3  0  2  0  4  6  8  10  !2  Figure 4.3: The effect of different values of (α2 , α3 ) defined in (4.9) on the global system average performance specified in (4.5). do not converge to constant values, it is because transmission policy pt converges to a correlated equilibrium set when t → ∞ and the correlated equilibrium set has more than one correlated equilibrium policy.  4.5  Summary  We have proposed implementation of cognitive femtocell base stations for resource block allocation in the downlink of a eNB macrocell 3GPP LTE system. By considering the eNB as the primary base station, the HeNBs are formulated as multiple secondary base stations competing for spectrum  112  System Performance in Static Enviroment 0.95 0.9  Performance (Worst Player)  0.85 RB Access Algorithm  0.8 0.75 0.7  Best Response  0.65 0.6 0.55 0.5  0  500  1000 Iteration Index  1500  2000  Figure 4.4: Performance comparison between RB access algorithm (Algorithm 5) and the “Best Response” algorithm. resources. The RB allocation problem is formulated in a static environments, using static game framework. An RB access algorithm is proposed to compute the correlated equilibrium policy in such a environment. We also prove that the RB access algorithm converges to the correlated equilibrium set of the formulated game. Numerical examples are used to verify the performances of the proposed algorithm.  113  Chapter 5  Application of Mechanism Design in Opportunistic Scheduling under Cognitive Radio Systems 1 The conventional opportunistic scheduling algorithm in cognitive radio networks does the scheduling among the secondary users based on the reported state values. However, such opportunistic scheduling algorithm can be challenged in a system where each secondary user belongs to a different independent agent and the users work in competitive way. In order to optimize his own utility, a selfish user can choose not to reveal his true information to the 1  This chapter is based on the following publication. J. W. Huang and V. Krishnamurthy, “Game Theoretical Issues in Cognitive Radio Systems,” Journal of Communications, vol. 4, no. 10, pp. 790-802, November 2009. (Invited Paper)  114  central scheduler. In this chapter, we proposed a pricing mechanism which combines the mechanism design with the opportunistic scheduling algorithm and ensures that each rational selfish user maximizes his own utility function, at the same time optimizing the overall system utility. The proposed pricing mechanism is based on the classic VCG mechanism and has several desirable economic properties. A mechanism learning algorithm is proposed for users to learn the mechanism and to obtain the Nash equilibrium policy. A numerical example shows the Nash equilibrium of the proposed algorithm achieves system optimality.  5.1  Background  An efficient spectrum assignment technology is essential to a cognitive radio system, which allows secondary users to opportunistically utilize the unoccupied spectrum holes based on agreements and constraints. These secondary users have to coordinate with each other in order to maintain the order and result in maximum spectrum efficiency. This motivates the development of spectrum access approaches in cognitive radio systems. An opportunistic scheduling in cognitive networks which assumes that the scheduling is aware of primary user transmissions is being considered in [75] and [92]. [17, 88, 95] consider a system where only partial information about the primary user activities is available. However, all the existing opportunistic scheduling approaches overlook the fact that the secondary users may be owned by different agents and they may work in competitive rather than cooperative manners. These selfish users can become so sophisticated that they lie about their states to 115  optimize their own utility at the cost of reducing the overall system performance. It requires mechanism design theory in order to prevent this from happening. Mechanism design is the study of designing rules for strategic, autonomous and rational agents to achieve predictable global outcome [63]. A milestone in mechanism design is the VCG mechanism, which is a generalization of Vickrey’s second price auction [89] proposed by Clark [20] and Groves [35]. The particular pricing policy of the VCG mechanism makes the true reporting values the dominant strategy of the buyers. Our goal is to model each user as a selfish agent whose aim is to optimize his own utility function and we try to find a pricing mechanism which can efficiently allocate the resources within the network. In this chapter, we consider a cognitive radio network with multiple secondary users. The central scheduler adopts opportunistic scheduling to schedule the users under an overall transmission power constraint. The scheduling is based on the reported states of each user. When the secondary users belong to different independent selfish agents, the users may lie about their true state values in order to optimize their own utilities, sometimes at the cost of reducing the overall system performance.  5.2  Opportunistic Scheduling in Cognitive Radio Systems  This section describes the conventional opportunistic scheduling algorithm in the uplink of a K secondary users cognitive radio system (Fig. 5.1). One or more secondary users can be scheduled for transmission at each scheduling. We assume the system has perfect information about the primary user 116  activities and the algorithm is applicable when the primary user is absent. As soon as any primary user appears on the desired spectrum band, secondary users will release the spectrum resources. Spectrum Hole  Scheduler  Secondary User 1  Secondary User 2  Secondary User K  Figure 5.1: A K secondary users cognitive radio where the central scheduler does the scheduling according to the opportunistic accessing algorithm.  