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DYNAMIC RESOURCE ALLOCATION AND OPTIMIZATION IN WIRELESS NETWORKS By YANG SONG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 @ 2010 Yang Song To my beloved parents and friends ACKNOWLEDGMENTS I would like to gratefully and sincerely thank my Ph.D. advisor Prof. Yuguang Fang for his invaluable guidance, understanding, patience, and most importantly, his continual faith and confidence in me during my Ph.D. studies at the University of Florida. I feel extremely fortunate to have him as my advisor, who is always willing to help me both academically and personally. Thanks for everything. I also owe my wholehearted gratitude to my Ph.D. committee members, Prof. Pramod P. Khargonekar, Prof. Sartaj Sahni, Prof. Tan Wong, and Prof. Shigang Chen, for their constructive suggestions and valuable comments on my Ph.D. research and dissertation. I have been fortunate to have many friends in WINET. I specially thank Chi Zhang, Xiaoxia Huang, Yun Zhou, Shushan Wen, Jianfeng Wang, Hongqiang Zhai, Yanchao Zhang, Feng Chen, Pan Li, Jinyuan Sun, Miao Pan, Rongsheng Huang, Yue Hao for many valuable discussions and good memories. Last but definitely not least, this work would not have been achieved without the support and understanding of my parents. They have always supported me in every choice I have chosen in my life. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................. ................. 4 LIST O FTABLES ..................... ................. 8 LIST OF FIGURES .................... ................. 9 ABSTRACT .................... ..................... 11 CHAPTER 1 INTRODUCTION .................... ............... 13 2 REVENUE MAXIMIZATION IN MULTIHOP WIRELESS NETWORKS ..... 17 2.1 Introduction ................... ................ 17 2.2 Related Work .................... .............. 19 2.3 Revenue Maximization with QoS (Quality of Service) Requirements 22 2.3.1 System Model .................... .......... 22 2.3.2 Problem Formulation .......................... 25 2.4 QoSAware Dynamic Pricing (QADP) Algorithm 27 2.5 Performance Analysis ............................. 30 2.5.1 Proof of Revenue Maximization .. .. 31 2.5.2 Proof of Network Stability ..... .... 39 2.5.3 Proof of QoS Provisioning ..... ... ... 40 2.6 S im ulations . .. 41 2.6.1 SingleHop Wireless Cellular Networks .... 41 2.6.2 MultiHop Wireless Networks .. ... 44 2.7 C conclusions . . 47 3 ENERGYCONSERVING SCHEDULING IN WIRELESS NETWORKS 48 3.1 Introduction . .. 48 3.2 System Model ............ .. ................. 50 3.3 Minimum Energy Scheduling Algorithm ... 54 3.3.1 Algorithm Description ......................... 54 3.3.2 ThroughputOptimality ............ .. .... ... .. 57 3.3.3 Asymptotic EnergyOptimality . ... 62 3.4 Sim ulations ... .. .... .. 64 3.5 Conclusions . . 67 4 CHANNEL AND POWER ALLOCATION IN WIRELESS MESH NETWORKS 70 4.1 Introduction ... .. .... .. 70 4.2 System Model .................. ............. 72 4.3 Cooperative Access Networks ... 74 4.3.1 Cooperative Throughput Maximization Game ........... 4.3.2 NETMA NegotiationBased Throughput Maximization Algorithm 4.4 NonCooperative Access Networks ..................... 4.5 An Extension to Adaptive Coding and Modulation Capable Devices . 4.6 Performance Evaluation .. ....................... 4.6.1 Legacy IEEE 802.11 Devices .. ................. 4.6.1.1 Example of small networks ................ 4.6.1.2 Example of large networks ................ 4.6.2 ACMCapable Devices ................ ....... 4.7 Conclusions ..................... ............. 5 CROSS LAYER INTERACTIONS IN WIRELESS SENSOR NETWORKS . 5.1 Introduction . . 5.2 Related W ork . . 5.3 A Constrained Queueing Model for Wireless Sensor Netwo 5.3.1 Network Model ........... 5.3.2 Traffic Model ............ 5.3.3 Queue Management ........ 5.3.4 SessionSpecific Requirements . 5.4 Stochastic Network Utility Maximization in 5.4.1 Problem Formulation ........ 5.4.2 The ANRA Cross Layer Algorithm . 5.4.3 Performance of the ANRA Scheme 5.5 Performance Analysis . 5.6 Case Study ................. 5.7 Conclusions and Future Work . Wireless Sensor . 100 . 102 )rks . SNetworks 6 THRESHOLD OPTIMIZATION FOR RATE ADAPTATION ALGORITHMS IN IEEE 802.11 W LANS . . 6.1 Introduction . . 6.2 Related W ork ...................... 6.3 Reverse Engineering for the ThresholdBased Rate / 6.4 Threshold Optimization Algorithm . 6.4.1 Learning Automata ............... 6.4.2 Achieving the Stochastic Optimal Thresholds 6.5 Performance Evaluation ................ 6.6 Conclusions and Future Work . ............ ............ Adaptation Algorithm . . ............ . . ............ . . 7 STOCHASTIC TRAFFIC ENGINEERING IN MULTIHOP COGNITIVE WIRELESS MESH NETW ORKS ................................. 157 Introduction . . Related W ork . . System M odel ..................... Stochastic Traffic Engineering with Convexity .. 157 160 163 167 100 . 104 S104 S105 S108 . 109 S110 . 111 S112 . 114 . 118 S125 . 128 S130 130 133 135 143 144 145 150 155 7.4.1 Form ulation . . 7.4.2 Distributed Algorithmic Solution with the Stochastic A approach . . 7.5 Stochastic Traffic Engineering without Convexity . 7.6 Performance Evaluation ................... 7.7 Conclusions.. ....................... 8 CONCLUSIONS .......................... REFERENCES ................... ........... BIOGRAPHICAL SKETCH ....................... PrimalDual 167 169 173 178 183 185 186 198 LIST OF TABLES Table page 21 Average admitted rates for multimedia flows ..... 44 22 Average admitted rates for multimedia flows ..... 47 41 Data rates v.s. SINR thresholds with maximum BER = 10 90 61 SNR v.s. BER for IEEE 802.11b data rates ... 152 71 Available paths for edge routers ..... .. 179 72 Convergence rates when Y is affected by all five primary users 182 73 Convergence rates when Y is not affected by any of the primary users 182 LIST OF FIGURES Figure page 21 An example of multihop wireless networks ...... ...... 23 22 A conceptual example of the network capacity region with two multimedia flows. The minimum data rate requirements reduce the feasible region of the optimum solution 23 24 25 A singlehop wireless cellular network with three users ....... Impact of different values of J on the performance of QADP ... Impact of different values of J on the average experienced delays . . . . 2 7 26 Queue backlog dynamics for all users ......... 27 Price dynamics in QADP for all users .. ....... 28 Virtual queue backlog updates in QADP for J = 50000 29 Average queue backlogs in the network for J = 50000 31 Network topology with interconnected queues ..... 32 Comparison of the energy consumption of the MaxW M ES algorithm . . 33 The average queue backlogs in the network for J = 50 34 The average queue backlogs in the network for J = 35 eight algorithm and 150 . 0 and 500 .. 35 Comparison of the lifetime of the MaxWeight algorithm and the MES algorithm 41 Hierarchical structure of wireless mesh access networks ............. 42 An illustrative example of multiple Nash equilibria ................. 43 Markovian chain of NETMA with two players .................... 44 Performance evaluation of the wireless mesh access network with N = 5 and c = 3 . . . 45 The trajectory of frequency negotiations in NETMA when N = 5 and c = 3 . 46 The trajectory of power negotiations in NETMA when N = 5 and c = 3 ..... 47 Performance evaluation of the wireless mesh access network with N = 20 and c = 3 . . . 48 Performance evaluation of the wireless mesh access network with/without the pricing scheme e . . . and 49 The trajectories of power utilization prices of each ACMcapable AP 98 410 The trajectories of actual power utilized by each ACMcapable AP ...... ..99 51 Topology of wireless sensor networks ... 106 52 Example network topology ............................. 125 53 Average network utility achieved by ANRA for different values of J ...... ..126 54 Average network queue size by AN RA for different values of J ... 127 55 Sample trajectories of the admitted rates of two sessions for J = 5000 127 56 Sample trajectories of the virtual queues for J = 5000 ... 128 57 Sample trajectories of the hop selections for J = 5000 ... 129 61 Average throughput v.s. Doppler spread values ... 153 62 The trajectories of 0, and Od in achieving the stochastic optimum values .. 154 63 Evolution of the probability vector P . 154 64 Evolution of the probability vector Pd ... 155 71 Architecture of wireless mesh networks ... 157 72 Example of cognitive wireless mesh network . ... 179 73 Cognitive wireless mesh networks with convexity . 181 74 Trajectory of the probability vector of router A's first path .. 183 75 Trajectories of the probability vectors on other paths ... 184 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMIC RESOURCE ALLOCATION AND OPTIMIZATION IN WIRELESS NETWORKS By Yang Song August 2010 Chair: Yuguang "Michael" Fang Major: Electrical and Computer Engineering Due to the hostile wireless medium and the limited resources in wireless networks, how well wireless networks can perform and how to make wireless networks provide better service are critical and challenging problems. These motivate our research in both theoretical analysis and protocol designs in timevarying wireless networks. In this dissertation, we aim to address the dynamic resource allocation and optimization problems in wireless networks, spanning wireless ad hoc networks, wireless mesh networks, wireless sensor networks, wireless local area networks, and cognitive radio networks. Our contributions can be summarized as follows. First, we propose an online dynamic pricing scheme which maximizes the network revenue subject to stability in multihop wireless networks with multiple QoSspecific flows. Secondly, we design a novel green scheduling algorithm in multihop wireless networks with stochastic arrivals and timevarying channel conditions which minimizes the energy expenditure subject to network stability. Thirdly, we develop a negotiationbased algorithm which attains an coptimal solution to the nonconvex joint power and frequency allocation problem in cooperative wireless mesh access networks. In addition, we analyze the existence and the inefficiency of the Nash equilibrium in noncooperative wireless mesh access networks and proposed pricing schemes to improve the equilibrium efficiency. Fourthly, we propose a queueing based model to capture the cross layer interactions in multihop wireless sensor networks and designed a joint rate admission control, traffic engineering, dynamic routing and scheduling scheme to maximize the overall network utility. Fifthly, we analyze the thresholdsbased rate adaptation algorithms in IEEE 802.11 WLANs from a reverse engineering perspective and propose a threshold optimization algorithm to enhance the performance of IEEE 802.11 WLANs. Finally, we investigate the stochastic traffic engineering problem in multihop cognitive radio networks and derive a distributed algorithm based on the stochastic primaldual approach for convex scenarios as well as a general solution based on the learning automata techniques for nonconvex scenarios. CHAPTER 1 INTRODUCTION WANETs are selfconfiguring and standalone networks of nodes connected by wireless links. They have attracted extensive attention as ideal networking solutions for scenarios where fixed network infrastructures are not available or reliable. In addition, multimedia transmissions have become an indispensable component of network traffic nowadays. However, the issue of QoS provisioning for multimedia transmissions is remarkably challenging in multihop wireless ad hoc networks. Time varying channel conditions among wireless links impose severe adverse impact on the QoS of multimedia transmissions. Therefore, while wired networks have mature solutions and established protocols for providing QoS, novel solutions which incorporate salient features of wireless transmissions need to be developed to support multimedia transmissions in multihop wireless networks. In addition, from the network provider's perspective, it is imperative to design a pricing mechanism which maximizes the network revenue while addressing the QoS requirements of users. Such a scheme should also ensure a networkwide stability under stochastic traffic arrivals and time varying channel conditions. In Chapter 2, we proposed a QoSaware dynamic pricing scheme which provably ensures the networkwide stability while attaining a solution which is arbitrarily close to the global maximum revenue, with a controllable tradeoff with the average delay in the network. Moreover, a weight assignment mechanism is devised to address the service differentiation issue for multiple flows in the network with different delay priorities. In Chapter 3, we investigate the issue of Green Computing in wireless networks, which is an important concern raised from computer scientists and engineers that attempts to utilize the computing resources efficiently while introducing minimum impact on the environment. In multihop wireless networks, we first identify that the wellestablished MaxWeight, or backpressure scheduling algorithm, is not energyoptimal. The scheduling process of the MaxWeight algorithm neglects the enormous energy consumption in retransmissions which are not negligible in wireless networks with fading channels. To address this, we proposed a minimum energy scheduling algorithm which significantly reduced the overall energy expenditure compared to the original MaxWeight algorithm. The energy consumption induced by the proposed algorithm can be pushed arbitrarily close to the global minimum solution. Moreover, the improvement on the energy efficiency is achieved without losing the throughputoptimality. A wireless mesh network is characterized by a multihop wireless backbone connecting wired Internet entry points, or gateways, and wireless access points (AP) which provide network access to end users. In a WMN, how to assign multiple channels and power levels to each AP is of great importance to maximize the overall network utilization and the aggregated throughput. In Chapter 4, we investigate the nonconvex throughput maximization problem in WMNs for both cooperative and noncooperative scenarios. In the cooperative case, we model the interactions among all APs as an identical interest game and present a decentralized negotiationbased throughput maximizing algorithm for the joint frequency and power assignment problem. We prove that this algorithm converges to the optimal frequency and power assignment solution, which maximizes the overall throughput of the wireless mesh network, with arbitrarily high probability. In the case of noncooperative APs, we prove the existence of Nash equilibria and show that the overall throughput performance is noticeably inferior to the cooperative scenario. To bridge the performance gap, we develop a pricing scheme to combat the selfish behaviors of noncooperative APs. The overall network performance in term of aggregated throughput is significantly improved. Recent years have witnessed a surge in research and development of WSNs for their broad applications in both military and civilian operations. Before the wide deployment of wireless sensor networks, a systematic understanding about the performance of multihop wireless sensor networks is desired. However, finding a suitable and accurate analytical model for wireless sensor networks is particularly challenging. An appropriate model should reflect the realistic network operations with emphasis on the distinguishing features of wireless sensor networks such as the challenge of automatic load balancing among multiple sink nodes, the task of dynamic network scheduling and routing under time varying channel conditions, and the instantaneous decisionmaking on the number of admitted packets in order to ensure the networkwide stability. To capture the cross layer interactions of multihop wireless sensor networks, in Chapter 5, we proposed a constrained queueing model to investigate the joint rate admission control, dynamic routing, adaptive link scheduling, and automatic load balancing in wireless sensor networks through a set of interconnected queues. To demonstrate the effectiveness of the proposed constrained queueing model, we investigate the stochastic network utility maximization problem in multihop wireless sensor networks. Based on the proposed queueing model, we develop an adaptive network resource allocation scheme which yields a nearoptimal solution to the stochastic network utility maximization problem. The proposed scheme consists of multiple layer components such as joint rate admission control, traffic splitting, dynamic routing, and an adaptive link scheduling algorithm. Our proposed scheme is essentially an online algorithm which only requires the instantaneous information of the current time slot and hence remarkably reduces the computational complexity. IEEE 802.11 WLAN has become the dominating technology for indoor wireless Internet access. In order to maximize the network throughput, IEEE 802.11 devices, i.e., stations, need to adaptively change the data rate to combat the time varying channel environments. However, the specification of rate adaptation algorithms is not provided by the IEEE 802.11 standard. This intentional omission encourages the studies on this active area where a variety of rate adaptation algorithms have been proposed. Due to its simplicity, the thresholdsbased rate adaptation algorithm is predominantly adopted by vendors. The data rate increases if a certain number of consecutive transmissions are successful. Although widely deployed, the obscure objective function of this type of rate adaptation algorithms, commonly based on the heuristic up/down mechanism, is less comprehended. In Chapter 6, we study the thresholdsbased rate adaptation algorithm from a reverse engineering perspective. The implicit objective function, which the rate adaption algorithm is maximizing, is unveiled. Our results provide an analytical model from which the heuristicsbased rate adaptation algorithm, such as ARF, can be better understood. Moreover, we propose a threshold optimization algorithm which dynamically adapts the up/down thresholds. Our algorithm provably converges to the set of stochastic optimum thresholds in arbitrary stationary yet potentially fastvarying channel environments and the performance in term of throughput is enhanced remarkably. In Chapter 7, we investigate the stochastic traffic engineering (STE) problem in multihop cognitive radio networks. More specifically, we are particularly interested in how the traffic in the multihop cognitive radio networks should be steered, under the influence of random returns of primary users. It is worth noting that given a routing strategy, the corresponding networks performance, e.g., the average queueing delay encountered, is a random variable. In multihop cognitive radio networks, this distinguishing feature of randomness, induced by the unpredictable behaviors of primary users, must be taken into account in protocol designs. We formulate the STE problem in a stochastic network utility maximization framework. For the case where convexity holds, we derive a distributed cross layer algorithm via the stochastic primaldual approach, which provably converges to the global optimum solution. For the scenarios where convexity is not attainable, we propose an alternative decentralized algorithmic solution based on the learning automata techniques. We show that the algorithm converges to the global optimum solution asymptotically under mild conditions. Finally, Chapter 8 concludes this dissertation. CHAPTER 2 REVENUE MAXIMIZATION IN MULTIHOP WIRELESS NETWORKS 2.1 Introduction In recent years, multihop wireless networks have made significant advance in both academic and industrial aspects. Besides traditional data services, multimedia transmissions become an indispensable component of network traffic nowadays. For example, people can watch live games while listening to online musical stations at the same time. Therefore, supporting multimedia services in multihop wireless networks effectively and efficiently has received intensive attention from the community. However, the issue of QoS provisioning for multimedia transmissions is remarkably challenging in multihop wireless networks. Time varying channel conditions among wireless links impose severe adverse impact on the QoS of multimedia transmissions. Therefore, while wired networks have mature solutions and established protocols for providing QoS, novel solutions which incorporate salient features of wireless transmissions, need to be developed to support multimedia transmissions in multihop wireless networks. Multimedia flows usually impose applicationspecific requirements on the minimum average attainable data rates from the network. Furthermore, different multimedia flows may have distinct rate requirements. For example, multimedia streams for high quality videoondemand (VoD) movie transmissions usually require larger minimum data rates on average than those of online music transmissions. Consequently, a natural question arises that how the network resource should be allocated such that all the minimum rate constraints are satisfied simultaneously. In addition, due to the nature of wireless transmissions, decentralized solution with low complexity is strongly desired. In this chapter, we investigate the resource allocation problem in multihop wireless networks, from a network administrator's perspective. For each flow, the network charges a certain amount of admission fee in order to build up a systemwide revenue. The price imposed on each flow is subject to adaptation in order to obtain the optimum revenue. Hence, a dynamic pricing policy is desired by the network administrator to maximize the overall network revenue subject to the stability of the network. To achieve this, we propose a QoSaware dynamic pricing algorithm, namely, QADP, which provably accumulates a network revenue that is arbitrarily close to the optimum solution while maintaining network stability under stochastic traffic arrivals and time varying channel conditions. Meanwhile, the minimum date rate requirements from all multimedia flows are satisfied simultaneously. Besides minimum data rates, multimedia flows, especially wireless video transmissions, usually impose additional requirements on maximum endtoend delays. For example, a multimedia stream for video surveillance may need a lower data transmission rate compared to high quality videoondemand movie transmissions, whereas a much more stringent delay requirement is imposed. Therefore, the network inclines to allocate more network resource to those delayimperative multimedia transmissions provided that the minimum data rate requirements of all flows are satisfied. Unfortunately, however, an ab solute guarantee for arbitrary delay requirements is extremely difficult, if not impossible, due to the lack of accurate delay analysis in multihop wireless networks. For example, [1] derives a lower bound on the delay performance of arbitrary scheduling policy in multihop wireless networks. However, the upper bound of delay is unspecified. In fact, it is shown that in wireless scenarios, even to decide whether a set of delay requirements can be supported by the network is an APhard problem and thus is intrinsically difficult to solve [2]. Even worse yet, time varying wireless channel conditions and stochastic traffic arrivals significantly exacerbate the hardness of sheer delay guarantees. In light of this, in this work, we alternatively aim to provide a service differentiation solution for the delay requirements of all flows. More specifically, the network provides a set of service levels, denoted by = 1, L where level one has the highest priority in the system with respect to delay guarantees. Note that L can be arbitrarily large. Each multimedia flow, according to the upper layer application, proposes a service level request to the network. For example, a background traffic for movie downloading is associated with level ten whereas a VoD online movie transmission demands a service level of two1 By meticulously assigning weights to multimedia flows, QADP algorithm provides a service differentiation solution in a way that the guaranteed maximum average endtoend delay for service level one traffic isj times less than that of level j transmissions, where j = 1, L. In other words, level one transmissions have the minimum upper bound for endtoend delay and thus represent the highest priority. Therefore, by following QADP, a QoSaware revenue maximization solution, subject to the stability of the network, is provided. QADP algorithm is inherently an online dynamic control based algorithm which is selfadaptive to the changes of statistical characteristics of traffic arrivals. Moreover, our scheme enjoys a decoupled structure and hence is suitable for decentralized implementation which is of great interest for protocol design in multihop wireless networks. Note that our results are applicable to some special interesting cases of network topology such as singlehop wireless cellular networks. The rest of this chapter is organized as follows. Section 2.2 compares our work with other existing solutions on network optimization with multiple flows. System model and problem formulation are provided in Section 2.3. Our proposed solution, i.e., QADP algorithm, is introduced in Section 2.4 followed by the performance analysis in Section 2.5. Simulation results are provided in Section 2.6 and Section 2.7 concludes this chapter. 2.2 Related Work There exists a rich literature on how the network resource should be allocated efficiently among multiple competitive transmitting flows. According to the network model, they can be roughly divided into two categories, i.e., fluid based approach and 1 We emphasize that this applicationtoservicelevel mapping is arbitrary and can be specified by the network before transmissions. queue based approach. In fluid based algorithms, e.g., [37], the characteristics of a particular flow are uniquely associated with fluid variables. For example, the flow injection rate is a commonly used fluid variable which is determined by the source node. However, whenever a change of the flow injection rate occurs, it is usually assumed that all the nodes on the path will perceive this change instantaneously. In addition, the knowledge of uptodate local information on intermediate nodes, e.g., shadow prices [4, 6, 7], is usually vital for the flow control algorithm implemented on the source node. Simply put, state information of a particular flow is usually assumed to be shared and known by all nodes on its path instantaneously and accurately, although some exceptions are discussed, e.g., [8]. In general, fluid based approach does not consider the practical queue dynamics in real networks. Moreover, fluid based algorithms usually utilize a dual decomposition framework, such as in [3, 4, 7] which heavily relies on convex optimization techniques [9]. Consequently, a fixed point solution is attained. However, in time varying environments such as wireless networks, a fixed operating point is hardly optimal2 On the contrary, queue based approach models the network as a set of interconnected queues. Each source node injects packets into the network which traverse through the network hop by hop until reach the destination. Every packet needs to wait for service in the queues of intermediate nodes along the path, which reflects the reality in practical networks. In addition, queue based approach usually adopts a dynamic control based solution where responsive actions are adjusted on the fly by which a long term average optimum is achieved. In this work, we utilize a queue based network model where an optimal dynamic pricing policy is developed. For more discussions on fluid based algorithms, refer to [10, 11] and the references therein. 2 In fact, in the work, we analytically show that the imposed price needs to be adjusted dynamically in order to achieve the optimum. The interconnected queue network model has attracted significant attention since the seminal work of [12] where the wellknown MaxWeight scheduling algorithm is proposed. Neely extends the results into a general time varying setting in [13, 14], based on which a pioneering stochastic network optimization technique is developed [15, 16]. For a comprehensive treatment on this area, refer to [17] and the references therein. Unfortunately, in the literature, few work has been devoted to addressing the issue of QoS provisioning in practical wireless settings, which is of special interest in supporting multimedia transmissions in multihop wireless networks. For example, [1820] investigate the delay constraints by assuming that the queue behaves as an M/M/1 queue. However, this may not be realistic due to the complex interaction of queues induced by the scheduling algorithm. In [2, 21, 22], delay guaranteed scheduling algorithms are studied. However, the derived delay bound is up to a logarithmic factor of the proposed delay requirement, given that the delay requirements satisfy certain perserver and persession conditions [2]. Finding a general policy that is able to achieve arbitrary feasible delay requirements is still an open problem. In addition, while existing solutions on prioritized transmissions are available, e.g., [23, 24], the question of how to achieve a minimum data rate guarantee concurrently, as well as maximizing a systemwide revenue, is unspecified. In [25, 26], lazy packet scheduling algorithms are proposed. However, they require the knowledge on future stochastic arrivals as a priori, while in our scheme, a.k.a., QADP algorithm, such information is not needed. Revenue maximization problem has been studied extensively in the literature for many different settings. Nevertheless, in general, either QoS provisioning is not particularly addressed[2730], or the network model is restricted to wired networks where the channel conditions of the system are assumed to be timeinvariant and remain unchanged [31, 32]. Our work is inspired by [33]. However, our work differs from [33] in the following crucial aspects. First, [33] studies a single hop network with only one access point (AP) while our work investigates a general multihop wireless ad hoc network where exogenous arrivals may enter the network via any node. Secondly, [33] assumes a Markovian user traffic demand while our work is applicable to arbitrary traffic demands. Thirdly and most importantly, [33] does not consider the issue of QoS provisioning for multiple transmission flows, which is the main focus of our work. To the best of our knowledge, this work is the first work to address the problem of revenue maximization, subject to network stability while providing QoS differentiations to qualitysensitive traffics such as multimedia video transmissions. 2.3 Revenue Maximization with QoS (Quality of Service) Requirements 2.3.1 System Model We consider a static multihop wireless network represented by a directed graph g = (AF, ), illustrated in Figure 21, where A is the set of nodes and S is the set of links. The numbers of nodes and links in the network are denoted by N and E, respectively. A link is denoted either by e e S or (a, b) e S where a and b are the transmitter and the receiver of the link. Time is slotted, i.e., t = 0, 1, .... For link (a, b), the instantaneous channel condition at time slot t is denoted by Sa,b(t). For example, Sa,b(t) can represent the time varying fading factor on link (a, b) at time t. Denote S(t) as the channel condition vector on all links. We assume that S(t) remains constant during a time slot. However, S(t) may change on slot boundaries. We assume that there are a finite but arbitrarily large number of possible channel condition vectors and S(t) evolves following a finite state irreducible Markovian3 chain with well defined steady state distribution. However, the steady state distribution itself and the transition probabilities are unknown. At each time slot t, given the channel state vector S(t), the network controller chooses a link schedule, denoted by I(t), from a feasible set Ts(t), which is restricted by factors such as underlying interference model, duplex constraints 3 Note that the Markovian channel state assumption is not essential and can be relaxed to a more general setting as in [14]. or peak power limitations. For a wireless link (a, b), the link data rate Pa,b(t) is a function of I(t) and S(t). We denote y(t) as the vector of link rates of all links at time slot t. S" Flow 1 Flow 2 Flow 3 Figure 21. An example of multihop wireless networks There are C commodities, a.k.a., flows, in the network, where each commodity, say c, c = 1, 2, .. C, is associated with a routing path Pc = {c(0), c(1), c(Vc)} where c(0) and c(jc) are the source and the destination node of flow c while c(j) denotes the jth hop node on its path. Without loss of generality, we assume that every node in the network initiates at most one flow4 However, multiple flows can intersect at any node in the network. Each node maintains a separate queue for every flow that passes through it. For each flow c, denote Ac(t) as the exogenous arrival to the transport layer of node c(0) during time slot t. We assume that the stochastic arrival process, i.e., Ac(t), has an expected average rate of Ac. More specifically, we have St1 lim 1 E(Ac(T))= Ac Vc. (21) too t r=o For a single queue, define the overflow function [13] as 1 t1 g(B) = lim sup Pr(Q(r) > B) (22) t*oo t 4 If a node initiates more than one flow in the network, we can replace this node with multiple duplicate nodes and the following analysis still holds. where Q(r) is the queue backlog at time r. We say the queue is stable if [13, 14] lim g(B) > 0. (23) Boo A network is stable if all the queues in this network are stable. Denote A = {A,, Ac} as the arrival rate vector of the network. Note that all the arrival rate vectors are defined in an average sense. A flow control mechanism is implemented where during time slot t, an amount of Rc(t) traffic is admitted to the network layer for flow c. Apparently, we have Rc(t) < Ac(t). For simplicity, we assume that there are no reservoirs to hold the excessive traffic, i.e., incoming packets are dropped if not admitted. However, we emphasize that our analysis can be applied to general cases where transport layer reservoirs are deployed to buffer the unadmitted traffic. The network capacity region, a.k.a., the network stability region, denoted by Q, is defined as all the admission rate vectors that can be supported by the network, in the sense that there exists a policy that stabilizes the network under this admission rate. Moreover, we are particularly interested in multimedia transmissions where each flow c has specific QoS requirements. To be specific, each flow c has a minimum data rate requirement ac as well as an applicationdependent prioritized service level request, denoted by c, 1 < c < L. Denote a = {ac, ... ac} and = {, c} as the minimum rate vector and the service level request vector of the network where has the highest priority in terms of the delay guarantees provided by the network. In addition, we assume that the minimum rate vector, i.e., a, is inside of the network capacity region Q. Since that if a is inherently not feasible, we cannot expect to find any policy to meet those demands and the only solution is to increase the network's informationtheoretic capacity by traditional methods such as adding more channels, radios, enabling network coding, or utilizing MIMO techniques with multiple antennas. Nevertheless, in this work, we restrict ourselves to the specific question of how to find a simple yet optimal policy for the network if such requirements are indeed theoretically attainable. We emphasize that, however, the answer to this question is by no means straightforward due to the intractability of quantifying the underlying network capacity region. Moreover, time varying channel conditions and stochastic exogenous arrivals make the problem even more challenging. It is our main objective to develop an optimum policy without knowing the network capacity region and the statistical characteristics of random arrival processes as well as time varying channels. We will formulate the QoSaware revenue maximization problem next. 2.3.2 Problem Formulation Denote the queue backlog of node n for flow c as Qc(t). Note that Q (,) 0 since whenever a packet reaches the destination, it is considered as leaving the network. The queue updating dynamic of Qc(t) is given as follows. Forj = 1, Kc 1, we have Qco)(t + 1) < [QC()(t) o(t)] c + ),(t) (24) and forj = 0, Qgo)(t + 1) = [QCQ)(t) It),c(t)] + R(t) (25) where [x]+ denotes max(x, 0) and pnc((t), it(t) represent the allocated data rate of the incoming link and the outgoing link of node n, by the scheduling algorithm, with respect to flow c. Note that (24) is an inequality since the previous hop node may have less packets to transmit than the allocated data rate p'c)(t). During time slot t, pc(t) is charged for flow c as the per unit flow price. The functionality of the price is not only to control the admitted flows, but also, more importantly, to build up a systemwide revenue from the network's perspective. We further assume that each flow is associated with a particular user and thus we will use flow and user interchangeably. Every user c is assumed to have a concave, differentiable utility function Oc(Rc(t)) which reflects the degree of satisfaction by transmitting with data rate Rc(t). At time slot t, user c selects a data rate which optimizes the net income, a.k.a., surplus, i.e., Rc(t) = argmax, (Oc(r) r x p(t)) Vc = 1, C. (26) Without loss of generality, we assume that Oc(r) = log(1 +r). (27) However, we stress that the following analysis can be extended to other heterogeneous forms of utility functions straightforwardly. Note that the fairness issue of multiple flows can be solved by choosing utility functions properly. For example, a utility function of log(r) represents the proportional fairness among competitive flows. For more discussions, refer to [4] and [10]. From the network administrator's perspective, the overall networkwide revenue is the target to be maximized. Meanwhile, the stability of the network as well as the QoS requirements from multimedia flows need to be addressed. Formally speaking, the objective of the network is to find an optimal policy to QoSAware Revenue Maximization Problem: t1 maximize D = lim inf1 Y () (28) t*oo t r=0 s.t. (a) the network is stable, (b) the minimum data rate requirements, i.e., a, are satisfied, (c) the guaranteed maximum endtoend delays for multiple multimedia flows are prioritized according to the service levels of t, where O(t) = E( Rc(t)pc(t)) (29) C is the expected overall network revenue during time slot t, with respected to the randomness of arrival processes and channel variations. Reduced Feasible Region a2 Figure 22. A conceptual example of the network capacity region with two multimedia flows. The minimum data rate requirements reduce the feasible region of the optimum solution However, the QoSaware revenue maximization problem is inherently challenging due to unawareness of future random arrivals and stochastic time varying channel conditions. Even worse yet, the QoS requirements significantly complicate the problem, as illustrated in Figure 22. In unconstrained cases, the feasible region of the optimum solution A* is essentially the whole network capacity region. However, as shown in Figure 22, the original optimum solution may not even be feasible under the minimum data rate requirements. Besides, finding a policy which achieves the new constrained optimum solution, denoted by X*, is a nontrivial task as well. In addition, due to the multihop nature of wireless transmissions, decentralized solutions with low complexity are remarkably favorable. To address the aforementioned concerns, in next section, we will propose an optimal online policy, namely, QADP, which provably generates a network revenue that is arbitrarily close to the optimum solution of (28). Meanwhile, the imposed requirements of the problem, i.e., (a), (b) and (c), are achieved on the fly. In the following, we first assume that the arrival rate vector A lies outside of the capacity region Q for all time slots, i.e., a heave traffic scenario. 2.4 QoSAware Dynamic Pricing (QADP) Algorithm In this section, we propose an online policy, i.e., QADP algorithm, which solves the QoSaware revenue maximization problem in (28). We first introduce some systemwide parameters to facilitate our analysis. Define Rmax as the upper bound of admitted traffic of flow c during one time slot, i.e., Rc(t) < Rcax, Vc, t. For example, R cx can represent the hardware limitation on the maximum volume of traffic that a node can admit during one time slot, or simply the peak arrival rate within one time slot if such information is available. Let pmax be the maximum data rate on any link of the network, which may be determined by factors such as the number of antennas, modulation schemes and coding policies. In addition, for each flow c, we introduce a virtual queue Yc(t) which is initially empty, and the queue updating dynamic is defined as Y,(t + 1) = [Y(t) R(t)]+ +a Vc. (210) Note that virtual queues are easy to implement. For example, the source node of flow c, i.e., c(0), can maintain a software based counter to measure the backlog updates of virtual queue Yc(t). In addition, for each flow c, we define 1 2 S= N(pmax)2 + (Rcax)2 ( ac) Vc. (211) 2 Denote 01, Oc as the weights which will be calculated and assigned to all flows, where C denotes the number of flows in the network. Let J be a tunable5 positive large number determined by the network. In addition, we assume a maximum value of the allocated weight, denoted by Omax, i.e., Oc < Oax, Vc. The proposed QADP algorithm is given as follows. QADP ALGORITHM: Part I: Weight Assignment For all multimedia transmissions, find the flow with the minimum value of ac x fc, c = 1, C, say, flowj. For each flow c, assign an associated weight, denoted 5 The impact of J on the performance of QADP algorithm will be clarified shortly. by Oc, which is calculated by 0max x ;x OC = 0x Vc= 1, ... ,C. (212) Oac X c Part II: Dynamic Pricing For every time slot t, the source node of flow c, i.e., c(O), measures the value of Qc(0)(t) and Yc(t). If Q(,0)(t) > YC(t), the instantaneous admission price is set as ec (cW)(t Y(t)) pc(t) = (213) and pc(t) = 0 otherwise. Part III: Scheduling For every time slot t, find a link schedule I*(t), from the feasible set Ts(t), which solves max ab(t a,b (214) I(t)ETs( (a,b)GE where a,b = max (Oc(Qf(t) Qb(t))) (215) c:(a,b)ePc if 3c, such that (a, b) e Pc, and a,b = 0 otherwise. END It is worth noting that Part I of QADP can be precalculated before actual transmissions. The weight assignment can be implemented either by the network controller which knows the QoS requirements of all flows, or by mutual information exchanges among multiple multimedia flows, in a decentralized fashion. The value of Oc represents the QoSwise "importance" of flow c and remains unchanged unless the QoS requirements from flows, i.e., (a, t), are updated, by which a new weight calculation is triggered. The dynamic pricing part is the crucial component of QADP By following (213), not only the incoming admitted rates can be regulated effectively, but also the overall average network revenue can be maximized, as will be shown shortly. Note that after the weight assignment, in order to compute pc(t), the source node of flow c, i.e., c(0), which is considered as the edge node of the network, requires only local information, i.e., current backlogs of the source data queue and the virtual queue. It is interesting to observe that if Q (o)(t) < Yc(t), the admission is free! Intuitively, a small value of Qg(0)(t) indicates a deficient arrival rate of flow c. On the contrary, a large value of Yc(t) means that the average "service rate" is less than the average "arrival rate" and thus the virtual queue is building up. Note in (210) that this indicates that the average of Rc(t) falls below the arrival rate, i.e., ac. Therefore, when Q(,0)(t) < Yc(t), the network provides free admission to encourage more incoming packets in order to satisfy the QoS constraints. We will make this intuition precise and rigorous in the following section. The third part of QADP is a weighted extension to the wellknown MaxWeight scheduling algorithm [12, 13, 34]. Instead of the exact difference of queue backlogs, we deliberately select the weighted difference of queue backlogs as the weight of a particular link in the scheduling algorithm. Intuitively, if a flow is assigned with a larger value of Oc, the links associated with it will have a higher possibility of being selected for transmissions by QADP. Therefore, by assigning proper values of Oc to flows with different QoS requirements, a service differentiation can be achieved which provides more flexibility to previous schemes in the literature, e.g., [13, 15, 16]. In addition, as indicated by (213), a higher priority needs to pay at a higher price. Note that to calculate (214), QADP needs to solve a complex optimization problem which requires a global information on channel states, i.e., S(t). However, availed of the prosperous development of distributed scheduling schemes, such as [3538], the difficulty of centralized computation can be circumvented, which provides QADP algorithm the amenability for decentralized implementations. 2.5 Performance Analysis In this section, we provide the main result on the performance of QADP algorithm. Theorem 2.1. Define D* as the optimum solution of (28). For QADP algorithm, we have (a) Revenue Maximization 1 t1 lim inf O(T) > D* (216) t+oo t J =0 where K is a constant and is given by K = Oc6c (217) C and 6c is defined in (211). (b) Network Stability The network is stable under QADP algorithm, i.e., for every queue in the network, (23) is satisfied. (c) QoS Provisioning By following QADP algorithm, any feasible minimum data rate requirements a can be satisfied. In addition, the guaranteed maximum average endtoend delays for multimedia flows with service level j arej times larger than that of level one transmissions. It is of great importance to observe that in (216), the achieved performance of QADP algorithm can be pushed arbitrarily close to the optimum solution D* by selecting a sufficiently large value of J. The proof of Theorem 2.1 is provided in the rest of this section. 2.5.1 Proof of Revenue Maximization Recall that the minimum rate vector a is assumed to lie inside the capacity region Q. Therefore, there exists a small positive number > 0 such that a + 1 e Q where 1 is a unity vector with dimension C. Lemma 1. For any feasible input rate vector, there exists a stationary6 randomized policy, denoted by RAND, which generates E (p / c Pn WO(t)) = 0 Vn, c, t (218) 6 Stationary means that the probabilistic structure of the randomized policy does not change with different values of queue backlogs. and E(()(t))> ac+ Vt, c (219) where O, (t) is the exogenous arrival on node n for flow c during time slot t. Proof. (Sketch) The proof of Lemma 1 utilizes standard techniques as in [13, 15, 16]. The basic idea is to reduce the exponentially large dimension of extreme points to a finite set with dimension E + 1 by Caratheodory's Theorem. Then a randomized link schedule selection is implemented among all reduced E+1 extreme points. The detailed proof is omitted. We emphasize that Lemma 1 is only an existence proof in the sense that the randomized algorithm RAND cannot be implemented in practice. This is because that RAND requires a prior knowledge on the network capacity region, i.e., the underlying steady state distribution of Markovian channels, and hence is computationally prohibitive. However, the existence of RAND plays a crucial role for the performance analysis of QADP algorithm, which, in contrast, is an online adaptive policy and does not require the statistical characteristics of the stochastic arrivals and time varying channel conditions. Recall that a virtual queue Yc(t) is introduced for every flow c and the queue updating dynamic is given by (210). As a result, the minimum data rate requirement is converted to a queue stability problem since if the virtual queue Yc(t) is stable, the average service rate, i.e., the time average of Rc(t), needs to be greater than the average arrival rate, i.e., ac. Define Z(t) = [Q(t); Y(t)] as all the real data queues and virtual queues at time slot t. If we can ensure that the network is stable with respect to Z(t), the backlogs of real queues are bounded and the minimum rate requirements are achieved at the same time. Define a systemwide potential function (PF) as (220) PF(Z(t))= PFC(Z(t)) C where PFC(Z(t)) = ( c(Q()2 2 n Oc(YCt))2) Note that PF(Z(t)) is a scalarvalued nonnegative function. Define A(Z(t)) = E(PF(Z(t 1)) PF(Z(t))Z(t)) as the drift of the potential function PF(Z(t)). For flow c, we take the square of both sides of (24) and (25) and obtain7 (Qo) (t + 1))2 < (Q )()2  (/c),c(t)) 2Qc2 (t) (/t ),c( Q P ),()) forj = 1, ,K 1, and (o)(t + 1))2 < (Q )(t))2 Rc(t)) forj = 0. Similarly, for (210), we have (Ye(t + 1))2 < ( Y(t))2 + (Rc(t))2 + (ac)2 ac). 7 We use the fact that for nonnegative real numbers a, b, c, d, if a < [b a2 < b2 + c2 + d2 2b(c d), as given in Lemma 4.3 on [17]. (221) (222) (223) (224) d, then (p( ),c(t)) (Rc(t))2 2 Q(c)(t) (pL),c(t) 2 Yc(t)(R(t)  By combining the inequalities above, we have PF(Z(t + 1)) PF(Z(t)) < E O+ 0 Q ((t)R(t) Oc Yc(t)( Rc(t) a) SOQ(t) (f(t) p ,(t)) (225) n where =c = 0 N(Ipmax)2 (Rmax)2 + (ac)2. (226) Note that (225) is summed over the whole network. If node n is not on the path of flow c, pc(t) = (t) = 0. Moreover, pn(t) = 0 for the source node of flow c and p~ (t) = 0 for the destination node of flow c. Next, we sum over all flows to derive the networkwide potential function difference as PF(Z(t + 1)) PF(Z(t)) < K B Ot)_ n,c(t)) n,c S Q Yc(t)R(t) (t)(R(t) a) C C where K = c Ec. Therefore, we have A(Z(t)) < K OcQ(t)E ( o(t) pinc(t) Z(t)) n,c + YOCQW(o)(t)E(Rc(t)Z(t)) C OcYc(t)E(Rc(t) aZ(t)). (227) C Next, for a positive constant J, we subtract both sides of (227) by JE (Yc Rc(t)pc(t)lZ(t)) and have A(Z(t)) JE( R(t)p,(t)Z(t) < K OC Q(t)E (po (t) C(t)l Z(t)) n,c +YOCQco)(t)E(Rc(t) Z(t)) C Oc Y(t)E(Rc(t) cZ(t)) C JE Rc(t)pc(t)Z(t). (228) Note that (228) is general and holds for any possible policy. For an arbitrarily small positive constant 0 < e < emax define the ereduced network capacity region, Qe, as all possible input rate vectors such that Q, = {Alc + e Vc} (229) where Q is the original network capacity region. We will discuss about how to obtain emax shortly. Define D* as the optimum value of the reduced problem to (28) where Q is replaced by Qe. It can be verified that [14] lim D* > D* (230) where D* is the optimum value of the original revenue maximization problem in (28), i.e., the target of QADP algorithm. Specifically, we denote r, (0), r,(1), .. r*(t), and p*,c(0), p,c(1), p,c(t),.. as the optimum sequences of admitted rates and prices, for flow c, which achieve D*. Define ?E as the time average of the optimum sequence of r,(0), r,(1), rc(t), . Therefore, following the definition of (229), we have r + c E Q. By Lemma 1, we claim that there exists a randomized policy, denoted by RAND, which yields E(,p '(t) n(t) (t)) = Vc,n =c(0) (231) and E( (t) n,(t)) = Vc, n z c(O) (232) and E(rc(t) + ) > ac + Vc. (233) Denote the RHS of (228) as Without loss of generality, we assume e < c. Therefore, for randomized policy RAND, we have VRAND Ke Oc O(t)+ OCY(t) n,c C JE (rc(t) + )P(tlZ(t) (234) ( C where p ,c(t) is the corresponding price which induces a rate of rc(t) + by (26). Lemma 2. QADP algorithm minimizes the RHS of (228) over all possible policies. Proof. The scheduling part of QADP algorithm in Section 2.4 is a weighted version of (G not)  (t)) MaxWeight algorithm [12, 13, 34] which maximizes En,c OcQ(t)( () c(t)) Z(a,b)E la,b(t)ba,b if 3c, such that (a, b) e Pc, for every time slot t. The dynamic pricing part of QADP is essentially maximizing S(Oc(Yc(t) Q (o)(t))Rc(t) +JRc(t)pc(t)) (235) C for every time slot. By (26) and (27), we see that QADP finds an optimum price p*(t) which maximizes 1 M = O(Yc(t) Q~0)(t))( 1) + J( p(t)). (236) (D Wt Define W = Oc(Qg(O)(t) YC(t)). Case 1: If W > 0, we have M = J + W (Jp(t) 1 W ( ). Pc M Take the first order derivative, we have V V J M' = J. (Pc(t))2 The second order derivative brings us W (Pc(t))3 <0. Therefore, M is a concave function and the optimum value is achieved by (t) W Case 2: If W < 0, M is a decreasing function with respect to pc(t). Therefore, p*(t) 0. Following Lemma 2, we conclude that for QADP algorithm, < WRAND < K JE ( ,c(t)  C c Oc QC)(t) .OCYC(t) \n,c C 0e)p (*)Z(t) . (241) We take expectation with the distribution of Z(t), on both sides of (241). By the fact that Ey(E(X Y)) E(X), we have E(PF(Z(t + 1))) E(PF(Z(t))) SK+ JO(t) E c( \ nc SOCYCt)) JE ( (r'c(t) C c (237) (238) (239) (240) A(Z(t)) JE Rc(t)pc(t)lZ(t) < WQADP e)p. (t) (242) Note that (242) is satisfied for all time slots. Therefore, we take a sum on time slots r = 0,... T 1 and have E(PF(Z(T))) E(PF(Z(0))) < TK T1 S ~cE ( OCQ,(,T) + r=0 n,c c T1O \ (243) by T and( ) rearrange terms to obtain T=0 c Divide (243) by T and rearrange terms to obtain T1 T=0 C Y (T)) 4 ) (243) (244) where the nonnegativity of the potential function is utilized. We first take lim infT on both sides of (244) and have8 1 liminf E T O CT) Too T ' \ En Tn )\ n r STl lim inf T JE Too T TO ( ( ( (T) ,c "7  0C YC(T,))  C)P T) (245) T1 < K lim inf 1 JO(T). S Too T T0 8 We assume that the initial queue sizes of real data queues and virtual queues are bounded. T1 TT=0 1 T1 E(PF(Z(O))) _ K T JO(Y ) T =0 Note that the first term of (245) is nonnegative and thus we have lim inf O(T) ST1 > nf E Y (rc (T) + 0p ) . 7 T T=0 ( C Recall that the analysis above applies to arbitrarily small 0 < e < emax. We observe that I T1 lim lim infT E ( lT) + 0)P c(( ) D*. (246) c+O T+ T> 7 T=0 C Therefore, by (230), taking the limit of e 0 yields the performance bound of (216) in Theorem 2.1. 2.5.2 Proof of Network Stability To prove the stability of the network, we take lim sup on (244) and obtain 1T1 lim sup 1E I OQ(r) + OCYC()) T+oo T TO n,c c < K J lim sup 1 O() (247) ( T+ TO 1 since the second term of (244) is also nonnegative. Note that if it satisfies that O(t) < Omax for all time slot t, (247) becomes T1 lim sup 1 \(E O CQC() S OYC)Y(7) T_ o T nc c < K JOmax (248) Recall that the above analysis applies to any 0 < e < max. In addition, by the definition of (229), we have max P max maxa where = 1, ... C (249) I where pmax is the maximum possible data rate on a link. Moreover, by (26) and (27), we have Omax = C where C is the number of flows in the network. Therefore, we have lim supI TE C QC(\ Cy(\) K +JC Too T0 n, Using the fact that X(t) < Y(t) Vt = lim supX(t) < lim sup Y(t), (250) we have, for every flow c, the average queue length on its routing path is bounded by 1 T K JC lim sup E ( QC(J ()) < max (251) T+oo 70 max Finally, by applying Markov Inequality, we conclude that all data queues in the network are stable, based on the fact that the RHS of (251) is bounded. 2.5.3 Proof of QoS Provisioning Similar to (251), for every virtual queue Yc(t), we can obtain T1 1 K + JC lim supT E(Y,(T)) < KJCx (252) Too T 0^  r=0 Therefore, by similar analysis and the definition of virtual queues, we conclude that the virtual queues are stable and thus the minimum data rate requirements imposed by multimedia flows are achieved. Next, we show that QADP indeed provides a service differentiation solution on the guaranteed maximum endtoend delays for all multimedia flows. Denote the actual experienced average delay of flow c as w~. By Little's Law, wc is approximated9 by 9 Note that we consider a heavy loaded network. Propagation delays are assumed to be negligible compared to queueing delays and thus are omitted. average overall queue length on the path of flow c C average incoming rate of flow c lim supT, E ( Q ( o )) MU (253) lim supT, yT 1 E(Rc(T)) In light of the stability of virtual queue Yc(t), we have lim supTX, 1 y J E(Rc(T)) > ac. Hence, we can obtain S< maxJC (254) Equivalently speaking, to differentiate the guaranteed maximum delay bound, we need to find a set of weights such that Ocajct = Od dd (255) is satisfied for any pair of multimedia flows c and d. Therefore, it is straightforward to verify that the weight assignment algorithm in QADP indeed provides a service differentiation solution where the guaranteed maximum endtoend delays of multimedia flows are distinguished according to applicationdependent service level requests, which completes the proof of Theorem 2.1. The performance of QADP algorithm will be evaluated numerically in Section 2.6. 2.6 Simulations 2.6.1 SingleHop Wireless Cellular Networks We first consider a singlehop wireless cellular network with downlink multimedia transmissions, as shown in Figure 23. The base station (BS) is associated with three users with infinite backlogged traffic. A separate queue is maintained by the base station for every user. In addition, at each time slot, BS can only transmit to one particular user. A wireless link is assumed to have three equally possible channel states, i.e., Good, Medium, Bad. The corresponding transmission rates for three channel states are 20, 15 and 10 bits per slot, respectively. Therefore, the base station encounters a complex opportunistic scheduling problem where the revenue maximization problem, network stability as well as the QoS requirements need to be addressed simultaneously. Base Station 1 2 3 UseUser 2r 3 Figure 23. A singlehop wireless cellular network with three users Without loss of generality, we assume that the minimum average rate requirements for user 1, 2, 3 are a = [1, 2, 3] bits per slot. In addition, to provide service differentiation, the network offers three prioritized service levels, e.g., Platinum, Gold and Silver, where level Platinum possesses the highest priority in terms of endtoend delay upper bound. We assume that user 1 has a service level request for Platinum while user 2 and 3 demand for level Gold and Silver, respectively, according to the upper layer applications. Other system parameters are assumed to be Rlax = 20 bits per slot for all flows and Omax = 100. We next implement QADP algorithm for different values of J where J = [50, 100, 500, 1000, 5000, 10000, 20000, 50000, 100000]. Every experiment is simulated for 500000 time slots. Figure 24 depicts the system revenue, i.e., the solution of (28) by QADP algorithm, with respect to different values of J. As shown in (216) and demonstrated pictorially in Figure 24, the achieved system revenue by QADP converges gradually to the optimum solution as J grows. Note that the values of system revenues are almost indistinguishable when J > 50000. Figure 25 illustrates the actual experienced average delays of all three flows with different values of J, where the delays of user 1 to user 3 are compared from left to right. It is worth noting that, not only the maximum guaranteed endtoend delays are distinguished analytically by QADP, but also the Impact of Different Values of J 50 100 500 1000 5000 100002000050000100000 Values of J Figure 24. Impact of different values of J on the performance of QADP Impact of Different Values of J 60 User 1 User 2 50 User3 0 40 40 u 50 100 500 1000 5000 10000200005000000000 Values of J Figure 25. Impact of different values of J on the average experienced delays actual experienced average delays are prioritized for all three flows with distinct service level requests. More specifically, user 1, i.e., the Platinum user, enjoys a delay which is less than half of that of user 2 and one third of that of user 3, for all values of J, as demonstrated in Figure 25. However, as shown by Figure 24 and Figure 25 jointly, while a larger value of J yields an improvement on the performance of QADP, the endtoend delays of all three flows are augmented concurrently. Therefore, by tuning J, a tradeoff between optimality and average delays can be achieved. The time average admitted rates of multimedia flows are shown in Table 21. We observe that in all cases, the average rates of flows exceed the minimum data rate requirements specified by a. The sample paths of price adaptations and queue backlog evolutions are illustrated in Figure 27 and Figure 26, respectively, for the first 100 time slots with J = 50000. Table 21. Average admitted rates for multimedia flows Flow 1 Flow 2 Flow 3 J = 50 4.01 2.71 3.07 J = 100 4.04 2.72 3.09 J = 500 4.05 2.91 3.11 J = 1000 4.31 3.07 3.16 J = 5000 4.51 3.57 3.42 J= 10000 4.63 3.94 3.62 J = 20000 4.75 4.28 4.09 J = 50000 5.09 4.71 4.61 J = 100000 5.13 4.88 4.85 Note that, in Figure 27, the prices imposed for three users are dynamically adjusted at every time slot. It is worth noting that whenever a data queue in Figure 26 has a tendency to build up, the price imposed by QADP algorithm, as shown in Figure 27, rises correspondingly, which in turn discourages the excessive admitted rate and thus all queues in the network remain bounded. As a result, the stability of the network is achieved. 2.6.2 MultiHop Wireless Networks We next consider a multihop wireless network with a topology shown in Figure 21. There are three multimedia flows exist in the network, denoted by Flow 1, 2, 3. The routing paths of flows are specified by P1 = {A, B, C, D}, P2 = {F, G, C, D} and P3 = {E, F, G, H}. Without loss of generality, we assume a twohop interference model which represents the general IEEE 802.11 MAC protocols [36, 37]. Other configurations are the same as the singlehop scenario described above except that the possible link rates are assumed to be 40, 30, 20 bits per slot for three channel conditions. We observe that in this network topology, link C > D and link F > G are shared by two different flows. Therefore, in the scheduling part of QADP, the particular flow with a larger weighted queue backlog difference should be selected. In Figure 28, we specifically depict the dynamics of three virtual queues for the first 400 time slots with J = 50000. Unlike the singlehop case in Figure 26, the virtual Figure 26. Queue backlog dynamics for all users 0.7 e price for user 1 0.6 price for user price for user 3 S0.3 0.2 ii 0 20 40 60 80 100 Time Slots Figure 27. Price dynamics in QADP for all users 50 4Queue 1 Virtual Queue 2 40 ...0o Queue 2 Virtual Queue 3 Queue3 'Virtual Queue 1  30 0 C O I 20 0: 00 0o 0 20 40 60 80 100 Time Slots queues behave remarkably different in this multihop scenario. It is worth noting that while the virtual queues of user 1 and 3 have relatively low occupancies, the virtual queue associated with user 2 suffers a larger average backlog. Intuitively, due to the underlying twohop interference model, link G > C needs to be scheduled exclusively in the network for successful transmissions. In other words, link G > C is the bottleneck of the network. Therefore, to ensure networkwide stability, a much more stringent regulation is enforced on the admitted rate of flow 2. As a consequence, although remains bounded, the virtual queue of flow 2 accumulates more backlogs compared to other competitive flows. In addition, we compare the time average queue backlogs of all data queues in the network, from left to right, in Figure 29. We can observe that the data queues on the path of flow 1 have fewer average backlogs due to the highest 20 >15 10 0 50 100 150 200 250 300 350 400 Time Slots Figure 28. Virtual queue backlog updates in QADP for J = 50000 70 60 Q1 20 50 I 40 4F IN' 1CII 0;! Queues Figure 29. Average queue backlogs in the network for J = 50000 priority with respect to the service level, i.e., Platinum. On the contrary, the queues on the path of flow 3 have larger backlogs compared to other two flows. It is noticeable that Q3 and Qj have considerably larger average queue sizes. This is because that QF has to share the link rate of F > G with Q2. Nevertheless, Q3 possesses a smaller share of bandwidth than Q2 due to the lower prioritized service level associated with flow 3. Even worse yet, both link F > G and G > H have less opportunity to be scheduled due to their locations and the underlying interference model. Therefore, the average backlogs on the path of user 3 has higher occupancies compared to other two competitive flows. The average data rates are provided in Table 22. Note that the minimum average rate requirements of all multimedia flows are satisfied simultaneously, as expected. The tradeoff between optimality and average delay, which is controlled by different values of J, as well as the network service differentiation in terms of delays are analogous to the singlehop scenario discussed above. Duplicated simulation figures are omitted. Table 22. Average admitted rates for multimedia flows Flow 1 Flow 2 Flow 3 J = 50 3.04 2.08 3.88 J = 100 3.12 2.03 3.98 J = 500 3.17 2.04 4.60 J = 1000 3.30 2.03 5.04 J = 5000 4.13 2.01 6.45 J = 10000 4.36 2.02 7.32 J = 20000 5.08 2.03 8.12 J = 50000 6.43 2.01 9.18 J= 100000 7.63 2.01 9.89 2.7 Conclusions We consider a multihop wireless network where multiple QoSspecific multimedia flows share the network resource jointly. To maximize the overall network revenue, we propose a dynamic pricing based algorithm, namely, QADP, which achieves a solution that is arbitrarily close to the optimum, subject to network stability. Moreover, a weight assignment mechanism is introduced to address the service differentiation issue for multimedia flows with different delay priorities. In tandem with the virtual queue technique which provides minimum average data rate guarantees, the QoS requirements of multiple multimedia streams are addressed effectively. In this work, the applicationdependent service level requests and the users' utility functions are assumed to be attained truthfully. The mechanism design for strategyproof user information acquisition seems interesting and needs further investigation. Moreover, in this work, we assume that the information of channel states, i.e., S(t), is available for QADP algorithm, which is either acquired by the central base station or approximated by the distributed local scheduling algorithms. As a future work, an online channel probing mechanism needs to be incorporated into QADP, as suggested in [39]. CHAPTER 3 ENERGYCONSERVING SCHEDULING IN WIRELESS NETWORKS 3.1 Introduction There has been a lot of interest over the past few years in characterizing the network capacity region as well as designing efficient scheduling algorithms in multihop wireless networks. Due to the stochastic traffic arrivals and timevarying channel conditions, supporting high throughput and high quality communications in multihop wireless networks is inherently challenging. To utilize the scarce wireless bandwidth resource effectively, scheduling algorithms which can dynamically allocate the network resource, i.e., select active links, are investigated intensively in the community. For example, MaxWeight algorithm, a.k.a., backpressure algorithm, has been extensively studied in the literature, e.g., [5, 14, 34, 37, 40], following the seminal work of [12]. The MaxWeight algorithm enjoys the merit of selfadaptability due to its online nature. In addition, MaxWeight algorithm is known to be throughput optimal [17]. That is to say, the MaxWeight algorithm can stabilize the network under arbitrary traffic load that can be stabilized by any other possible scheduling algorithms. Therefore, the MaxWeight algorithm attracts significant attention and becomes an indispensable component for link scheduling in network protocol designs, e.g., [4143]. While the throughputoptimality of the MaxWeight algorithm is well understood, the energy consumption induced by the MaxWeight algorithm is less studied in the literature. However, due to the scarcity of energy supplies in wireless nodes, it is imperative to study the energy consumption of the scheduling algorithm which is of special interest in energyconstrained wireless networks such as wireless sensor networks. Is the throughput optimal MaxWeight scheduling algorithm also energy optimal? In this work, we show that the answer to this question is no. The reason is that the vast energy consumption during packet retransmissions are completely neglected by the MaxWeight algorithm. For example, in [16], an energy optimal control scheme is proposed where a minimum power expenditure is achieved. However, as in other related works, e.g., [17], the wireless channels in [16] are assumed to be errorfree, i.e., all the transmissions are assumed to be successful. Nevertheless, in practice, wireless channels are errorprone and data transmissions are subject to random failures due to the hostile channel conditions. Therefore, before a packet can be successfully removed from the transmitter's queue, several transmissions may have occurred, including the original attempt and the posterior retransmissions, which deplete a significant amount of energy for the transmitter. However, in the traditional MaxWeight algorithm, such energyconsuming retransmissions induced by channel errors are overlooked. Intuitively, from energysaving perspective, the possibility that a particular link is selected for transmissions should rely on not only the queue difference between the transmitter and the receiver, which is the design rationale of the traditional MaxWeight algorithm, but also the potential energy consumption of retransmissions induced by erroneous channels. We will make this intuition precise and rigorous in the following sections. In this work, we propose a minimum energy scheduling (MES) algorithm which consumes an amount of energy that can be pushed arbitrarily close to the global minimum solution. In addition, the energy efficiency attained by the MES algorithm incurs no loss of throughputoptimality. The proposed MES algorithm significantly reduces the overall energy consumption compared to the traditional MaxWeight algorithm and remains throughput optimal. Therefore, our proposed MES algorithm is more favorable for dynamic link scheduling and network protocol designs in energyconstrained wireless networks such as wireless sensor networks. The rest of this chapter is organized as follows. Section 3.2 describes the system model used in this work. The proposed MES algorithm is introduced in Section 3.3 where the performance analysis is provided. Simulation results are shown in Section 3.4 and Section 3.5 concludes this chapter. 3.2 System Model We consider a static multihop wireless network denoted by a directed graph (A/, C) where A is the set of nodes and C denotes the set of links in the network. We use X to represent the cardinality of set X. Time is slotted by t = 0, 1, 2, .. and in every time slot, the instantaneous channel state of a link (a, b) C is denoted by Sa,b(t) where a and b are the transmitter and the receiver of the link. In this work, we use S(t) to denote the channel state vector of the whole network. Note that S(t) remains constant within one time slot, however, it is subject to changes on time slot boundaries. We assume that S(t) has a finite but potentially large number of possible values and evolves following an irreducible Markovian chain with well defined steady state distributions. Nevertheless, the steady state distribution and the transition probabilities are unknown to the network. Given an instantaneous channel state Sa,b(t), the transmission of a packet on link (a, b) is successful with a probability of pa,b(t), if link (a, b) is active and suffers no interference from concurrent transmissions. From the network's perspective, at each time slot t, an interferencefree link schedule, denoted by I(t), is selected from a feasible set Q(t), which is constrained by the underlying interference model, e.g., Khop interference model [44], as well as other limitations such as duplex constraints and peak power limitations. Denote Ub(t) as the nominal link rate if link (a, b) is selected by the network and the channel is errorfree, i.e., pa,b(t) = 1. The actual data rate of link (a, b) is hence represented by Ua,b(t) = b(t)Pa,b(t). We assume that Ua,b(t) is upper bounded by a constant umax for all (a, b) e In practice, umax can be determined by the number of antennas equipped in a single node as well as the coding/modulation schemes available to the network. We denote u(t) as the link rate vector of the network at time t. Apparently, u(t) is a function of both I(t) and S(t). The network consists of t flows indexed by f = 1, 2, .. T. Each flow f is associated with a routing path Rf = [no, n, . nf] where nf,j = 0, f denotes the nodes on the path of flow f. At each time slot t, the stochastic exogenous arrivals of flow f, i.e., the number of new packets that are initiated by node no, is denoted by Af(t). For the ease of exposition, we assume that Af(t) is i.i.d. for every time slot with an average rate of Af. In addition, the arrival processes of all flows are assumed to be independent. We further assume that the maximum number of new packets generated by a flow during one time slot is upper bounded, i.e., Af(t) < Amax, Vf, t. We emphasize that the i.i.d. assumption incurs no loss of generality and our model can be extended to cases where Af(t) is nonstationary in a straightforward fashion, as in [14]. Every node in the network maintains a separate queue for each flow that passes through it. Denote Q (t) as the queue backlog at time t, for node n, where f is one of the flows that traverses through n, i.e., ne cRf. We assume that Qf(t) = 0, Vt if n = n. That is to say, if a packet reaches the destination, we consider the packet as leaving the network immediately. Define the overflow function of Qf as 1 t1 O(M) = l sup 1 Pr(Qj'() > M). (31) The queue is stable if [14] lim O(M) 0 (32) Moo and the network is stable if all individual queues are stable concurrently. For notation brevity, we define the network admission rate vector as A = {An,f, Vn, f} where Anf is the average exogenous arrival rate of flow f on queue1 Q,. We have Anf = Af if n = no and An,f = 0 otherwise. Denote A as the network capacity region [12], namely, the set of all feasible admission rate vectors, i.e., A, that the network can support, in the sense that there exists a scheduling algorithm which stabilizes the network under traffic load A. It is shown in [12] and [14] that A is convex, closed and bounded. 1 Note that with a slight abuse of notation, we use Qf to denote both the queue itself and the number of packets in the queue. Without loss of generality, we consider an energy consumption model as follows. Recall that the successful transmission probability of link (a, b) is pa,b(t). Therefore, for a particular packet to be transmitted from a to b, on average, a number of ' P,,b(t) transmissions are needed in order to "erase" this packet from the queue of node a. From the transmitter's perspective, however, every transmission, either the original attempt or retransmissions, costs the same amount of energy. Denote aa,b as the energy needed to transmit a packet on link (a, b). For example, aa,b can be proportional to the distance between node a and b. Therefore, in order to transmit a packet successfully on (a, b), node a needs to spend a total energy of on average. For the receiver b, o r(t)e. For the receiver b, we assume that the energy depletion on successful packets receptions are dominant, i.e., the energy spent for overhearing and short ACK messages are neglected. Denote 3a,b as the energy consumed for a successful packet reception in demodulation and decoding on node b. With a data rate2 of Ua,b(t) on link (a, b), the overall energy spent during time slot t is given by Ga,b(t) = Ua,b(t) Pa,bt a,b \Pa,b(t) 0 = ,b(t) (a,,b + Pa,bPa,b(t)). (33) Note that due to the stochastic nature of wireless channels, Ga,b is a random variable. We stress that the simple energy consumption model above is not essential and our analysis can be extended to other more complex forms of energy models straightforwardly, as will be shown in the next section. 2 The unit of data rate in this work is defined as packets/slot. It is worth noting that other units such as bits/slot are also applicable. In addition, we assume that the energy consumption of the whole network during one single time slot is upper bounded, i.e., SGa,b(t) < Gmax, Vt. (34) (a,b)GC Therefore, to minimize the energy consumption, the objective of the network is to find a scheduling algorithm which solves Minimum Energy Scheduling Problem: T1 minimize C = limsup G (t) (35) T*oo T 't t=0 s.t. the network remains stable, and G(t)= E (a. hab(t) Pab) +ab (36) is the expected overall network energy consumption during time slot t, with respect to the randomness of arrival processes and channel variations. Note that ha,b(t) is the actual number of successfully transmitted packets on link (a, b) during time slot t and3 ha,b(t) < Ua,b(t) In the next section, we will propose a minimum energy scheduling (MES) algorithm which minimizes the average network energy consumption asymptotically subject to network stability. In addition, the proposed MES algorithm is throughput optimal, in the sense that the MES algorithm can ensure the network stability for all feasible network admission rate vectors in the network capacity region. Restated, the set of feasible arrival rates supported by the MES algorithm is the superset of all other possible 3 The inequality holds when node a has less packets to transmit than the allocated data rate Uab(t). scheduling algorithms, including those with a priori knowledge on the futuristic arrivals and channel conditions. 3.3 Minimum Energy Scheduling Algorithm 3.3.1 Algorithm Description The minimum energy scheduling (MES) algorithm is given as follows. MES ALGORITHM: At every time slot t: Every link (a, b) e finds the flow f* which maximizes max (2Q;(t) Pab(t) 2Q(t) Jab (37) f:(a,b)GRf ( Pa,b~t) b where J is a positive constant which is tunable as a system parameter. Every link (a, b) E calculates the link weight as Hab(t) = 2Q(t) Ja,b 2Q(t) Jb (38) Pabt) where [x]+ denotes max(x, 0). For the network, a link schedule I*(t) is selected which solves max Uab(t)Hab(t) (39) I(t)GQ(t) (a,b) E END Note that the proposed MES algorithm is different from the MaxWeight algorithm proposed in [12, 13]. In the original MaxWeight algorithm, the weight of a particular link (a, b) is the queue difference between node a and b. However, in the MES algorithm, as indicated by (38), the link weight Hab(t) is related to the potential energy consumption on this link under the current channel condition as well. More specifically, the link weight in the MES algorithm is the queue difference subtracted by a weighted energy consumption factor, i.e., J .b(t + /,b where J represents the weight. Intuitively, if the current channel is unfavorable, i.e., Pa,b(t) is small, the energy required for a successful transmission is remarkably large due to the retransmissions and thus the link should be selected less likely. Therefore, the link weight in the MES algorithm, i.e., Ha,b(t), can be viewed as a balance between the queue difference and the energy consumption under the current channel condition. More specifically, in the original MaxWeight algorithm, if a link has a larger queue difference, the link is more likely to be selected for transmissions. However, in the MES algorithm, both the queue difference and the energy expenditure are taken into consideration. In addition, as indicated by (37), by replacing J ((t + 3a,b with other metrics, our MES algorithm can incorporate other forms of energy consumption models straightforwardly. Observe that if J = 0, the MES algorithm reduces to the original MaxWeight algorithm, i.e., the energy consumption during retransmissions are omitted. Therefore, the MaxWeight algorithm is a special case of our MES algorithm. On the other hand, if we let J > oc, the performance of the MES algorithm can be pushed arbitrarily close to the global minimum solution, as will be shown analytically in Section 3.3.3. However, as illustrated in (37) and (38), when J increases, the network becomes more reluctant to transmissions, for the sake of energy conservation, unless the accumulated queue backlogs are significantly large. Intuitively, a larger average queue size induces a longer experienced delay for transmissions. Therefore, the system parameter J is essentially a control knob which provides a tradeoff between the energyoptimality and the experienced delay in the network. Note that similar to the traditional MaxWeight algorithm, the calculation in (39) is centralized. However, following [12], much progress has been made in easing the computational complexity and deriving decentralized solutions for the centralized MaxWeight algorithm, e.g., [3538, 40, 4249]. It is worth noting that solving (39) is equivalent to finding the maximum weight independent set in the conflict graph, which is combinatorial in nature and thus is intrinsically difficult to solve. A natural heuristic, denoted by GreedyMax, is described as follows. First, the link with the highest weight in the network is selected. Next, the link with the second highest weight which does not involve conflicts with any previously selected links, is selected and the iteration continues until no link can be added. Based on GreedyMax, a novel prepartition based approach is introduced in [35]. The authors prove that if the topology of the network satisfies certain conditions, a.k.a., local pooling factor conditions, GreedyMax can achieve the same throughput as MaxWeight. Furthermore, tree based topology are shown to satisfy the local pooling conditions. In light of this, [35] utilizes graph algorithms to partition the whole network into trees where each tree is allocated an orthogonal channel. As a result, the whole network can operate with simple GreedyMax algorithm and achieve the same throughput as MaxWeight. This line of research is further simplified by [36] where the author proves that a local greedy maximal algorithm can obtain the same performance as the global GreedyMax algorithm. Specifically, in each time slot, a link only needs to compare its weight with local neighboring links to decide a feasible transmission schedule. Therefore, in tandem with the tree prepartition method in [35], the complex scheduling algorithm in (39) can be implemented in a fully distributed fashion. Another feasible direction is to utilize the distributed random access approximation schemes, e.g., [37, 38, 42]. For example, in [42], each node in the network utilizes an IEEE 802.11 MAC protocol where the channel access probability is dynamically adjusted in accordance to the link weight, i.e., Hb(t) in our scenario. The effectiveness of such random access based distributed approximations is studied extensively in [43] and [42] We note that, although distributed implementation is not the focus of this work, our proposed MES algorithm can be approximated well by the solutions suggested in the above papers. If an admission control mechanism is implemented in the network to regulate the overwhelming arrivals, in order to achieve an average minimum rate provision for each flow, we can utilize the concept of virtual queues introduced in [17, 34], where for each flow f, we define a virtual queue yf(t) which is initially empty, i.e., Y(0) = 0, Vf (310) and the virtual queue updating dynamic is defined as y,(t+ 1) = [y(t) R(t)]++ 6 Vf (311) where Rf(t) is the admitted traffic for flow f at time slot t and 6f is a feasible minimum rate requirement of flow f. Intuitively, if each of the virtual queues in the system is stable, the average arrival rate should be less than the average departure rate of the virtual queue and hence we have T1 lim Rf(t) > 6f, Vf (312) Too T Z t t=0 which is exactly the desired minimum rate requirement for flow f. Moreover, the virtual queues are easy to implement. For example, the source node of flow f can maintain a software based counter to measure the backlog updates of virtual queue Yf(t). Therefore, the minimum rate requirements can be incorporated into the proposed scheme straightforwardly. 