5.2.1  System States Description  bk denotes the level of date in the buffer of user k and bk ∈ with L representing the size of the buffer. The composition of buffer state of all the users is b = {b1 , b2 , . . . , bK } ∈ B which B is the buffer state space. We use ˆbk to represent the buffer state that the kth user reports to the central scheduler and ˆbk ∈ {0, 1, . . . , L}. Note here that in a truth telling system each user reports the true state value and we have ˆbk = bk , k = 1, 2, . . . , K. Assume each user has a normalized transmission power, the signal to noise ratio (SNR) of user k is denoted as ck . The channel status of a user is assumed to be circularly symmetric complex Gaussian random variables 117  and it is independent from the channel states of other users. After applying quantization, the SNR state is ck ∈ {1, 2, . . . Qck } where Qck denotes the maximum quantization level. The composition of the SNR states of all the K secondary users is denoted as c = {c1 , c2 , . . . , cK } ∈ C where C is SNR state space. Let the symbols per second transmission rate of user k be wk and its bits per symbol rate be mk (different values of mk lead to different modulation schemes), its instantaneous throughput can be expressed as ρk = wk mk 1 − pe (ck , mk )  sk  .  (5.1)  Here, sk is the average packet size in bits. pe (ck ) is the transmission bit error rate (BER), it is a function of the SNR and modulation mode of user k. In the case of an uncoded M-ary quadrature modulation (QAM), the BER pe (ck ) can be approximated as [24]: pe (ck , mk ) = 0.2 × exp  −1.6ck . 2 mk − 1  (5.2)  After applying quantization to the instantaneous throughput ρk , we have ρk ∈ {0, 1, 2, . . . , Qρ } for k = 1, 2, . . . , K where Qρ indicating the maximum quantization level of the throughput. In the case that the primary users appear on the system, the throughput state of all cognitive radio user switch to 0, ρk = 0, and no secondary user is allowed to transmit. Similarly, ρˆk is used to indicate the instantaneous throughput state that user k reports to the central scheduler and ρˆk ∈ {0, 1, 2, . . . , Qρ }. For notation convenience, in this chapter we use θk = {ρk , bk } to rep118  resent the true states of user k and θˆk to represent the reported states of user k. θ−k is used to indicate the true states of all the remaining K − 1 users, excluding user k, while θˆ−k is the corresponding reported values. We use Θ = {θk , θ−k } to denote the combined states of all the K users and ˆ = {θˆk , θˆ−k } to denote the combined reported states of all the users. Θ  5.2.2  Conventional Opportunistic Accessing Scheme  Opportunistic accessing scheme is commonly used by central scheduler to assign certain users for transmission with the aim to maximize the total system utility subject to the system constraint. This section simply uses the results of the opportunistic scheduling algorithm but does not describe the details of the algorithm. The decentralized channel access algorithm described in Chapter 2.2.1 is an example of the implementation of the opportunistic scheduling algorithm. The information that central scheduler has about each user includes the reported buffer state and reported throughput state, which is {ˆ ρk , ˆbk } for k = 1, 2, . . . , K. The central scheduler opportunistic accessing scheme is based on the reported buffer and throughput states. Define A as a feasible set, it is a subset of the union of all the users, and A ⊆ {1, 2, . . . , K}. A feasible set is a set which satisfies the system constraint. There are several choices of the system constraint, however, in this chapter we choose it to be the overall transmission power in the system. The constraint is specified as: the overall transmission power in the system should be equal or less than the system power limit P . As we specified in Section 5.2.1, each user uses a unit transmission power to transmit. Thus, the transmission power constraint on the system can be converted to the constraint on the  119  total number of users who are transmitting, that is the number of users who are transmitting simultaneously should be less or equal to the system power limit P . The conventional opportunistic accessing scheme with transmission power constraint defines the optimal feasible set of chosen users A∗ in the following way: A∗  Uk (ˆ ρk , ˆbk ),  = arg max  (5.3)  k∈A  s.t.  |A| ≤ P.  (5.4)  |A| is used to denote the number of users in set A and Uk (ˆ ρk , ˆbk ) is the corresponding utility of user k with throughput ρˆk and buffer state ˆbk . One typical choice of the utility function is Uk (ˆ ρk , ˆbk ) = ρˆkˆbk [57]. Central scheduler calculate A∗ which optimizes the system performance subjects to the system constraint. Note if both set A1 and A2 lead to the same optimal result, A∗ will be chosen randomly between them. The conventional opportunistic accessing scheme works when the central scheduler has the true information of each user, which means the {ˆ ρk , ˆbk } that the users report to the central scheduler are the true values, ρˆk = ρk and ˆbk = bk . However, the conventional opportunistic algorithm may be challenged when the secondary users become so sophisticated and are able to reconfigure themselves to make efficient use of the local resources (e.g., manage their own reporting data and have the most efficient data transmission). It is important to design a mechanism to optimize the overall system performance while ensuring the profit of each secondary user.  120  5.3  The Pricing Mechanism  This section applies the VCG pricing mechanism to the opportunistic scheduling algorithm, the new mechanism enforces the truth revealing property of each user. A mechanism learning algorithm is also proposed to implement the mechanism. It is shown that the Nash equilibrium of the proposed algorithm ensures each user reports its values truthfully.  5.3.1  The Pricing Mechanism  Different from the centralized conventional opportunistic scheduling algorithm, the proposed pricing mechanism is a distributed algorithm where each user tries to maximize his own utility function by choosing the right state values to report to the central scheduler. The buffer and throughput that user k chooses to report to the central scheduler is a solution of the following optimization problem: {ρˆk , ˆbk } := max vk (θk , θˆk , θˆ−k ) θˆk  ˆ  = max αρk bk × ρˆk ,ˆ bk  j∈A∗ ,j=k j∈A  αρˆj bj  αρˆj ˆbj  Ik∈A∗ + Ik∈A / ∗,  (5.5)  with the set A∗ and A defined in the following ways: A∗  :=  ρˆj ˆbj ,  arg max  s.t.  j∈A  A  :=  ρˆj ˆbj ,  arg max  ρˆk =0  j∈A,j=k  |A| ≤ P ; s.t.  |A|k∈A / ≤ P.  In the above optimization problem, α is a system fixed constant and it is α > 1, I{.} is an indication function whose value is 1 when the condition is 121  true, otherwise, it is 0. vk (θk , θˆk , θˆ−k ) is the utility function of user k, which is the function of the true states itself, the reported states of user k and the reported states of all the remaining users. If a user is not scheduled for transmission, his utility function equals to 1. If a user is scheduled for transmission, his utility function equals to the first part of (5.5), which is: ˆ  j∈A∗ ,j=k  vk (θk , θˆk , θˆ−k ) =  αρk bk ×  vk (θk , θˆk , θˆ−k ) =  1 if k ∈ / A∗ .  j∈A  αρˆj bj  αρˆj ˆbj  if k ∈ A∗ ;  (5.6) (5.7)  The first part of (5.6) αρk bk can be interpreted as the gain of user k per unit of subcarrier with throughput state ρk and buffer state bk . The second term  ρ ˆj ˆ bj j∈A∗ ,j=k α ˆ αρˆj bj j∈A  can be interpreted as the number of unit of subcarrier  that user k will be allocated and it is a function of the state of the remaining users in the system. In other words, the inverse of the second term could be interpreted as the price that user k has to pay to the system if it is scheduled for transmission. Each user select {ˆ ρk , ˆbk } to report to the central scheduler in the aim to maximize its own utility. There is one condition necessary in order to achieve an efficient allocation policy among selfish agents [21, 66]: if a user k (k = 1, 2, . . . , K) reports a false state values θˆk = θk results in the same value of the utility function as that of if it reports the true value, which is vk (θk , θˆk , θˆ−k ) = vk (θk , θk , θˆ−k ), θˆk = θk , then the user will choose to report the true values. We name this as the truth preferred rule in this chapter. The interpretation of this rule is that when lying about the states does not bring any benefit  122  to a user, a user would prefer telling the truth. The pricing mechanism we propose above is based on the VCG mechanism, where we modified the conventional summation form of the utility function into a product form. Such pricing mechanism can be easily related to a practical cognitive radio system and interpret the utility function in terms of physical parameters in a practical system.  5.3.2  Economic Properties of the Pricing Mechanism  The pricing mechanism we proposed above still maintains the same desirable economic properties as that of VCG mechanism, these properties are specified as follows [22, 23]: 1. The mechanism is incentive-compatible in ex-post Nash equilibrium. The best response strategy is to reveal the true state information θˆk = θk even after they have complete information about other users θ−k . 2. The mechanism is individually rational. A selfish agent will join the mechanism rather than choosing not to, because the value of the utility function is non-negative. 3. The mechanism is efficient. Since all the users will truthfully reveal their state information, the opportunistic scheduling algorithm carried out by the central scheduler will maximize the system performance. The detailed proof of these economic properties is shown as follows. Proposition 5.3.1 The mechanism is incentive-compatible in ex-post Nash equilibrium. 123  A mechanism is incentive-compatible in ex-post Nash equilibrium if the best strategy for a user is to report truthfully even with complete state information of other users. Proof: Suppose all users except user k report their true state values, θˆ−k = θ−k . Assume the optimal allocations set when the kth user reports truthfully is A∗0 , we are going to analyze the utility function of user k by reporting θˆk = θk . If the kth user reports θˆk = θk but the optimal allocation is the same as that when he reports the true values, A∗ = A∗0 , this will then result in equal value of the utility function, which is vk (θk , θˆk , θˆ−k ) = vk (θk , θk , θˆ−k ). According to the truth preferred rule, user k will choose to report the true values. / A∗0 , but if Consider the case that if user k reports the true values θk , k ∈ user k reports θˆk = θk , k ∈ A∗ . The relation between the reported values and the true values is: ρˆkˆbk > ρk bk . With reporting value θˆk , the utility function is given as (5.6). Since k ∈ / A∗0 , this implies that ρk bk ≤ minj,j∈A (ˆ ρj ˆbj ), and it will in turn result in vk (θk , θˆk , θˆ−k ) ≤ 1. Thus, user k will choose to report truthfully θk in this case. Now we are going to consider the case that if user k reports the true values θk , k ∈ A∗0 , but if user k reports θˆk = θk , k ∈ / A∗ . The relation between the reported values and the true values is: ρˆkˆbk < ρk bk . The value of the utility function of user k with reporting values θˆk equals