3.3.2 ThroughputOptimality We first show that the proposed MES algorithm is throughput optimal, as given in the following theorem. Theorem 3.1. The proposed MES algorithm is throughput optimal, i.e., for an arbitrary network admission rate vector A which is inside of the network capacity region A, MES stabilizes the network under A. Proof. We first provide the queue updating equation of Qf. Note that a packet is removed from the transmitter's queue if and only if it is received by the receiver successfully. Therefore, the queue updating dynamic is given by Q(t + 1) < [Q( ut(t) + u ,(t) + An,(t) (313) where uu(t) and uf(t) are the allocated data rate on the outgoing link and the incoming link of node n, with respect to flow f. Note that uu(t) = uLt(t)p(tt) and pout(t) is the current successful transmission probability on this link. Note that Unt(t) = 0, Vt if node n is the destination node of flow f and un,(t) = 0, Vt if node n is the source node of flow f. Furthermore, An,f(t) = Af(t) if n is the source node of flow f and An,f(t) = 0 otherwise. From (313), we have (Qf(t + 1))2 (Qnf(t))2 < ((umax)2 + (Umax max)2) 2Q(t) (t(t) (t) A(t)) (314) Next, we sum (314) over the whole network on all data queues and obtain (Qfn(t + 1))2 (Qn(t))2 n,f n,f < B 2 Q(t) ( (t) u(t) An(t)) (315) n,f where B = FITI ((umax)2 + (max + Amax)2) (316) is a constant. Denote Q(t) = {Qf(t), Vf, n} as the instantaneous queue backlogs in the network. We take the conditional expectation with respect to Q(t) on (315) and have E (Qf(t+ 1))2Q(t) E Y(Qf(t)) Q(t) n,f / \n,f < B 2 2 Q/(t)E (ut(t) u,(t) ,n(t) (t)) . n,f Define t u' out GmES(t) = E h ot (t) n,f in ) in G MES {+ \ out{+\ P\nf outn,f t / n, Q\i\ Q = hnE (, otn + h"n.(t)Pn. Q n~f v ^n~f) / / where ae (I f,) denotes the energy needed for a packet transmission (reception) on the outgoing (incoming) link of node n, with respect to flow f. In addition, we define out GMES(t) = E out (t) nt+\ nf n, in ,inf n,f Ot) + ht n t) nn n,f Pn.f t) as the expected networkwide energy consumption during time slot t, by following the proposed MES algorithm. Apparently, we have E (GES(t)) = GMES(t). Next, we add both sides by JGES(t) where J is a positive constant, and have E 2(Q Q(t 1)) Q(t) E n(Q,'(t)) Q(t) n,f /n,f +JGMESt) < B 2 Q (t)E (uft(t) i(t) Af(t)Q(t)) n,f < B 2 (t)E (ut(t) u ,(t) An,, f(t)Q(t)) n, f out +JE un, ft( + un,(t) IQ(t) (317) n,f Pn, f ) + ) Denote the R.H.S. of (317) as 0. It is of great importance to observe that, from the algorithm description of the MES algorithm above, at every time slot t, the MES algorithm essentially minimizes the R.H.S. of (317) over all possible scheduling algorithms. Since A lies in the interior of the network capacity region A, it immediately follows that there exists a small positive constant c > 0 such that A + = {(A1,1 +c), , (An,f + ), } E A. (318) By invoking Corollary 3.9 in [17], we claim that there exists a randomized scheduling policy, denoted by RA, which stabilizes the network while providing a data rate of E uyf(t) U (t)Q(t) =Anf+C, Vt (319) where uO,(t), uin(t) are the link data rates induced by the randomized policy RA. Therefore, we have ERA = B 2c f Qn5(t) JE (E n (uot) + U ,(t),) IQ(t) (320) Note that the last term in (320) is the actual energy consumption by RA algorithm during time slot t. Following (34), we have 0RA < B 2c Qf(t) JGmax. (321) n,f In light of (317), we have E (E,,(Q(t + 1))21Q(t)) E (Cn, (Qf(t)) 2o (t)) JGMES(t) < MES < RA < B 2e n, Q (t) + JGmax. (322) Next, we take the expectation with respect to Q(t) on (322) and have E (,(Q (t 1))2) E (: ,(Q (t))2) +JGMES(t) < B 2eE (En,r Q(t)) + JGmax. (323) Note that the above inequality holds for any time slot t. Hence, we sum over from time slot 0 to T 1 and obtain E (E,r(Q (T))2) E (E,.r(Qo(0))2) +J GMES(t) < TB 2c Eo1 E E (Q(t)) + TJGmax. (324) Next, we rearrange terms and divide (324) by T and have T1 2c4 E (Q(t)) t=0 n,f max E (Ynr(Q((O))2) < B JGmx T 1 T E (,(Qf(T))2 (325) JT T GMES(t) T (325) t=0 Note that the last two terms of (325) are both nonpositive. By taking lim sup,,, on both sides of (325), we attain 1 1 B + JGmax lim sup E ((t)) < 2 < t=0 n,f Therefore, for every individual queue Q,, we have S 1 B + JGmax lim sup E (Q((t)) < < (327) to Finally, by invoking Markov inequality, we have 1 B +1 JGmax lim sup T Pr (Q(t) > M) < 2M (328) T+oo T n 2zM t0 Therefore, by taking limMv,, we obtain the stability result of the MES algorithm and thus completes the proof. 3.3.3 Asymptotic EnergyOptimality In this section, we show that the MES algorithm yields an asymptotic optimal solution to (35), i.e., the average energy consumption induced by the MES algorithm can be arbitrarily close to the global minimum solution of (35), by selecting a sufficiently large value of J. Theorem 3.2. Define CMES and C* as the average energy consumption induced by the MES algorithm and the optimal (minimum) solution of (35), respectively. The performance of the MES algorithm is given by B CMES < C* + B (329) where B is defined in (316). Therefore, by choosing J oo, the performance of the MES algorithm can be pushed arbitrarily close to the optimum solution C*. Proof. First, denote the optimal sequence of link rates, which generates the optimum solution C*, as u*(O), u*(1), . u*(t), ... Next, let us consider a deterministic policy, denoted by DE, which allocates exactly the optimum link data rates on every time slot t. Similar to (322), we have E (,,(Qf(t + 1))2 Q(t)) E (Yn,(Qf(t)) 2IQ(t)) +JGoES(t) < EMES < DE (330) where SDE SB 2 Qt)E (ot(t) u, un(t) An, f(t)Q(t)) n,f out +JE u o p) u + t(t) Q(t) (331) n, f Pn, f It is worth noting in (331), only the data rates are replaced by the ones generated by the DE algorithm whereas all other values remain the same. Note that in this case, the optimal data rates u*pt(t) and u.in(t) are known as a priori by the DE algorithm and thus are constants with respect to Q (t). We take the expectation on both sides of (330) and have E (Q(t 1+)) ) \ n,f +JGMES(t) < B 2E Qf(t)E (u*"t JE u tn n,f \ n,ff +JE un, f(t) t n, f Pnu` E (Q (t))2 \n,f u(t) U (t) A) f n in We emphasize that, however, in this scenario, at arbitrary time slot t, the value of nf ( uf(t ,f (333) could be either nonpositive or nonnegative. In other words, the relationship of (319) does not hold. To circumvent this, we first sum (332) over time slots t = 0, T T 1 and divide it by T. Thus, we attain 1 T1 J GMES(t) < B t=0 1 0 J E u ,,ft (t) ) T nT t=o ^(EQf())2) gT1 t0 where (334) (332) uin t in Un, f n, ) E(t) = E Qf(t)E (u t(t) (t) An, (t))) . \ nf ,) By taking lim supT ,, we have T1 Jlimsup GES(t)< B JC* T t= Tl 2 im sup(). (335) Too t=0 Note that lim(AB)= lim(A)lim(B) (336) if lim(A) and lim(B) exist and are bounded. Recall that in the previous section, we have shown that T1 iimsup E(Q(t)) (337) T t=0 n,f exists and is finite. Moreover, since u*"t(t) and u. n(t) are the optimum solution to (35), the network stability is achieved under A. Therefore, we have T1 T1 1 1 lim sup ufut(t) > lim sup un '7(t) + An Too Y Too T n t=o0 t=o since otherwise, the network stability cannot be ensured by u*. Finally, we conclude that I T1 lim sup E(t) (338) Too / T t=0 is nonnegative and thus the last term of (335) can be omitted. Consequently, by dividing J on both sides of (335), we have T1 1 B lim sup GMES(t) < C*. (339) Too J J t=0 Therefore, Theorem 3.2 holds. 3.4 Simulations We consider a multihop wireless network illustrated in Figure 31. There are three flows in the network, denoted by Flow 1, 2, 3. The routing paths of flows are specified by R, = {A, B, C, D}, R2 = {F, G, C, D} and R3 = {E, F, G, H}. The exogenous arrival processes are Bernoulli processes with an average rate of 5 packets per slot for all three flows. Without loss of generality, we assume a twohop interference model which represents the general IEEE 802.11 MAC protocols [36, 37]. The nominal link rate of a wireless link is assumed to be 20 packets per slot. For a particular link, there are three equally possible channel states, i.e., Good, Medium, Bad where the corresponding successful transmission probabilities are 0.8, 0.6, 0.3, respectively. ....... c .Flow 1 D (r/" Flo Flow 2 Flow 3 Figure 31. Network topology with interconnected queues For the ease of exposition, we assume that aa,b = a,b = 50J for all links in the network [50]. We investigate the average energy consumption induced by the MaxWeight algorithm and the proposed MES algorithm with different values of J, i.e., 50, 100,200,500,800, 1000, 2000, 5000, 8000, 10000, 12000, 14000, 16000, 18000, 20000, 25000. Every simulation is executed for 10000 time slots. Figure 32 depicts the average energy consumption per time slot for the MaxWeight algorithm and the MES algorithm with different values of J. As illustrated in Figure 32, when J = 0, the MES algorithm reduces to the original MaxWeight algorithm and yields the same amount of energy expenditure. However, as J increases, the energy consumption induced by the MES algorithm decreases remarkably. In addition, as shown analytically in Theorem 3.2, the MES algorithm approaches to the global minimum energy expenditure gradually as J increases. To achieve a better understanding on the impact of larger values of J, we compare the average queue backlogs of all data 260 ******** HEEHEEHEEUE 700000000000000 , Energy Consumption by the MaxWeight Algorithm j 2500 2400 0 2300 02200 Energy Consumption by r_ the MES Algorithm S2100 S2000 1900 1800 0 0.5 1 1.5 2 2.5 Different Values of J x 10o Figure 32. Comparison of the energy consumption of the MaxWeight algorithm and the MES algorithm queues, with J = 50, 150, 350 and 500, in Figure 33 and Figure 34, where the queues in the network are indexed by 1 to 9 in the order of QA, QA, Q~, Q2, Q2, Q2, Qj, Qj and Qj. It is worth noting that, as J grows, the average queue backlogs in the network increases correspondingly. Following Little's Law, larger queue backlogs yield longer network delays. Therefore, while the networkwide energy expenditure is noticeably reduced, a larger value of J yields a longer average delay in the network. As a consequence, a tradeoff between the energyoptimality and the experienced delay can be attained by tuning J properly. So far, throughout this work, we have assumed that the energy in each node is constrained yet sufficiently large. Next, we investigate the performance of the MES algorithm in multihop wireless networks where each node has a limited and finite amount of energy. More specifically, the same network in Figure 31 is considered, however, we assume that each node in the network has a battery with an initial energy of 1 Joule. We compare the performance of the MaxWeight algorithm with the MES algorithm for different values of J, in terms of the network lifetime, which is defined as the time instance when the first node in the network depletes the battery completely. The values of J are 50, 200, 500, 800, 1000, 2000, 5000, 8000, 10000, 15000 and 20000. J=50 120 0) 0 100 U S40 0) 20 1 2 3 4 5 6 7 8 9 Indices of Queues J= 150 120 o 100 U 40 0) 1 2 3 4 5 6 7 8 9 Indices of Queues Figure 33. The average queue backlogs in the network for J 50 and 150 Figure 35 pictorially compares the network lifetime induced by the MaxWeight algorithm and that of the MES algorithm for different values of J. Similar to the previous scenario, when J is small, the MES algorithm yields similar performance, in terms of the network lifetime, compared to the MaxWeight algorithm. However, when J increases, the MES algorithm significantly outperforms the traditional MaxWeight algorithm. It is worth noting that when J = 20000, the MES algorithm prolongs the network lifetime by more than twice as much as that of the MaxWeight algorithm! By the same token, a tradeoff between the network lifetime and the experienced delay of the network can be controlled effectively by tuning the value of J. 3.5 Conclusions In this chapter, we investigate the energy consumption issue of the scheduling algorithms in multihop wireless networks. We show that the traditional MaxWeight algorithm, which is well known to be throughput optimal, is not energy optimal due to J = 350 120 M 80 o 60 1 2 3 4 5 6 7 8 9 Indices of Queues J = 500 120 60 S2000 250 1 0 2 3 4 5 6 08 1 8 Indices of QueuesJ x Figure 3. CompThe average queue backlogs in the network falgor J ithm350 and the MES algorithm the overlook of the energy consumption induced by inevitable packet retransmissions. In light of this, we propose a minimum energy scheduling (MES) algorithm which s3500 < ..... , significantly reduces the energy consumption compared to the original MaxWeight algorithm. In addition, we analytically show that the MES algorithm is essentially energy Diffoptimal in the sense that the average energy expenditure of the MES algorithm can beValues of J Figure 35. Comparison of the lifetime of the MaxWeight algorithm and the MES algorithm the overlook of the energy consumption induced by inevitable packet retransmissions. In light of this, we propose a minimum energy scheduling (MES) algorithm which significantly reduces the energy consumption compared to the original MaxWeight algorithm. In addition, we analytically show that the MES algorithm is essentially energy optimal in the sense that the average energy expenditure of the MES algorithm can be pushed arbitrarily close to the global minimum solution. Moreover, the improvement on the energy efficiency is achieved without losing the throughputoptimality. Therefore, the proposed MES algorithm is of great importance for network protocol designs in energyconstrained multihop wireless networks such as wireless sensor networks. CHAPTER 4 CHANNEL AND POWER ALLOCATION IN WIRELESS MESH NETWORKS 4.1 Introduction Metropolitan wireless mesh networks gain enormous popularity recently [51]. The deployment of wireless mesh networks not only facilitates the data communication by removing cumbrous wires and cables, but also provides a means of Internet access scheme, which is a further step towards the goal of "communicating anywhere anytime". No matter where the location is or the purpose that the wireless mesh network is deployed, the same conceptual layered architecture is utilized. Figure 41 illustrates the hierarchical structure of wireless mesh networks. The peripheral nodes are the access points (AP) which provide wireless access for the end users, or clients. Each AP is associated with a mesh router. They can be manufactured in a single device with two separate functional radios [52] [53], or simply connected with Ethernet cables [54]. The mesh routers are capable of communicating with each other via the wireless backbone. The central node is a gateway mesh router which functions as an information exchange between the wireless mesh access network and other networks such as the Internet. Both the routing algorithmic design and channel assignment for backbone mesh routers are interesting issues and attract tremendous attention [5557]. AP Access Point 0 G Gateway Node D R Wireless Mesh Router Figure 41. Hierarchical structure of wireless mesh access networks In this work, we investigate an important issue which needs to be solved in wireless mesh networks. As in Figure 41, the AP and its associated clients form a regular WLAN cell, which operates with the de facto IEEE 802.11 standards. The throughput of one cell depends on the signaltointerferenceplusnoise ratio (SINR) experienced at the receiver where the interference mainly comes from the other operating cells. For example, if each of the cells operates with IEEE 802.11 b standard, we can utilize a different frequency band such as IEEE 802.11 a or WiMAX [58], for the intercell communication among mesh routers and hence causes no interference to intracell transmissions. However, the cochannel interference from other operating cells is inevitable due to the limitation of available transmission channels, e.g., 3 nonoverlapping channels in our example. Most current offtheshelf APs are capable of adjusting the transmission rate according to the measured channel condition which is indicated by transmission bit error rate (BER). Given a particular modulation scheme, BER is uniquely determined by the SINR experienced by the receiver of the link. Generally speaking, higher SINR value yields lower BER and higher data rate. Therefore, the mutual interference dramatically degrades the transmission rate of each cell and the aggregated throughput of the whole network [59]. Each AP attempts to tune the physical parameters such as operating frequency1 and transmission power in order to maximize the SINR and hence the throughput. In our work, we investigate the issue of maximizing the overall throughput of the network, defined as the summation of throughput of all cells, by finding the optimal frequency and transmission power allocation strategy. Also, due to the concern of scalability and computational complexity, we prefer a decentralized solution to the throughput maximization problem. Unfortunately, the throughput maximization problem is challenging. For example, the frequency and power selected by one AP affects the SINR of other APs, and vice versa. Worse yet, if the APs belong to different regulation entities, the noncooperative APs may only want to maximize their own cell's throughput rather than the overall one. Therefore, the throughput maximization problem is coupled and finding the optimum 1 We will use frequency and channel interchangeably. solution is not straightforward. Moreover, traditional siteplanning methods in cellular networks are not feasible either. For example, the network administrator may want to add more APs when more users are joining the network or disable some APs where the associated users fail to pay the bill. The network topology is not static, although the changes take place slowly. Therefore, the demand for adaptability and light computation burden requires a decentralized solution for the throughput maximization problem. In this work, we analyze the throughput maximization problem for both cooperative and noncooperative scenarios. In the cooperative case, we model the interaction among all APs as an identical interest game and present a decentralized negotiationbased throughput maximizing algorithm for the joint frequency and power assignment. We show that this algorithm converges to the optimal frequency and power assignment strategy, which maximizes the overall throughput of the wireless mesh access network, with arbitrarily high probability. In the cases of noncooperative APs, we prove the existence of Nash equilibria and show that the overall throughput performance is usually inferior to the cooperative cases. To bridge the performance gap, we propose a linear pricing scheme to combat with the selfish behaviors of noncooperative APs. The rest of this chapter is organized as follows. Section 4.2 outlines the system model we considered in the work. The cooperative wireless mesh access networks and the noncooperative counterpart are investigated in Section 4.3 and Section 4.4, respectively. An extension of our model is discussed in Section 4.5 and the performance evaluation is provided in Section 4.6. Finally, Section 4.7 concludes this chapter. 4.2 System Model In this work, we consider a wireless mesh access network illustrated in Figure 41. Each AP and corresponding clients form a cell. Without loss of generality, we assume that all the cells operate with IEEE 802.11b standard and the interference exclusively comes from the cells with same frequency. Furthermore, the distance between cells are sufficiently large in the sense that the accumulated interference experienced at the receiver only affects the SINR value and not block the whole transmission. We assume that the channels are slowvarying additive white Gaussian noise (AWGN) channels. The channel gains of each pair of nodes are assumed to be constant over the time period of interest. As we are interested in the maximum achievable throughput, we consider the worst case where all APs are saturated. In other words, the APs always have packets to transmit and they can communicate with each other via the backbone mesh routers with negligible delay. Also, we assume that the APs are transmitters and clients are receivers due to the dominance of downlink traffic, as assumed2 in [61] [62] and [63]. We only focus on the joint frequency and power allocation where the contention behavior is less relevant and thus omitted. Therefore, we can simplify our model as that all the APs are transmitting data to the associated clients consistently. We assume that each AP is capable of adjusting the operating frequency and power as well as acquiring the SINR values measured at the client by short ACK messages. Let us first consider the simplest case where there is only one cell in the wireless mesh access network, i.e., a single WLAN. In the following two sections, we assume that the APs have predetermined and fixed modulation and coding schemes. In other words, upon receiving the SINR value3 measured by the client, denoted by 7, the AP tunes the physical parameters in order to maximize the throughput, which is defined as R*(7) = maxRi x (1 Pe(y, Ri)) (41) Rf where Ri is the raw data rate specified by the IEEE 802.11 standard and R*, i.e., the throughput of this cell, is a nondecreasing function of received SINR 7. Pe is the error 2 The dominance of the downlink traffic is verified by the experimental measurements in [60] as well. 3 Although there is no interference in this case, we adopt SINR instead of SNR for notation consistency. probability of the transmission channel, which is a function of SINR value providing the modulation and coding scheme [64]. Apparently, if there is only one cell in the mesh access network, the AP will boost the power as much as possible to increase the value of 7 and thus the throughput is maximized. We now consider the cases where N cells coexist in the wireless mesh access network. Let pi and f denote the power and frequency for the ith AP, respectively. We use p = [pi, p2, PN] and f = [fl, f2, fN] to represent the power and frequency assignment vector for all N APs. Therefore, for each cell i, the value of SINR4 i.e., 7i, is a function of (p, f). The throughput of one cell depends not only on the power level and frequency of itself, but also those of other APs in the network. Therefore, the throughput maximization problem is coupled and by no means straightforward. In the following two sections, we will discuss the scenarios where the APs are cooperative and noncooperative, respectively, under the assumption that the modulation and coding schemes of APs are preloaded and fixed. In Section 4.5, we will extend our analysis by considering the scenario where the APs are capable of adaptive coding and modulation. The performance evaluation of all scenarios are provided by simulations in Section 4.6. 4.3 Cooperative Access Networks In this section, we consider the scenarios where all APs in the wireless mesh access network are cooperative. The transmission power of APs are quantized into discrete power levels for simplicity. From the system point of view, we want to find a joint frequency and power level assignment such that the overall throughput in the whole 4 Throughout the chapter, the term SINR of the cell represents the average SINR among all the clients in the cell, which can be obtained by a moving average of the reported SINR value. network is maximized. Our objective function can be written as N N Unetwork(p, f)= R*(7) = R (p, f) (42) i= l i= l where R, is defined in (41). However, finding the optimal frequency and power assignment which maximizes (42) is nontrivial. The interdependency makes the problem coupled and difficult to solve by traditional optimization methods [65]. A combination of (p, f) is named a profile and a naive approach to solve the problem is to investigate all profiles exhaustively. However, this is impossible in practice. For example, in a mediumsize wireless mesh access network with 20 APs where each has 3 frequency channels and 10 power levels, the search space is (3 x 10)20 profiles! Obviously, the centralized algorithms are not favorable in the wireless mesh access network due to the scalability concern. Next, we will introduce a decentralized negotiationbased throughput maximization algorithm, from a gametheoretical perspective. 4.3.1 Cooperative Throughput Maximization Game The APs in the wireless mesh access networks are considered as players, i.e., decision makers of the game. We model the interaction among APs as a Cooperative Throughput Maximization Game (CTMG), where each player has an identical objective function Ui, as N Ui(p, f) Unetwork(P, f) R (p, f) Vi. (43) j= 1 For each player i, all possible frequency and power level pairs form a strategy space ;, which has a size of c x /, where c is the number of frequency channels available and I is the number of feasible power levels. Define Q = 0 x 02 X X VN. (44) Then, the N players autonomously negotiate about the joint frequencypower profile in Q in order to find the optimal profile which maximizes (43). However, due to the interdependency among N players caused by mutual interference, one question of interest is that whether this negotiation will eventually meet an agreement, a.k.a., a Nash equilibrium. The importance of Nash equilibria lies in that a possible steady state of the system is guaranteed. If the game has no Nash equilibrium, the negotiation process never stops and oscillates in an everlasting fashion. In addition, we are concerning about what the performance of the steady states would be, if exist, in terms of overall throughput of the whole network. We provide answers to these questions in the following. Lemma 3. The CTMG is a potential game. A potential game is defined as a game where there exists a potential function P such that P(a', ai) P(a", ai) = U,(a', ai) Ui(a", ai) Vi, a', a" (45) where U, is the utility function for player i and a', a" are two arbitrary strategies in 0i. More specifically, we have a' = [pl, f,'] and a" = [p", f,"]. The notation of a_i denotes the vector of choices made by all players other than i. Potential games have been broadly applied in modeling the interactions in communication networks [66]. The popularity is on account of the nice properties of potential games, such as  Potential games have at least one Nash equilibrium.  All Nash equilibria are the maximizers of the potential function, either locally or globally. There are several learning schemes available which are guaranteed to converge to a Nash equilibrium, such as better response and best response [67] [68]. For detailed description about potential games, readers are referred to [67] and [69], which investigate the potential game theory in engineering context. We observe that in the cooperative case, each player has the same utility function as in (43), which is the overall throughput of the network. Apparently, one potential function of the game is the common utility function itself, i.e., P = U = U2 =... = UN. (46) In fact, the games where all players share the same utility function are called identical interest games [70], which is a special case of potential games and hence all the properties of potential games can be applied directly. In the literature, both best response and better response are popular learning mechanisms that have been utilized in potential games [7173]. At each step of the best response approach, one of the players investigates its strategy space and chooses the one with maximum utility value. This updating procedure is carried out sequentially. The primary drawback of the best response is the computational complexity, which grows linearly with the cardinality of the strategy space. An improvement of the best response is the socalled better response, where at each step, the player updates as long as the randomly selected strategy yields a better performance. The dramatically reduced computation is the tradeoff with the convergence speed. Both the best response and the better response dynamics are guaranteed to converge to a Nash equilibrium in potential games [66]. However, there may be multiple Nash equilibria in a potential game and the performance of different equilibria may vary dramatically. Therefore, although the best response and the better response could guarantee the convergence, they may reach an undesirable Nash equilibrium with inferior performance. A B D C 1 2 2 1 1 2 1 2 Figure 42. An illustrative example of multiple Nash equilibria Let us consider an illustrative example in Figure 42. There are four labeled APs in the network. A and B are close to each other, and so are C and D. Without loss of generality, we assume that the APs have the same power and only adjust the operating frequencies in an order of A > B > C > D to avoid the interference. The adaptation continues with the best response mechanism until a Nash equilibrium is reached. Suppose that there are two frequency channels available, say 1 and 2. First, A randomly selects one channel, say 1. B will pick 2. Next, C has the chance to update. Since C is closer to B than A, channel 1 will be selected. Finally, D will choose channel 2. By inspection, we claim that profile 1 2 2 1 is a Nash equilibrium since no player is willing to update its strategy unilaterally. Meanwhile, we observe that another profile 1 2 1 2 is also a Nash equilibrium. Obviously, the second Nash equilibrium generates much less interference than the first Nash equilibrium and hence yields superior performance in terms of overall throughput. However, the best response only leads to the less desirable Nash equilibrium. In fact, the existence of multiple Nash equilibria is observed in [71] by simulations. However, the authors fail to specify which one would be the steady state of their game due to the limitation of the best response, even in a statistical fashion. Recall that the Nash equilibria are the maximizers of the potential function in potential games, converging to an inferior Nash equilibrium analogously indicates being trapped at a local optimum of the potential function. However, it is the global optimum, i.e., the optimal Nash equilibrium, that is the desirable steady state which we are yearning for. Next, we introduce a negotiationbased throughput maximization algorithm (NETMA) which can converge to the optimal Nash equilibrium with arbitrarily high probability. 4.3.2 NETMA NegotiationBased Throughput Maximization Algorithm We assume that the APs are homogeneous and each has a unique ID for routing purpose. Each AP maintains two variables Dpre and Dcr. The AP has the knowledge of its current throughput and records it in Dcu,. Whenever there is a change of throughput caused by exterior interference5 the AP sets Dpre = Dcur and resets Dcur with the newly measured throughput. When the wireless mesh access network enters the negotiation phase6 NETMA is executed. The detailed procedure of NETMA is provided as follows. NETMA:  Initialization: For each AP, a pair of frequency and power level is randomly selected. Set Dpre = D,,r, the current throughput. Repeat: 1. Randomly choose one of the AP, say k, as the updating one, i.e., each AP updates with probability 1/N. 2. For the updating AP k: (a) Randomly chooses a pair of frequency and power level, say f' and p', from the strategy space Ok. Then the AP computes the current throughput with f' and p' and records it into D,,r. (b) Broadcasts a short notifying message which contains its unique IDk to all the other APs in the mesh access network. 3. For each AP other than k, say j: (a) If the 7j value changes, records the previous throughput into Dpre and the current throughput into D,,r. Remains unchanged otherwise. (b) Upon receiving the notifying message, a threevalue vector of [Dpre, D,,r, IDj] is sent back to the kth AP. 4. After receiving all the threevalue vectors by counting the identifiers IDj, the kth AP computes the sum throughput before and after f' and p' are selected, which are denoted by Ppre and Pcur. 5 We assume the channel is slowvarying and the change of throughput for a single cell is due to the mutual interference only. 6 To reduce the negotiation overhead, a negotiation phase can be initiated by the network administrator after a new contracted user joins or a current user terminates the service, or on a daily basis. 5. For a smoothing factor 7 > 0, the kth AP keeps f' and p' with probability ePa/T 1 1= e 1(47) ePcur/T + ePpre/7 1 + e(PprePcu)/17 6. The kth AP broadcasts another short notifying message, which indicates the end of updating process and a specific number 6, to all the other APs. Until: The stopping criteria F is met. Note that in step 6, the specific format of 6 depends on the predefined stopping criterion F. For example, * If the stopping criterion is the maximum number of negotiation steps, 6 is a counter which adds one after each updating process. If the stopping criterion is that no AP has updated for a certain number of steps, 6 is a binary number where 1 means updating. If the stopping criterion is that the difference between sum throughput obtained in consecutive steps are less than a predefined threshold c, 6 is the calculated sum throughput after each updating process. We can have other stopping criteria Fs and corresponding formats of 6 as well. The NETMA algorithm is inspired by the work in [74], where a similar algorithm was first introduced in the context of stream control in MIMO interference networks. The distinguishing feature of this type of negotiation algorithms, from the better response and the best response, is the randomness deliberately introduced on the decision making in step 5. The rationale can be illustrated in Figure 42 intuitively. If there is no randomness in decision making i.e., 7 = 0, the four APs may get trapped at a low efficiency Nash equilibrium 1 2 2 1. However, with the randomness caused by nonzero 7, they may reach an intermediate state 1 2 2 2 and arrive at the optimum Nash equilibrium 1 2 1 2 eventually. Moreover, the updating rule in step 5 also implies that if f' and p' yield a better performance, i.e., Ppre Pcur < 0, the kth AP will keep them with high probability. Otherwise, it will change with high probability. The steady state behavior of NETMA is characterized in the following theorem. Theorem 4.1. NETMA converges to the optimal Nash equilibrium in CTMG with arbitrarily high probability. Proof. The proof of Theorem 4.1 follows similar lines of the proof in [74] and [75]. Figure 43. Markovian chain of NETMA with two players First, we observe that the joint frequencypower negotiation generates an Ndimensional Markovian chain. Figure 43 illustrates the Markovian chain introduced by NETMA with two players, say A and B. Let x and y be the choices for each player, where x e CA and y e OB. In other words, player A can choose a frequencypower pair from [xI, xcx] and player B can choose from [yi, ycx/]. Note that at an arbitrary time instant, only one of the players can update. In Figure 43, for example, state (xi, yi) can only transit to a state either in the same row or the same column, not anywhere else. This is true for every state in the Markovian chain. Let Si, denote the state of (xi, yi). We have Pr(Sm,,Si,) = eP(Sm.n)/7 2xcx x(eP(sm.n)l/+eP(Sij) ' if m = i or n = j 0, otherwise. (48) where 7 is the smoothing factor in step 5 of NETMA and P(Si,) is the value of the potential function, i.e., (421), at the state of Sj. Let us derive the stationary distribution Pr* for each state. We examine the balanced equations. Writing the balance equations [76] at the dashed line, we obtain cxl cxl Pr*(Si,l) x Pr(Sl,kS,) = Pr*(Sl,k) x Pr(S1i,S1,k). (49) k=2 k=2 By substituting (49) with (48), we have cx/ Pr*(Si,) x e P(Sl,k)/ Pr*(eSl.) x ePsl)l /_+ep~sl,1/  k 2k=2 1" l,k eP(Sx ,)/, eP(5 ,k)/ Observing the symmetry of equation (410) as well as the Markovian chain, we note that the set of equations in (410) are all balanced if for arbitrary state S in the strategy space Q, the stationary distribution is Pr*(S) = Cep()/' (411) where /K is a constant. By applying the probability conservation law [77] [76], we obtain the stationary distribution for the Markovian chain as eP(3) / Pr*(S) =e (412) Cs5so eP(S;)/ for arbitrary state S e 2. In addition, we observe that the Markovian chain is irreducible and periodic. Therefore, the stationary distribution given in (412) is valid and unique. Let S* be the optimal state which yields the maximum value of potential function P, i.e., S* = argmaxs,,P(S,). (413) From (412), we have lim Pr*(S*)=1 (414) T0 which substantiates that NETMA converges to the optimal state in probability. Finally, the analogous analysis can be straightforwardly extended to an Ndimensional Markovian chain and thus completes the proof. In NETMA, there is no central computational unit required. The joint frequencypower assignment is achieved by negotiations among cooperative APs and the maximum overall throughput is achieved with arbitrarily high probability. The autonomous behavior and decentralized implementation make NETMA suitable for large scale wireless mesh access networks. Moreover, NETMA has fast adaptability for the topology change of the wireless mesh access networks. NETMA does not depend on any rate adaption algorithms, nor on any underlying MAC protocols. In our simulation in Section 4.6, we use IEEE 802.11 b as the MAC layer protocol. However, it can be easily extended to arbitrary MAC protocol with multirate multichannel capability, such as IEEE 802.11 a. In addition, even with the existence of exterior interference source, such as coexisting WLANs, NETMA works properly as well since the objective of NETMA is to maximize the overall throughput of the network in the current wireless environments. The tradeoff between algorithmic performance and convergence speed is controlled by parameter 7 in step 5, where large 7 represents extensive space search with slow convergence. On the contrary, small represents limited space search with fast convergence. Note that the smoothing factor 7 here is analogous to the concept of temperature in simulated annealing [78]. Therefore, it is advisable that at the beginning period of the negotiation, the value of 7 is set with a large number and keeps deceasing as the negotiation iterates. We choose r = 10/k2 in our simulations, where k denotes the negotiation step. In step 1, we require that each AP updates with a probability of 1/N. For example, we can utilize a random token mechanism where each updating AP randomly selects an AP as the next updating AP, i.e., passing the token. Note that in NETMA, even an erroneous operation happens, for example, two APs update at the same time in our case, it only prolongs the convergence time for NETMA yet does not affect the final output of NETMA. This is because that such an error, as verified in [72] via extensive simulations, has no influence on the statistically monotonicincreasing tendency of the potential function. 4.4 NonCooperative Access Networks In the previous section, we discuss the scenarios where all APs in the wireless mesh access network are cooperative, and the overall throughput is maximized by negotiations among autonomous APs using the NETMA mechanism. However, cooperation is not always attainable. Although the functionality of relaying packets for each other can be achieved by incentive mechanisms such as [79], the adjustable parameters inside each cell cannot be enforced and effectively controlled. The N APs may belong to distinct selfinterested users and they care about exclusively their own throughput rather than the overall aggregated throughput. In other words, the utility function of each selfish user is Ui = R*(i) (415) where Rf is the throughput of the ith cell, defined in (41). Analogous to CTMG, we can formulate the interaction among N selfish APs as a Noncooperative Throughput Maximizing Game (NTMG) where each AP is attempting to find the frequencypower pair which maximizes its own SINR value as well as the corresponding throughput. As in the cooperative case, each player's utility function depends on the frequency and power of itself as well as those of others. However, NTMG is no longer an identical interest game. Lemma 4. In NTMG, all the APs will transmit with the maximal power at the Nash equilibrium, if exists. Proof. The proof of Lemma 4 is straightforward. For a single player, we have 'i =pigi (416) YkET,(f) Pkgki + NiV, where gd is the channel gain from cell i's transmitter to j's receiver and N, is the Gaussian noise at the i's receiver. T,(f,) denotes the set of cells which operate at the same frequency f, other than cell i. Note that given other players' strategies, yi is a monotonic increasing function of p, and so is U,. Assume at a Nash equilibrium of NTMG, the kth AP has a power level of Pk satisfying 0 < pk < Pmax, where pmax denotes the maximum power defined by MAC layer. The kth AP is inclined to increase its power Pk in order to yield a higher value of U,, which contradicts the definition of Nash equilibrium. Thus, at the Nash equilibrium of NTMG, if exists, all the APs will operate at the same power level, i.e., pmax. Based on Lemma 4, the NTMG can be viewed as a simplified game where each player has the same power and only adjusts the frequency to minimize the interference. Moreover, according to (415) (416) and the assumption of uniform environment, the NTMG is equivalent to the following simplified game where each player has the utility function7 as U, = ( Pmax9ki + N) (417) kEG,(f,) and U, is a function of frequency assignment vector f exclusively. As in the cooperative case, the frequency selection among N players is mutually dependent. The question arises that whether this frequency adjusting dynamic converges, or equivalently, whether NTMG has a Nash equilibrium. We provide the answer of this question in the following theorem. 7 The negative sign comes from the convention that utility functions are the ones to be maximized. Theorem 4.2. There exists at least one Nash equilibrium in NTMG. Proof. Let us first consider the simplified game. For each player, the utility function is given as U, = ( pgki+ N,) (418) k.F, (f,) = ( Pk9gk,i x6( fk) Ni) (419) k#i,kcN where J 1, ifk =0 6(k) 1, if k (420) 0, otherwise. We conjecture that one of the feasible potential functions is P= xE pkgki. (421) iEN kGE (f) The verification is as follows. 2P = Pkk, ieN kc(F(fi) kGT,( f) j i jEN kG j( J) + Pk9kj} kGe(f),k#i = Y.Pkgki6(fk fi) {k#ikGN + pig,(fi ) + E Pk9kji (422) j ijCN kcG (fj),k i Note that Pkgi(fk fi) = pgik6(f fk) (423) for any pair of i, k. We have P= Pkgk,6(fk f) + XQ(i) (424) k i,kGN ) where Q(i)= Pkj (425) j ijeCN kc.Fe (),k/ i and Q(i) is independent of f;. Therefore, for arbitrary two frequencies a' and a" of player i, we have QOa(i) = a,,(i) (426) and P(a', a_i) P(a", a_i) {kCE (al) Pkgki x QX ( i)} { ZEC (al') pkgk, x Qa X i()} U,(a', a_,) U,(a",a_i). (427) Therefore, according to the definition in (45), the simplified game is a potential game and has at least one Nash equilibrium. Thus, the existence of Nash equilibrium in NTMG is obtained from the equivalence derived from Lemma 4. As shown in Lemma 4, at the equilibrium, the noncooperative APs will always transmit at the maximum power level. This seem to be the best choice for each one of the APs. However, it is usually not a favorable strategy from a socialwelfare point of view. To bridge the performance gap, we propose a linear pricing scheme to combat with the selfish behaviors, i.e., the players are forced to pay a tax proportional to the utilized resources. For example, we could impose a price to all selfish APs for the power they utilize. Hence for each AP, the utility function becomes U, = R (i) Appi (428) where A' represents the power utilization price specific for the ith AP and pi is its transmission power. Therefore, the more power AP uses, the more tax it has to pay. By imposing power prices properly, a more desirable equilibrium may be induced, from a socialwelfare point of view. We define the corresponding game as a Noncooperative Throughput Maximization Game with Pricing (NTMGP). Let us first investigate the impact of prices on the behaviors of players. If Ai = 0, where no price is imposed, the ith AP will transmit at the maximum power and causes extra interference to other APs. However, if we impose an unbearably high price, say AP = oo, the AP would rather not to transmit at all. Based on these observations, we propose a heuristic linear pricing scheme to improve the overall throughput in noncooperative wireless mesh access networks. To enforce the scheme, we introduce a pricing dictator unit (PDU) into the network which determines the prices for all APs and informs them timely. In addition, we assume that the PDU has the monitoring capability and is aware of the operating frequencies of each cell. There are two prices charged by the PDU for each noncooperative AP Besides the power utilizing price Ai, a frequency switching price A' is imposed on the ith AP whenever it changes the operating frequency. The price setting process is described as follows. Price setting process: Phase I: The PDU sets A' = .. AN = 0 and A = .. AN' = 0 and all APs play NTMG until converges, i.e., a Nash equilibrium is reached. The PDU collects the current throughput information from each cell, denoted by Mi, where / is the index of the cell. Phase II: The PDU sets = A" = oo. For each AP indexed by i = 1, .. N : 1. The PDU sets Ai = oo for the ith AP and let the APs play the NTMGP. Upon convergence, the PDU collects the overall throughput, say V,, in the current price setting. 2. Calculate the power utilizing price for the ith AP as Vi YE l"i Mi A = P1 M (429) Pmax 3. Reset A, = 0. Output: Power utilizing price vector A~ = [', .. AN] Frequency switching price vector Af = [oo, o] In the price setting process above, the PDU imposes zero prices for all APs initially. As a consequence, all APs will transmit with pmax at the equilibrium, as shown in Lemma 4. Upon convergence, the PDU fixes the frequency switching price to infinity which discourages the noncooperative APs from switching channels thereafter. In (429), =1,ji Mj is the sum throughput for all cells other than i, when the ith AP transmits with the maximal power due to the zero power price. Similarly, V, is the sum throughput of other cells when the ith AP is silent due to the unaffordable power price. Therefore, in (429), the power utilization price charged for the ith AP, a.k.a., A', can be viewed as a compensation to the impact it causes on the overall throughput of other cells. The more power it utilizes, the more severe it affects the other players and thus the more it pays, as illustrated in (428). Hence, by imposing taxes deliberately, the selfish behaviors of noncooperative APs are effectively discouraged and a more desirable equilibrium can be induced, in term of overall throughput of the whole network. The proof of existence of Nash equilibrium in NTMGP is straightforward. Note that the utility function of (428) is quasiconcave with respect to power. The existence of pure strategy Nash equilibrium follows directly from the results of [80]. We will present the detailed performance evaluation of CTMG, NTMG and NTMGP in Section 4.6. 4.5 An Extension to Adaptive Coding and Modulation Capable Devices So far, we have assumed that the throughput of a cell is given in the form of (41), which is not a continuous function. To be precise, in both Section 4.3 and Section 4.4, we are confined to the traditional IEEE 802.11 family devices where the modulation and coding schemes are predetermined and fixed. For example, Table 41 provides a mapping between SINR values and corresponding rates [81], the feasible data transmission rate is a discrete set and is usually much less than the theoretical channel capacity. We name such devices as legacy IEEE 802.11 devices. Table 41. Data rates v.s. SINR thresholds with maximum BER = 105 Rate(Mbps) Minimum SINR (dB) 1 2.92 2 1.59 5.5 5.98 11 6.99 However, thanks to the advance of coding techniques, the maximum data transmission rate can be largely closed to the theoretical Shannon capacity in AWGN channels [82]. Note that achieving this requires variablerate transmissions by matching to the instantaneous SINR, which can be implemented in practice through adaptive coding and modulation (ACM) techniques [83]. This motivates us to extend our results to ACMcapable devices. More specifically, we are considering more advanced and powerful APs which attempt to tune the frequency and power in order to maximize C,(7i) = W, log,2(1 +i) (430) where C, is the Shannon capacity of the ith cell, and W, is the bandwidth. Note that in this scenario, the maximum achievable transmission rate, as denoted by the Shannon capacity, is a continuous variable with respect to y,, rather than discrete, as exemplified in Table 41. It is worth noting that the only difference in this scenario is the alternative objective function of each AP. Therefore, all the results we have obtained so far are extendable8 to this special scenario with merely a change of the objective function. Furthermore, due to the continuity of the objective function in (430), the aforementioned heuristic pricing scheme in Section 4.4 can be improved. Without loss of generality, we assume that all the cells have unity bandwidth. Restated, we assume that the PDU is a centralized device which knows the channel environment sufficiently by greedy acquiring. In addition, the PDU is assumed to have monitoring capability and is aware of the operating frequency of each cell. The tailored pricing scheme for this ACMcapable scenario is described as follows. * First, the PDU sets A = .. A = 0 and A' = A^ = 0 and all APs play selfishly, until a Nash equilibrium is achieved. The PDU sets \} = .. A = oo. For each AP indexed by i = 1, N, the PDU sets 8 More specifically, in the cooperative case, we replace (42) with Unetwork(P, f) Ni_1 C,('yi) whereas in the noncooperative case, (415) and (428) are replaced by U, = C,(i) and U, C,(Qy) Appi, respectively. kac Pk X gkk X gik S, In 2 x (1 +Yk) x (YE, P, X 9gk + Nk)2 kET, 5 Pk X gkk X gik 2 In2 x (1 + ,) x (P kk)2 kEFT, 7k 5ik X 72 =gik k (431) In 2 x (1 + k) Pk X gkk' Then the PDU informs each AP the corresponding prices, i.e., A = oo and A, calculated in (431). Note that the major difference in this pricing scheme lies in (431), where the A, charged for the ith AP is the Pigouvian price levied to discourage its reckless power increase. After obtaining the prices imposed by the PDU, each AP fixes the operating frequency and calculates the optimum power p,, which maximizes U, = C, pi,A according to the following steps. SU X (432) 9pi pi p where ac, gii .(433) 8pi In 2 x (Ek e, Pk X gki Ni) X (1 yi) By setting (432) equal to zero, we have 1 k= PkE X gki + N, Pmax34) i In 2 x A, gIi pmin where [x]b denotes max{min{b, x}, a} and pmx, Pmin represent the maximum and minimum power of the device, respectively. Note that Ck ,, Pk x gki + Ni can be easily measured when the ith AP sets its transmission power to zero. Therefore, finding the optimum power of each AP requires only local information and can be implemented in a distributed fashion. In (431), we observe that the prices imposed on APs depend on the power vector p, and vice versa. Therefore, after each AP optimizes its power according to (434), the PDU measures the new value of power, adjusts the price following (431), and then announces the new price again. The pricing setting process in (431) and the utility maximization process in (434) are executed in an iterative manner, until convergence. The whole pricing scheme is summarized as follows. Pricing Scheme: 1. The PDU initially sets zero prices. As a consequence, all APs will converge at a Nash equilibrium with maximum power. 2. The PDU sets infinity for the frequency switching price which prevents the whole network from unstable frequency oscillations. 3. The PDU sets and announces the power prices for each AP according to (431). 4. Informed by the PDU, each AP optimizes its transmission power following (434). 5. Go back to step 3 until the iteration converges. The performance evaluation of this tailored pricing scheme is illustrated in the next section. 4.6 Performance Evaluation 4.6.1 Legacy IEEE 802.11 Devices In this subsection, we first investigate the performance of NETMA, NTMG and NTMGP with legacy devices, i.e., the scenarios we considered in Section 4.3 and Section 4.4. We assume the wireless mesh access network has N homogeneous APs. The other simulation parameters are summarized as follows. * Each AP has a maximum power pmax = 100mW and a minimum power pmi = 10mW and 10 different power levels as [10mW, 20mW, 100mW]. The noise experienced at each receiver is assumed identical and has a power of 2mW. * All APs use IEEE 802.11 b standard as the MAC protocol. In other words, each AP has four feasible data rate, 1, 2, 5.5, 11 Mbps and 3 nonoverlapping channels, i.e., c = 3. Without loss of generality, we assume that the received power is inversely proportional to the square of the Euclidian distance. The smoothing factor 7 decreases as r = 10/k2, where k is the negotiation step. The stopping criteria for NETMA and NTMG are the maximum number of iterations, denoted by w. For the sake of simplicity, we utilize a tabledriven rate adaption algorithm provided in Table 41. Note that our results can also be applied to arbitrary propagation models, rate adaptation algorithms and underlying multichannel multirate MAC protocols. 4.6.1.1 Example of small networks We first consider a small wireless mesh access network with 5 APs, i.e., N = 5. All APs are randomly located in a square of 10by10 area. The global optimum solution is obtained by enumerating all feasible strategies, i.e., (3 x 10)5 profiles, as the performance benchmark. We first investigate the cooperative scenario where NETMA mechanism is applied. Next, the noncooperative scenario is considered and each AP operates at the maximum power and adjusts the frequency only. The stopping criteria for both NETMA and NTMG are the maximum number of iterations where W = 200. The performance comparison is shown in Figure 44. N=5, c=3 34 S32 30 S28 NETMA S...NTMG 26 H S24 22 20 1 0 20 40 60 80 100 120 140 160 180 200 Iteration Figure 44. Performance evaluation of the wireless mesh access network with N = 5 and c=3 N=5, c 3 o AP 1 28 AP2 I AP4 S26 I AP5 N 24 I 22 I S16  o i I I 14 12 1 0 20 40 60 80 100 120 140 160 180 200 Iteration Figure 45. The trajectory of frequency negotiations in NETMA when N = 5 and c = 3 N=5, c 3 *AP1 90 1 AP2 H si *.,AP4 80 I *AP5 50* I I I S 0 20 40 60 80 100 120 140 160 180 200 Iteration Figure 46. The trajectory of power negotiations in NETMA when N = 5 and c = 3 As indicated by the OP curve, the global optimum obtained by enumeration approach functions as the upper bound of the overall throughput. In Figure 44, we observe that NETMA gradually catches up with the global optimum as negotiations go. As expected, the noncooperative APs yield remarkably inferior performance in terms of overall throughput, depicted by the NTMG curve. The inefficiency is due to the selfish behavior that APs transmit at the maximum power and are regardless of the interference. The existence of Nash equilibrium in both CTMG and NTMG are substantiated by the convergence of curves in Figure 44. Figure 45 and Figure 46 depict the trajectories of frequency negotiations and power level negotiations in NETMA, respectively. At the initialization, each AP randomly picks a frequency and a power level and negotiates with each other following NETMA mechanism, until the optimum Nash equilibrium is achieved. Note that when the frequency vector and power vector converge in Figure 45 and Figure 46, the corresponding overall throughput obtained by NETMA catches the global optimum in Figure 44 simultaneously. 4.6.1.2 Example of large networks We now consider a large wireless mesh access network with 20 APs. The enumeration approach is no longer feasible in this scenario due to the enormous strategy space. The 20 APs are randomly scattered in a dbyd square, where the side length d is a tunable parameter in simulations. We investigate both cooperative and noncooperative cases represented by NETMA and NTMG curves, where the maximum number of iterations is set to w = 1000. Figure 47 pictorially depicts the performance inefficiency of NTMG caused by the noncooperative APs which transmit at the maximum power. The average throughput per AP is calculated by averaging the results of 50 simulations, for each value of the side length d. In Figure 47, it is worth noting that as the side length d gets bigger, the performance gap between NETMA and NTMG reduces. The reason is that when the area is large, the impact of mutual interference is less severe and so is the performance deterioration. However, when the network is crowded, i.e., d is small, the selfish behaviors are remarkably devastating. To alleviate the throughput degradation by the noncooperative APs, we implement the linear pricing scheme introduced in Section 4.4. The throughput improvement is illustrated as NTMGP in Figure 47. It is noticeable that by utilizing the proposed pricing scheme, the efficiency of Nash equilibrium is dramatically enhanced, especially for crowded networks. Therefore, the selfish incentives of the noncooperative APs have been effectively suppressed. 4.6.2 ACMCapable Devices In this subsection, we investigate the performance of our model with ACMcapable devices. Since the major difference lies in the improved pricing scheme tailored for this N =20, c= 3 10 95 7 65 B 9 ........ .. N** 'p..' NETMA 100 110 120 130 140 150 160 170 180 190 200 Side Length Figure 47. Performance evaluation of the wireless mesh access network with N = 20 and c = 3 specific scenario, we will only provide the performance evaluation of the tailored pricing scheme, i.e., NTMGP, in order to avoid duplicate results. We consider a populated network where 30 APs are randomly scattered in an 100mby100m square, i.e., N = 30. All other simulation parameters are the same as in the previous subsection except that the power is a continuous variable in this scenario. We first investigate the performance in terms of overall achievable rate of the network when no pricing scheme is applied, as a performance benchmark. Afterwards, the tailored pricing scheme is implemented to improve the equilibrium efficiency, a.k.a., the overall performance at the equilibrium. As shown in Figure 48, the overall achievable rate of the network is dramatically improved by the pricing scheme. The PDU adapts the announced price for each AP according to the optimum power, which is calculated by the previous announced price, in an iterative fashion. The aggregated achievable rate converges as the price setting iteration goes, as depicted by Figure 48. Figure 49 shows the trajectories of the power utilization prices for each AP The counterpart of actual utilized power is illustrated in Figure 410, where the curves start at ,,ax and evolve with the announced price in Figure 49. As observed in Figure 48, to achieve a better performance, the price setting process needs to be executed iteratively, comparing with the "one shot" heuristic 55 54 S53 149 0 10 20 30 40 50 60 70 80 90 100 Iteration Figure 48. Performance evaluation of the wireless mesh access network with/without the pricing scheme Figure 49. The trajectories of power utilization prices of each ACMcapable AP pricing scheme proposed for legacy devices. Therefore, the further improvement of the equilibrium efficiency is achieved as a tradeoff of communication overheads. However, it is worth noting that even at the first iteration, where the prices are determined by pmax, the induced equilibrium yields remarkable superior performance than the case where no pricing scheme is applied. 4.7 Conclusions In this chapter, we investigate the throughput maximization problem in wireless mesh networks. The problem is coupled due to the mutual interference and hence challenging. We first consider a cooperative case where all APs collaborate with each other in order to maximize the overall throughput of the network. A negotiationbased throughput maximization algorithm, a.k.a., NETMA, is introduced. We prove that 54 f = 5 f Figure 410. The trajectories of actual power utilized by each ACMcapable AP NETMA converges to the optimum solution with arbitrarily high probability. For the noncooperative scenarios, we show the existence and the inefficiency of Nash equilibria due to the selfish behaviors. To bridge the performance gap, we propose a pricing scheme which tremendously improves the performance in terms of overall throughput. The analytical results are verified by simulations. In addition, we extend our model and analytical results to the scenarios where more advanced APs are utilized, i.e., the devices with the adaptive coding and modulation capability. In this scenario, we propose a tailored pricing scheme which remarkably improves the overall performance in an iterative fashion. CHAPTER 5 CROSS LAYER INTERACTIONS IN WIRELESS SENSOR NETWORKS 5.1 Introduction Wireless sensor networks have attracted significant attention in both industrial and academic communities in the past few years, especially with the advances in lowpower circuit design and small size energy supplies which significantly reduce the cost of deploying large scale wireless sensor networks. The sensor networks can sense and measure the physical environment, e.g., temperature, speed, sound, radiation and the movement of the object etc. In addition, wireless sensor networks have become an important solution for military applications such as information gathering and intrusion detections. Other implementations of the wireless sensor networks include the healthcare body sensor networks, vehiculartoroadside communication networks, multimedia sensor networks and underwater communication networks. For more discussions on the wireless sensor networks, refer to the survey papers such as [84] and [85]. Since the sensor nodes in the network are usually placed in hostile environments, wireless transmissions among sensor nodes are strongly preferred. In addition, due to the restrained size of wireless sensor nodes, the computational capability of a single node is limited. Therefore, the measured information is usually transmitted to a remote data processing center (DPC) for further data analysis. Furthermore, due to the unreliable wireless links, multiple data sinks may exist in the network which collect the measured data and transmit the packets to the DPC node securely and reliably, possibly through the Internet. Before the wide deployment of wireless sensor networks, a systematic understanding on the performance of the multihop wireless sensor networks is desired. However, finding a suitable and accurate analytical model for wireless sensor networks is particularly challenging. First, the time varying channel conditions among wireless links 100 significantly complicate the analysis for the network performance in terms of throughput and experienced delay, even in an average sense [1, 86, 87]. Secondly, due to the unpredictability of the behavior of the monitored object, the exogenous traffic arrival to the network, i.e., the number of newly generated packets, is a stochastic process. Therefore, to ensure the stability of the network, i.e., to keep the queues in the network constantly finite, the analytical model of wireless sensor networks should comprise a rate admission control mechanism which can dynamically adjust the number of admitted packets into the network. Thirdly, due to the hostile wireless communication links, a dynamic routing scheme should be included in the analytical model. Moreover, the model should capture the complex issue of wireless link scheduling which is significantly challenging due to the mutual interference of wireless transmissions. Lastly, in order to fully explore the network resource and to mitigate the network congestion, an appropriate analytical network model should be able to dynamically deliver packets through multiple data sinks and thus an automatic load balancing solution can be achieved. In the existing literature, most of the proposed models for wireless sensor networks rely on the fluid model [6], where a flow is characterized by a source node and a specific destination node, e.g., [3, 4, 6, 7]. However, this model is not applicable to the cases where the generated packets can be delivered to any of the sink nodes, i.e., the destination node is one of the sinks and is selected dynamically. Moreover, this fluid model neglects the actual queue interactions within the wireless sensor network. In this work, to study the cross layer interactions of the multihop wireless sensor networks, we propose a constrained queueing model where a packet needs to wait for service in a data queue. More specifically, we investigate the joint rate admission control, dynamic routing, adaptive link scheduling and automatic load balancing solution to the wireless sensor network through a set of interconnected queues. Due to the wireless interference and the underlying scheduling constraints, at a particular time slot, only a subset of queues can be scheduled for transmissions simultaneously. To demonstrate the effectiveness of the proposed constrained queueing model, we investigate the stochastic network utility maximization (SNUM) problem in multihop wireless sensor networks. Based on the proposed queueing model, we develop an adaptive network resource allocation (ANRA) scheme which is a cross layer solution to the SNUM problem and yields a (1 c) nearoptimal solution to the global optimum network utility where e > 0 can be arbitrarily small. The proposed ANRA scheme consists of multiple layer components such as joint rate admission control, traffic splitting, dynamic routing as well as adaptive link scheduling. In addition, the ANRA scheme is essentially an online algorithm which only requires the instantaneous information of the current time slot and hence significantly reduces the computational complexity. The rest of the chapter is organized as follows. Section 5.2 briefly summarizes the related work in the literature. The constrained queueing model for the cross layer interactions of wireless sensor networks is proposed in Section 5.3. The stochastic network utility maximization problem of the wireless sensor network is investigated in Section 5.4, where a cross layer solution, a.k.a., the ANRA scheme is developed. The performance analysis of the ANRA scheme is provided in Section 5.5. An example which demonstrates the effectiveness of the ANRA scheme is given in Section 5.6 and Section 5.7 concludes this chapter. 5.2 Related Work To capture the cross layer interactions of multihop wireless sensor networks, several analytical models have been proposed in the literature. For example, in [37, 88], the multihop network resource allocation problem has been studied through a fluid model. Each flow, or session, is characterized by a source and a destination node where single path routing or multipath routing schemes are implemented. Most of the work rely on the dual optimization framework which decomposes the complex cross layer interactions into separate sublayer problems by introducing dual variables. For example, 102 the flow injection rate, controlled by the source node of the flow, is calculated by solving an optimization problem with the knowledge of the dual variables, a.k.a., shadow prices [6, 7], of all the links that are utilized. However, there are several drawbacks for the fluid based model. First, to calculate the optimum flow injection rate, the information along all paths should be collected in order to implement the rate admission control mechanism. In a dynamic environment such as wireless sensor networks, this process of information collection may take a significant amount of time which inevitably prolongs the network delay. Secondly, the optimization based solutions usually pursue fixed operating points which are hardly optimal in dynamic wireless settings with stochastic traffic arrivals and time varying channel conditions. Thirdly, the fluid model usually assumes that the changes of the flow injection rates are "perceived" by all the nodes along its paths instantaneously. The actual queue dynamics and interactions are neglected. In contrast, following the seminal paper of [12], many solutions have been focused on the queueing model for studying the complex interactions of communication networks. Neely et.al. extend the results of [12] into wireless networks with time varying channel conditions. For a more complete survey of this area, refer to [17]. The key component of the queue based solutions in these papers is the MaxWeight scheduling algorithm [12, 14]. Intuitively, at a time slot, the network picks the set of queues which (1) can be active simultaneously and (2) have the maximum overall weight. It is wellknown that the MaxWeight algorithm is throughputoptimal in the sense that any arrival rate vector that can be supported by the network can be stabilized under the MaxWeight scheduling algorithm. In addition, the MaxWeight algorithm is an online policy which requires only the information about current queue sizes and channel conditions. However, one notorious drawback of the MaxWeight algorithm is the delay performance. The reason is that in order to achieve the throughputoptimality, the MaxWeight algorithm explores a dynamic routing solution where long paths are utilized even under a light traffic load. This phenomenon is substantiated via simulations 103 by [89]. In [89], the authors propose a variant of the MaxWeight algorithm where the average number of hops of transmissions is minimized. Therefore, when the traffic is light, the proposed solution provides a much lower delay than the traditional MaxWeight algorithm. However, as a tradeoff, the induced network capacity region in [89] is noticeably smaller than that of the original MaxWeight algorithm. Consequently, it is difficult to provide a minimum rate guarantee on all the sessions in the network. Our work is inspired by [89]. With respect to [89], however, our work innovates in the following ways. First, different from [89], we focus on a heavyloaded wireless sensor network. Therefore, our solution incorporates a rate admission control mechanism which is not considered in [89]. Secondly, rather than minimizing the overall number of hops, we maximize the overall network utility which can also ensure the fairness among competitive traffic sessions. Thirdly, we specifically provide a minimum average rate guarantee for every session to ensure the QoS requirement. Fourthly, instead of a single destination scenario as considered in [89], we extend the model to cases where multiple data sink nodes are available. Each source node can deliver the packets to any of the sinks. Moreover, the dynamic routing and the issue of automatic load balancing is realized by the network on the fly. Finally, while [89] treats different sessions equally when minimizing the overall number of hops, our model priorities all the sessions with different QoS requirements. Therefore, a more flexible solution with service differentiations can be achieved. We will present the constrained queueing model in the next section. 5.3 A Constrained Queueing Model for Wireless Sensor Networks 5.3.1 Network Model We consider a multihop wireless sensor network represented by a directed graph G = {N, L} where N and L denote the set of vertices and the set of links, respectively. We will use the notation of IAI to represent the cardinality of set A, e.g., the number of nodes in the network is N and IL is the number of links. The time is slotted as t = 0, 1, and at a particular time slot t, the instantaneous channel data rate of link (m, n) e L is denoted by pm.n(t). In other words, link (m, n) can transmit a number of m .n(t) packets during time slot t. Note that the value of pm,n(t) is a random variable. Denote p(t) as the network link rate vector at time slot t. In this work, we assume that p(t) remains constant within one time slot but is subject to changes at time slot boundaries. The value of p(t) is assumed to be evolving following an irreducible and periodic Markovian chain with arbitrarily large yet finite number of states. However, the steady state distributions are unknown to the network. At time slot t, the network selects a feasible link schedule, denoted by I(t) = {/(t), /(t), ... IIL(t)} where /1(t) = 1 if link / is selected to be active and /1(t) = 0 otherwise. The set of all feasible link schedules is denoted by Q(t) which is determined by the underlying scheduling constraints such as interference models and duplex constraints. Therefore, selecting an interferencefree link schedule in the network graph g is equivalent to the process of attaining an independent set in the associated conflict graph g, where the vertices are the links in g and a link exists in 0 if the two original links in g cannot transmit simultaneously. 5.3.2 Traffic Model There are a number of S1 source nodes in the wireless sensor network which consistently monitor the surroundings and inject exogenous traffic to the network. The generated packets need to be delivered to the remote data processing center (DPC) in a multihop fashion. To simplify analysis, we assume that each source node is associated uniquely with a session. The set of source nodes is denoted by S = {n0, no, nos} where ns, s = 1, .. S is the source node of session s. It is worth noting that the following analysis can be extended straightforwardly to the scenarios where each source node may generate multiple sessions. There are DI number of sinks in the network which are connected to the remote data processing center via the Internet. In other words, the sink nodes can be 105 viewed as the gateways of the wireless sensor network. Denote the set of sinks as D = {da, d2, dlDo}. In this work, we consider a general scenario where the data packets from a source node can be delivered to the DPC via any of the sink node in D. Therefore, different from the existing literature such as [3, 88, 89], the source nodes do not specify the particular destination node for the generated packets. The selection of the destination node is achieved by the network via dynamic routing schemes. The network topology considered in this work is illustrated in Figure 51. Data Processing Center (DPC) O Internet Internet ...... :  ,,. 0 , F Sink Node 'I Source Node Figure 51. Topology of wireless sensor networks For a particular node in the network, say node n, we denote Od as the number of minimum hops from node n to the dth data sink in set D. Define n = d = ,... D (51) d as the minimum value of O for node n, i.e., the minimum number of hops from node n to a sink node in set D. We assume that node n is aware of the value of n, as well as those values of the neighboring nodes, which are attainable via precalculations by traditional routing mechanisms such as Dijkstra's algorithm. 106 At time slot t, the exogenous arrival of session s, i.e., the number of new packets1 generated by the source node of session s, is denoted by As(t). We assume that there is an upper bound for the number of new packets within one time slot, i.e., As(t) < Amax, Vs, t. For ease of exposition, we assume that As(t) is i.i.d. over time slots with an average rate of As. However, the data rates from multiple source nodes can be arbitrarily correlated. For example, if the wireless sensor network is deployed for monitoring purpose, it is very likely that a movement of the object will trigger several concurrent updates of the nearby sensors. Denote the vector A = {A1, f l Alsl} as the network arrival rate vector. The network capacity region A is thus defined as all the feasible2 network arrival vectors that can be supported by the network via certain policies, including those with the knowledge of futuristic traffic arrivals and channel rate conditions. In this work, we consider a heavyloaded traffic scenario where the network arrival vector A is outside of the network capacity region. Therefore, in order to achieve the network stability, a rate admission control mechanism is implemented at the source nodes. More specifically, at time slot t, we only admit a number of Xs(t) packets into the network from the source node of session s, i.e., nO. Apparently, we have Xs(t) < As(t), Vs, t. (52) In addition, we assume that each session has a continuous, concave and differentiable utility function, denoted by Us(Xs(t)), which reflects the degree of satisfaction by 1 We assume that the packets have a fixed length. For scenarios with variable packet lengths, the unit of data transmissions can be changed to bits per slot and the following analysis still holds. 2 Note that additional constraints may be imposed. For example, the constraints on the minimum average rate and the maximum average power expenditure can be enforced. For more discussions, please refer to [17]. 107 transmitting Xs(t) number of packets. It is worth noting that by selecting proper utility functions, the fairness among competitive sessions can be achieved. For example, if Us(X5(t)) = Iog(Xs(t)), a proportional fairness among multiple sessions can be enforced [4, 10, 11]. 5.3.3 Queue Management For each node n in the network, there are I N number of queues that are maintained and updated. The queues are denoted by Qn,h, where h = ,, . IN 1. Note that I N 1 is the maximum number of hops for a loopfree routing path in the network. The packets in the queue of Qn,h are guaranteed to reach one of the sink nodes in set D within h hops, as will be shown in Section 5.4. It is interesting to observe that for a newly generated packet by session s, the source node, i.e., n,, can place it in any of the queues of Q, ,h, where h = ,o, I INI 1, for further transmission. That is to say, consecutive packets from the source node n, may traverse through different number of hops before reaching a destination sink node in set D. Therefore, when a new packet is generated, the source node needs to make a decision on which queue the packet should be placed, namely, traffic splitting decision. In addition, the decision should be made promptly on an online basis with low computational complexity. With a slight abuse of notation, we use Qn,h to denote the queue itself and Qn,h(t) to represent the number of queue backlogs3 in time slot t. For a single queue, say Qn,h, it is stable if [13, 14] lim g(B) > 0 (53) Boo where Tl g(B) = lim sup TPr(Qnh(t)> ). too The network is stable if all the individual queues in the network are stable. 3 In the unit of packets. 108 For a link (n,j) e L, we require that the packets from Qn,h can be only transmitted to Q,h1, if exists. Therefore, the queue updating dynamic for Qn,h is given by OUnh ) ,(t) U (t) + m ('h +lt) Xh(t)nnO (54) (nj)GL (m,n)EL s where [A]+ denotes max(A 0) and un (t) represents the allocated data ate for the transmissions of Qn,h Qj,h1 on link (n,j), at time slot t, and N1 n,j (t) = Unj(t) h=pn where unj(t) = pnj(t) if Inj(t) = 1, i.e., link (n,j) is scheduled to be active during time slot t, and un(t) = 0 otherwise. The notation of xh(t) denotes the number of packets that are admitted to the network for session s and are stored in queue Qn,h for future transmissions. The indicator function 6A = 1 if event A is true and 6A = 0 otherwise. Note that the inequality in (54) incorporates the scenarios where the transmitter of a particular link has less packets in the queue than the allocated data rate. We assume that during one time slot, the numbers of packets that a single queue can transmit and receive are upper bounded. Mathematically speaking, we have SUmn '(t) < u., Vn, h, t, (55) (m,n)EL and Sj (t) < uot, Vn, h, t. (56) (nj)CL 5.3.4 SessionSpecific Requirements In this work, we consider a scenario where each session has a specific rate requirement as. Therefore, to ensure the minimum average rate, we need to find a policy 109 that lim Xs(t) > a, Vs. (57) T T t=0 In addition, we assume that each session in the network has an average hop requirement /s. More specifically, define INII Ms(t) = hXh(t) (58) h= no where INII SXh(t) =X(t),Vs, t h= no We require that for each session s, Tl lim 5 Ms(t) < 3, Vs. (59) t=0 Note that the average hop for a particular session s is related to the average delay experienced and the average energy consumed for the packet transmissions of session s. Therefore, by assigning different values of as and /s, a prioritized solution among multiple competitive sessions can be achieved for the network resource allocation problem. 5.4 Stochastic Network Utility Maximization in Wireless Sensor Networks In the previous section, we propose a constrained queueing model to investigate the performance of multihop wireless sensor networks. The model consists of several important issues from different layers, including the rate admission control problem, the dynamic routing problem as well as the challenge of adaptive link scheduling. To better understand the proposed constrained queueing model, in this section, we will examine the stochastic network utility maximization problem (SNUM) in multihop wireless sensor networks. As a cross layer solution, an Adaptive Network Resource Allocation (ANRA) scheme is proposed to solve the SNUM problem jointly. The proposed ANRA scheme is an online algorithm in nature which provably achieves an asymptotically optimal average 110 overall network utility. In other words, the average network utility induced by the ANRA scheme is (1 c) of the optimum solution where c > 0 is a positive number that can be arbitrarily small, with a tradeoff with the average delay experienced in the network. 