to 1 according to the expression (5.7). The utility function if user k reports truthfully, vk (θk , θk , θˆ−k ), is given according to that in (5.6). k ∈ A∗0 implies that ρk bk ≥ ρˆj ˆbj with j ∈ A , j ∈ / A∗ , so we have vk (θk , θk , θˆ−k ) ≥ 1. Thus, 124  user k will choose to report its true values as well. Proposition 5.3.2 The mechanism is individually rational. A mechanism is individually rational if there is an incentive for users to join the system rather than opting out of it. Assume that the utility a user achieves by not joining the mechanism is 0, then we only have to prove the utility a user gains in this pricing mechanism is always ≥ 0. Proof: The expression of the utility function of any user k is given in (5.5). If a user is not scheduled for transmission the utility function equals to 1. If it is scheduled, the utility function is always greater or equal to 0 as the utility function is an exponentiation function of a constant α, α > 1. Proposition 5.3.3 The mechanism is efficient. This implies that the opportunistic scheduling algorithm carries out by the central scheduler schedules according to: A∗  = arg max  Uk (ρk , bk ),  (5.8)  k∈A  s.t.  |A| ≤ P.  (5.9)  Proof: Using the result from Proposition 5.3.1, each user reports truthfully under this pricing mechanism, the opportunistic scheduling algorithm is able to achieve system efficiency.  5.3.3  Mechanism Learning Algorithm  In the case that the users in the cognitive radio are not fully acknowledged with the pricing mechanism, we propose a mechanism learning algorithm 125  for each user to learn the mechanism and obtain the Nash equilibrium state. The mechanism learning algorithm is shown in Algorithm 6. Algorithm 6 Mechanism Learning Algorithm 1: Initialization: ρ ˆ0 and dˆ0 for k=1, 2, . . . , K; k  Main Iteration 3: for k = 1 : K do 4: {ˆ ρn+1 , ˆbn+1 } := k k  k  2:  5: 6: 7: 8: 9: 10: 11: 12: 13:  maxρˆk ,ˆbk αρk bk ×  ˆn ρ ˆn j bj j∈A∗ ,j=k α ρ ˆn ˆ bn α j j j∈A  Ik∈A∗ + Ik∈A / ∗.  With A∗ and A defined by: A∗ := arg maxA j∈A ρˆnjˆbnj , s.t. |A| ≤ P ; A := arg maxA,ρk =0 j∈A,j=k ρˆnjˆbnj , s.t. |A|k∈A / ≤ P; end for n = n + 1; The iteration terminates when the parameters dˆnk , ρˆnk converge; else return back to Step 3.  The mechanism learning algorithm can be summarized as follows. First, each user randomly choose a initial reporting state {ˆ ρ0k , dˆ0k }. The main iteration is from Step 2 to Step 9 where each user optimizes the reporting state {ˆ ρn+1 , ˆbn+1 } iteratively with the states of other users fixed. The rek k porting state of each user is chosen based on maximizing its own utility function. Note here that at each step, each user follows the truth preferred rule. The algorithm terminates when the reporting states of all the users are converged.  126  5.4  Numerical Results  In this section, we are going to provide a simulation result to illustrate the performance of the mechanism learning algorithm proposed in Section 5.3.3. We simulation a 30 users cognitive radio system with each user has 5 buffer states and 10 throughput states. The transmission power constraint on the system is P = 3 which is equivalent to that the maximum number of users transmit simultaneously is 3. The constant α is set to be 2 during simulation. ρˆ0k and ˆb0k are generated randomly during initialization. Result with Mechanism Learning Algorithm 35 Buffer State Throughput State 30  25  MSE  20  15  10  5  0  2  4  6  8 10 12 14 Number of Iterations  16  18  20  Figure 5.2: The MSE of the reported buffer states and throughput states using the mechanism learning algorithm. The result is of a 30 users system with L = 5, Qρ = 10 and P = 3. In Fig. 5.2, the x-axis represents the number of iterations and y-axis represents the Mean Squared Error (MSE) of the reported buffer states and throughput states. With a slight abuse of notations, we let ρˆn =  127  {ˆ ρn1 , . . . , ρˆnK } and ˆbn = {ˆbn1 , . . . , ˆbnK }, the MSE of the reported buffer states and throughput states can be written mathematically as: M SE(ˆ ρn ) = M SE(ˆbn ) =  1 × K  K k=1  1 × K  (ˆ ρnk − ρnk )2 ;  (5.10)  (ˆbnk − bnk )2 .  (5.11)  K  k=1  The result is obtained by applying the mechanism learning algorithm proposed in Section 5.3.3. In the figure, the solid curve and the dash curve show the mean squared error (MSE) of the reported buffer states and throughput states, respectively. We can see from the figure the MSE converge to 0 after 11 iterations, at which state, all the users in the system report truthfully ˆ = Θ. This simulation result shows numerically that the mechanism and Θ learning algorithm is able to converge the reported states to the true states.  5.5  Summary  In this chapter, we combine the mechanism design into the opportunistic scheduling used in cognitive radio networks so that each secondary user optimizes his own utility while the optimality of the overall system performance is ensured. The pricing mechanism we impose on each secondary user is based on the VCG mechanism and we interpret the price with system parameters in a practical cognitive radio system. We also prove the several desirable economic properties that the mechanism has. A mechanism learning algorithm is used to implement the pricing mechanism and the convergence of such algorithm is shown numerically.  