5.4.1 Problem Formulation Recall that every session s possesses a utility function Us(Xs(t)) which is continuous, concave and differentiable. Without loss of generality, in the rest of this chapter, we will assume that Us(X,(t)) = Iog(X,(t)). Therefore, in light of the stochastic traffic arrival as well as the time varying channel conditions, our objective is to develop a policy which maximizes Stochastic Network Utility Maximization (SNUM) Problem SE(Us(X,(t))) (510) S s.t. * The network remains stable. * The average rate requirements of all S1 sessions, denoted by a = {ac, .. Qsl}, are satisfied. The average hop requirements of all S1 sessions, denoted by = {/1, .. s\}, are satisfied. Note that if the underlying statistical characteristics of the stochastic traffic arrivals and the time varying channel conditions are known, the SNUM problem is inherently a standard optimization problem and thus is easy to solve. However, due to the unawareness of the steady state distributions, the SNUM problem is remarkably challenging. In addition, in wireless sensor networks, dynamic algorithmic solutions with low computational complexity are strongly desired. In the following, we propose an ANRA scheme to solve the SNUM problem asymptotically. The ANRA scheme is a cross layer solution which consists of joint rate admission control, traffic splitting, dynamic routing as well as adaptive link scheduling components. Moreover, the ANRA algorithm can achieve an automatic load balancing solution by utilizing different sink nodes corresponding to the variations of the network conditions. The ANRA algorithm is an online algorithm in nature which requires only the state information of the current time slot. We show that the ANRA algorithm achieves a (1 c) optimal solution where c can be arbitrarily small. Therefore, the proposed ANRA algorithm is of particular interest for dynamic wireless sensor networks with time varying environments. 5.4.2 The ANRA Cross Layer Algorithm Before presenting the proposed ANRA scheme, we introduce the concept of virtual queues [16, 17, 34] to facilitate our analysis. Specifically, for each session s, we maintain a virtual queue Ys, which is initially empty, and the queue updating dynamic is defined as Ys(t + 1) = [Ys(t) Xs(t)]+ as, Vs, t. (511) Similarly, we define another virtual queue for every session s, denoted by Zs, and the queue dynamic is given by Zs(t + 1) = [Z(t) ]+ + Ms(t), Vs, t, (512) where Ms(t) is defined in (58). Note that the virtual queues are software based counters which are easy to maintain. For example, the source node of each session can calculate the values of virtual queues Ys(t) and Zs(t) and update the values accordingly following (511) and (512). In addition, we introduce a positive parameter J which is tunable as a system parameter. The impact of J on the algorithm performance will be discussed shortly. The proposed ANRA cross layer algorithm is given as follows. ADAPTIVE NETWORK RESOURCE ALLOCATION (ANRA) SCHEME: Joint Rate Admission Control and Traffic Splitting (at time t): 112 For each source node, say n, there are a number of queues, i.e., Qn,oh, h = ,no, I INI 1. Find the value of h which minimizes Zs(t)h + Qn,h(t) (513) where ties are broken arbitrarily. Denote the optimum value of h as h*. The source node n0 admits a number of new packets as Xs(t) = min ( ), A(t)) (514) where X5() 2 (Z(t)h* + Qn,h(t)) 2Ys(t) (515) For traffic splitting, the source node no will deposit all Xs(t) packets in Qno,h*. Joint Dynamic Routing and Link Scheduling (at time t): For each link (m, n) e L, define a link weight denoted by Wm,n(t), which is calculated as Wm,n(t) max (Qm,h(t) Qn,h(t)) (516) h= Note that if Qn,h1 does not exist, the transmissions from queue Qm,h to Qn,h1 are prohibited. At time slot t, the network selects an interferencefree link schedule I(t) which solves max pmn(t)Wm,n(t). (517) I(t)en (t) If link (m, n) is active, i.e., lmn(t) = 1, the queue of Qm. is selected for transmissions where h = argmaxh,, ~~N1 (Qm,h(t) Qn,h(t)). (518) END Note that (517) is similar to the original MaxWeight algorithm, introduced in [12] and generalized in [13, 14, 34]. The dynamic routing and link scheduling are addressed jointly by solving (517), which requires centralized computation. However, following [12], many work have been focused on the distributed solutions of (517). Although the distributed computation issue is not the focus of this work, we emphasize that our proposed ANRA scheme can be approximated well by existing distributed solutions such as [3638, 41, 42, 45, 46, 48]. 113 For the packets placed at queue Qno,h, at most h hops of transmissions are needed in order to reach one of the sink nodes in set D. This can be verified straightforwardly due to the requirement that a transmission from Qm,h to Qn,h1 can occur if and only if h 1 > ,. Moreover, the joint rate admission control and the optimum traffic splitting components of ANRA can be implemented by the source node in a distributed fashion. Note that in order to calculate the instantaneous admitted rate, the source node of session s needs only to know the local queue backlog information. Moreover, the decision of traffic splitting requires only local queue information as well. Therefore, at every time slot, the joint rate admission control and traffic splitting decision can be made on an online basis in accordance to the time varying conditions of local queues. Furthermore, we will show that this simple adaptive strategy does not incur any loss of optimality. The achieved network utility induced by the ANRA scheme can be pushed arbitrarily close to the optimum solution. Next, we will characterize the global optimum utility in the network and provide the main performance results of the proposed ANRA scheme. 5.4.3 Performance of the ANRA Scheme In this section, we first characterize the global optimum solution of the SNUM problem in (510). Define U* as the global maximum network utility that any scheme can achieve, i.e., the optimum solution of (510). In order to achieve U*, it is naturally to consider more complicated policies such as those with the knowledge of futuristic arrivals and channel conditions. However, in the following theorem, we show that, somewhat surprisingly, the global optimum solution of the SNUM problem can be achieved by certain stationary policies, i.e., the responsive action is chosen regardless of the current queue sizes in the network and the time slot that the decision is made. Recall that we have assumed that for each session, As(t) is i.i.d. over time slots. Denote A(t) as the vector of instantaneous arrival rates of all sessions, at time slot t. Let A be the set of all possible value of A(t). Note that for every element in A, i.e., Aa, a = 1, A, we have 0 Aa Amax where _ denotes the elementwise comparison. We use r, to represent the steady state distribution of Aa. Theorem 5.1. If the constraints in the SNUM problem are satisfied, the maximum network utility, denoted by U*, can be achieved by a class of stationary randomized policies. Mathematically, the value of U* is the solution of the following optimization problem, with the auxiliary variables pk and Rk, as ISI+1 max Y e Us(Z paRk) (519) a s k=l s.t. The constraints in (510) are satisfied. 4 R>k Aa. pk > 0, Va, k. lSIl Pak a 1, Va. Proof. We prove Theorem 5.1 by showing that for arbitrary policy which satisfies the constraints in the SNUM problem, the overall network utility is at most U*, which is the optimum utility attained by a class of stationary randomized policies. In other words, we need to show that ST1 Up = lim sup y Y Us(X,(t)) Too t t=0 s where UP is the average network utility under a policy P. For each state in A, say Aa, define Ra as the set of nonnegative rate vectors that are elementwise smaller than A,. Define CR, as the convex hull of set Ra. Therefore, any point in CR, can be considered as a feasible network admitted rate vector given that the current arrival rate vector is Aa. Note that every point in CR, is a vector with a dimension of ISby1. Therefore, it can be represented by a convex combination of at most S + 1 points, denoted by Rk, k = 1, IS + 1, according to Caratheodory's theorem. In light of this, we first consider a time interval from 0 to T 1. Denote N,(T) 115 as the set of time slots that A(t) = Aa. Therefore, we can rewrite (520) as U= limsup INa(T) Us pk R . Too a T s \k Due to the stationary assumption, we have 1 + UP = ~~~a Z lim sup Us p( RR . a s Tkoo 1 Note that we also assume that the utility function is continuous and bounded. Therefore, the compactness of the utility functions is assured. Next, we focus on a subsequence of time durations, denoted by T,, i = 1, .. ,oo. Denote Unet(T,) a U a s k=1 It is straightforward to verify that S= lim sup UPet(T). i0oo Due to the compactness of the utility functions, following BolzanoWeierstrass theorem [90], we claim that there exists a subsequence of T,, i = 1, ., oo, such that lim U, (sl p ( ) T ) R 5. k=l Denote p as the values which generate Ua, i.e., U = Us p k R . \k= 1 We have U = lim sup Unet(T) = Ta Us pR . ia so k=1 According to the definition of U* in (519), we conclude that UP < U*. 116 Intuitively, Theorem 5.1 indicates that the global maximum network utility can be achieved by certain randomized stationary policies. However, to calculate U*, the stationary policy needs to know the steady state distributions which are difficult to obtain in practice. In light of this, we propose an adaptive network resource allocation scheme, namely, ANRA, which is an online solution and does not require such statistical information as a priori. For notation succinctness, denote UA(t)= E (Us(X(t))) S as the expected network utility induced by the ANRA scheme. The performance of the ANRA algorithm, with a parameter J, is given as the following theorem. Theorem 5.2. For a given system parameter J, we have 1 T1 liminf UA(t) > U* (521) t=0 where B is a constant and is given by B = IN(INI 1) ((out0)2 (ui, Amax )2) + ((as)2 (/)) (Amax) (( )4 + S In addition, the constraints in the SNUM problem, i.e., (510), are satisfied simultane ously. Proof. The proof of Theorem 5.2 is deferred to Section 5.5. It is worth noting that if we let J > oc, the performance induced by the ANRA algorithm can be arbitrarily close to the global optimum solution U*. However, as a tradeoff, a larger value of J also yields a longer average queue size in the network. According to Little's Law, a larger queue size corresponds to a longer average delay experienced in the network. Therefore, by selecting the value of J properly, a tradeoff 117 between the network optimality and the average delay in the network can be achieved. We will discuss more about this issue in the next section. 5.5 Performance Analysis In this section, we provide a proof to Theorem 5.2 in the previous section. Recall that in (511) and (512), we introduce two virtual queues, i.e., Ys(t) and Zs(t) for each session s. Therefore, the average rate and the average hop requirements from all sessions are converted into the stability requirements for the virtual queues. For example, the virtual queue update of Y,(t) is given by (511). If the virtual queue Y, is stable, the average arrival rate should be less than the average departure rate of the queue, i.e., T1 a, < lim X,(t, T ooT (t) t=0 which is exactly the minimum average rate requirement imposed by session s. By the same token, the average hop requirement of session s is converted to the stability problem of the virtual queue Zs. Define Q(t) = (Q(t), Y(t), Z(t)), namely, all the data queues and the virtual queues in the network. Our objective is to find a policy which stabilizes the network with respect to Q while maximizing the overall network utility. We first take the square of (54) and have (Qn,h(t 1))2 < ( n,h(t))2 n,h 1 I ,m h+l( \t) + h(t) 6 ) + \ ,J (t) + Um n n L  xht)(n nO (nnj)L ( mn L s 2Qn,h(t) U( (t)mh(t) Xh(t)6n no (nj)EL (m,n)EL s Since we assume that each node generates at most one session, we have Xh(t)6,no < Amax, Vt, n. S 118 Note that if we allow that a node can initiate multiple sessions, we have Xh(t)6n=no < ISlAmax, Vt, n S where S1 is the number of sessions in the network. In light of (55) and (56), we have (Qn,h(t + 1))2 ( (tf))2 < (Utt)2 + 2Qn,h(t) U( n ,h ( (t) \( nj) L ( m,n) m, (nj)GL (m,n)EL 2 Uin Amax )  Xsh(t)n S We next sum (522) over all the data queues in the network, i.e., Qn,h, and have 1))2 2Qn,h(t) (n h ) (n,j)eL ( nnh ( )2 n,h E u, 'mh+l(t ( m,m nL (m,n)EL (523) Xh S s where Bi = INI(INI 1) ((ou) 2 (uin Amax)2). Note that Ms(t), defined in (58), satisfies Ms(t)< (INI 1)2Amax. Next, we take the square of (511) and (512) and thus have Y(t + 1))2 (Xs(t))2 (a)2 2Ys(t)(Xs(t) (/s)2 2Zs(t) (s (522) and (Z (t 1))2 < Z(t)2 1) ) < W (MA(t))2 Ms(t)). 119 no) S(Qn,h(t n,h < Y(t))2 Similarly, we sum over all the sessions and have (Ys(t)) < B2 2Z S Y )( t) (X(t) B2 = IS(Amax)2 Also, we obtain (zs(t S 1))2 Zs(t)) 2 B 2 Zs(t)(3s S where B3 = (s) IS (INI 1)4 (A ax)2. Define the systemwide Lyapuno function as Define the systemwide Lyapunov function as Y(YSt)2 s Next, we define the Lyapunov drift [14] of the system as A = E (L(Q(t 1)) L(Q(t)) IQ(t)). Define B = B1 + B2 B3, we have A < B 2 Q,h(t)E u (t) n,h (nj)CL 2E( Y(t)(Xs(t)0 s) Q(t) Ss E ^m,h, _h umnh(t) (m,n)eL s .2E Zs(t)(s  Ms(t)) Q(t)) . 120 (YS(t S 1) 2 where as) Ms(t)) S(Z(t))2. S (524) L(Q(t))= (Qn,h(t)) n,h Next, we subtract both sides by JE( Y Us(X,(t)) Q(t)) and have A JE ( U(X(t)) Q(t)) S ) 2 Q,h(t)E ( nE  n,h (nj) L (m 2E(z Qnoh(t)Xh(t) Q(t) ns h ) m ,h+ Um, n ,n) L  2E '(t) Q(t)) s Y(t)Xs(t) Q(t) 2E (Z(t)Ms(t) Q(t)) S 2 Zs(t)/3 S JE ( S Us(X1(t)) Q(t)) We rewrite the R.H.S. of (525) as R.H.S. = B + 2 Y(t)as s ? 2 Qn,h(t)E n,h E ( 2Ys(t) )5(t) S2Z,(t)Ms(t) S 2 Z,(t) , S 5 2Qn, h(t)Xh(t) s h JY Us(X,(t)) Q(t) . We observe that the dynamic routing and scheduling component of the ANRA scheme is actually maximizing SQn,h(t)E n,h u(jn (t) (nj)CL 5 uh '(t) Q( (m,n)CL In addition, the joint rate admission control and traffic splitting component of the ANRA scheme is essentially maximizing E ( 2Ys(t)Xs(t) with the constraints of S2Z,(t)M (t) s ? 5 2Qo h(t)Xh(t) s h Ji Us(Xs(t)) Q(t) (527) xh(t) h X,(t), Vs, t. 2 Ys(t)as (525) n,h I(t) Um (t) ( nj)L (m n)ChL (n j)GL (m,n)GL t)). (526) (528) To see this, we can decompose (527) to show that each session s only maximizes JU, (X(t)) + 2Y(t)Xs(t) 2Z,(t) hXh(t) 2Qnh(t)Xh(t). (529) h h Therefore, the proposed ANRA algorithm indeed minimizes the R.H.S. of (525) over all policies. Consider a reduced network capacity region, denoted by Ae, parameterized by > 0, as {AAn,h + A} (530) where A is the original network capacity region and T1 Anh = Tm X (n no. (531) t=0 s Define U* as the global optimum network utility achieved in the reduced capacity region. Apparently, we have lim,,o U* > U*. In addition, denote X,(0), Xh(1), S, X(t), ... as the optimum rate sequence which yields U,. Define X, as the average of the optimum rate sequence of session s, in the reduced capacity region. It is straightforward to verify that X, + is in the original network capacity region A. Therefore, following a similar analysis as in [1315, 91], we claim that there exists a randomized policy, denoted by R, which generates n,t m,'h+(t) Xh* 32) E ( un(t) t) (t)) > + C (532) (nj)GL (m,n)EL if n is one of the source nodes and E n (t) u Mhl(t) > (533) (nj)GL (m,n)s L for other nodes. Furthermore, policy R ensures Et) >h +c h 122 and E (M,(t) +e < ,) where M,,(t) is generated by Xh (t). Due to the fact that the proposed ANRA scheme minimizes the R.H.S. of (525) overall all policies, including R, we have A JE( Us(X,(t)) Q(t) < B 2c ( Qnh(t) + ) Ys(t) \ nh S SZS(t)) s / We next take the expectation with respect to Q(t) and obtain L(Q(t + 1)) L(Q(t)) JE ( U(X(t)) JEF Us Xh (t) ). \s h/ We sum over time slots 0, T 1 and have L(Q(T)) L(Q(O)) 1) T L(Q(T)) L(Q(0)) JE Us(X,(t)) < TB t=0 s JE Us( Xsh:(t) t=0 h (534) (535) since E (ZQn,h(t) + Y (t) ZZ(t)") \n,h s s / is always nonnegative. Next, we divide the both sides of (535) by T and rearrange terms to have T JE Us(Xs(t))> JE Us Xs(t) + L(Q(0)) t= S t0 S h where the nonnegativity of the Lyapunov function is utilized. Since we assume that the initial queue backlogs in the system are finite and the virtual queues are initially empty, 123 JE Us( Xsh(t)+)Q \ s h taking c > 0 and lim infr,, yields the performance result of the ANRA algorithm stated in Theorem 5.2. We next show that the constraints of the SNUM problem are also satisfied. To illustrate this, we show that the queues in the network, including real data queues and virtual queues, are stable. Based on (534), we sum over time slots 0, T 1 and have T1 T1 S2eE Qnh(t + Y(t) + Zs(t) < L(Q(0))+ J(E Us(Xs(t)) TB. t=0 n,h s s t=0 O (536) Due to Xs(t) < Amax and the assumptions on the utility function, we claim that Us(t) is upper bounded and denote the maximum utility within one time slot as Umax, i.e., Us(t) < Umax, s, t. (537) Divide the both sides of (536) by T and we have T1 Y 2cE Qn,h(t) sY,(t) SZs(t) < L(Q(O)) JS Umax B. t=0 n,h s s( By taking lim supT ,o, we have T1 lim sup' E (t) Y(t) Zs(t)U) < x B (538) T t=0 n,h s s Note that the above analysis holds for any feasible value of c. Denote (p as the maximum value of c such that A, is not empty. Finally, we conclude that T1 lim sup T E Qnh(t) Y E (t) Zs(t)) < J < C. (539) t= s n,Sh s s The stability of the network follows immediately from Markov's Inequality and thus completes the proof. It is worth noting that as shown in (539), a large value of J induces a longer average queue size in the network. Therefore, a tradeoff between the algorithm performance of the ANRA scheme and the average delay experienced in the network can be controlled effectively by tuning the value of J. 5.6 Case Study In this section, we demonstrate the effectiveness of the ANRA algorithm numerically through a simple network shown in Figure 52. We stress that, however, this exemplifying study case reproduces all the challenging problems involved in the complex cross layer interactions in time varying environments, such as stochastic traffic arrivals, random channel conditions and dynamic routing and scheduling etc. As shown in Figure 52, the source nodes in the network are node A and B whereas the destination sink nodes are denoted by E and F. Sl dl S I B "" D  82 d2 Figure 52. Example network topology There are six nodes and twelve links in the network. Therefore, node A and B each maintains four queues, from hop 2 to hop 5, and node C and D each maintains five queues, from hop 1 to hop 5, in the buffer. At each time slot, a wireless link is assumed to have three equally possible data rates4 2, 8 and 16. The traffic arrivals are i.i.d. with three equally possible states, i.e., 0, 10 and 20. The minimum rate requirements of the 4 Note that the unit of data transmissions is packet per slot. 4 Note that the unit of data transmissions is packet per slot. 125 two sessions are 5 and 8 and the average hop requirements of the sessions are 30 and 10. Without loss of generality, we assume that at a given time slot, two links with a common node cannot be active simultaneously. For example, if link A > B is active, link B A, A C, C A, B D and D > B cannot be selected. Figure 53 depicts the average network utility achieved by the ANRA scheme for different values of J where each experiment is executed for 50000 time slots. We can observe from Figure 53 that the overall network utility rises as the value of J increases. However, the speed of utility improvement decreases and the achieved network utility converges to the global optimum utility U* gradually. To demonstrate the tradeoff of different values of J, in Figure 54, we show the average queue size in the network for J = 20, 50, 200, 500, 1000, 2000, 5000, 10000 and 20000. We can see that, as expected, the average queue size increases as the value of J gets larger. Note that the average queue size is related to the average delay in the network. Therefore, a tradeoff between the network optimality and the average experienced delay can be achieved by tuning the value of J. 24 22 .... ..... .... I' '. . I20 i 18 Z16 14 i 12 10~ * 0 20 50 200 500 1000 2000 5000 10000 20000 50000 Values of J Figure 53. Average network utility achieved by ANRA for different values of J In Figure 55, we illustrate the sample trajectories of the admitted rates of two sessions with J = 5000, for the first 50 time slots. We can observe that each session admits different amount of packets into the network adaptively following the time varying conditions of the network. In addition, we depict the trajectories of the four virtual queues with the same settings, in Figure 56, for the first 100 time slots. By comparing 126 20 so 2 5o00 1000 2000 5000 1ooO 200o Values ofJ Figure 54. Average network queue size by ANRA for different values of J Figure 55 and Figure 56 jointly, we can observe that for the minimum rate virtual queue, say Y1, whenever there is the tendency that the virtual queue is accumulating, as depicted in Figure 56, the corresponding admitted rate by session 1 increases in Figure 55. By the definition of the virtual queue, a larger backlog of Y1 indicates that the average departure rate of the virtual queue, i.e., the average admitted rate, is insufficient. Therefore, the source node of session 1 will attempt to increase the admitted rate and thus the backlog of the virtual queue will decrease accordingly where the stability of the virtual queue can be assured. 20 S Session 1 A Session 2 15 : l I 0i 0 A 0 10 20 30 40 50 Time Slots Figure 55. Sample trajectories of the admitted rates of two sessions for J 5000 127 0) m 60 in 06 O 40 20 Figure 56. Sample trajectories of the virtual queues for J = 5000 In Figure 57, the traffic splitting decisions of the two source nodes, i.e., the hop selections of the source nodes, are illustrated. We can observe that both source nodes incline to utilize the queues with the smaller number of hops. The queues with longer hops, e.g., h = 3 or 4, are used only when the queue backlogs in the queues with smaller hops are overwhelmed. In addition, we can see that on average, session 2 utilizes a smaller number of average hops than session 1. Recall that session 2 has a much more stringent constraint on the average number of hops than session 1, i.e., 10 v.s. 30. Therefore, the source node of session 2, i.e., n, inclines to deposit more packets on the queues with smaller hop counts. As a consequence, by assigning different values of rate and hop requirements, a service differentiation solution can be achieved by the ANRA scheme among multiple competitive sessions in the network. In addition, a nearoptimal network utility can be attained simultaneously. 5.7 Conclusions and Future Work In this chapter, we propose a constrained queueing model to capture the cross layer interactions in multihop wireless sensor networks. Our model consists of components from multiple layers such as rate admission control, dynamic routing and wireless link scheduling. Based on the proposed model, we investigate the stochastic network 128 4 ?  i SSession 1 I * 3.5 ' Session 2 i SII I II II I I I 3 i : II I luII 31 in ; I u I; SI I I I I 0.5 II II IU III 1. I ', I i Ii Ij I 0.5 nS 0 10 20 30 Time Slots 40 50 Figure 57. Sample trajectories of the hop selections for J = 5000 utility maximization problem in wireless sensor networks. As a cross layer solution, an adaptive network resource allocation scheme, a.k.a., ANRA algorithm, is proposed. The ANRA algorithm is an online mechanism which yields an overall network utility that can be pushed arbitrarily close to the global optimum solution. As a future work, energyaware distributed scheduling algorithms are to be studied and evaluated. In addition, the extension of our model to wireless sensor networks with network coding seems interesting and needs further investigation. 129 CHAPTER 6 THRESHOLD OPTIMIZATION FOR RATE ADAPTATION ALGORITHMS IN IEEE 802.11 WLANS 6.1 Introduction IEEE 802.11 WLAN has become the dominating technology for indoor wireless Internet access. While the original IEEE 802.11 DSSS only provides two physical data rates (1 Mbps and 2 Mbps), the current IEEE standard provides several available data rates based on different modulation and coding schemes. For example, IEEE 802.11 b supports 1 Mbps, 2 Mbps, 5.5 Mbps and 11 Mbps and IEEE 802.11 g provides 12 physical data rates up to 54 Mbps. In order to maximize the network throughput, IEEE 802.11 devices, i.e., stations, need to adaptively change the data rate to combat with the timevarying channel environments. For instance, when the channel is good, a high data rate which usually requires higher SNR can be utilized. On the contrary, a low data rate which is errorresilient might be favorable for a bad channel. The operation of dynamically selecting data rates in IEEE 802.11 WLANs is called rate adaptation in general. The implementation of rate adaptation algorithms is not specified by the IEEE 802.11 standard. This intentional omission flourishes the studies on this active area where a variety of rate adaptation algorithms have been proposed [92108]. The common challenge of rate adaptation algorithms is how to match the unknown channel condition optimally such that the network throughput is maximized. According to the methods of estimating channel conditions, rate adaptation algorithms can be divided into two major categories. The first one is called closedloop rate adaptation. Most schemes in this approach enable the receiver to measure the channel quality and sends back this information explicitly to the transmitter for rate adaptation. For example, the receiver records the SNR or RSSI value of the received packet and sends this information back to the transmitter via CTS or ACK packet. Consequently, the transmitter estimates the channel condition based on the feedback signal and adjusts the data rate accordingly. 130 By utilizing additional feedback mechanisms, the closeloop rate adaptation algorithms can achieve a better performance than the openloop counterpart. However, in practice, the closeloop rate adaptation algorithms are rarely used in commercial IEEE 802.11 devices. This is because that the extra feedback information needs to be conveyed reliably and hence an inevitable modification on the current IEEE 802.11 standard is needed. This noncompatibility hinders the closeloop rate adaptation algorithms from practical implementations in current offtheshelf IEEE 802.11 products. The second category of rate adaptation algorithms, which is predominantly adopted by the vendors, is labeled as openloop algorithms. The widely utilized Auto Rate Fallback algorithm, a.k.a., ARF, falls into this category. As many other openloop rate adaptation algorithms, ARF adjusts the date rate solely based on the IEEE 802.11 ACK packets. For example, in Enterasys RoamAbout IEEE 802.11 card [109], two consecutive frame transmission failures, indicated by not receiving ACKs promptly, induces a rate downshift, while ten consecutive successful frame transmissions triggers a rate upshift [110]. Most commercialized IEEE products follow this up/down scheme [95, 96]. In this chapter, we focus on the openloop rate adaptation algorithms due to the practical merits. More specifically, we consider a thresholdbased rate adaptation algorithm which works as follows. If there are 0, consecutive successful transmissions, the data rate is upgraded to the next level. On the other hand, if Od consecutive transmissions failed, a rate downshift is triggered. Since ARF is merely a special case of this thresholdbased rate adaptation algorithm, our analysis can be applied to ARF and its variants as well. As a tradeoff with simplicity, there are several challenges existing for the thresholdbased rate adaptation algorithm. First, due to the trialanderror based up/down mechanism, it inherently lacks the capability of capturing short dynamics of the channel variations [96]. To tackle this issue, Qiao et.al. propose a fast responsive rate adaptation solution in [92]. By introducing a measure of "delay factor", the responsiveness of the thresholdbased rate adaptation algorithm can be guaranteed. The second challenge of the thresholdbased rate adaptation algorithm is the indifference to collisioninduced failures and noiseinduced failures. It is worth noting that in a multiuser WLAN, an ACK timeout can be ascribed to either an MAC layer collision, or an erroneous channel. However, the thresholdbased rate adaptation algorithm is unable to distinguish them effectively. Therefore, excessive collisions may introduce unnecessary rate degradations which significantly deteriorate the system performance. Attributing the actual reason of a transmission failure, or a packet loss, is named loss diagnosis [111] and has attracted tremendous attention from the community. For example, Choi et.al. [112] propose an algorithmic solution, which is specific to the thresholdbased rate adaptation algorithm, to mitigate the collision effect in multiuser IEEE 802.11 WLANs. Therefore, the performance deterioration by the "indifference to collisions" can be compensated effectively. While the first two challenges of the thresholdbased rate adaptation algorithm have been tackled effectively, the third major obstacle, namely, the optimal selection of the up/down thresholds, remains as an open problem. A systematic treatment on how to select the values of 0, and Od in the thresholdbased rate adaptation algorithm is lacking in the literature, although several heuristic solutions are proposed [93, 94]. The contribution of this work is twofold. First, we analytically investigate the behavior of the thresholdbased rate adaptation algorithm from a reverse engineering perspective. In other words, we answer the essential yet unresolved question, i.e., "What is the thresholdbased rate adaptation algorithm actually optimizing?", by unveiling the implicit objective function. As a result, several intuitive observations of the thresholdbased rate adaptation algorithm can be explained straightforwardly by inspecting this objective function. Therefore, our reverse engineering model provides an alternative means to understand the thresholdbased rate adaptation algorithm. Our work is a complement to the recent trend of reverse engineering studies on existing 132 heuristicsbased networking protocols, such as TCP (transport layer) [4, 113, 114], BGP (network layer) [115] and random access MAC protocol (data link layer) [116]. To the best of our knowledge, this is the first work of studying thresholdbased rate adaptation algorithms from a reverse engineering perspective. Our results explicitly show that the values of 0, and Od play an important role in the objective function and thus the network performance hinges largely on the selection of the up/down thresholds. In light of this, we propose a threshold optimization algorithm which dynamically tunes the up/down thresholds of the thresholdbased rate adaptation algorithm and provably converges to the stochastic optimum solution in arbitrary stationary random environments. We show that the optimal selection of the thresholds significantly enhances the system's performance. The rest of chapter is organized as follows. Section 6.2 briefly overviews the stateoftheart rate adaptation algorithms in the literature. The reverse engineering model of the thresholdbased rate adaptation algorithm is derived in Section 6.3. In Section 6.4, the threshold optimization algorithm is proposed. The performance evaluations are provided in Section 6.5 and Section 6.6 concludes this chapter. 6.2 Related Work RBAR [103] proposes an SNRbased closeloop rate adaptation algorithm where the rate decision relies on the feedback signal from the receiver. Specifically, the receiver estimates the channel condition and determines a proper rate via the RTS/CTS exchange. While consistently outperforms the openloop rate adaptation algorithms such as ARF, RBAR is incompatible with the current IEEE 802.11 standard by altering the CTS frames [96]. In [105], a hybrid rate adaptation algorithm with SNRbased measurements is proposed, where the measured SNR is utilized to bound the range of feasible settings and thus shortens the response time to channel variations. [97] is another example of the closeloop rate adaptation algorithms which attempts to improve the throughput by predicting the channel coherence time, which is nevertheless difficult 133 in practice. Chen et.al. introduce a probabilisticbased rate adaptation for IEEE 802.11 WLANs. In [100], a rateadaptive acknowledgement based rate adaptation algorithm is introduced. The basic idea is that by varying the ACK transmission rate, the appropriate rate information is conveyed to the transmitter. Wang and Helmy [108] propose a trafficaware rate adaptation algorithm which explicitly relates the background traffic to the rate selection problem. However, aforementioned solutions either rely on altering the frame structures, or do not conform to the de facto IEEE 802.11 standard in commercial usage1 CHARM [101] avoids the overhead of RTS/CTS by leveraging the channel reciprocity. However, modifications on the IEEE 802.11 standardized frame structures, e.g., beacons and probe signals, are still needed. In light of the complexity and the incompatibility of the closeloop rate adaptation algorithms, alternative openloop rate adaptation algorithms are proposed. Although usually providing inferior performance than the closeloop solutions, openloop algorithms soon become the predominant technique in commercial IEEE 802.11 devices due to the simplicity and the compatibility. The most popular openloop rate adaptation algorithm is the ARF protocol proposed by Ad and Leo [104] where a rate upshift is triggered by ten consecutive successful transmission while two consecutive failures induce a rate downshift. SampleRate [102] is another widely adopted openloop rate adaptation algorithm which performs arguably the best in static settings [96]. However, as pointed out by [96], SampleRate suffers from significant packet losses in fast changing channels. ONOE [106] is a creditbased mechanism included in MadWiFi drivers. The credit is determined by the number of successful transmissions, erroneous transmissions and retransmissions jointly. However, it is pointed in [98] that ONOE is less sensitive to individual packet failure and behaves overconservatively. 1 For example, they largely rely on the RTS/CTS signaling mechanism which is hardly used in practice. As mentioned previously, one of the drawbacks of the simple openloop rate adaptation algorithms is the inability to discriminate the collisioninduced losses and the noiseinduced losses. While several collisionaware rate adaptations are available in the literature [95, 96, 99, 107], they base on the closeloop solutions which utilize either RTS/CTS signaling or modifications on the standard. In [110], [112], Choi et.al. propose a collision mitigation algorithm for openloop ARF algorithm based on a Markovian modeling. However, the channel conditions are assumed to be constant in their work. In this work, on the contrary, we particularly focus on the threshold optimization to combat with fast channel fluctuations rather than collisions. Therefore, in tandem with [110, 112], our work provide a solution to jointly mitigate the channel fluctuations and the multiuser collisions. For example, the up/down thresholds, i.e., 0, and Od can be first calculated by our threshold optimization algorithm in Section 6.4. Next, they are subjected to further adjustments following [112] to mitigate the collision effects. 6.3 Reverse Engineering for the ThresholdBased Rate Adaptation Algorithm We consider a station in a multirate IEEE 802.11 WLAN. There are N stations in the WLAN where each station, say i, has a transmission probability of pi. Note that the equivalence of the ppersistent model and the IEEE 802.11 binary exponential backoff CSMA/CA model has been extensively studied in [116] and [117]. Throughout this work, we assume that the transmission probability of each station is fixed. The interaction of the data rate and the transmission probability remains as future research. Without loss of generality, we assume that the stations have a same transmission probability of p. In this work, we focus on the widely deployed open loop thresholdbased rate adaptation algorithm. Not surprisingly, the speed of channel variations has a great impact on the performance of the rate adaptation algorithm. While the original intention of the thresholdbased rate adaptation algorithm is to maximize the average throughput, a natural question arises that whether this is indeed the case. In this section, to better understand the impact of 0, and Od on the performance of the thresholdbased rate 135 adaptation algorithm, we investigate the thresholdbased rate adaptation algorithm via a reverse engineering approach. We assume that the RTS/CTS signals are turned off. In a time slot, say t, we denote the channel state2 as s(t) and denote the successful transmission probability, given the current transmission rate r(t) and the channel condition s(t), as Ps (s(t), r(t)) = p(1 p)" (1 e(s(t), r(t))) (61) where e denotes the frame error rate (FER) and is given by e(s(t), r(t))= 1 (1 P(s(t), r(t)))L (62) and Pe(s(t), r(t)) is the bit error rate (BER) which is determined by the current data rate, i.e., modulation scheme, and the current channel condition. L is the frame length of the packet. Similarly, we define PF(S(t), r(t)) = 1 Ps(s(t), r(t)) as the transmission failure probability at time t. It is worth noting that both Ps and PF are functions of the current data rate r(t) as well as the instantaneous channel condition s(t), which is random. Particularly, we assume that the thresholdbased rate adaptation algorithm will increase the data rate by an amount of 6 if there are 0, consecutive successful transmissions and decrease it by 6 if there are Od consecutive failures. Denote u = 0, 1 and d = Od 1 for notation succinctness. We define a binary indicator function ((t) where ((t) = 1 means that the transmission at time slot t is successful and ((t) = 0 otherwise. Mathematically, the updating rule of the thresholdbased rate adaptation algorithm can be written as 2 Note that the number of feasible channel states can be potentially infinite. 136 r(t + 1) = (r(t) + o r(tr jFC(tr r...r._C(t)i +(r(t) 6)Fr(t) oFc(t1)o F (td)o +r(t)Fo. (63) where 1, if event x is true Fx = (64) 0, if event x is false. For example, Fc(t)1 = 1 if the transmission at time t is successful and Fc(t)1 = 0 otherwise. The symbol of o.w. denotes the event that neither 9, consecutive successful nor Od consecutive failed transmissions happened. For simplicity, we assume that the maximum allowable data rate is sufficiently large and the minimum data rate is zero, i.e., not transmitting. Define h(t) = [r(t), r(t 1),... r(l), e(s(t), r(t)),... ,e(s(l), r(l))] (65) as the history vector. In addition, we define Z(t + 1) = {r(t + 1)h(t)} (66) where S is the expectation operator. Condition: (C.1) The channel states between two consecutive successful transmissions or two consecutive failed transmissions are independent random variables. We emphasize that the restrictive condition (C.1) is not our general assumption in this work. If (C.1) is satisfied, however, the derivation of the reverse engineering analysis can be presented in a more concise form, as will be shown shortly. 