128  Chapter 6  Discussion and Conclusions 6.1  Overview of Thesis  The thesis considered the topics of transmission adaptation, resource allocation, and mechanism design in the cross layers of wireless networks. In many ways, the thesis has highlighted how the game theoretic approach can be applied to optimize the performances in wireless communication systems. Specifically, the transmission adaptation problems in cognitive radio systems and WLANs are formulated as switching control games, the resource allocation problems in LTE systems are formulated as a static game, and a pricing mechanism design is proposed to design a truth revealing opportunistic scheduling algorithm. In some cases, the Nash equilibrium transmission policy is proved to have a specific structure, which makes the estimation and implementation of the optimal solutions highly efficient. Due to code division multiple access schemes, and multi-antenna communications, as well as other advances in signal processing and communica129  tions, users can share or compete for a wireless medium. Then the two very important issues arising in wireless network optimization are how to allocate the resource among users so an equilibrium system state is obtained and how to balance between multiuser interference and multiple access efficiency in medium access control. Chapter 2 and Chapter 3 considered these issues by computing the Nash equilibrium solutions which optimizes each user’s utility with its latency constraint. Chapter 4 considered these issues by studying the correlated equilibrium resource allocation solutions in static game formulation. Chapter 5 considered these issues from the mechanism design perspective. The thesis applies several mathematical tools to solve various issues in wireless communication systems. For example, switching control game is used in Chapter 2 and Chapter 3 to compute the Nash equilibrium transmission policies in cognitive radio systems and WLANs. The concept of supermodularity has been used together with dynamic programming to prove the monotone structural results on the Nash equilibrium transmission policy. Several learning algorithms, including gradient-based stochastic approximation algorithm, regret matching algorithm, and Q-learning algorithm have been deployed extensively in the thesis for estimating/learning transmission and resource allocation policies. One of the main focus of this thesis is optimization methodologies, specifically, we focus on its application in wireless communication networks. However, some of the structural results and learning algorithms can find application in resource allocation problems in other areas, such as operation research. In the remaining of the chapter, various aspects of results and 130  algorithms will be discussed and an outline of possible directions for future research will be presented.  6.2  Discussion on Results and Algorithms  The thesis contains several structural results and learning algorithms, which are discussed below. The first part of the thesis, which consists of Chapter 2 and Chapter 3, was on distributed transmission adaptation is both cognitive radio systems and WLANs. The analysis of such a problem is done using dynamic game theoretic approach. In particular, we formulate the transmission adaptation problem as a switching control game and the Nash equilibrium solutions can be obtained by solving a sequence of Markov decision processes. Such approach offers robust, distributed solutions that can be estimated in real time via very efficient adaptive learning algorithms. The major results in the first part of the thesis include the structure of the Nash equilibrium transmission policies. Especially, it is proved in Chapter 2 and Chapter 3 that under certain conditions, the Nash equilibrium policy of each user is a randomization of two pure policies and each of the policies is a threshold function with respect to its buffer state. Optimality of the monotone transmission policy converts the problem of estimating the Nash equilibrium policy into estimating few parameters, which can be solved via gradient-based stochastic approximation algorithm with tracking capability. It is shown via simulation that as users adapt their policies using the proposed algorithm, the system converges to the Nash equilibrium. The second part of the thesis, which consists of Chapter 4, concerns the application of correlated equilibrium concept in wireless communication 131  systems. Specifically, it formulates the resource allocation problem among cognitive radio base stations in an LTE system under a static environments as a static game. In the chapter, an RB access algorithm is proposed to compute the correlated equilibrium solutions under the static game formulation. It is proved theoretically that the RB access algorithm is ensured to converge to the correlated equilibrium set of the game. Chapter 5 is the third part of the thesis. It combines the mechanism design into the conventional opportunistic scheduling in cognitive radio networks so that each secondary user optimizes its own utility while optimizing the overall system performance. The pricing mechanism proposed is based on VCG mechanism. It is improved that the proposed mechanism has several desirable economic properties. Furthermore, a mechanism learning algorithm is used to implement such pricing mechanism and the convergence of the algorithm is shown numerically.  6.3  Summary  The thesis has provided structural results for cross-layer transmission adaptation in wireless networks in several contexts, using many powerful mathematical tools that have many application in electrical computer engineering. The problems considered include studying switching control game theoretic approach toward transmission adaptation in cognitive radio system and a WLAN; investigating the correlated equilibrium resource allocation solutions in an LTE system; and designing a pricing mechanism which eliminates the malicious behaviours in the conventional opportunistic access algorithm. The analytical tools utilized in the thesis include static game, stochastic 132  game, switching control game, the Lagrange multiplier method, and mechanism design. Furthermore, most of the learning algorithms in the thesis are gradient-based stochastic approximation algorithms with tracking capability. Wireless networks application of the work in the thesis range from cognitive radio network, to WLANs, to LTE systems.  6.4  Future Research  This thesis has developed some systematic methods for deriving cross-layer transmission adaptation and scheduling algorithms. In most cases, the results are also applicable to more generale resource allocation problems in wireless networks. The theme of this thesis has been to derive structural results, efficient learning algorithms, and mechanism algorithms for relatively general models. In other words, the analysis in this thesis has assumed slightly abstract frameworks. For future research, the work in this thesis can be extended in several directions. Either the theoretical results can be strengthened/generalized or further analysis can be carried out for more specific network models. In what follows, we propose a few research problems related to the work in this thesis.  6.4.1  How to Avoid the Curse of Dimensionality in Stochastic Games?  Potential applications notwithstanding, there remain substantial hurdles in the application of dynamic stochastic games as a modeling tool in practice. Discrete-time stochastic games with a finite number of states are central to the analysis of strategic interactions among selfish users in dynamic en-  133  vironments. The usefulness of discrete-time games, however, is limited by their computational burden; in particular, there is a “curse of dimensionality”. In a discrete-time dynamic stochastic game, each game player is distinguished by and individual state at each time slot. The system state is a vector encoding the number of players with each possible value of the individual state variable. E.g., the system state in Chapter 2 comprises the channel and buffer state information from all the players. At each time slot, a given player selects an action to maximize its expected discounted payoffs; its subsequent state is determined by its current individual state, its chosen action, and a random shock. The selection will depend on player’s individual state and system state. The computation entailed in selecting such an action quickly becomes infeasible as the number of players and individual states grows. How to efficiently reduce the state space is yet an issue to solve before the implementation of stochastic games in practical wireless communication systems. [26] aims to reduce the dimensionality by exploring the alternative of continuous-time stochastic games with a finite number of states and show that the continuous time has substantial advantages.  6.4.2  Transmission Scheduling with Partially Observable States  This thesis has made the assumption that perfect quantized channel state information is available to each user. In many situations, channel states can only be observed in noise and perfect quantized channel information is difficult to obtain. Under a Markov decision process formulation, in the case when the channel state can not be obtained perfectly, the transmis-  134  sion scheduling problem can be formulated as a partially observable MDP (POMDP). In Chapter 2 and Chapter 3, the transmission adaptation problem is formulated as a switching control game. Switching control game is a special type of dynamic game where the transition probability in any given state depends on only one player. It is known that the Nash equilibrium for such a game can be computed by solving a sequence of Markov decision processes. Whether or not we can extend the POMDP formulation to solve a switching control game with imperfect channel state information is one direction of the future research.  6.4.3  More Analytical Results on Correlated Equilibrium in Stochastic Games  A correlated equilibrium is an outstanding solution concept in game theory and it is a generalization of the Nash equilibrium concept. Correlated equilibrium arises from situations where users select their strategy according to some distribution with inherent coordination. If all users have no incentive to unilaterally deviate, the distribution comprises a correlated equilibrium. In Chapter 4 of this thesis, we have looked into the correlated equilibrium resource allocation solutions in an LTE system under a static environment where static game theoretic framework has been applied. A decentralized RB access algorithm is proposed to compute the correlated equilibrium solutions in a static game formulation. We proved theoretically that the RB access algorithm always converges to the correlated equilibrium set. The study of more analytical result on the correlated equilibrium solutions in stochastic games, e.g., structural results on the correlated equilibrium poli-  135  cies, prove the convergence of the algorithms deployed therein theoretically, is yet an area to explore.  6.4.4  Other Types of Mechanism Design  The pricing mechanism used in Chapter 5 is based on the well-known VCG mechanism. The development of other types pricing mechanisms or reputation based mechanism can be one direction of the future work. In a reputation based mechanism, system use reputation as a tool to motivate cooperation between users and indicate a good behaviour within the network. To be more precise, a user can be assigned a reputation value determined by its neighbors. Based on how “good” this value, a user can be used or not in a given service provision. 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In Proceedings of IEEE International Conference on Multimedia and Expo (ICME), July 2006.  