137 First, we obtain S{Fr(c )= 1 (t1)= 1 ... c(.tu)=1h(t)} = Pr {[F(t)= = 1, F(t1) = = 1, ... Fr(tu)1 = lh(t)} (67) If condition (C.1) is satisfied, (67) can be further decomposed as S {FtC(t)=1C(t1)=1 ... c(tu)1 h(t)} = Pr {c(t) = 1lh(t)} x Pr {r(t1)=1= 1lh(t)} x Pr {Fr(t_u)= lh(t)} = ]Ps(s(t k), r(t k)) (68) k= Similarly, we have S (r(t)=o0 ((t1)=O ... (td)O I h(t)} = Pr {r(t)o = 1, Fr(t1)=o = 1, Fr(tu)o = lh(t)} (69) If (C.1) is assumed to be valid, we can obtain S ({F(t)=O (c(t 1)0 ... F (td)= I h(t)} d = (1 Ps(s(t k),r(t k)) (610) k=0 For notation succinctness, we will temporarily assume that (C.1) is satisfied. The condition will be relaxed after the implicit objective function is revealed. Therefore, we 138 can write (66) as Z(t + 1) = S {r(t + 1)lh(t)} (r(t) 6) Ps(s(O k=0 d +(r(t) ) (1 k=0 +r(t) 1 P P( d  (1 Ps(s(t k=0 r(t) +6 [Ps(s(t (k=0 fl(1 k=0  Ps(s(t k),r(t k)) Ps(s(t k), r(t k)) (t k),r(t k)) k),r(t k)) k), r(t k)) Let us revisit (63), which can be rewritten as +(r(t) 6)F(t)=oFc(t1)=o F((td)= +r(t)F.. r(t)) r(t) + 6(t) (612) x (r(t) (r(t) 6)Fc(t) oF(t1) F r(td) o r(t)Fro.w. r(t) I. (613) 139 k),r(t k)) (611) r(t + 1) = r(t) + x ((r(t) where ((t) 6)F (t)=1 FC(t_u)= 6)Fc(t)1Fc(t1)i F(t ) It should be noted that S{((t)h(t)} = Ps(s(t k),r(t k)) k=0 d (i Ps(s(t k),r(t k)). (614) k=0 Therefore, we observe that (612) is a form of stochastic approximation with respect to (611). Next, we present the reverse engineering theorem for the thresholdbased rate adaptation algorithm. Theorem 6.1. The thresholdbased rate adaptation algorithm of (63) is a stochastic approximation which solves an implicit objective function U(t), with a constant stepsize of 6, where U(t) is in the form of U(t) = r(t) Ps(s(t k), r(t k)) k=0 d (1i Ps(s(t k),r(t k)) }+C (615) k=0 if condition (C. 1) is satisfied and /C is a constant with respect to rate r(t). Proof. Theorem 1 follows directly from the previous analysis. Note that the thresholdbased rate adaptation algorithm can be written as r(t + 1) = r(t) + (t) (616) where ((t) is the stochastic gradient and satisfies aU S{~ (t)lh(t)} = r(t)h(t) (617) ar(t) Hence Theorem 1 holds. If condition (C.1) does not hold, we can replace U f Ps(s(t k),r(t k)) k=0 140 and d (1 Ps(s(t k),r(t k)) k=0 in (615) with Pr {FC(t)1, F(t,) h(t)} (618) and Pr {Fc(t)= 0,.. (td)= h(t)} (619) respectively and the theorem remains valid. It is worth noting that the objective function U(t) is a timevarying function which is determined by the data rate as well as the channel conditions of the past 7 time slots where 7 = max(O,, Od). Note that the data rate within the last 7 time slots always remains unchanged. However, the channel fluctuations affect the successful transmission probability Ps and thus alter the objective function U(t). The partial derivative of the objective of U(t) given h(t), i.e., u d (t = Ps(s(t k), r(t k)) (1 Ps(s(t k), r(t k)) (620) k=0 k=0 is also timevarying. If the probability that the last O, transmissions are all successful is greater than the probability that the last Od transmissions are all failures, the station tends to increase the data rate and vice versa. The speed of rate increasing or deceasing is determined by the difference of these two probabilities. In other words, the partial derivative in (620) could be either positive or negative which corresponds to a rate upshift or a rate downshift. The absolute value of the instantaneous derivative determines the speed of rate changing. A direct computation of the partial derivative in (620) is challenging, if not impossible, due to the uncertainty induced by the unpredictable stochastic channel. Therefore, the thresholdbased rate adaptation algorithm, described in (63), utilizes an alternative stochastic approximation approach with the unbiased estimation of (t), i.e., ((t) in (613), which significantly reduces the computational complexity since only local information based on ACKs is required. This nature of simplicity and practicability facilitates the popularity of the thresholdbased rate adaptation algorithm and its variants such as ARF To achieve a better understanding of (615), let us consider the following extreme cases from a reverse engineering standpoint. * (u = 0, d = 0) = (0, = Od = 1): In this case, the thresholdbased rate adaptation algorithm increases the data rate if the current transmission is successful and decreases otherwise. From (615), we can observe that this aggressive algorithm merely compares the successful probability with the failure probability of the current time slot and expects that the next time slot will remain the current channel condition. (u = +oc, d = 0) = (0, = +oC, Od = 1): From (615), we can see that the derivative is always negative since the first part of (620) is zero. Therefore, the thresholdbased rate adaptation will keep decreasing data rate until the minimum data rate is reached, i.e., zero, which is consistent with the intuition. (u = 0, d = +oc) = (0, = 1, Od = +00): In this scenario, the second part of (620) is always zero and thus the algorithm will keep upgrading data rate until the maximum data rate is achieved. (u = +oc, d = +oc) = (0, = oo, Od = +00): The derivative of (620) is always zero and hence the data rate never changes. Hence, by inspecting (615) and (620) directly, we provide an alternative means to understand the behavior of the thresholdbased rate adaptation algorithm. In (63), we have assumed a constant stepsize 6 while in current offtheshelf IEEE 802.11 devices, a discrete set of data rates are provided. However, continuous rate adaptation can be achieved by controlling the transmission power jointly or deploying AdaptiveCodingandModulation (ACM) capable devices. Therefore, our reverse engineering model, while fits in continuous rate scenarios, provides an approximate model for discrete rate adaptation scenarios. We believe that the reverse engineering result in this work provides a first step towards a comprehensive understanding on the 142 goodyetsimple rate adaptation algorithm designs. In addition, based on the unveiled implicit objective function of (615), the interactions of rate adaptations among multiple IEEE 802.11 stations can be investigated in a game theoretical framework. It is immediate to observe from (615) and (620) that the selection of 0, and Od has significant impact on the performance of the rate adaptation algorithm. Ideally, the rate adaptation algorithm attempts to find the data rate which maximizes the expected throughput in the next time slot, i.e., r' = argmaxr x Ps(s(t + 1), r). (621) Define V= r x Ps(s(t+ 1), r). (622) Therefore, if we can estimate the value of av and relate it by OV (t) (623) Or then the thresholdbased rate adaptation algorithm is indeed optimizing the expected throughput via the stochastic approximation approach. However, to achieve a derivative estimation of a general nonMarkovian system is nontrivial [118] [119]. Moreover, if the channel appears totally random, e.g., nonstationary and fast fading, there exists no effective optimizationbased solution unless certain level of knowledge on the channel randomness is available. Therefore, in the next section, we will consider a stationary random channel environment. Nevertheless, the stochastic channel can be slowvarying or fastvarying following arbitrary probabilistic distributions. We propose a threshold optimization algorithm which provably converges to the stochastic optimum values of 0, and Od and the overall performance of the network is remarkably enhanced. 6.4 Threshold Optimization Algorithm In this section, we model the stochastic channel as a stationary random process denoted by s(t). It is worth noting that if the channel is quasistatic, i.e., s(t) is a 143 piecewise constant function, the probingbased rate adaptation algorithms, e.g., SampleRate [102], can achieve good performance. It is interesting to observe that even in a quasistatic environment, for different channel conditions, the optimal values of 0, and Od may be different. One feasible way to find the optimum values of 0, and Od with different channel conditions, in a quasistatic environment, is the samplepath based policy iteration approach introduced in [120] and [121], based on the Markovian model of the thresholdbased rate adaptation algorithm proposed in [110]. However, it is arguable that the channel environment is stochastic and timevarying in nature. Therefore, it is not unusual that the channel condition has already changed before the optimization algorithm has reached a steady state solution. The algorithm will be thereby consistently chasing after a moving object and thus the thresholds are always chosen suboptimally. This is more severe in a fast fading stochastic environment. In light of this, we alternatively pursue the stochastic optimum values of the thresholds in a timevarying and potentially fast changing channel environment. That is to say, we attempt to find the set of values for 0, and Od which maximize the expected system performance with respect to the random channel. Before elaborating further, we briefly outline the learning automata techniques based on which our threshold optimization algorithm is proposed. 6.4.1 Learning Automata Learning automata techniques are first introduced in the control community where in many scenarios, the system is timevarying and stochastic in nature. Therefore, stochastic learning approaches are desired to address the stochastic control problem in random systems. The basic idea of learning automata techniques can be described as follows. We consider a stochastic environment and a set of finite actions available for the decision maker. Each selected action induces an output from the random environment. However, due to the stochastic nature, the outputs for a given input may be different at different time instances. Based on observations, a learning automata algorithm is expected to find an action which is the stochastic optimum solution in the sense that the expected system's objective is maximized. At each decision instance, the decision maker selects one of the actions according to a probability vector. The selected action is fed to the stochastic environment as the input and a random output is attained. The gist of learning automata techniques lies in the provable convergence to the coptimal solution, as will be defined later, in arbitrary stationary random environment. Thanks to the practicality and applicability, learning automata techniques have been broadly studied in various aspects of the communication and networking literature such as [122][123] [124] [125]. In this work, we propose a learning automata based threshold optimization algorithm which finds the stochastic optimum values of the up/down thresholds efficiently in any stationary yet potentially fastvarying random channel environment. The detailed implementations are introduced next. 6.4.2 Achieving the Stochastic Optimal Thresholds We consider an IEEE 802.11 station as the decision maker which adjusts the values of 0, and Od. Without loss of generality, we assume that the maximum value of 0, and Od are m, and md, i.e., the feasible set of 0, is A, = {1, m} and that of Od is given by Ad = {1, md}. The station maintains two probability vectors P, and Pd for A, and Ad, respectively. The kth element in P,, i.e., P,, denotes the probability that u, is set to k. pk is defined analogously. Simply put, at each decision instance, the threshold optimization algorithm randomly determines the values of L, and Od according to P, and Pd. Next, the probability vectors, i.e., P, and Pd, are updated and the iteration continues until convergence, i.e., Prm = 1 and Pn = 1 where m* and n* denote the stochastic optimal values of O, and Od, respectively. Define a time series denoted by t = [to, t1, .. ] where to denotes the starting time and other elements represent the exact time instances when the data rate is changed. 145 Denote T(j) = [tj_, tj], j = 2, .. as the jth time period, or time duration. It is worth noting that within a particular time period, say j, the value of the data rate, the values of thresholds, i.e., 0, and Od are all fixed numbers. We denote these perioddependent parameters as3 r(j), Ou(j) and Od(j). With this observation, we utilize the time series t as the time series of decision instances. More specifically, for example, at time tj, a rate change is triggered either by Ou(j) consecutive successful transmissions or Od(j) consecutive failed transmissions. In tandem with the data rate up/down shift, new values of the thresholds, i.e., 0,(j + 1) and d(j + 1) are determined by the threshold optimization algorithm according to the probability vectors Pu and Pd. Besides Pu and Pd, we introduce additional vectors for A, and Ad, namely, the counting vectors Cu, Cd, the surplus vectors S,, Sd, the estimation vectors Du, Dd and the comparison vectors Z,, Zd. The definitions of parameters of the algorithm are provided as follows. Algorithm: Parameters4:  Pu (Pd): The probability vector for 0, (Od) over Au (Ad).  m. (md): The maximum value of O, (0d), or equivalently, the cardinality of Au (Ad).  CL (Cd): The counting vector of O, (0d) where the kth element, i.e., Ck (Cdk), denotes the times that k has been selected as the value of 9, (Od). Su (Sd): The surplus vector of 9, (0d) where the kth element, i.e., SI (S), denotes the accumulated throughput with Ou = k (Od = k). 3 Note that with a slight abuse of notation, we use j to denote the jth time duration. 4 We present the analogous definitions in the parenthesis. 146 SD (Dd): The estimation vector of 0, (Od) where the kth element, i.e., Dk (Dk), is calculated by DI = (Dk = ). R: The resolution parameter which is a positive integer and is tunable by the station. 6 (6d): The stepsize parameter of Q, (Od) and is given by 6, = R (bd = R). c ((cd): The perturbation vector of L, (Od) where the kth element, i.e., k ( k)), is a zero mean random variable which is uniformly distributed in [ P +P ] P where p is a system parameter and is controllable by the station. The notation of [a, b]y represents [max(a, y), min(b, x)]. SZ (Zd): The comparison vector of L, (Od) where the kth element, i.e., Zk (Zd), is given by Zk = Dk + Pk (Zd = Dk + k). B: The predefined convergence threshold, e.g., 1, which is determined by the station. J: A running parameter which records the updated maximum achieved throughput during one time period. At time to: Initialization: The station sets Pu = [pi, pi,,* pm] where pi = for all 1 < < m. Similarly, Pd is given by [pi, .. pi, Pm] where pi = for all 1 < i < md. Initializes C,, Cd, Su, Sd, DL and Dd to zeros. Randomly selects the values of O6(1) and Od(1), say m, n, according to Pu(1) and Pd(l). Transmits with ,O = m and Od = n until T(1) ends, i.e., a data rate change is triggered. Records the average throughput within T(1) as J. At time tj(j > 1): Do: Records the average throughput during T as 0(j). If O(j) > J, sets J = 0(j) and remains J unchanged otherwise. 147 Updates the mth element in the surplus vector S, and the nth element in Sd by adding the measured normalized throughput of the last time period, as Sm SM +0) and S, = S, + ) Updates the counting vectors by adding one to the mth counter in C, and the nth counter in Cd, as CL = C, + 1 and C = Cn + 1. Updates the mth element in DL and the nth element in Dd by D, = s and d, c " For every element in Z, and Zd, updates Zk = Dk + P, k = 1, me and Zk = Dk + k = 1, ... md. Finds the element in Z, with the highest5 value of ZLk, k = 1, m me, say, the ifth element in A,. Similarly, finds the element in Zd which has the highest Zk, k = 1, .. md, say, the nth element in Ad. Updates the probability vectors of Pu and Pd as Pk = max(PL ,,0) ifkkf/m,k= 1,.. ,m. Pk = 1 Pk if k = (624) and Pdk = max(P k d,0) if k/n,k= 1, ,md Pd = 1 Pdifk= (625) where = 6 1 and d= 1 muxR mdxR With the updated probability vectors Pu and Pd, new values of Ou and Od, i.e., Ou(j + 1) and Od(J + 1), are selected. Starts the transmissions in T(j + 1) with O,(j + 1) and Od(j + 1). Until: max(PP) > B and max(Pd) > B where B is the predefined convergence threshold. 5 Note that a tie can be easily broken by a random selection. 148 End The proposed threshold optimization algorithm is similar to the stochastic estimator learning automata proposed in [126]. The key feature of this genre of learning automata is the randomness deliberately introduced by p and Note that although p and O are zero mean random variables, their variances are dependent on the values of C, and Ck, respectively. Specifically, the variances approach to zeros with the increase of the number of times that the corresponding values of 9, and Od are selected. As a consequence, the threshold optimization algorithm inclines to more reliable stochastic estimates and thus possesses a faster convergence behavior than other learning algorithms [126]. The values that have been selected less frequently still have the chance of being considered as optimal. However, the missing probability diminishes to zero along iterations. In the algorithm, the resolution parameter R controls the stepsize of probability adjustment in the algorithm. A smaller value of R produces a finegrained probability adjustment yet unavoidably prolongs the convergence time. The convergence threshold B determines the stopping criteria of the algorithm. Therefore, a tradeoff between optimality and convergence rate can be adjusted by tuning the value of B. The steady state behavior of the proposed threshold optimization algorithm is provided in the following theorem. Theorem 6.2. The proposed threshold optimization algorithm is coptimal for any stationary channel environment with arbitrary distribution. Mathematically, for any arbitrarily small e > 0 and y > 0, there exists a t' satisfying Pr{1 P  < e} > 1 7 Vt > t' (626) and Pr{ll P* < e} > 1 yVt > t' (627) where m* and n* are the stochastic optimal values of O, and Od, respectively. 149 The proof of Theorem 6.2 follows similar lines as in [126] and is omitted for brevity. For more discussions on the stochastic estimator algorithms, refer to [127] and [128]. In the next section, we will demonstrate the efficacy of our proposed threshold optimization algorithm via simulations. 6.5 Performance Evaluation In this section, we evaluate the performance of the proposed threshold optimization algorithm with simulations. For comparisons, we first implement the heuristicsbased threshold adjustment algorithms in [93] and [94]. In [93], the downshift threshold Od is fixed to 1 and the default value of 0, is 10. After 0, successful transmissions, a data upshift is triggered and if the first transmission after the rate upshift is successful, the algorithm assumes that the link quality is improving rapidly [93]. Consequently, 0, is set to a small number, e.g., 0, = 3, in order to capture the fast improving channel. Otherwise, the algorithm assumes that the channel is changing slowly and thus a larger value of 0, is desired, e.g., 0, = 10. We denote this threshold adaptation scheme as DLA in our simulations. Another wellknown threshold adjusting scheme, namely, AARF, is proposed in [94]. Similarly, the rate downshift threshold is fixed to Od = 2 empirically. However, for O,, a binary exponential backoff scheme is applied. If the first transmission after a rate upshift failed, the data rate is switched back to the previous rate and the value of 0, is doubled with a maximum value of 50. The value of 0, is reset to 10 whenever a rate downshift is triggered. To simulate the indoor office environment for IEEE 802.11 WLANs, we simulate a Rayleigh fading channel environment. In other words, we assume a flat fading environment. However, it could be either a fast fading or slow fading channel. The Doppler spread (in Hz) corresponds to the channel fading speed where a large Doppler spread value represents a fast fading channel and a small Doppler spread indicates a slowvarying channel. It should be noted that the Doppler spread value describes the time dispersive nature of the wireless channel [129], which is inversely proportional to 150 the channel coherence time. More specifically, we relate the channel coherence time Tc and the Doppler spread value fm as [129] 0.423 T =423 (628) fm Therefore, a larger Doppler spread value indicates a small channel coherence time which represents a fast fading scenario. We conduct the simulations under various channel fading conditions via different Doppler spread values. We consider the IEEE 802.11 b PHY specification, i.e., the available data rates are 1 Mbps, 2 Mbps, 5.5 Mbps and 11 Mbps and the RTS/CTS signalling scheme is turned off. Since the objective of our proposed threshold optimization algorithm is to combat with the channel variation, the collision effect is omitted. Therefore, we consider a WLAN with one station consistently transmitting packets to the AP However, note that [112] provides a complementary solution to mitigate the collision effect and thus our algorithms can work collectively as a joint solution. We emphasize that this simplification does not induce any loss of generality since although seemingly simple, it produces all the challenging problems involved in rate adaptation algorithms in a timevarying stochastic channel environment. The data traffic is generated using constant bit rate UDP traffic sources and the frame size is set to 1024 octets. The power of the transmitter and the power of thermal noise are set to 50 mW and 1 mW, respectively. The SNRBER relation is given by Table 61 which is derived from [130]. We vary the Doppler spread value to simulate the various channel fading speeds. For each value of Doppler spread value, the simulation is executed for 1000 seconds and the average throughput of the following algorithms6 which are commonly based on the 6 The performance comparison of ARF and other openloop rate adaptation algorithms such as SampleRate and ONOE has been studied extensively in [96] and [98] for various channel conditions. thresholdbased up/down mechanism, are compared: (1), OP the optimum throughput attained by assuming an oracle which foresees the variation of channel and adapts the data rate optimally. This curve is attained as a performance comparison benchmark; (2), ARF the thresholdbased rate adaptation algorithm with Ou = 10 and Od = 2; (3), DLA  the dynamic threshold adjustment algorithm proposed in [93]; (4), AARF the threshold adjustment algorithm proposed in [94]; (5), TOA the threshold optimization algorithm proposed in this work. The algorithms are compared with each other in terms of the average system throughput (in Mbps). For TOA, without loss of generality, we assume that m, = md = 10 and the resolution parameter R is 1. The convergence threshold B is 0.999 and p is 1. The performance curves of the aforementioned algorithms are plotted in Figure 61. In Figure 61, we observe that except the OP curve, all other rate adaptation algorithms suffer from performance degradations with large Doppler spread values. Recall that a large Doppler spread value indicates a fast fading channel environment and hence the average throughput deteriorates due to the incompetency of capturing short channel fluctuations. Among which, ARF and AARF provide worst performance with a slight difference. DLA performs better due to the capability of switching between Table 61. SNR v.s. BER for IEEE 802.11 b data rates SNR BPSK QPSK CCK CCK (dB) (1Mbps) (2Mbps) (5.5 Mbps) (11 Mbps) 1 1.2E5 5E3 8E2 1E1 2 1E6 1.2E3 4E2 1E1 3 6E8 2.1E4 1.8E2 1E1 4 7E9 3E5 7E3 5E2 5 2.3E10 2.1E6 1.2E3 1.3E2 6 2.3E10 1.5E7 3E4 5.2E3 7 2.3E10 1E8 6E5 2E3 8 2.3E10 1.2E9 1.3E5 7E4 9 2.3E10 1.2E9 2.7E6 2.1E4 10 2.3E10 1.2E9 5E7 6E5 152 2 4 6 8 10 12 14 Doppler Spread Value (Hz) Figure 61. Average throughput v.s. Doppler spread values the large value and the small value of 0, for different channel conditions. Our proposed threshold optimization algorithm, as demonstrated in Figure 61, consistently outperforms other openloop rate adaptation algorithms and bridges the performance gap with the OP curve. This superiority becomes remarkably significant in fast fading stochastic channel environments, i.e., scenarios with large Doppler spread values. While other algorithms are jeopardized by the unmature rate changes due to the uncertainty caused by the random and fastvarying channel environment, our proposed threshold optimization algorithm, based on the learning automata techniques, is able to find the stochastic optimum values of 0, and Od which maximize the expected system performance. Therefore, our scheme is particularly suitable for fast changing yet statistically stationary random channel environments where other openloop rate adaptation solutions usually provide unsatisfactory performance. To illustrate the process of finding the stochastic optimum values of ~, and Od, in Figure 62, we provide the trajectories of 0, and Od in a sample simulation run with a fixed Doppler spread value of 10. It is observable that starting from the initial point, the threshold optimization algorithm adapts the values of 0, and Od on the fly along with the rate changes. The algorithm soon finds the stochastic optimum values and 0, and Od converge to the optimum solutions effectively. The evolutions of the probability vectors, 153 7 6 5 4 3 3 0 20 40 60 80 100 Number of Transmissions Figure 62. The trajectories of 0, and Od in achieving the stochastic optimum values i.e., P, and Pd, are demonstrated in Figure 63 and Figure 64, respectively. Note that, in Figure 63, the sixth curve which represents the value of P6 soon excels others and approaches to 1. Correspondingly, the value of O, converges to 6 rapidly as depicted in Figure 62. In Figure 64, the second curve approaches to unity gradually while others diminish to zeros. As a consequence, in Figure 62, the value of Od converges to the stochastic value, i.e., 2. In this sample run of simulation, the stochastic optimum values of 9, and Od, which maximizes the expected throughput of the system, is given by 8, = 6 and Od = 2. The proposed threshold optimization algorithm finds these values effectively and efficiently, while providing superior performance than other nonadaptive thresholdbased rate adaptation algorithms, as demonstrated in Figure 61. 9o P1 P S , . Figure 63. Evolution of the probability vector Pu 154 Figure 64. Evolution of the probability vector Pd 6.6 Conclusions and Future Work In this chapter, we investigate the thresholdbased rate adaptation algorithm which is predominantly utilized in practical IEEE 802.11 WLANs. Although widely deployed, the obscure objective function of this type of rate adaptation algorithms, commonly based on the heuristic up/down mechanism, is less comprehended. In this work, we study the thresholdbased rate adaptation algorithm from a reverse engineering perspective. The implicit objective function, which the rate adaptation algorithm is maximizing, is unveiled. Our results provide, albeit approximate, an analytical model from which the thresholdbased rate adaptation algorithm, such as ARF, can be better understood. In addition, we propose a threshold optimization algorithm which dynamically adapts the up/down thresholds, based on the learning automata techniques. Our algorithm provably converges to the stochastic optimum solutions of the thresholds in arbitrary stationary yet potentially fast changing random channel environment. Therefore, by combining our work with the threshold adaptation scheme in [112], where the thresholds are adjusted to mitigate the collision effects, a joint collision and random channel fading resilient solution can be attained. In this work, to emphasize on the impact of random channel variations, we restrict ourselves to the scenario where all stations have a fixed and known transmission probability p. The interaction of the transmission probabilities and the data rates seems 155 Number of Transmissions interesting and needs further investigation. Due to the competitive nature of channel access, a stochastic game formulation may be utilized. 156 CHAPTER 7 STOCHASTIC TRAFFIC ENGINEERING IN MULTIHOP COGNITIVE WIRELESS MESH NETWORKS 7.1 Introduction The past decade has witnessed the emergence of new wireless services in daily life. One of the promising techniques is the metropolitan wireless mesh networks (WMN), which are envisioned as a technology which advances towards the goal of ubiquitous network connection. Figure 71 illustrates an example of wireless mesh network. The wireless mesh network consists of edge routers, intermediate relay routers as well as the gateway node. Edge routers are the access points which provide the network access for the clients. The relay routers deliver the traffic aggregated at the edge routers to the gateway node, which is connected to the Internet, in a multihop fashion. *................ ... ........... Figure 71. Architecture of wireless mesh networks While the current deployed wireless mesh networks provide flexible and convenient services to the clients, the performance of a mesh network is still constrained by several each other and thus the network performance is devastated. To address this problem, Figure 71. Architecture of wireless mesh networks While the current deployed wireless mesh networks provide flexible and convenient services to the clients, the performance of a mesh network is still constrained by several limitations. The first barrier is due to the multihop nature of the wireless mesh network, where the nodes in geographic proximity generate severe mutual interference among each other and thus the network performance is devastated. To address this problem, several scheduling schemes have been proposed in the literature [131]. Recently, a novel codingbased scheme which may produce an interferencefree wireless mesh 157 network, is proposed [132]. Another example of the interferencefree network is the CDMAbased wireless mesh networks [133] where by assigning orthogonal codes for each link, the network throughput is remarkably improved. The second hinderance for the network performance is the limited usable frequency resource. In current wireless mesh networks, the unlicensed ISM bands are most commonly adopted for backbone communications. Not surprisingly, the wireless mesh network is largely affected by all other devices in this ISM band, e.g., nearby WLANs and Bluetooth devices. Moreover, the limited bandwidth of the unlicensed band cannot satisfy the increasing demand for the bandwidth due to the evolving network applications. Ironically, as shown by a variety of empirical studies [134], the current allocated spectrum is drastically underutilized. As a consequence, the urge to explore the unused whitespace of the spectrum, which can significantly enhance the performance of the wireless mesh networks, attracts tremendous attention in the community [135139]. Cognitive radios are proposed as a viable solution to the frequency reuse problem [131]. The cognitive devices are capable of sensing the environment and adjusting the configuration parameters automatically. If the primary user, i.e., the legitimate user, is not using the primary band currently, the cognitive devices, namely, secondary users, will utilize this whitespace of the spectrum. Incorporating with the established interferencefree techniques such as [132] and [18], the throughput of the wireless mesh network can be dramatically enhanced. The protocol design for cognitive wireless mesh networks (CWMN), or more generally, multihop cognitive radio networks, is an innovative and promising topic in the community [140] and has been less studied in the literature. In this work, we consider a cognitive wireless mesh network where the unlicensed band, e.g., ISM band, is utilized by the mesh routers for the backbone transmission. Moreover, each router is a cognitive device and hence is capable of 158 sensing and exploiting the unused primary bands for transmissions whenever the primary users are absent. In this work, we investigate an important yet unexplored issue in the cognitive wireless mesh networks, namely, the stochastic traffic engineering problem. More specifically, we are particularly interested in how the traffic in the multihop cognitive radio networks should be steered, under the influence of random behaviors of primary users. It is worth noting that given a routing strategy, the corresponding network's performance, e.g., the average queueing delay encountered, is a random variable. The reason is that the available bandwidth for a particular link depends on the appearance of all the affecting primary users. If all the primary users are vacant, a link can utilize all available frequency trunks collectively by utilizing advanced physical layer techniques, e.g., OFDMA. However, if all the primary users are present, the only available frequency space is the unlicensed ISM band and thus the traffic on this link will experience longer delay than the previous case. In other words, the performance of a traffic engineering solution hinges intensely on the unpredictable random behaviors of the primary users. We emphasize that in multihop cognitive radio networks, this distinguishing feature of randomness, induced by the random behaviors of primary users, must be taken into account in protocol designs. Due to the location discrepancy, it is possible that some node is affected by many primary users while others are not. As a consequence, if we route the traffic via this particular node, the transmissions are more likely to be corrupted by the returns of the primary users. Apparently, a favorable solution is more inclined to steer the traffic from those "severelyaffected area", to the paths which are less affected by the primary users. We will make this intuitive approach precise and rigorous in this work. To our best knowledge, this work is the first work on the traffic engineering problem in multihop cognitive radio networks, with a special focus on the impact of random behaviors from the primary users. 159 The rest of this chapter is organized as follows. The related work is reviewed in Section 7.2 and Section 7.3 provides the system model of our work. The stochastic traffic engineering problem with convexity is investigated in Section 7.4. In Section 7.5, we extend our framework to the nonconvex stochastic traffic engineering problem. Performance evaluation is provided in Section 7.6, followed by concluding remarks in Section 7.7. 7.2 Related Work Traditional traffic engineering (TE) algorithms are proposed as the solution to the traffic management of the network in a costefficient manner. Different from the traditional QoS routing, the traffic engineering solution not only guarantees a certain QoS level for each flow, but also optimizes a global performance metric over the whole network, by splitting the ingress traffic optimally among several available paths. The multipath routing is usually supported by the MultiProtocol Label Switching (MPLS) techniques where the explicit routing path for a packet is predetermined rather than being computed in a hopbyhop fashion. For a pair of source and destination nodes, the set of available paths, a.k.a., label switched paths (LSP), are established and managed by signalling protocols such as RSVPTE [141] and CRLDP [142] or manually configuration. The traditional traffic engineering solution evolves to the stochastic traffic engineering (STE) solution when uncertainty exists in the network, e.g., the random returns of the primary users in our scenario. TE solutions require consistent route changes which are unfavorable in that the network will be overwhelmed by the oscillations induced by the unpredictable behaviors of the primary users. In light of this stability concern, STE solution alternatively pursues an optimum multipath routing strategy such that the expected utility of the network is maximized. The stochastic traffic engineering with uncertainties are discussed in the literature such as in [143] and [144]. However, the previous works usually assume a probability distribution of the uncertainty, while in our scenario, the behaviors of the primary users are completely unpredictable 160 from the mesh network's point of view. Distinguishing from the previous works, we propose an algorithmic solution which requires no prior knowledge about the distribution of the uncertainty, in a stochastic network utility maximization framework. It is worth noting that our work differ from the traditional statedependent traffic engineering solution as well. For example, in [145], a state dependent traffic engineering solution is proposed. However, the authors assume that the system state, i.e., the current value of uncertainty, is fully observable. In our approach, we do not assume that the ingress node has the perfect knowledge of the current appearance of the network. We will discuss this issue in detail in Section 7.4. Recently, cognitive wireless mesh networks (CWMN) have attracted great attention in the literature. In [135], the channel assignment is discussed in a CWMN. In [136], a clusterbased cognitive wireless mesh network framework is proposed. The infrastructurebased cognitive network is discussed in [137] with a focus on the cooperative mobility and the channel selection schemes. The spectrum sensing and channel selection are jointly considered in a unified framework in [138]. In addition, the IEEE 802.16h is in the process of incorporating the cognitive radios into the WiMAX mesh networks [139]. However, none of the previous works considers the stochastic traffic engineering problem. Therefore, a systematic study of the impacts of the random returns from the primary users, on the network routing performance is lacking in the existing literature. In [146], the joint congestion control and traffic engineering problem is considered. He et.al. propose a distributed algorithm to balance the user's utility and the system's objective. However, the authors assume the environment is fixed and does not consider the randomness which is the distinguishing yet usually overlooked feature in cognitive radio networks. [147] and [148] discuss the routing issue in cognitive radio networks yet the impact of random returns of primary users is not investigated. Hou et.al. [149] [150] [151] [152] formulate the joint routing, power and subband allocation problem in cognitive radio networks as a mixedinteger programming. However, the channels' bandwidths are assumed to be fixed, i.e., the random behaviors of primary users are still neglected. Our work is partially inspired by [153]. However, our work differs from theirs in three crucial aspects. First, by targeting the stochastic traffic engineering problem, our model differs from the joint power scheduling and rate control work in [154]. Secondly, in [153, 154], the authors only consider a singlepath scenario while our work extends to a multipath routing network where the network traffic can be steered. Thirdly and most importantly, [153, 154] require that the current system state is fully observable at the decision maker. To achieve this, the authors assume a centralized mechanism which knows all the channel states of all the links over the network. However, our work differs from [153, 154] significantly in that we do not require that the current system's state is known, which is of great practical interest since in multihop cognitive wireless mesh networks, the decision makers, i.e., the edge routers in our scenario, cannot be aware of the appearance of all primary users in the whole network as a priori. Moreover, our schemes enjoy a decentralized implementation, in contrast to centralized mechanisms in [153, 154], by utilizing the feedback signals and local information only. In our previous work of [20], we proposed a routing optimization scheme to combat with the randomness of instantaneous traffic in noncognitive wireless mesh networks. With respect to [20], this work differs in the following ways. First, in the wireless mesh networks considered in [20], the capacity of each wireless link is assumed to be fixed, i.e., timeinvariant. However, in cognitive wireless mesh networks, due to the unpredictable appearance of primary users, the bandwidth of each wireless link is random. Secondly, in [20], the quality of service (QoS) requirement is not considered. Nevertheless, in this chapter, we particularly address the QoS concern of each user, e.g., the expected accumulated delay on the paths cannot exceed a userspecific delay tolerance, as will be elaborated in Section 7.4. Thirdly and most importantly, the analysis in [20] was based upon the assumption that the users all have convex utility functions. In this work, we extend the techniques to address the scenarios 162 with nonconvex utility functions. We will discuss the aforementioned issues further in the following sections. 7.3 System Model We consider a multihop wireless mesh network illustrated in Figure 71 where an uplink traffic model is considered, i.e., all edge routers aggregate the traffic from clients and deliver to the gateway node via the intermediate relay routers. To ensure connectivity, we utilize the ISM 2.4G band as the underlying common channel for the wireless mesh network. In addition, each link can utilize the opportunistic channels, i.e., secondary bands to increase the link's achievable data rate whenever the primary user is vacant. We assume that there exists1 IMI primary users. Each primary user possesses a licensed frequency channel and each mesh router is a cognitive node which has the capability of sensing the current wireless environment. We model the multihop cognitive wireless mesh network as a directional graph g where the vertices are the nodes. We also denote link (i,j) as link e, e c E where t(e) = / and r(e) = j represent the transmitter and the receiver of link e. We first consider a particular link denoted by (m, n). The instantaneous available frequency bands, at time t, for a node / is denoted by I,(t), which is determined by the current presence of the primary users. Besides the underlying ISM band, the communication between m and n can further utilize all secondary bands within Im(t) q In(t), if available. The current cognitive radio devices benefit largely from the softwaredefined radio (SDR) techniques with advanced coding/modulation capabilities. For example, by utilizing the multicarrier modulation, e.g., OFDMA, a cognitive radio device can utilize all the disjoint available frequency band simultaneously [149, 150, 155157]. At the transmitter, a software based radio combines waveforms for different subbands and thus transmit signal at these subbands simultaneously. 