149  Appendix A  Proof of Theorem 2.3.2 A.1  Proof of Theorem 2.3.2  Consider the case where the delay constraint parameter Dk in (2.8) is chosen so that a Nash equilibrium policy of the optimization problem exists. When the Nash equilibrium is obtained, the delay constraint (2.8) will hold with equality. This feature implies that by introducing the Lagrange multiplier λk , we can get the Nash equilibrium policy when the objective function (2.7) is minimized and the equality of the delay constraint (2.8) is obtained. According to Algorithm 1, the transmission policy and value matrix of  150  the kth user are updated by the following steps: πk∗ (s, s ∈ Sk )n  =  arg minn c(s, ak ) + λk · dk (s, ak ) + πk (s) |S|  β  (n−1)  s =1  vkn (s, s ∈ Sk ) =  (s )  (A.1)  c(s, ak ) + λk · dk (s, ak ) +  min  πk (s)n  β  P(s |s, ak )vk  |S| s =1  (n−1)  P(s |s, ak )vk  (s ) .  (A.2)  In order to prove that the optimal transmit action policy πk∗ (s, s ∈ Sk )n is monotone nondecreasing on the buffer state bk , we have to show that the right-hand side of (A.1) is a submodular function of (bk , ak ). According to assumption A 2.3.2, c(s, ak ) + λk · dk (s, ak ) is a submodular function of |S| s =1 P(s  (bk , ak ), thus, we only need to demonstrate that  (n−1)  |s, ak )vk  (s )  is also submodular in the pair (bk , ak ). As mentioned before, the overall state space S is the union of all the K sub-spaces: S = S1 ∪S2 · · ·∪SK . Thus, we can write  |S| s =1 P(s  (n−1)  |s, ak )vk  (s )  as a summation of K terms according to the state space partition. In order to show the submodularity of  |S| s =1 P(s  (n−1)  |s, ak )vk  (s ), we only need to  prove the submodularity property of each term. The decomposition is given as follows: |S| s =1  K  P(s  (n−1) |s, ak )vk (s  (n−1)  )= i=1 s ,s ∈Si  P(s |s, ak )vk  (s ).  (A.3)  If we consider the ith term of the right-hand side of (A.3), using the state  151  transition probability expression from (2.4), we have (n−1)  s ,s ∈Si  P(s |s, a)vk  K  (s )  K  = l=1  P(hl |hl ) ·  (n−1)  l=1,l=k  P(bl |bl ) · P(bk |bk , ak ) · vk  (s ).  The proof of Theorem 2.3.2 requires the result from Lemma A.1.1, whose proof will be given afterwards. Lemma A.1.1 Under the assumptions of Theorem 2.3.2, vk (s) is nondecreasing on bk when the state lives in state space Si , that is s ∈ Si , i = 1, 2, . . . , K. Continuing with the proof of the theorem, the assumption says that  L bk =l  P(bk |bk , ak )  is submodular in (bk , ak ) for any l. According to the definition of submodularity, this assumption can be mathematically written in the following way, for all the l and s ∈ Si : L  bk =l  L − P(bk |b− k , ak ) +  bk =l  L  ≤  bk =l  + P(bk |b+ k , ak )  L + P(bk |b− k , ak ) +  bk =l  − P(bk |b+ k , ak ).  (A.4)  Based on the result from Lemma A.1.1, we can apply Lemma 4.7.2 from [76] to our model, which yields L  bk =0  L − P(bk |b− k , ak )  · vk (s ) +  L  bk =0  bk =0  + P(bk |b+ k , ak ) · vk (s ) ≤  L + P(bk |b− k , ak ) · vk (s ) +  152  bk =0  − P(bk |b+ k , ak ) · vk (s )  (A.5)  for s ∈ Si . The summation of (A.5) over h for [h , b ] ∈ Si yields  s ,s ∈Si s ,s ∈Si  − n−1 + n−1 P(s |s, b− (s ) + P(s |s, b+ (s ) ≤ k , ak )vk k , ak )vk + n−1 − n−1 P(s |s, b− (s ) + P(s |s, b+ (s ) . k , ak )vk k , , ak )vk  This is the definition of submodularity of  (n−1)  s ,s ∈Si  P(s |s, ak )vk  (s ) in  (bk , ak ). Furthermore, the positive weighted sum of submodular functions is also submodular, which establishes the submodularity of the right-hand side of function (A.1). Thus, the optimal transmit action policy πk∗ (s, s ∈ Sk )n is monotone nondecreasing on the buffer state bk for a fixed Lagrange multiplier λk . According to [11], the constrained optimal transmission scheduling policy is a randomized mixture of two pure policies, which can be computed with two different Lagrange multipliers. As discussed above, it has been shown that each of these two pure policies is nondecreasing on the buffer occupancy state. Thus, the mixed policy is also nondecreasing on the buffer state, which concludes the proof.  A.2  Proof of Lemma A.1.1  We use backward induction to prove that vk (s) is nondecreasing on bk with state s ∈ Si for i = 1, 2, . . . , K.  (n−1)  First we assume that vk  (s) is nondecreasing on bk for s ∈ Si , i =  1, 2, . . . , K at the time instant (n − 1). The value matrix at time instant n is  updated according to (A.2). Denoting the optimal action policy as πk∗ , the  153  update of value matrix can be written more explicitly as: vkn (s, s ∈ Sk )  = c(s, πk∗ (s)) + λk · dk (s, πk∗ (s)) + K  (n−1)  β i=1 s ,s ∈Si  P(s |s, πk∗ (s))vk  (s ).  (A.6)  Because of assumption A4, we know that c(s, πk∗ (s)) + λk · dk (s, πk∗ (s)) is nondecreasing on bk for any h, f and a∗ . By applying the result from Lemma 4.7.2 in [76], we can prove that  (n−1)  s ,s ∈Si  P(s |s, πk∗ (s))vk  (s ) is  nondecreasing on bk for any h, f and πk∗ (s). Thus, vkn (s) is nondecreasing on bk for s ∈ Sk , k = 1, 2, . . . , K at time instant n. For any s ∈ Si , i = k, the value vector is updated according to (A.7). In this case, the instantaneous cost is independent of bk , thus the nondecreasing (n−1)  property of vkn (s) on bk when s ∈ Si , i = k is preserved from vk vkn (s, s ∈ Si , i = k) =  (s).  c(s, πi∗ (s)) + λk · dk (s, πi∗ (s) + β  |S|  s =1  (n−1)  P(s |s, πi∗ (s))vk  (s ).  (A.7)  When the time horizon of the Markovian problem is very large, that is (0)  n → ∞, the initial value matrix vk (s) no longer affects vkn (s). The chosen (0)  (0)  value of vk (s) is arbitrary. In our case, we can initialize vk (s) to satisfy the condition stated in Lemma A.1.1, which concludes the proof.  154  Appendix B  Proof of The Convergence of Algorithm 3 1. By following the definition of πkn (s, s ∈ Sk ) from Algorithm 3 we can deduce that (n+1)  Vk  (n+∆)  2. When πkn (s, s ∈ Sk ) = πk (n+∆)  of (B.1), if Vkn = Vk  ≤ Vkn .  (B.1) (n+∆)  (s, s ∈ Sk ), we have Vkn = Vk (n+∆)  , then πkn (s, s ∈ Sk ) = πk  . In view  (s, s ∈ Sk ) for any  ∆ =1 , 2, 3, . . . . 3. The payoff function of the matrix game with s ∈ S1 is of the form − c(s, ak ) + λm k · dk (s, ak ) + β term of this payoff function  |S| s =1 P(s  |s, ak )vkn (s ). Please note the first  − c(s, ak ) + λm k · dk (s, ak ) is independent  of Vkn . By using the result from [31] Lemma 6.3.2, πkn (s, s ∈ Sk ) equals to some extreme optimal action in a submatrix game with payoff function 155  − c(s, ak ) + λm k · dk (s, ak ) . As there are only finite number of extreme optimal action candidates for πkn (s, s ∈ Sk ), there exist n and ∆ ≥ 1 such (n+∆)  that πkn (s, s ∈ Sk ) = πk (n−1)  4. If Vkn = Vk  (n+∆)  (s, s ∈ Sk ), which in turn implies Vkn = Vk  .  , πkn (s, s ∈ Sk ) is the optimal action policy for user k  (k ∈ {1, . . . , K}) of the game with a fixed Lagrange multiplier λm k . Based on the 1 − 4 observations above, we conclude that Algorithm 3 will converge in a finite number of iterations with fixed Lagrange multipliers λm k=1,2,...,K .  156  Appendix C  Proof of Theorem 3.4.1 C.1  Proof of Theorem 3.4.1  We choose delay constraint parameters Dk,k=1,2,...,K in the optimization problem (3.13) carefully to ensure the existence of an optimal policy. The optimal policy can be obtained when the objective function (3.11) is maximized and the equality of the delay constraint (3.12) is held. According to Algorithm 3, for ∀s ∈ S, the optimal policy and value matrix of kth user are updated by the following steps: πkn (s)  = arg min n  πk (s)  +β  |S| s =1  (n+1)  vk  (s)  − c(s, ak ) + λm k · dk (s, ak )  P(s |s, ak )vkn (s ) ;  (C.1)  n = −c(s, πkn (s)) + λm k · dk (s, πk (s))  +β  |S|  s =1  P(s |s, πkn (s))vkn (s ).  157  (C.2)  A sufficient condition for the optimal transmission action policy πkn (s) to be monotone nondecreasing in the buffer state bk is that the right hand side of (C.1) is a submodular function of (bk , ak ). According to assumptions A 3.4.2 and A 3.4.3, −c(s, ak ) + λm k · dk (s, ak ) is a submodular function of (bk , ak ), |S| s =1 P(s  thus, we only need to demonstrate  |s, ak )vkn (s ) is also submodular  in the pair (bk , ak ). Recall the overall state space S is the union of all the K sub-spaces, S = |S| s =1 P(s  S1 ∪ S2 · · · ∪ SK . Thus, we can write  |s, ak )vkn (s ) as a summation  of K terms according to the partition of the state space, where the ith term (i = 1, 2, . . . , K) is denoted by Qi (s, ak ). By using the property of the state transition probability (3.15) and Assumption A 3.4.4, we have the following result: Qi (s, ak ) = s ,s ∈Si K  l=1  P(s |s, ak )vkn (s ) =  K  P(hl |hl )  l=1  K  P(ml |ml )  l=1,l=k  In order to show the submodularity of  P(bl |bl )P(bk |bk + ak )vkn (s ). |S| s =1 P(s  (C.3)  |s, ak )vkn (s ), we only need  to prove the submodularity property of Qi (s, ak ) for i = 1, 2, . . . , K. The proof of Theorem 3.4.1 needs the result from Lemma C.1.1 , whose proof will then be given after the proof of the Theorem. Lemma C.1.1 Under the conditions of Theorem 3.4.1, Qi (s, ak ) is of the ¯ i (bk − ak , {h, m, b−k }). Function Q ¯ i (x, y) is integer form Qi (s, ak ) = Q convex function of x for any given y = {h, m, b−k }. ¯ i (x, y) is integer convex in x, which can According to Lemma C.1.1, Q be written as: 158  ¯ i (x + ∆, y) − Q ¯ i (x , y) Q ≤  ¯ i (x, y) ¯ i (x + ∆, y) − Q Q  (C.4)  for x ≥ x and ∆ ≥ 0. Substitute in the above equation with x = bk − ak , x = bk − ak and ∆ = bk − bk for bk ≥ bk and ak ≥ ak , we obtain Qi ({bk , ak }, y) − Qi ({bk , ak }, y) ≤ Qi ({bk , ak }, y) − Qi ({bk , ak }, y),  (C.5)  which is the definition of submodularity of Qi (s, ak ) in bk and ak . Furthermore, the linear combination of submodular functions still remains the submodular property. Thus,  |S| s =1 P(s  |s, ak )vkn (s ) is submodular in the  pair (bk , ak ). Based on the above results, we can prove the submodularity of the right hand side of equation (C.1). This proves the optimal transition action policy πk∗ (s)n is monotone nondecreasing on the buffer state bk under fixed positive Lagrange constant. According to [11], the optimal randomized policy with a general constraint is a mixed policy comprising of two pure policies that can be computed under two different Lagrange multipliers. As discussed above, we have already shown that each of these two pure policies is nondecreasing on the buffer state occupancy. Thus, the mixed policy also owns the nondecreasing property on the buffer state. This concludes the proof.  159  C.2  Proof of Lemma C.1.1  Let us first assume that vkn (s ) is an integer convex function of bk . Then, according to Proposition 2 of [82] and A 3.4.4 in Section 3.4.2 along with (C.3), it can be concluded that Qi (s, ak ) is an integer convex function of bk − ak .  Thus, we only need to prove that vkn (s ) is an integer convex function of  bk , which can be proved via backward induction. The value vector can be updated by the value iteration algorithm as follows: (n)  vk (s) = −c(s, ak ) + λm k · dk (s, ak ) + β  |S| s =1  P(s |s, ak )vkn−1 (s ).  (C.6)  If vkn−1 (s ) is integer convex in bk , as well as the A 3.4.2 and A 3.4.3 from Section 3.4.2, it can be proved that vkn (s) is integer convex in bk by using the key Lemma 61 from [5]. The convergence of the value iteration algorithm is not affected by the initial values. Choose vk0 (s ) to be an integer convex function of bk , vkn (s) is integer convex in bk follows by induction. This concludes the proof.  160  

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