1 The symbol of X represents the cardinality of the set X. 163 While at the receiver, a software based radio decomposes the combined waveforms and thus receives signal at these subbands simultaneously [151, 152, 156, 157]. In this work, we assume a spectrum sensing scheme available that each node can sense the presence of the primary users in range, such as [131, 158], although the time of random returns cannot be predicted. A link will utilize all the available vacant bands and that the cognitive radios are fullduplex and can transmit at different bands concurrently [151, 152, 156, 157]. We further assume that some scheduling mechanism is in place or some physical layer mechanisms are utilized such that the nodes cannot interfere with each other during the transmissions. For example, in a multichannel multiradio wireless mesh network, the channels can be assigned properly that the transmissions do not interfere with the neighboring nodes [132, 159]. Other examples are the OFDMA/CDMA based wireless mesh networks [133, 160] where the interference among nodes can be eliminated by assigning orthogonal subcarriers/codes. We emphasize that this assumption is only for the sake of modeling simplicity and does not incur any loss of generality, as will be clarified shortly. It is worth noting that the available bandwidth of each link in the cognitive wireless mesh network is a random variable. For example, at time instance tl, node m has three secondary bands available, i.e., Im(ti) = {/0, /1, 2, Is} and In(t1) = {/0, 2, ,/, 14 } due to the location discrepancy, where band 0 is the underlying unlicensed ISM band and 1, 2, 3, 4, 5 are the licensed bands of primary users. The current bandwidth of link (m, n) is represented by Wm,n(tl) = BWo + BW2 3 where BW, is the bandwidth of band i. At another time instance t2, the primary user 2 returns and the bandwidth of link (m, n) becomes Wmn(t2) = BWo + BW3. In other words, the bandwidth of links are random variables which are determined by the unpredictable appearance of the primary users. We model this randomness induced by the primary users as a stationary random process with arbitrary distribution. The system is assumed to be timeslotted. In each time slot n, the system state is assumed to be independent and is denoted by a state vector s = {51, ... 61m }, s e S, where 5, = 1 denotes the absence of the ith primary user and 0 otherwise. We denote the stationary probability distribution of state s as Ts. For the ease of exposition, we assume that the primary users are static. However, we emphasize that our model can be extended to mobile primary users scenarios straightforwardly. For example, if a primary user is moving following a Markovian walk model with well defined steady state distribution, the following analysis still applies. Without loss of generality, we express the link capacity in the form of CDMAbased networks, i.e., the capacity of a wireless link e e E, given the system state s, is denoted by cs, which is given by [4, 161] c W = Ws l Iog2(1 + Ky~), where Ws is the bandwidth of link e in state s and 7y is the current SINR value of link e. The constant T is the symbol period and will be assumed to be one unit without loss of generality [4]. The constant K = iog(ER) where 01 and 02 are constants depending on the modulation scheme and BER denotes the bit error rate. We will assume K = 1 in this work for simplicity [161]. Note that our network model can be incorporated into other types of networks such as MIMO, OFDM with TDMA or CSMA/CA based MAC protocols by modifying the form of the capacity accordingly, which represents the achievable data rate in general. For example, if we consider a schedulingbased MAC protocol where each link obtains a time share of the channel access, the achievable data rate is given by c c = cq x ,' where be is the fraction of time that the link is active following the scheduling scheme and ce is the nominal Shannon capacity of the link. There are LI unicast sessions in the network, denoted by set L, where each session / has a traffic demand di. We associate each session with a unique user. Therefore, we will use session I and user I interchangeably. For each session I e L, we denote the source node and destination node as 5(I) and D(I), respectively. Recall that we assume an uplink traffic model and thus all the source nodes are edge routers and the destination node is the gateway. Furthermore, to improve the reliability and 165 dependability, we allow multipath routing schemes. We denote the available2 set of acyclic paths from 5(1) to D(I) by P/ and the kth path is represented by P /. We introduce a parameter rk as the flow allocated in the kth path of session I. The overall flow of user I, represented by x1, is given as x= r (71) k= 1 0 where [x]b denotes max{min{b, x}, a}. Define an EbyIPi matrix H1 where the element H".k = 1 if link e is on the kth path of P/ and 0 otherwise. Hence, H = {H1, [ HIL} represents the network topology. Note that the traffic splitting and the source routing are executed on the source node 5(1). For each link e e E, there is an associated cost function, denoted by I(fe, cs) where fe is the accumulated flow on link e. We assume the function I/ is an increasing, differentiable and convex function of fe for a fixed ce. For example, if we assume Il(fe, ci) = 1 when cs > fe, the cost essentially represents the delay for a unit flow on link e under the M/M/1 assumption. Note that in our scenario, even the accumulated flow fe is fixed, the value of cost function is random due to the statedependent variable ci. From the network's perspective, the stochastic traffic engineering solution will distribute the aggregated flow among multiple paths optimally, in the sense that the overall network utility is maximized. In next section, we will formulate the stochastic traffic engineering problem in a stochastic network utility maximization framework [4] and provide a distributed solution which requires no priori information about the underlying probability distribution, i.e., 7s, of the system states. 2 The available set of multiple paths can be obtained by signalling mechanisms such as RSVPTE[141] or preconfigured manually. In this work, we assume a predetermined set of acyclic paths. The protocol design for acquiring such paths is beyond the scope of this work. 166 7.4 Stochastic Traffic Engineering with Convexity 7.4.1 Formulation In the standard network utility maximization framework, each user has a utility function Ul(xl), which reflects the degree of satisfaction of user / by transmitting at a rate of x1, e.g., Ul(x/) = log(x/). In this section, we assume the utility functions to be concave and differentiable. The nonconvex utility functions are considered in Section 7.5. Note that the fairness issue can be embodied in the utility functions [4]. For example, in the seminal paper [113], the logutility functions are adopted to achieve the proportional fairness among different flows. Define a feasible stochastic traffic engineering solution as r = [rl, .. riI] where r/ [r/1, r' ]. We can formulate the stochastic traffic engineering problem as P: max U (Y rk) ICL kGcP, s.t. r1k < dl V/ E L (72) kcG 7, r/k e, c's) < bi VI e L (73) sGS k fe < 7,csVe E (74) scS fe = S ',krk Ve GE (75) /EL keP, cs = We 1og2(1 + Kys) Ve E (76) T where e e P represents the links along the kth path of user I. The variable in P, is the vector of r. The first set of constraints reflect that the overall data rates of all paths cannot exceed the traffic demand dl. The second set of constraints indicate that for 167 each user /, the expected cost has to be no more than a predefined constraint bi. The third set of constraints represent that the aggregated flow on link e cannot exceed the average link capacity. Apparently, if the underlying probability distribution of each state 7s is known as a priori, Pi is a deterministic convex optimization problem and thus easy to solve. However, in practice, the accurate measurement of probability distribution is a nontrivial task. In [20], we utilized a stochastic approximation based approach to circumvent the difficulty of estimating the probability distribution. In the following, we will extend this technique and develop a tailored distributed algorithm to address the issues of timevarying link capacities as well as the userspecific QoS requirements, which are of particular interest in multihop cognitive wireless mesh networks. First, define the Lagrangian function of P1 as L(r, A, v) /IL keP I kePL Svi bi T (s e (f, Cs)) I/EL sGS kEPI eEPk Y.ie(fe SscD) eEGE sES e6]E sGS = s Y Ul( r) + A (d r )+ vl bl sIc Ik kk kc sGS /GL \ keP, keP 5r ( ik (vi(fe, cPe))} kIEPE ) +eG P E) v k< eP, / eE )J 168 Define M5(A, p, v) = sup Y U (U/(Z rk) + A/(d r/) + vib r>0 ILO kEIP, keP, rk(l(fe, Cs )+ le)) + Jes (77) kEPi eP, / eGE Let f be the optimum solution of (77). We will discuss how to obtain f shortly. The dual function of P1 is obtained by g(A, p, v) = TsMA(A,, v). (78) scS Thus, the dual problem of P1 is given by P2 min g(A,j, v). (79) A,'P,v>O 7.4.2 Distributed Algorithmic Solution with the Stochastic PrimalDual Approach In this subsection, we propose a distributed algorithmic solution of Pi, or equivalently p2, based on the stochastic primaldual method. In order to reach the stochastic optimum solution, the dual variables A, p and v are updated according to the following dynamics A(n + 1) = [Ai(n) ai(n)(C(n)] V/E IL (710) pe(n+ 1) = [Pe(n) ae(nn)e(n)] Ve E (711) vl(n + 1) = [v(n) b(n)pi(n)]+ V/ E L (712) where [x]+ denotes max (0, x) and n is the iteration number. a (n), ae(n) and ab(n) are the current stepsizes while (1(n), _e(n) and p/(n) are random variables. More precisely, they are named the stochastic subgradient of the dual function g(A, p) and the following 169 requirements need to be satisfied ({(,(n)lA(1), A(n)} = Ox,g(A, p, v) V/I L (713) {(e(n)i(1),' ,~ (n)} = 9,g(A, p, v) Ve E (714) ({p/(n)v(1), ... v(n)} = O,g(A, p, v) V/I L (715) where ((.) is the expectation operator and A(1), A(n), p(1), p(n) and v(1), v(n) denote the sequences of solutions generated by (710), (711) and (712), respectively. By Danskin's Theorem [162], we can obtain the subgradients as (,(n) = d TIk(n) V/I L (716) kEP )e(n) = c(n) fe(n) Ve E (717) p,(n) = bi Tk(n) I I(fe(n), c(n)) V/ E L (718) kIEPI eGeP where /k is the optimum solution of (77). Note that ce(n) denotes the instantaneous channel capacity on link e at iteration n. We next show how to calculate Ms(A, p, v) in (77), i.e., finding the optimum solution, denoted by r, which maximizes ( r + A(di rk) + vib, /IL \ kEP, kElP, r/k(V/(fec )+ e)) +ece (719) kIEPi eEPk eGE Note that when updating the primal variable, i.e., r, the link costs are deterministic which are obtained via the feedback signal, e.g., ACK messages. Therefore, by utilizing the same stochastic subgradient approach, we have rkn + 1) = [rk(n) + a(n)l(n)] d(720) YI \"I/ "/ O 720 170 where (n) k rIk( A (le vlle(fe, cS)) (721) ee pk is the stochastic subgradient measured at time n. Theorem 7.1. The proposed algorithm converges to the global optimum of Pz with probability one, if the following constraints of stepsizes are satisfied: (1) a(n) > 0, (2) Z0 a0(n) = oo, and (3) 0(a(n))2 < oo, V/I L and e E, where a represents ae, a/, ab and ar generally. Proof. First, let us revisit the updating equations of (710), (711) and (712). Note that in the stochastic subgradient approach, the measured values of (1(n), e(n) and pl(n) are considered as the instantaneous observation of the real gradients, denoted by (1(n), (e(n) and p/(n), respectively. We consider the relationship of (1(n) and (/(n) for instance. The observation value, i.e., (1(n), can be rewritten as (1(n) = (i(n) 6(i(n)) + 6((n)) i(n) + (n) = i(n) + ((/(n)) (/(n) + (/(n) (( (n)) = (/(n) + C/(n) + (/(n) (722) where is the expectation operator and (n) = ((i/(n)) (<(n) (723) 1(n) = /(n) ((C/(n)). (724) Note that /(n) is the difference between the expectation of the observations and the real gradient. Hence, it is the biased estimation error term. Next, we examine that Q(V(n)1/(" 1), (0)) = 0 Vn. (725) Therefore, the series of ((n) is a martingale difference sequence [163]. The relationship of (722) indicates that the observation value is the real gradient disturbed by a biased estimation error as well as a martingale difference noise. We next investigate the convergence conditions of the stochastic primaldual approach. For C/(n), the following requirement ae(n) (Ci(n)) ((n) < co (726) n=0 is satisfied due to the stationary assumption. Similarly, 2 (((/(n) ) = (((/(n) ((i/(n)))2) (727) is bounded as well. The similar analysis can be extended to _e(n), pi(n) and rT(n) in (711), (712) and (720) straightforwardly. Therefore, the standard conditions are satisfied and the convergence result of Theorem 1 follows [8]. It is worth noting that the aforementioned distributed algorithm enjoys the merit of distributed implementation from an engineering perspective. With the current values of dual variables, each source node 5(I) optimizes (719) according to (721) and (720). The information needed is either locally attainable or acquirable by the feedback along the paths. For example, the channel states of the intermediate nodes along paths can be piggybacked by the endtoend acknowledgement messages from the destination node, i.e., the gateway node in our scenario. The source node updates the A/ and v1 according to (710) and (712) where the needed information is calculated by (716) and (718), respectively. For each link e, the current status of (717) is measured. Next, the value of /i is updated following (711). The iteration continues until an equilibrium point is reached. Note that our framework can incorporate the wireless lossy network scenarios by replacing the flow rate with the effective flow rate in the leakypipe flow model [164]. 172 7.5 Stochastic Traffic Engineering without Convexity Thus far, we have considered the scenarios where all the users have concave utility functions. However, in practice, several network applications may possess a nonconcave utility function. For example, in a data streaming application, the user is satisfied if the achieved data rate exceeds a threshold, where the utility function is a step function and thus the convexity does not preserve. Therefore, the proposed stochastic primaldual approach in Section 7.4 cannot be applied here. It is worth noting that we can still formulate the stochastic traffic engineering problem as in Pi except that the optimization problem is a stochastic nonconvex programming, which is NPhard in general, and computationally prohibitive to solve even in a centralized fashion [165]. In the following section, we will propose an algorithmic solution to the nonconvex stochastic traffic engineering problem, based on the learning automata techniques. Moreover, we analytically show that the proposed algorithm will converge to the global optimum solution asymptotically, in a decentralized fashion. We first convert the compact strategy space of each user into a discretized set denoted by R. More specifically, each user, say /, maintains a probability vector p/,k for each path k e P/. The segment of [0, rm] is quantized into Q sections where rm is the maximum allowed transmission rate on any path. In other words, the continuous variable rk is transformed into a discrete random variable, r within a discretized set R with Q 1 elements. The data rate is randomly selected from R according to the probability vector of p/,k where the qth element, pk, q = 0, .. ,Q, denotes the probability that the /th user transmits with a rate of r/k = q x r on the kth path of P/. Associated with each probability vector P/,k, there is a weighting vector w/,k with the same dimension of 1 x (Q + 1). The probability vector p/,k is uniquely determined by the weighting vector w/,k by the softmax function [166], e Vk pq Q V, k, q (728) q=0 ewl,k 173 where wqk is the qth element of w/,k, q = 0,... Q. Next, we formulate an identical interest game where the players are the L source nodes and the common objective function is the overall network utility, i.e., the summation of the utility functions. In addition, for each source node, a team of learning automata [167] is constructed. At each time step, every source node picks the data rates on its own paths according to the probability vectors, which are determined by the weighting vectors. Based on the feedback signal E, which will be defined shortly, each source node adjusts the weighting vectors and the iteration continues. The executed algorithm on every source node, say /, is provided as follows. Algorithm: Repeat: For every path, say k, randomly selects a transmission rate rik from R, according to the current probability vector plk(n) where n denotes the current time slot. After receiving the feedback signal E(n) from the gateway node, if the cost constraint is satisfied, the weighting vector w/,k is updated as e /k , W,k(n 1)= [(n) + (n)wk(n)] for q for q(729) wpk(n  )=[ Wk(n) + /(0 J;k(n) Otherwise, the weighting vector remains the same. The probability vector pl,k is then updated, following equation (728). Until: max(plk(n + 1)) > B where B is a predefined convergence threshold. In the algorithm, r(n) is the learning parameter of the algorithm satisfying 0 < r(n) < 1. C is a sufficiently large yet finite number which keeps the weighting vector bounded. The sequence of ~k(n) is a set of i.i.d. random variables with zero mean and a variance of o2(n). The global feedback signal E(n) is calculated by the gateway node and sent back to all source nodes, as (n) = ) (730) where J is a number to normalize the feedback signal. For example, we can set J to the maximum value of overall utility till n and update this value on the fly. Therefore, the value of E(n) lies within [0, 1]. Ul is the nonconvex utility function for the /th user. Note that the utility functions of all users are assumed to be truly acquired by the gateway node [168]. In practice, the value of E(n) can be circulated efficiently by established multicast algorithms such as [169]. Based on the feedback, the learning automata team adjusts the weighting vector in a decentralized fashion. In addition, note that B is the predefined convergence threshold, e.g., B = 0.999, which provides a tradeoff between the performance of the algorithm and its convergence speed. Before analyzing the steady state behavior of the proposed algorithm, we first discuss the following concepts. Denote the maximum network utility of the original traffic engineering problem, i.e., Pi, as O*. Next, we define the final outcome of the proposed algorithm as 0'. We say that the algorithm provides an caccurate solution, if for any arbitrarily small C > 0, there exists a Q' such that I* O'I < VQ > Q' (731) A potential game [67] is defined as a game where there exists a potential function V such that V(a', a_) V(a", a_/) = Ul(a', a_/) Ul(a", a_) V/, a', a" (732) where UL is the utility function for player I and a', a" are two arbitrary strategies in its strategy space. The notation of a_/ denotes the vector of choices made by all players other than /. 175 A weighted potential game [67] is defined as a game where there exists a potential function V such that (V(a', al) V(a", al)) x hi = Ul(a', a_) Ul(a", al) (733) for all /, a', a" where hi > 0. According to the definitions, it is apparent that the formulated identical interest game is a special case of weighted potential games. In the following theorem, we will provide the convergence behavior of a more general setting for weighted potential games, and hence the result applies to our specific scenario naturally. Theorem 7.2. For an Nperson weighted potential game where each person represents a team of learning automata, the proposed algorithm can converge to the global optimum solution asymptotically, which is an caccurate solution to the original stochastic traffic engineering problem, for sufficient small value of r and o. Proof. The proof follows similar lines as in [167]. However, we extend the result to a more general setting where the underlying Nperson stochastic game is a weighted potential game. Therefore, the proof in [167] can be viewed as a special case. Define V as the potential function of the game. Note that the selected rate is determined by the probability vector which is generated uniquely by the weighting vector. Therefore, we can view the weighting vector as the variable in this case and the objective function is given by z = ((Vw). In the updating procedure of (729), signal E is replaced by V. We first verify that (Oz )((V w) e wq pkVw ( kn), r k ). aWk SW kWq =0e Yk Q e WfJ 176 Note that eWk (v(1 )w(n)) Q4 e Wfuk q=0 ,k S= epk e )(Vw(n), r)k q q=0 e ,k Wq W q q=oe0 we q=0 e k az Next, it is straightforward to verify that the standard conditions in [170] (Chap.6, Theorem 7) are satisfied. We omit the verifications since they are the similar procedures as in [167]. Thus, we conclude that the above dynamic weakly converges to the following SDE [170, 171] dw = Vz + adW (734) for a sufficiently small 7 > 0 where a is the standard deviation of the i.i.d. random variables 'fk and W is a standard Wiener Process. Note that the SDE (734) falls into the category of Langevin equation [172] which is wellknown that the probability measure concentrates on the global maximum solution of z for a sufficiently small a [167, 172]. Therefore, we conclude that in the weighted potential game scenario, the proposed algorithm will converge to the global optimum of the objective function, for the quantized data rate setting. The association of the caccurate solution to the original stochastic traffic engineering problem of P, follows the result of [173]. Note that Theorem 2 establishes the convergence behavior of the aforementioned algorithm with no additional requirement for the problem structure. In contrast, we propose a stochastic primaldual approach in Section 7.4, which requires the underlying problem to be a stochastic convex programming. The stochastic primaldual approach cannot be applied efficiently otherwise. However, the aforementioned learningbased algorithm is suitable for almost every aspect such as stochastic 177 nonconvex programming and stochastic mixedinteger programming. The asymptotically convergence result still holds. Therefore, in this work, the proposed algorithms provide two different exemplifying methods for protocol designs under the stochastic environment. However, it is worth noting that for the latter approach, the tradeoff for general applicability is the convergence speed. In other words, in order to achieve an accurate result, i.e., when c is small, the convergence speed may be slow. The actual convergence speed depends on the values of c, Q, 7, a as well as the inherent structure of the problem and hence is difficult to quantify. Fortunately, in practical applications, achieving a "good enough" result is sometimes satisfactory. This tradeoff can be achieved by utilizing diminishing values of 7 and a, as demonstrated in the simulated annealing literature [78] and [174]. To sum up, if the stochastic traffic engineering problem possesses a nice property of convexity, the algorithm based on the stochastic primaldual approach in Section 7.4 is recommended due to its nice decomposed structure and computationally efficient solution. However, if the problem is nonconvex in nature, the learning automata based algorithm can be utilized to achieve an approximate solution. The tradeoff between the accuracy and convergence speed can be tuned by adjusting the values of 7 and a. 7.6 Performance Evaluation In this section, we present a simple yet illustrative example to demonstrate the theoretical results. We consider a cognitive wireless mesh network3 depicted in Figure 72. There are three edge routers as the source nodes, denoted by A, B, C, which transmit to the gateway node GW via the relay routers X, Y and Z. Among all feasible 3 Figure 72 only shows the links on the available paths obtained by the signalling mechanisms or manually configurations. The actual physical topology of the network can be potentially larger. 178 paths, we select the following available paths for edge routers, as summarized in Table 71. Relay Routers Edge Routers Figure 72. Example of cognitive wireless mesh network Table 71. Available paths for edge routers A P,: {A X GW} P : {A > X  Y GW} P: {A X Y Z GW B PA: {B X GW} P: {B Y GW} P: { B Y X GW} 4: {B Y Z GW} P : {B Z GW} C PI: {C Z GW} P : {C Z Y GW} P : {C Z  Y X GW} There are five primary users in the area, denoted by 1, 2, 3, 4 and 5 where each one has a primary band of 10MHz. The common ISM band is assumed to be 10MHz. The return probability of the primary users is given as w = [0.2, 0.3, 0.4, 0.3, 0.3]. The transmitting power of each node is fixed as 100mW and the noise power is assumed to be 3mW. We consider a model where the received power is inversely proportional to the square of the distance. Note that the transmitting power is uniformly spread on all available bands. In addition, we explicitly specify the affecting primary users for a particular node. We use {i,j, k, } to represent that a particular node is affected by primary user i,j, k, For example, node X is labeled with {1, 2} which indicates that 179 the transmission of node X will devastate the transmissions of primary user 1 and 2 if the corresponding primary band is utilized. Note that the central node, namely, Y, is most severely affected by all primary users. Intuitively, to achieve an expected optimum solution, the stochastic traffic engineering algorithms are inclined to steer the traffic away from Y. We will demonstrate this detour effect next. We first consider the cognitive wireless mesh network with convexity, e.g., Ul(xl) = log x1 to achieve a proportional fairness among the flows [161]. The link cost is assumed to be in the form of I((fe, ce) = which reflects the delay experienced for a unit flow on link e under the M/M/1 assumption [76]. Note that if fe > Ce, the cost is +o. We set the traffic demand of all edge routers as dl = 30Mbps while the cost budget is bl = 5. The step sizes are chosen as a = 1/n where n is the current iteration step. Figure 73A illustrates the trajectories of the rate variables and Figure 73B shows the convergence of the network overall utility as well as the individual utility functions4 We observe that while the rate variables converge as the iterations go, the overall objective, i.e., the sum of the individual utilities, approaches to the global optimum indicated by the dashed line, which is attained by calculating the steady state distribution following the return probability w. In addition, Table 72 provides the rate on each path after convergence for a sample run of the algorithm. For comparison, we provide the convergence rates when node Y is switched from the most affected node to the least affected node, i.e., node Y can utilize all five primary bands all the time, in Table 73. From Table 72 and 73, it is interesting to note that, in the first scenario, each user allocates a relatively small amount of flow on the paths which traverse through node Y. On the contrary, when node Y is less affected, all the flows allocate noticeably larger data rates on paths that traverse through 4 Note that Figure 73B also reflects the evolution of the throughput of each edge router logarithmically. 180 Convergence of Rate Variables 11  S loalOp....... riu 11 r  r 0 10 20 30 40 50 60 70 80 90 100 Iteration A Trajectories of Rate Variables Convergence of Overall Utility 10 Overall Utility U, 7 F 2 0 10 20 30 40 50 60 70 80 90 100 Iteration B Trajectories of Utility Functions Figure 73. Cognitive wireless mesh networks with convexity Y despite the fact that node Y is the central node which is least favorable by traditional traffic engineering solutions. Therefore, our proposed stochastic traffic engineering algorithm is of particular interest for multihop cognitive wireless mesh networks due to the capability of steering the traffic away from the severely affected areas automatically, without a prior knowledge of the underlying probabilistic structure, in a distributed fashion. 181 Table 72. Convergence rates when Y is affected by all five primary users P1. P': P2. P : P : P1. PB. 15.3035 3.2458 2.7362 7.3725 1.9584 2.1752 2.6458 5.3489 17.3051 2.3792 2.5675 Table 73. Convergence rates when Y is not affected by A P : 9.6413 PA: 11.1335 P : 9.1861 B P : 5.0101 PB: 6.2625 PB: 4.9711 PI: 4.9921 PB: 8.7878 C P: 15.3118 P : 9.1042 PF: 5.6121 any of the primary users We next consider a cognitive wireless mesh network with nonconvex utility functions. Specifically, we consider the utility function as 1, if x > 2Mbps U(x) 2Mbps. 0, if xi < 2Mbps. while other settings are the same as in the previous scenario. Additionally, we utilize diminishing values of r and a as 7  Without loss of generality, we set 1/n and o = 1/n, where n is the iteration step5 . 100 and the quantization level Q 20. The 5 By utilizing diminishing parameters, a tradeoff between the performance and the convergence speed can be achieved by tuning the decreasing speed [78]. 182 (735) maximum allowed rate rm is assumed to be 10. Therefore, the discretized data rate set is given by R = [0.5, 1.0, 1.5, 9.0, 9.5, 10.0]. Figure 4 illustrates the evolution of the probability vector of PA,1. Note that as the iterations evolve, the probability of pA i.e., the probability that router A chooses the twentieth data rate (rm in this case), excels others and approaches to 1 asymptotically. We plot the evolutions of the probability vectors of other paths in Figure 5 collectively. For each path, the probability of selecting one particular data rate soon excels others. We observe that router A selects the twentieth, the first and the fourth date rate on its three paths asymptotically. Meanwhile, router B inclines to choose the eighth, the second, the first, the fourth and the twentieth data rate on its paths. The steady state data rates for router C is the twentieth, the first and the first element in R, as depicted in Figure 5. It is interesting to notice that all the routers automatically detour the traffic from the severely affected node Y by allocating more data rate on other paths. Router A (Path 1) 20 A,1 4o 02 0 Iteration Figure 74. Trajectory of the probability vector of router A's first path 7.7 Conclusions In this chapter, we investigate the stochastic traffic engineering (STE) problem in cognitive wireless mesh networks. To harness the randomness induced by the unpredictable behaviors of primary users, we formulate the STE problem in a stochastic network utility maximization framework. For the cases where convexity holds, we derive a distributed algorithmic solution via the stochastic primaldual approach, which provably converges to the global optimum solution. For the scenarios where convexity is not 183 Router A (Path 3) Router B (Path 3) Router B (Path 4) Router B (Path 5) Router C (Path 1) Iteration Iteration Iteration Iteration Router C (Path 2) Router C (Path 3) 1,2 P4C, 8o 2o 50 100 150 20 0 50 100 150 200 o 50 100 150 2000 Iteration Iteration Figure 75. Trajectories of the probability vectors on other paths attainable, we propose an alternative decentralized algorithmic solution based on the learning automata techniques. We show that the algorithm converges to the global optimum solution asymptotically, under certain conditions. In our work, we restrict ourself in a single gateway scenario. The extension to the multiple gateway scenario seems interesting and needs further investigation. In addition, in this work, we consider a cooperative case where all the edge routers attempt to maximize the overall network performance. In the cases where the edge routers are noncooperative, each player is interested in its own utility rather than the social welfare. Stochastic game theory provides a feasible tool to address the noncooperative case, which remains as future research. We also assume a negligible delay for the feedback signal while in a more general case, the impact of feedback delay needs further investigation. One feasible solution is to utilize the distributed robust optimization framework [175] where the worst case performance is maximized given that the feedback delay/error is within a reasonable range. Our work initiates a first step to investigate the impact of unpredictable returns of primary users, on the stochastic traffic engineering problem in cognitive wireless mesh networks. Router A (Path 2) Router B (Path 1) Router B (Path 2) CHAPTER 8 CONCLUSIONS In this dissertation, we provide efficient and effective solutions to a number of dynamic resource allocation and optimization problems in wireless networks with timevarying channel conditions. For wireless ad hoc networks, we first develop a dynamic pricing scheme which can achieve a networkwide stability while maximizing the accumulated revenue with a controllable tradeoff with the average delay. In addition, a service differentiation solution can be achieved by prioritizing the flows according to their QoS requirements. We also propose a minimum energy scheduling algorithm which minimizes the overall energy consumption in the network without decreasing the throughput region. For wireless mesh networks, we investigate the joint frequency and power allocation problem among multiple access points in a game theoretical framework. The proposed negotiationbased algorithm can achieve the optimal frequency and power allocation with arbitrarily high probability. The price of anarchy in noncooperative wireless mesh networks is also studied. For wireless sensor networks, we propose a constrained queueing model to capture the complex cross layer interactions among multiple sensor nodes. We also study the stochastic network utility maximization problem and propose a cross layer solution which solves the problems of admission control, load balancing, dynamic scheduling and routing in a collective fashion. For wireless local area networks, we study the thresholdsbased rate adaptation algorithms from a reverse engineering perspective, where the implicit objective function is revealed. 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Verdu, "Distributed robust optimization for communication networks," IEEE INFOCOM, 2008. 197 BIOGRAPHICAL SKETCH Yang Song received his Bachelor and Master of Science degrees from Dalian University of Technology and the University of Hawaii in 2004 and 2006, respectively. Since August 2006, he has been working as a Research Assistant with the Wireless Networks Laboratory in the Department of Electrical and Computer Engineering at the University of Florida. His research topics include routing, scheduling and cross layer design in wireless networks, optimization and control in complex networks, applied network economics, efficient algorithmic design in information networks, and social computing. 198 PAGE 1 DYNAMICRESOURCEALLOCATIONANDOPTIMIZATIONINWIRELESSNETWORKSByYANGSONGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010 PAGE 2 c2010YangSong 2 PAGE 3 Tomybelovedparentsandfriends 3 PAGE 4 ACKNOWLEDGMENTS IwouldliketogratefullyandsincerelythankmyPh.D.advisorProf.YuguangFangforhisinvaluableguidance,understanding,patience,andmostimportantly,hiscontinualfaithandcondenceinmeduringmyPh.D.studiesattheUniversityofFlorida.Ifeelextremelyfortunatetohavehimasmyadvisor,whoisalwayswillingtohelpmebothacademicallyandpersonally.Thanksforeverything.IalsoowemywholeheartedgratitudetomyPh.D.committeemembers,Prof.PramodP.Khargonekar,Prof.SartajSahni,Prof.TanWong,andProf.ShigangChen,fortheirconstructivesuggestionsandvaluablecommentsonmyPh.D.researchanddissertation.IhavebeenfortunatetohavemanyfriendsinWINET.IspeciallythankChiZhang,XiaoxiaHuang,YunZhou,ShushanWen,JianfengWang,HongqiangZhai,YanchaoZhang,FengChen,PanLi,JinyuanSun,MiaoPan,RongshengHuang,YueHaoformanyvaluablediscussionsandgoodmemories.Lastbutdenitelynotleast,thisworkwouldnothavebeenachievedwithoutthesupportandunderstandingofmyparents.TheyhavealwayssupportedmeineverychoiceIhavechoseninmylife. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2REVENUEMAXIMIZATIONINMULTIHOPWIRELESSNETWORKS ..... 17 2.1Introduction ................................... 17 2.2RelatedWork .................................. 19 2.3RevenueMaximizationwithQoS(QualityofService)Requirements ... 22 2.3.1SystemModel .............................. 22 2.3.2ProblemFormulation .......................... 25 2.4QoSAwareDynamicPricing(QADP)Algorithm ............... 27 2.5PerformanceAnalysis ............................. 30 2.5.1ProofofRevenueMaximization .................... 31 2.5.2ProofofNetworkStability ....................... 39 2.5.3ProofofQoSProvisioning ....................... 40 2.6Simulations ................................... 41 2.6.1SingleHopWirelessCellularNetworks ................ 41 2.6.2MultiHopWirelessNetworks ..................... 44 2.7Conclusions ................................... 47 3ENERGYCONSERVINGSCHEDULINGINWIRELESSNETWORKS ..... 48 3.1Introduction ................................... 48 3.2SystemModel ................................. 50 3.3MinimumEnergySchedulingAlgorithm ................... 54 3.3.1AlgorithmDescription ......................... 54 3.3.2ThroughputOptimality ......................... 57 3.3.3AsymptoticEnergyOptimality ..................... 62 3.4Simulations ................................... 64 3.5Conclusions ................................... 67 4CHANNELANDPOWERALLOCATIONINWIRELESSMESHNETWORKS 70 4.1Introduction ................................... 70 4.2SystemModel ................................. 72 4.3CooperativeAccessNetworks ........................ 74 5 PAGE 6 4.3.1CooperativeThroughputMaximizationGame ............ 75 4.3.2NETMANegotiationBasedThroughputMaximizationAlgorithm 78 4.4NonCooperativeAccessNetworks ...................... 84 4.5AnExtensiontoAdaptiveCodingandModulationCapableDevices .... 90 4.6PerformanceEvaluation ............................ 93 4.6.1LegacyIEEE802.11Devices ..................... 93 4.6.1.1Exampleofsmallnetworks ................. 94 4.6.1.2Exampleoflargenetworks ................. 96 4.6.2ACMCapableDevices ......................... 96 4.7Conclusions ................................... 98 5CROSSLAYERINTERACTIONSINWIRELESSSENSORNETWORKS ... 100 5.1Introduction ................................... 100 5.2RelatedWork .................................. 102 5.3AConstrainedQueueingModelforWirelessSensorNetworks ...... 104 5.3.1NetworkModel ............................. 104 5.3.2TrafcModel .............................. 105 5.3.3QueueManagement .......................... 108 5.3.4SessionSpecicRequirements .................... 109 5.4StochasticNetworkUtilityMaximizationinWirelessSensorNetworks .. 110 5.4.1ProblemFormulation .......................... 111 5.4.2TheANRACrossLayerAlgorithm ................... 112 5.4.3PerformanceoftheANRAScheme .................. 114 5.5PerformanceAnalysis ............................. 118 5.6CaseStudy ................................... 125 5.7ConclusionsandFutureWork ......................... 128 6THRESHOLDOPTIMIZATIONFORRATEADAPTATIONALGORITHMSINIEEE802.11WLANS ................................ 130 6.1Introduction ................................... 130 6.2RelatedWork .................................. 133 6.3ReverseEngineeringfortheThresholdBasedRateAdaptationAlgorithm 135 6.4ThresholdOptimizationAlgorithm ....................... 143 6.4.1LearningAutomata ........................... 144 6.4.2AchievingtheStochasticOptimalThresholds ............ 145 6.5PerformanceEvaluation ............................ 150 6.6ConclusionsandFutureWork ......................... 155 7STOCHASTICTRAFFICENGINEERINGINMULTIHOPCOGNITIVEWIRELESSMESHNETWORKS ................................. 157 7.1Introduction ................................... 157 7.2RelatedWork .................................. 160 7.3SystemModel ................................. 163 7.4StochasticTrafcEngineeringwithConvexity ................ 167 6 PAGE 7 7.4.1Formulation ............................... 167 7.4.2DistributedAlgorithmicSolutionwiththeStochasticPrimalDualApproach ................................ 169 7.5StochasticTrafcEngineeringwithoutConvexity .............. 173 7.6PerformanceEvaluation ............................ 178 7.7Conclusions ................................... 183 8CONCLUSIONS ................................... 185 REFERENCES ....................................... 186 BIOGRAPHICALSKETCH ................................ 198 7 PAGE 8 LISTOFTABLES Table page 21Averageadmittedratesformultimediaows .................... 44 22Averageadmittedratesformultimediaows .................... 47 41Dataratesv.s.SINRthresholdswithmaximumBER=10)]TJ /F5 7.97 Tf 6.59 0 Td[(5 ........... 90 61SNRv.s.BERforIEEE802.11bdatarates .................... 152 71Availablepathsforedgerouters ........................... 179 72ConvergencerateswhenYisaffectedbyallveprimaryusers ......... 182 73ConvergencerateswhenYisnotaffectedbyanyoftheprimaryusers ..... 182 8 PAGE 9 LISTOFFIGURES Figure page 21Anexampleofmultihopwirelessnetworks .................... 23 22Aconceptualexampleofthenetworkcapacityregionwithtwomultimediaows.Theminimumdataraterequirementsreducethefeasibleregionoftheoptimumsolution ........................................ 27 23Asinglehopwirelesscellularnetworkwiththreeusers .............. 42 24ImpactofdifferentvaluesofJontheperformanceofQADP ........... 43 25ImpactofdifferentvaluesofJontheaverageexperienceddelays ........ 43 26Queuebacklogdynamicsforallusers ....................... 45 27PricedynamicsinQADPforallusers ....................... 45 28VirtualqueuebacklogupdatesinQADPforJ=50000 .............. 46 29AveragequeuebacklogsinthenetworkforJ=50000 .............. 46 31Networktopologywithinterconnectedqueues ................... 65 32ComparisonoftheenergyconsumptionsoftheMaxWeightalgorithmandtheMESalgorithm .................................... 66 33TheaveragequeuebacklogsinthenetworkforJ=50and150 ......... 67 34TheaveragequeuebacklogsinthenetworkforJ=350and500 ........ 68 35ComparisonofthelifetimeoftheMaxWeightalgorithmandtheMESalgorithm 68 41Hierarchicalstructureofwirelessmeshaccessnetworks ............. 70 42AnillustrativeexampleofmultipleNashequilibria ................. 77 43MarkovianchainofNETMAwithtwoplayers .................... 81 44PerformanceevaluationofthewirelessmeshaccessnetworkwithN=5andc=3 .......................................... 94 45ThetrajectoryoffrequencynegotiationsinNETMAwhenN=5andc=3 ... 95 46ThetrajectoryofpowernegotiationsinNETMAwhenN=5andc=3 ..... 95 47PerformanceevaluationofthewirelessmeshaccessnetworkwithN=20andc=3 .......................................... 97 48Performanceevaluationofthewirelessmeshaccessnetworkwith/withoutthepricingscheme .................................... 98 9 PAGE 10 49ThetrajectoriesofpowerutilizationpricesofeachACMcapableAP ...... 98 410ThetrajectoriesofactualpowerutilizedbyeachACMcapableAP ....... 99 51Topologyofwirelesssensornetworks ....................... 106 52Examplenetworktopology ............................. 125 53AveragenetworkutilityachievedbyANRAfordifferentvaluesofJ ....... 126 54AveragenetworkqueuesizebyANRAfordifferentvaluesofJ ......... 127 55SampletrajectoriesoftheadmittedratesoftwosessionsforJ=5000 ..... 127 56SampletrajectoriesofthevirtualqueuesforJ=5000 .............. 128 57SampletrajectoriesofthehopselectionsforJ=5000 .............. 129 61Averagethroughputv.s.Dopplerspreadvalues .................. 153 62Thetrajectoriesofuanddinachievingthestochasticoptimumvalues .... 154 63EvolutionoftheprobabilityvectorPu ........................ 154 64EvolutionoftheprobabilityvectorPd ........................ 155 71Architectureofwirelessmeshnetworks ...................... 157 72Exampleofcognitivewirelessmeshnetwork ................... 179 73Cognitivewirelessmeshnetworkswithconvexity ................. 181 74TrajectoryoftheprobabilityvectorofrouterA'srstpath ............. 183 75Trajectoriesoftheprobabilityvectorsonotherpaths ............... 184 10 PAGE 11 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDYNAMICRESOURCEALLOCATIONANDOPTIMIZATIONINWIRELESSNETWORKSByYangSongAugust2010Chair:YuguangMichaelFangMajor:ElectricalandComputerEngineering Duetothehostilewirelessmediumandthelimitedresourcesinwirelessnetworks,howwellwirelessnetworkscanperformandhowtomakewirelessnetworksprovidebetterservicearecriticalandchallengingproblems.Thesemotivateourresearchinboththeoreticalanalysisandprotocoldesignsintimevaryingwirelessnetworks.Inthisdissertation,weaimtoaddressthedynamicresourceallocationandoptimizationproblemsinwirelessnetworks,spanningwirelessadhocnetworks,wirelessmeshnetworks,wirelesssensornetworks,wirelesslocalareanetworks,andcognitiveradionetworks. Ourcontributionscanbesummarizedasfollows.First,weproposeanonlinedynamicpricingschemewhichmaximizesthenetworkrevenuesubjecttostabilityinmultihopwirelessnetworkswithmultipleQoSspecicows.Secondly,wedesignanovelgreenschedulingalgorithminmultihopwirelessnetworkswithstochasticarrivalsandtimevaryingchannelconditionswhichminimizestheenergyexpendituresubjecttonetworkstability.Thirdly,wedevelopanegotiationbasedalgorithmwhichattainsanoptimalsolutiontothenonconvexjointpowerandfrequencyallocationproblemincooperativewirelessmeshaccessnetworks.Inaddition,weanalyzetheexistenceandtheinefciencyoftheNashequilibriuminnoncooperativewirelessmeshaccessnetworksandproposedpricingschemestoimprovetheequilibriumefciency.Fourthly,weproposeaqueueingbasedmodeltocapturethecrosslayer 11 PAGE 12 interactionsinmultihopwirelesssensornetworksanddesignedajointrateadmissioncontrol,trafcengineering,dynamicroutingandschedulingschemetomaximizetheoverallnetworkutility.Fifthly,weanalyzethethresholdsbasedrateadaptationalgorithmsinIEEE802.11WLANsfromareverseengineeringperspectiveandproposeathresholdoptimizationalgorithmtoenhancetheperformanceofIEEE802.11WLANs.Finally,weinvestigatethestochastictrafcengineeringprobleminmultihopcognitiveradionetworksandderiveadistributedalgorithmbasedonthestochasticprimaldualapproachforconvexscenariosaswellasageneralsolutionbasedonthelearningautomatatechniquesfornonconvexscenarios. 12 PAGE 13 CHAPTER1INTRODUCTION WANETsareselfconguringandstandalonenetworksofnodesconnectedbywirelesslinks.Theyhaveattractedextensiveattentionasidealnetworkingsolutionsforscenarioswherexednetworkinfrastructuresarenotavailableorreliable.Inaddition,multimediatransmissionshavebecomeanindispensablecomponentofnetworktrafcnowadays.However,theissueofQoSprovisioningformultimediatransmissionsisremarkablychallenginginmultihopwirelessadhocnetworks.TimevaryingchannelconditionsamongwirelesslinksimposesevereadverseimpactontheQoSofmultimediatransmissions.Therefore,whilewirednetworkshavematuresolutionsandestablishedprotocolsforprovidingQoS,novelsolutionswhichincorporatesalientfeaturesofwirelesstransmissionsneedtobedevelopedtosupportmultimediatransmissionsinmultihopwirelessnetworks.Inaddition,fromthenetworkprovider'sperspective,itisimperativetodesignapricingmechanismwhichmaximizesthenetworkrevenuewhileaddressingtheQoSrequirementsofusers.Suchaschemeshouldalsoensureanetworkwidestabilityunderstochastictrafcarrivalsandtimevaryingchannelconditions.InChapter 2 ,weproposedaQoSawaredynamicpricingschemewhichprovablyensuresthenetworkwidestabilitywhileattainingasolutionwhichisarbitrarilyclosetotheglobalmaximumrevenue,withacontrollabletradeoffwiththeaveragedelayinthenetwork.Moreover,aweightassignmentmechanismisdevisedtoaddresstheservicedifferentiationissueformultipleowsinthenetworkwithdifferentdelaypriorities. InChapter 3 ,weinvestigatetheissueofGreenComputinginwirelessnetworks,whichisanimportantconcernraisedfromcomputerscientistsandengineersthatattemptstoutilizethecomputingresourcesefcientlywhileintroducingminimumimpactontheenvironment.Inmultihopwirelessnetworks,werstidentifythatthewellestablishedMaxWeight,orbackpressureschedulingalgorithm,isnotenergyoptimal.TheschedulingprocessoftheMaxWeightalgorithmneglectsthe 13 PAGE 14 enormousenergyconsumptioninretransmissionswhicharenotnegligibleinwirelessnetworkswithfadingchannels.Toaddressthis,weproposedaminimumenergyschedulingalgorithmwhichsignicantlyreducedtheoverallenergyexpenditurecomparedtotheoriginalMaxWeightalgorithm.Theenergyconsumptioninducedbytheproposedalgorithmcanbepushedarbitrarilyclosetotheglobalminimumsolution.Moreover,theimprovementontheenergyefciencyisachievedwithoutlosingthethroughputoptimality. AwirelessmeshnetworkischaracterizedbyamultihopwirelessbackboneconnectingwiredInternetentrypoints,orgateways,andwirelessaccesspoints(AP)whichprovidenetworkaccesstoendusers.InaWMN,howtoassignmultiplechannelsandpowerlevelstoeachAPisofgreatimportancetomaximizetheoverallnetworkutilizationandtheaggregatedthroughput.InChapter 4 ,weinvestigatethenonconvexthroughputmaximizationprobleminWMNsforbothcooperativeandnoncooperativescenarios.Inthecooperativecase,wemodeltheinteractionsamongallAPsasanidenticalinterestgameandpresentadecentralizednegotiationbasedthroughputmaximizingalgorithmforthejointfrequencyandpowerassignmentproblem.Weprovethatthisalgorithmconvergestotheoptimalfrequencyandpowerassignmentsolution,whichmaximizestheoverallthroughputofthewirelessmeshnetwork,witharbitrarilyhighprobability.InthecaseofnoncooperativeAPs,weprovetheexistenceofNashequilibriaandshowthattheoverallthroughputperformanceisnoticeablyinferiortothecooperativescenario.Tobridgetheperformancegap,wedevelopapricingschemetocombattheselshbehaviorsofnoncooperativeAPs.Theoverallnetworkperformanceintermofaggregatedthroughputissignicantlyimproved. RecentyearshavewitnessedasurgeinresearchanddevelopmentofWSNsfortheirbroadapplicationsinbothmilitaryandcivilianoperations.Beforethewidedeploymentofwirelesssensornetworks,asystematicunderstandingabouttheperformanceofmultihopwirelesssensornetworksisdesired.However,ndinga 14 PAGE 15 suitableandaccurateanalyticalmodelforwirelesssensornetworksisparticularlychallenging.Anappropriatemodelshouldreecttherealisticnetworkoperationswithemphasisonthedistinguishingfeaturesofwirelesssensornetworkssuchasthechallengeofautomaticloadbalancingamongmultiplesinknodes,thetaskofdynamicnetworkschedulingandroutingundertimevaryingchannelconditions,andtheinstantaneousdecisionmakingonthenumberofadmittedpacketsinordertoensurethenetworkwidestability.Tocapturethecrosslayerinteractionsofmultihopwirelesssensornetworks,inChapter 5 ,weproposedaconstrainedqueueingmodeltoinvestigatethejointrateadmissioncontrol,dynamicrouting,adaptivelinkscheduling,andautomaticloadbalancinginwirelesssensornetworksthroughasetofinterconnectedqueues.Todemonstratetheeffectivenessoftheproposedconstrainedqueueingmodel,weinvestigatethestochasticnetworkutilitymaximizationprobleminmultihopwirelesssensornetworks.Basedontheproposedqueueingmodel,wedevelopanadaptivenetworkresourceallocationschemewhichyieldsanearoptimalsolutiontothestochasticnetworkutilitymaximizationproblem.Theproposedschemeconsistsofmultiplelayercomponentssuchasjointrateadmissioncontrol,trafcsplitting,dynamicrouting,andanadaptivelinkschedulingalgorithm.Ourproposedschemeisessentiallyanonlinealgorithmwhichonlyrequirestheinstantaneousinformationofthecurrenttimeslotandhenceremarkablyreducesthecomputationalcomplexity. IEEE802.11WLANhasbecomethedominatingtechnologyforindoorwirelessInternetaccess.Inordertomaximizethenetworkthroughput,IEEE802.11devices,i.e.,stations,needtoadaptivelychangethedataratetocombatthetimevaryingchannelenvironments.However,thespecicationofrateadaptationalgorithmsisnotprovidedbytheIEEE802.11standard.Thisintentionalomissionencouragesthestudiesonthisactiveareawhereavarietyofrateadaptationalgorithmshavebeenproposed.Duetoitssimplicity,thethresholdsbasedrateadaptationalgorithmispredominantlyadoptedbyvendors.Thedatarateincreasesifacertainnumberof 15 PAGE 16 consecutivetransmissionsaresuccessful.Althoughwidelydeployed,theobscureobjectivefunctionofthistypeofrateadaptationalgorithms,commonlybasedontheheuristicup/downmechanism,islesscomprehended.InChapter 6 ,westudythethresholdsbasedrateadaptationalgorithmfromareverseengineeringperspective.Theimplicitobjectivefunction,whichtherateadaptionalgorithmismaximizing,isunveiled.Ourresultsprovideananalyticalmodelfromwhichtheheuristicsbasedrateadaptationalgorithm,suchasARF,canbebetterunderstood.Moreover,weproposeathresholdoptimizationalgorithmwhichdynamicallyadaptstheup/downthresholds.Ouralgorithmprovablyconvergestothesetofstochasticoptimumthresholdsinarbitrarystationaryyetpotentiallyfastvaryingchannelenvironmentsandtheperformanceintermofthroughputisenhancedremarkably. InChapter 7 ,weinvestigatethestochastictrafcengineering(STE)probleminmultihopcognitiveradionetworks.Morespecically,weareparticularlyinterestedinhowthetrafcinthemultihopcognitiveradionetworksshouldbesteered,undertheinuenceofrandomreturnsofprimaryusers.Itisworthnotingthatgivenaroutingstrategy,thecorrespondingnetworksperformance,e.g.,theaveragequeueingdelayencountered,isarandomvariable.Inmultihopcognitiveradionetworks,thisdistinguishingfeatureofrandomness,inducedbytheunpredictablebehaviorsofprimaryusers,mustbetakenintoaccountinprotocoldesigns.WeformulatetheSTEprobleminastochasticnetworkutilitymaximizationframework.Forthecasewhereconvexityholds,wederiveadistributedcrosslayeralgorithmviathestochasticprimaldualapproach,whichprovablyconvergestotheglobaloptimumsolution.Forthescenarioswhereconvexityisnotattainable,weproposeanalternativedecentralizedalgorithmicsolutionbasedonthelearningautomatatechniques.Weshowthatthealgorithmconvergestotheglobaloptimumsolutionasymptoticallyundermildconditions.Finally,Chapter 8 concludesthisdissertation. 16 PAGE 17 CHAPTER2REVENUEMAXIMIZATIONINMULTIHOPWIRELESSNETWORKS 2.1Introduction Inrecentyears,multihopwirelessnetworkshavemadesignicantadvanceinbothacademicandindustrialaspects.Besidestraditionaldataservices,multimediatransmissionsbecomeanindispensablecomponentofnetworktrafcnowadays.Forexample,peoplecanwatchlivegameswhilelisteningtoonlinemusicalstationsatthesametime.Therefore,supportingmultimediaservicesinmultihopwirelessnetworkseffectivelyandefcientlyhasreceivedintensiveattentionfromthecommunity.However,theissueofQoSprovisioningformultimediatransmissionsisremarkablychallenginginmultihopwirelessnetworks.TimevaryingchannelconditionsamongwirelesslinksimposesevereadverseimpactontheQoSofmultimediatransmissions.Therefore,whilewirednetworkshavematuresolutionsandestablishedprotocolsforprovidingQoS,novelsolutionswhichincorporatesalientfeaturesofwirelesstransmissions,needtobedevelopedtosupportmultimediatransmissionsinmultihopwirelessnetworks. Multimediaowsusuallyimposeapplicationspecicrequirementsontheminimumaverageattainabledataratesfromthenetwork.Furthermore,differentmultimediaowsmayhavedistinctraterequirements.Forexample,multimediastreamsforhighqualityvideoondemand(VoD)movietransmissionsusuallyrequirelargerminimumdataratesonaveragethanthoseofonlinemusictransmissions.Consequently,anaturalquestionarisesthathowthenetworkresourceshouldbeallocatedsuchthatalltheminimumrateconstraintsaresatisedsimultaneously.Inaddition,duetothenatureofwirelesstransmissions,decentralizedsolutionwithlowcomplexityisstronglydesired.Inthischapter,weinvestigatetheresourceallocationprobleminmultihopwirelessnetworks,fromanetworkadministrator'sperspective.Foreachow,thenetworkchargesacertainamountofadmissionfeeinordertobuildupasystemwiderevenue.Thepriceimposedoneachowissubjecttoadaptationinordertoobtaintheoptimumrevenue.Hence,a 17 PAGE 18 dynamicpricingpolicyisdesiredbythenetworkadministratortomaximizetheoverallnetworkrevenuesubjecttothestabilityofthenetwork.Toachievethis,weproposeaQoSawaredynamicpricingalgorithm,namely,QADP,whichprovablyaccumulatesanetworkrevenuethatisarbitrarilyclosetotheoptimumsolutionwhilemaintainingnetworkstabilityunderstochastictrafcarrivalsandtimevaryingchannelconditions.Meanwhile,theminimumdateraterequirementsfromallmultimediaowsaresatisedsimultaneously. Besidesminimumdatarates,multimediaows,especiallywirelessvideotransmissions,usuallyimposeadditionalrequirementsonmaximumendtoenddelays.Forexample,amultimediastreamforvideosurveillancemayneedalowerdatatransmissionratecomparedtohighqualityvideoondemandmovietransmissions,whereasamuchmorestringentdelayrequirementisimposed.Therefore,thenetworkinclinestoallocatemorenetworkresourcetothosedelayimperativemultimediatransmissionsprovidedthattheminimumdataraterequirementsofallowsaresatised.Unfortunately,however,anabsoluteguaranteeforarbitrarydelayrequirementsisextremelydifcult,ifnotimpossible,duetothelackofaccuratedelayanalysisinmultihopwirelessnetworks.Forexample,[ 1 ]derivesalowerboundonthedelayperformanceofarbitraryschedulingpolicyinmultihopwirelessnetworks.However,theupperboundofdelayisunspecied.Infact,itisshownthatinwirelessscenarios,eventodecidewhetherasetofdelayrequirementscanbesupportedbythenetworkisanNPhardproblemandthusisintrinsicallydifculttosolve[ 2 ].Evenworseyet,timevaryingwirelesschannelconditionsandstochastictrafcarrivalssignicantlyexacerbatethehardnessofsheerdelayguarantees.Inlightofthis,inthiswork,wealternativelyaimtoprovideaservicedifferentiationsolutionforthedelayrequirementsofallows.Morespecically,thenetworkprovidesasetofservicelevels,denotedby`=1,,Lwherelevelonehasthehighestpriorityinthesystemwithrespecttodelayguarantees.NotethatLcanbearbitrarilylarge.Eachmultimediaow,accordingtotheupperlayerapplication,proposesaservicelevelrequesttothe 18 PAGE 19 network.Forexample,abackgroundtrafcformoviedownloadingisassociatedwithleveltenwhereasaVoDonlinemovietransmissiondemandsaserviceleveloftwo1.Bymeticulouslyassigningweightstomultimediaows,QADPalgorithmprovidesaservicedifferentiationsolutioninawaythattheguaranteedmaximumaverageendtoenddelayforservicelevelonetrafcisjtimeslessthanthatofleveljtransmissions,wherej=1,,L.Inotherwords,levelonetransmissionshavetheminimumupperboundforendtoenddelayandthusrepresentthehighestpriority.Therefore,byfollowingQADP,aQoSawarerevenuemaximizationsolution,subjecttothestabilityofthenetwork,isprovided.QADPalgorithmisinherentlyanonlinedynamiccontrolbasedalgorithmwhichisselfadaptivetothechangesofstatisticalcharacteristicsoftrafcarrivals.Moreover,ourschemeenjoysadecoupledstructureandhenceissuitablefordecentralizedimplementationwhichisofgreatinterestforprotocoldesigninmultihopwirelessnetworks.Notethatourresultsareapplicabletosomespecialinterestingcasesofnetworktopologysuchassinglehopwirelesscellularnetworks. Therestofthischapterisorganizedasfollows.Section 2.2 comparesourworkwithotherexistingsolutionsonnetworkoptimizationwithmultipleows.SystemmodelandproblemformulationareprovidedinSection 2.3 .Ourproposedsolution,i.e.,QADPalgorithm,isintroducedinSection 2.4 followedbytheperformanceanalysisinSection 2.5 .SimulationresultsareprovidedinSection 2.6 andSection 2.7 concludesthischapter. 2.2RelatedWork Thereexistsarichliteratureonhowthenetworkresourceshouldbeallocatedefcientlyamongmultiplecompetitivetransmittingows.Accordingtothenetworkmodel,theycanberoughlydividedintotwocategories,i.e.,uidbasedapproachand 1Weemphasizethatthisapplicationtoservicelevelmappingisarbitraryandcanbespeciedbythenetworkbeforetransmissions. 19 PAGE 20 queuebasedapproach.Inuidbasedalgorithms,e.g.,[ 3 7 ],thecharacteristicsofaparticularowareuniquelyassociatedwithuidvariables.Forexample,theowinjectionrateisacommonlyuseduidvariablewhichisdeterminedbythesourcenode.However,wheneverachangeoftheowinjectionrateoccurs,itisusuallyassumedthatallthenodesonthepathwillperceivethischangeinstantaneously.Inaddition,theknowledgeofuptodatelocalinformationonintermediatenodes,e.g.,shadowprices[ 4 6 7 ],isusuallyvitalfortheowcontrolalgorithmimplementedonthesourcenode.Simplyput,stateinformationofaparticularowisusuallyassumedtobesharedandknownbyallnodesonitspathinstantaneouslyandaccurately,althoughsomeexceptionsarediscussed,e.g.,[ 8 ].Ingeneral,uidbasedapproachdoesnotconsiderthepracticalqueuedynamicsinrealnetworks.Moreover,uidbasedalgorithmsusuallyutilizeadualdecompositionframework,suchasin[ 3 4 7 ]whichheavilyreliesonconvexoptimizationtechniques[ 9 ].Consequently,axedpointsolutionisattained.However,intimevaryingenvironmentssuchaswirelessnetworks,axedoperatingpointishardlyoptimal2.Onthecontrary,queuebasedapproachmodelsthenetworkasasetofinterconnectedqueues.Eachsourcenodeinjectspacketsintothenetworkwhichtraversethroughthenetworkhopbyhopuntilreachthedestination.Everypacketneedstowaitforserviceinthequeuesofintermediatenodesalongthepath,whichreectstherealityinpracticalnetworks.Inaddition,queuebasedapproachusuallyadoptsadynamiccontrolbasedsolutionwhereresponsiveactionsareadjustedontheybywhichalongtermaverageoptimumisachieved.Inthiswork,weutilizeaqueuebasednetworkmodelwhereanoptimaldynamicpricingpolicyisdeveloped.Formorediscussionsonuidbasedalgorithms,referto[ 10 11 ]andthereferencestherein. 2Infact,inthework,weanalyticallyshowthattheimposedpriceneedstobeadjusteddynamicallyinordertoachievetheoptimum. 20 PAGE 21 Theinterconnectedqueuenetworkmodelhasattractedsignicantattentionsincetheseminalworkof[ 12 ]wherethewellknownMaxWeightschedulingalgorithmisproposed.Neelyextendstheresultsintoageneraltimevaryingsettingin[ 13 14 ],basedonwhichapioneeringstochasticnetworkoptimizationtechniqueisdeveloped[ 15 16 ].Foracomprehensivetreatmentonthisarea,referto[ 17 ]andthereferencestherein.Unfortunately,intheliterature,fewworkhasbeendevotedtoaddressingtheissueofQoSprovisioninginpracticalwirelesssettings,whichisofspecialinterestinsupportingmultimediatransmissionsinmultihopwirelessnetworks.Forexample,[ 18 20 ]investigatethedelayconstraintsbyassumingthatthequeuebehavesasanM/M/1queue.However,thismaynotberealisticduetothecomplexinteractionofqueuesinducedbytheschedulingalgorithm.In[ 2 21 22 ],delayguaranteedschedulingalgorithmsarestudied.However,thederiveddelayboundisuptoalogarithmicfactoroftheproposeddelayrequirement,giventhatthedelayrequirementssatisfycertainperserverandpersessionconditions[ 2 ].Findingageneralpolicythatisabletoachievearbitraryfeasibledelayrequirementsisstillanopenproblem.Inaddition,whileexistingsolutionsonprioritizedtransmissionsareavailable,e.g.,[ 23 24 ],thequestionofhowtoachieveaminimumdatarateguaranteeconcurrently,aswellasmaximizingasystemwiderevenue,isunspecied.In[ 25 26 ],lazypacketschedulingalgorithmsareproposed.However,theyrequiretheknowledgeonfuturestochasticarrivalsasapriori,whileinourscheme,a.k.a.,QADPalgorithm,suchinformationisnotneeded. Revenuemaximizationproblemhasbeenstudiedextensivelyintheliteratureformanydifferentsettings.Nevertheless,ingeneral,eitherQoSprovisioningisnotparticularlyaddressed[ 27 30 ],orthenetworkmodelisrestrictedtowirednetworkswherethechannelconditionsofthesystemareassumedtobetimeinvariantandremainunchanged[ 31 32 ].Ourworkisinspiredby[ 33 ].However,ourworkdiffersfrom[ 33 ]inthefollowingcrucialaspects.First,[ 33 ]studiesasinglehopnetworkwithonlyoneaccesspoint(AP)whileourworkinvestigatesageneralmultihopwirelessad 21 PAGE 22 hocnetworkwhereexogenousarrivalsmayenterthenetworkviaanynode.Secondly,[ 33 ]assumesaMarkovianusertrafcdemandwhileourworkisapplicabletoarbitrarytrafcdemands.Thirdlyandmostimportantly,[ 33 ]doesnotconsidertheissueofQoSprovisioningformultipletransmissionows,whichisthemainfocusofourwork.Tothebestofourknowledge,thisworkistherstworktoaddresstheproblemofrevenuemaximization,subjecttonetworkstabilitywhileprovidingQoSdifferentiationstoqualitysensitivetrafcssuchasmultimediavideotransmissions. 2.3RevenueMaximizationwithQoS(QualityofService)Requirements 2.3.1SystemModel WeconsiderastaticmultihopwirelessnetworkrepresentedbyadirectedgraphG=(N,E),illustratedinFigure 21 ,whereNisthesetofnodesandEisthesetoflinks.ThenumbersofnodesandlinksinthenetworkaredenotedbyNandE,respectively.Alinkisdenotedeitherbye2Eor(a,b)2Ewhereaandbarethetransmitterandthereceiverofthelink.Timeisslotted,i.e.,t=0,1,.Forlink(a,b),theinstantaneouschannelconditionattimeslottisdenotedbySa,b(t).Forexample,Sa,b(t)canrepresentthetimevaryingfadingfactoronlink(a,b)attimet.DenoteS(t)asthechannelconditionvectoronalllinks.WeassumethatS(t)remainsconstantduringatimeslot.However,S(t)maychangeonslotboundaries.WeassumethatthereareanitebutarbitrarilylargenumberofpossiblechannelconditionvectorsandS(t)evolvesfollowinganitestateirreducibleMarkovian3chainwithwelldenedsteadystatedistribution.However,thesteadystatedistributionitselfandthetransitionprobabilitiesareunknown.Ateachtimeslott,giventhechannelstatevectorS(t),thenetworkcontrollerchoosesalinkschedule,denotedbyI(t),fromafeasiblesetS(t),whichisrestrictedbyfactorssuchasunderlyinginterferencemodel,duplexconstraints 3NotethattheMarkovianchannelstateassumptionisnotessentialandcanberelaxedtoamoregeneralsettingasin[ 14 ]. 22 PAGE 23 orpeakpowerlimitations.Forawirelesslink(a,b),thelinkdataratea,b(t)isafunctionofI(t)andS(t).Wedenote(t)asthevectoroflinkratesofalllinksattimeslott. Figure21. Anexampleofmultihopwirelessnetworks ThereareCcommodities,a.k.a.,ows,inthenetwork,whereeachcommodity,sayc,c=1,2,,C,isassociatedwitharoutingpathPc=fc(0),c(1),,c(c)gwherec(0)andc(c)arethesourceandthedestinationnodeofowcwhilec(j)denotesthejthhopnodeonitspath.Withoutlossofgenerality,weassumethateverynodeinthenetworkinitiatesatmostoneow4.However,multipleowscanintersectatanynodeinthenetwork.Eachnodemaintainsaseparatequeueforeveryowthatpassesthroughit.Foreachowc,denoteAc(t)astheexogenousarrivaltothetransportlayerofnodec(0)duringtimeslott.Weassumethatthestochasticarrivalprocess,i.e.,Ac(t),hasanexpectedaveragerateofc.Morespecically,wehave limt!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E(Ac())=c8c.(2) Forasinglequeue,denetheoverowfunction[ 13 ]as g(B)=limsupt!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0Pr(Q()>B)(2) 4Ifanodeinitiatesmorethanoneowinthenetwork,wecanreplacethisnodewithmultipleduplicatenodesandthefollowinganalysisstillholds. 23 PAGE 24 whereQ()isthequeuebacklogattime.Wesaythequeueisstableif[ 13 14 ] limB!1g(B)!0.(2) Anetworkisstableifallthequeuesinthisnetworkarestable. Denote=f1,,Cgasthearrivalratevectorofthenetwork.Notethatallthearrivalratevectorsaredenedinanaveragesense.Aowcontrolmechanismisimplementedwhereduringtimeslott,anamountofRc(t)trafcisadmittedtothenetworklayerforowc.Apparently,wehaveRc(t)Ac(t).Forsimplicity,weassumethattherearenoreservoirstoholdtheexcessivetrafc,i.e.,incomingpacketsaredroppedifnotadmitted.However,weemphasizethatouranalysiscanbeappliedtogeneralcaseswheretransportlayerreservoirsaredeployedtobuffertheunadmittedtrafc.Thenetworkcapacityregion,a.k.a.,thenetworkstabilityregion,denotedby,isdenedasalltheadmissionratevectorsthatcanbesupportedbythenetwork,inthesensethatthereexistsapolicythatstabilizesthenetworkunderthisadmissionrate.Moreover,weareparticularlyinterestedinmultimediatransmissionswhereeachowchasspecicQoSrequirements.Tobespecic,eachowchasaminimumdataraterequirementcaswellasanapplicationdependentprioritizedservicelevelrequest,denotedby`c,1`cL.Denote=f1,,Cgand```=f`1,,`Cgastheminimumratevectorandtheservicelevelrequestvectorofthenetworkwhere`1hasthehighestpriorityintermsofthedelayguaranteesprovidedbythenetwork.Inaddition,weassumethattheminimumratevector,i.e.,,isinsideofthenetworkcapacityregion.Sincethatifisinherentlynotfeasible,wecannotexpecttondanypolicytomeetthosedemandsandtheonlysolutionistoincreasethenetwork'sinformationtheoreticcapacitybytraditionalmethodssuchasaddingmorechannels,radios,enablingnetworkcoding,orutilizingMIMOtechniqueswithmultipleantennas.Nevertheless,inthiswork,werestrictourselvestothespecicquestionofhowtondasimpleyetoptimalpolicyforthenetworkifsuchrequirementsareindeedtheoreticallyattainable.We 24 PAGE 25 emphasizethat,however,theanswertothisquestionisbynomeansstraightforwardduetotheintractabilityofquantifyingtheunderlyingnetworkcapacityregion.Moreover,timevaryingchannelconditionsandstochasticexogenousarrivalsmaketheproblemevenmorechallenging.Itisourmainobjectivetodevelopanoptimumpolicywithoutknowingthenetworkcapacityregionandthestatisticalcharacteristicsofrandomarrivalprocessesaswellastimevaryingchannels.WewillformulatetheQoSawarerevenuemaximizationproblemnext. 2.3.2ProblemFormulation DenotethequeuebacklogofnodenforowcasQcn(t).NotethatQcc(c)0sincewheneverapacketreachesthedestination,itisconsideredasleavingthenetwork.ThequeueupdatingdynamicofQcn(t)isgivenasfollows.Forj=1,,c)]TJ /F3 11.955 Tf 11.96 0 Td[(1,wehave Qcc(j)(t+1)[Qcc(j)(t))]TJ /F8 11.955 Tf 11.96 0 Td[(outc(j),c(t)]++inc(j),c(t) (2) andforj=0, Qcc(j)(t+1)=[Qcc(j)(t))]TJ /F8 11.955 Tf 11.96 0 Td[(outc(j),c(t)]++Rc(t) (2) where[x]+denotesmax(x,0)andinn,c(t),outn,c(t)representtheallocateddatarateoftheincominglinkandtheoutgoinglinkofnoden,bytheschedulingalgorithm,withrespecttoowc.Notethat( 2 )isaninequalitysincetheprevioushopnodemayhavelesspacketstotransmitthantheallocateddatarateinc(j),c(t). Duringtimeslott,pc(t)ischargedforowcastheperunitowprice.Thefunctionalityofthepriceisnotonlytocontroltheadmittedows,butalso,moreimportantly,tobuildupasystemwiderevenuefromthenetwork'sperspective.Wefurtherassumethateachowisassociatedwithaparticularuserandthuswewilluseowanduserinterchangeably.Everyusercisassumedtohaveaconcave,differentiableutilityfunctionOc(Rc(t))whichreectsthedegreeofsatisfactionbytransmittingwithdatarateRc(t).Attimeslott,usercselectsadataratewhich 25 PAGE 26 optimizesthenetincome,a.k.a.,surplus,i.e., Rc(t)=argmaxr(Oc(r))]TJ /F6 11.955 Tf 11.95 0 Td[(rpc(t))8c=1,,C.(2) Withoutlossofgenerality,weassumethat Oc(r)=log(1+r).(2) However,westressthatthefollowinganalysiscanbeextendedtootherheterogeneousformsofutilityfunctionsstraightforwardly.Notethatthefairnessissueofmultipleowscanbesolvedbychoosingutilityfunctionsproperly.Forexample,autilityfunctionoflog(r)representstheproportionalfairnessamongcompetitiveows.Formorediscussions,referto[ 4 ]and[ 10 ]. Fromthenetworkadministrator'sperspective,theoverallnetworkwiderevenueisthetargettobemaximized.Meanwhile,thestabilityofthenetworkaswellastheQoSrequirementsfrommultimediaowsneedtobeaddressed.Formallyspeaking,theobjectiveofthenetworkistondanoptimalpolicyto QoSAwareRevenueMaximizationProblem: maximizeD=liminft!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0O() (2) s.t. (a) thenetworkisstable, (b) theminimumdataraterequirements,i.e.,,aresatised, (c) theguaranteedmaximumendtoenddelaysformultiplemultimediaowsareprioritizedaccordingtotheservicelevelsof```, where O(t)=E(XcRc(t)pc(t))(2) istheexpectedoverallnetworkrevenueduringtimeslott,withrespectedtotherandomnessofarrivalprocessesandchannelvariations. 26 PAGE 27 Figure22. Aconceptualexampleofthenetworkcapacityregionwithtwomultimediaows.Theminimumdataraterequirementsreducethefeasibleregionoftheoptimumsolution However,theQoSawarerevenuemaximizationproblemisinherentlychallengingduetounawarenessoffuturerandomarrivalsandstochastictimevaryingchannelconditions.Evenworseyet,theQoSrequirementssignicantlycomplicatetheproblem,asillustratedinFigure 22 .Inunconstrainedcases,thefeasibleregionoftheoptimumsolutionisessentiallythewholenetworkcapacityregion.However,asshowninFigure 22 ,theoriginaloptimumsolutionmaynotevenbefeasibleundertheminimumdataraterequirements.Besides,ndingapolicywhichachievesthenewconstrainedoptimumsolution,denotedby ,isanontrivialtaskaswell.Inaddition,duetothemultihopnatureofwirelesstransmissions,decentralizedsolutionswithlowcomplexityareremarkablyfavorable.Toaddresstheaforementionedconcerns,innextsection,wewillproposeanoptimalonlinepolicy,namely,QADP,whichprovablygeneratesanetworkrevenuethatisarbitrarilyclosetotheoptimumsolutionof( 2 ).Meanwhile,theimposedrequirementsoftheproblem,i.e.,(a),(b)and(c),areachievedonthey. Inthefollowing,werstassumethatthearrivalratevectorliesoutsideofthecapacityregionforalltimeslots,i.e.,aheavetrafcscenario. 2.4QoSAwareDynamicPricing(QADP)Algorithm Inthissection,weproposeanonlinepolicy,i.e.,QADPalgorithm,whichsolvestheQoSawarerevenuemaximizationproblemin( 2 ). 27 PAGE 28 Werstintroducesomesystemwideparameterstofacilitateouranalysis.DeneRmaxcastheupperboundofadmittedtrafcofowcduringonetimeslot,i.e.,Rc(t)Rmaxc,8c,t.Forexample,Rmaxccanrepresentthehardwarelimitationonthemaximumvolumeoftrafcthatanodecanadmitduringonetimeslot,orsimplythepeakarrivalratewithinonetimeslotifsuchinformationisavailable.Letmaxbethemaximumdatarateonanylinkofthenetwork,whichmaybedeterminedbyfactorssuchasthenumberofantennas,modulationschemesandcodingpolicies.Inaddition,foreachowc,weintroduceavirtualqueueYc(t)whichisinitiallyempty,andthequeueupdatingdynamicisdenedas Yc(t+1)=[Yc(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Rc(t)]++c8c.(2) Notethatvirtualqueuesareeasytoimplement.Forexample,thesourcenodeofowc,i.e.,c(0),canmaintainasoftwarebasedcountertomeasurethebacklogupdatesofvirtualqueueYc(t).Inaddition,foreachowc,wedene c=N(max)2+(Rmaxc)2+1 2(c)28c.(2) Denote1,,Castheweightswhichwillbecalculatedandassignedtoallows,whereCdenotesthenumberofowsinthenetwork.LetJbeatunable5positivelargenumberdeterminedbythenetwork.Inaddition,weassumeamaximumvalueoftheallocatedweight,denotedbymax,i.e.,cmax,8c.TheproposedQADPalgorithmisgivenasfollows. QADPALGORITHM: PartI:WeightAssignment Forallmultimediatransmissions,ndtheowwiththeminimumvalueofc`c,c=1,,C,say,owj.Foreachowc,assignanassociatedweight,denoted 5TheimpactofJontheperformanceofQADPalgorithmwillbeclariedshortly. 28 PAGE 29 byc,whichiscalculatedby c=maxj`j c`c,8c=1,,C.(2) PartII:DynamicPricing Foreverytimeslott,thesourcenodeofowc,i.e.,c(0),measuresthevalueofQcc(0)(t)andYc(t).IfQcc(0)(t)>Yc(t),theinstantaneousadmissionpriceissetas pc(t)=vuut cQcc(0)(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Yc(t) J(2) andpc(t)=0otherwise. PartIII:Scheduling Foreverytimeslott,ndalinkscheduleI(t),fromthefeasiblesetS(t),whichsolves maxI(t)2S(t)X(a,b)2Ea,b(t)a,b(2) where a,b=maxc:(a,b)2Pc(c(Qca(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qcb(t)))(2) if9c,suchthat(a,b)2Pc,anda,b=0otherwise. END ItisworthnotingthatPartIofQADPcanbeprecalculatedbeforeactualtransmissions.TheweightassignmentcanbeimplementedeitherbythenetworkcontrollerwhichknowstheQoSrequirementsofallows,orbymutualinformationexchangesamongmultiplemultimediaows,inadecentralizedfashion.ThevalueofcrepresentstheQoSwiseimportanceofowcandremainsunchangedunlesstheQoSrequirementsfromows,i.e.,(,```),areupdated,bywhichanewweightcalculationistriggered. ThedynamicpricingpartisthecrucialcomponentofQADP.Byfollowing( 2 ),notonlytheincomingadmittedratescanberegulatedeffectively,butalsotheoverallaveragenetworkrevenuecanbemaximized,aswillbeshownshortly.Notethataftertheweightassignment,inordertocomputepc(t),thesourcenodeofowc,i.e.,c(0),whichisconsideredastheedgenodeofthenetwork,requiresonlylocalinformation,i.e.,currentbacklogsofthesourcedataqueueandthevirtualqueue.Itisinteresting 29 PAGE 30 toobservethatifQcc(0)(t)Yc(t),theadmissionisfree!Intuitively,asmallvalueofQcc(0)(t)indicatesadecientarrivalrateofowc.Onthecontrary,alargevalueofYc(t)meansthattheaverageservicerateislessthantheaveragearrivalrateandthusthevirtualqueueisbuildingup.Notein( 2 )thatthisindicatesthattheaverageofRc(t)fallsbelowthearrivalrate,i.e.,c.Therefore,whenQcc(0)(t)Yc(t),thenetworkprovidesfreeadmissiontoencouragemoreincomingpacketsinordertosatisfytheQoSconstraints.Wewillmakethisintuitionpreciseandrigorousinthefollowingsection. ThethirdpartofQADPisaweightedextensiontothewellknownMaxWeightschedulingalgorithm[ 12 13 34 ].Insteadoftheexactdifferenceofqueuebacklogs,wedeliberatelyselecttheweighteddifferenceofqueuebacklogsastheweightofaparticularlinkintheschedulingalgorithm.Intuitively,ifaowisassignedwithalargervalueofc,thelinksassociatedwithitwillhaveahigherpossibilityofbeingselectedfortransmissionsbyQADP.Therefore,byassigningpropervaluesofctoowswithdifferentQoSrequirements,aservicedifferentiationcanbeachievedwhichprovidesmoreexibilitytopreviousschemesintheliterature,e.g.,[ 13 15 16 ].Inaddition,asindicatedby( 2 ),ahigherpriorityneedstopayatahigherprice.Notethattocalculate( 2 ),QADPneedstosolveacomplexoptimizationproblemwhichrequiresaglobalinformationonchannelstates,i.e.,S(t).However,availedoftheprosperousdevelopmentofdistributedschedulingschemes,suchas[ 35 38 ],thedifcultyofcentralizedcomputationcanbecircumvented,whichprovidesQADPalgorithmtheamenabilityfordecentralizedimplementations. 2.5PerformanceAnalysis Inthissection,weprovidethemainresultontheperformanceofQADPalgorithm. Theorem2.1. DeneDastheoptimumsolutionof( 2 ).ForQADPalgorithm,wehave 30 PAGE 31 (a) RevenueMaximization liminft!11 tt)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0O()D)]TJ /F6 11.955 Tf 13.15 8.09 Td[(K J(2) whereKisaconstantandisgivenby K=Xccc(2) andcisdenedin( 2 ). (b) NetworkStability ThenetworkisstableunderQADPalgorithm,i.e.,foreveryqueueinthenetwork,( 2 )issatised. (c) QoSProvisioning ByfollowingQADPalgorithm,anyfeasibleminimumdataraterequirementscanbesatised.Inaddition,theguaranteedmaximumaverageendtoenddelaysformultimediaowswithserviceleveljarejtimeslargerthanthatoflevelonetransmissions. Itisofgreatimportancetoobservethatin( 2 ),theachievedperformanceofQADPalgorithmcanbepushedarbitrarilyclosetotheoptimumsolutionDbyselectingasufcientlylargevalueofJ.TheproofofTheorem 2.1 isprovidedintherestofthissection. 2.5.1ProofofRevenueMaximization Recallthattheminimumratevectorisassumedtolieinsidethecapacityregion.Therefore,thereexistsasmallpositivenumber~>0suchthat+~12where1isaunityvectorwithdimensionC. Lemma1. Foranyfeasibleinputratevector###,thereexistsastationary6randomizedpolicy,denotedbyRAND,whichgenerates E)]TJ /F8 11.955 Tf 5.47 9.69 Td[(outn,c)]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c)]TJ /F8 11.955 Tf 11.95 0 Td[(#cn(t)=08n,c,t(2) 6Stationarymeansthattheprobabilisticstructureoftherandomizedpolicydoesnotchangewithdifferentvaluesofqueuebacklogs. 31 PAGE 32 and E(#cc(0)(t))c+~8t,c(2) where#cn(t)istheexogenousarrivalonnodenforowcduringtimeslott. Proof. (Sketch)TheproofofLemma 1 utilizesstandardtechniquesasin[ 13 15 16 ].ThebasicideaistoreducetheexponentiallylargedimensionofextremepointstoanitesetwithdimensionE+1byCaratheodory'sTheorem.ThenarandomizedlinkscheduleselectionisimplementedamongallreducedE+1extremepoints.Thedetailedproofisomitted. WeemphasizethatLemma 1 isonlyanexistenceproofinthesensethattherandomizedalgorithmRANDcannotbeimplementedinpractice.ThisisbecausethatRANDrequiresapriorknowledgeonthenetworkcapacityregion,i.e.,theunderlyingsteadystatedistributionofMarkovianchannels,andhenceiscomputationallyprohibitive.However,theexistenceofRANDplaysacrucialrolefortheperformanceanalysisofQADPalgorithm,which,incontrast,isanonlineadaptivepolicyanddoesnotrequirethestatisticalcharacteristicsofthestochasticarrivalsandtimevaryingchannelconditions. RecallthatavirtualqueueYc(t)isintroducedforeveryowcandthequeueupdatingdynamicisgivenby( 2 ).Asaresult,theminimumdataraterequirementisconvertedtoaqueuestabilityproblemsinceifthevirtualqueueYc(t)isstable,theaverageservicerate,i.e.,thetimeaverageofRc(t),needstobegreaterthantheaveragearrivalrate,i.e.,c.DeneZZZ(t)=[QQQ(t);YYY(t)]asalltherealdataqueuesandvirtualqueuesattimeslott.IfwecanensurethatthenetworkisstablewithrespecttoZZZ(t),thebacklogsofrealqueuesareboundedandtheminimumraterequirementsareachievedatthesametime. 32 PAGE 33 Deneasystemwidepotentialfunction(PF)as PF(ZZZ(t))=XcPFc(ZZZ(t))(2) where PFc(ZZZ(t))=1 2 Xnc(Qcn(t))2+c(Yc(t))2!.(2) NotethatPF(ZZZ(t))isascalarvaluednonnegativefunction.Dene (ZZZ(t))=E(PF(ZZZ(t+1)))]TJ /F6 11.955 Tf 11.96 0 Td[(PF(ZZZ(t))jZZZ(t))(2) asthedriftofthepotentialfunctionPF(ZZZ(t)). Forowc,wetakethesquareofbothsidesof( 2 )and( 2 )andobtain7 )]TJ /F6 11.955 Tf 5.48 9.69 Td[(Qcc(j)(t+1)2)]TJ /F6 11.955 Tf 5.48 9.69 Td[(Qcc(j)(t)2+)]TJ /F8 11.955 Tf 5.48 9.69 Td[(outc(j),c(t)2+)]TJ /F8 11.955 Tf 5.48 9.68 Td[(inc(j),c(t)2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Qcc(j)(t))]TJ /F8 11.955 Tf 5.48 9.68 Td[(outc(j),c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inc(j),c(t) forj=1,,c)]TJ /F3 11.955 Tf 11.95 0 Td[(1,and )]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qcc(j)(t+1)2)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qcc(j)(t)2+)]TJ /F8 11.955 Tf 5.48 9.68 Td[(outc(j),c(t)2+(Rc(t))2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Qcc(j)(t))]TJ /F8 11.955 Tf 5.48 9.69 Td[(outc(j),c(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Rc(t) (2) forj=0.Similarly,for( 2 ),wehave (Yc(t+1))2(Yc(t))2+(Rc(t))2+(c)2)]TJ /F3 11.955 Tf 9.3 0 Td[(2Yc(t)(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(c). (2) 7Weusethefactthatfornonnegativerealnumbersa,b,c,d,ifa[b)]TJ /F6 11.955 Tf 12.5 0 Td[(c]++d,thena2b2+c2+d2)]TJ /F3 11.955 Tf 11.96 0 Td[(2b(c)]TJ /F6 11.955 Tf 11.96 0 Td[(d),asgiveninLemma4.3on[ 17 ]. 33 PAGE 34 Bycombiningtheinequalitiesabove,wehave PFc(ZZZ(t+1)))]TJ /F6 11.955 Tf 11.95 0 Td[(PFc(ZZZ(t))c+cQcc(0)(t)Rc(t))]TJ /F8 11.955 Tf 11.95 0 Td[(cYc(t)(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(c))]TJ /F13 11.955 Tf 11.29 11.36 Td[(XncQcn(t))]TJ /F8 11.955 Tf 5.48 9.68 Td[(outn,c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inn,c(t) (2) where c=cN(max)2+(Rmaxc)2+1 2(c)2.(2) Notethat( 2 )issummedoverthewholenetwork.Ifnodenisnotonthepathofowc,inn,c(t)=outn,c(t)=0.Moreover,inn,c(t)=0forthesourcenodeofowcandoutn,c(t)=0forthedestinationnodeofowc.Next,wesumoverallowstoderivethenetworkwidepotentialfunctiondifferenceas PF(ZZZ(t+1)))]TJ /F6 11.955 Tf 11.95 0 Td[(PF(ZZZ(t))K)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xn,ccQcn(t)(outn,c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inn,c(t))+XccQcc(0)(t)Rc(t))]TJ /F13 11.955 Tf 11.96 11.36 Td[(XccYc(t)(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(c) whereK=Pcc.Therefore,wehave (ZZZ(t))K)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xn,ccQcn(t)E)]TJ /F8 11.955 Tf 5.48 9.68 Td[(outn,c(t))]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c(t)jZZZ(t)+XccQcc(0)(t)E(Rc(t)jZZZ(t)))]TJ /F13 11.955 Tf 19.26 11.36 Td[(XccYc(t)E(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(cjZZZ(t)). (2) 34 PAGE 35 Next,forapositiveconstantJ,wesubtractbothsidesof( 2 )byJE)]TJ 5.48 .72 Td[(PcRc(t)pc(t)jZZZ(t)andhave (ZZZ(t)))]TJ /F6 11.955 Tf 11.96 0 Td[(JE XcRc(t)pc(t)jZZZ(t)!K)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xn,ccQcn(t)E)]TJ /F8 11.955 Tf 5.48 9.69 Td[(outn,c(t))]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c(t)jZZZ(t)+XccQcc(0)(t)E(Rc(t)jZZZ(t)))]TJ /F13 11.955 Tf 11.29 11.36 Td[(XccYc(t)E(Rc(t))]TJ /F8 11.955 Tf 11.96 0 Td[(cjZZZ(t)))]TJ /F6 11.955 Tf 9.3 0 Td[(JE XcRc(t)pc(t)jZZZ(t)!. (2) Notethat( 2 )isgeneralandholdsforanypossiblepolicy. Foranarbitrarilysmallpositiveconstant0 PAGE 36 thatthereexistsarandomizedpolicy,denotedbyRAND,whichyields E(outn,c(t))]TJ /F8 11.955 Tf 11.95 0 Td[(inn,c(t))]TJ /F6 11.955 Tf 11.96 0 Td[(r,c(t))=8c,n=c(0)(2) and E(outn,c(t))]TJ /F8 11.955 Tf 11.96 0 Td[(inn,c(t))=8c,n6=c(0)(2) and E(r,c(t)+)c+~8c.(2) DenotetheRHSof( 2 )as.Withoutlossofgenerality,weassume~.Therefore,forrandomizedpolicyRAND,wehave RANDK)]TJ /F8 11.955 Tf 11.95 0 Td[( Xn,ccQcn(t)+XccYc(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE Xc(r,c(t)+)^p,c(t)jZZZ(t)! (2) where^p,c(t)isthecorrespondingpricewhichinducesarateofr,c(t)+by( 2 ). Lemma2. QADPalgorithmminimizestheRHSof( 2 )overallpossiblepolicies. Proof. TheschedulingpartofQADPalgorithminSection 2.4 isaweightedversionofMaxWeightalgorithm[ 12 13 34 ]whichmaximizesPn,ccQcn(t)(outn,c(t))]TJ /F8 11.955 Tf 12.45 0 Td[(inn,c(t))=P(a,b)2Ea,b(t)a,bif9c,suchthat(a,b)2Pc,foreverytimeslott.ThedynamicpricingpartofQADPisessentiallymaximizing Xc)]TJ /F8 11.955 Tf 5.48 9.69 Td[(c(Yc(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qcc(0)(t))Rc(t)+JRc(t)pc(t)(2) foreverytimeslot.By( 2 )and( 2 ),weseethatQADPndsanoptimumpricepc(t)whichmaximizes M=c(Yc(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qcc(0)(t))(1 pc(t))]TJ /F3 11.955 Tf 11.96 0 Td[(1)+J(1)]TJ /F6 11.955 Tf 11.96 0 Td[(pc(t)).(2) DeneW=c(Qcc(0)(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Yc(t)). 36 PAGE 37 Case1:IfW>0,wehave M=J+W)]TJ /F3 11.955 Tf 11.96 0 Td[((Jpc(t)+W1 pc(t)).(2) Taketherstorderderivative,wehave M0=W (pc(t))2)]TJ /F6 11.955 Tf 11.95 0 Td[(J.(2) Thesecondorderderivativebringsus M00=)]TJ /F6 11.955 Tf 24.36 8.09 Td[(W (pc(t))3<0.(2) Therefore,Misaconcavefunctionandtheoptimumvalueisachievedby pc(t)=r W J.(2) Case2:IfW0,Misadecreasingfunctionwithrespecttopc(t).Therefore,pc(t)=0. FollowingLemma 2 ,weconcludethatforQADPalgorithm, (ZZZ(t)))]TJ /F6 11.955 Tf 11.95 0 Td[(JE XcRc(t)pc(t)jZZZ(t)!QADPRANDK)]TJ /F8 11.955 Tf 11.96 0 Td[( Xn,ccQcn(t)+XccYc(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE Xc(r,c(t)+)^p,c(t)jZZZ(t)!. (2) WetakeexpectationwiththedistributionofZZZ(t),onbothsidesof( 2 ).BythefactthatEY(E(XjY))=E(X),wehave E(PF(ZZZ(t+1))))]TJ /F6 11.955 Tf 11.95 0 Td[(E(PF(ZZZ(t)))K+JO(t))]TJ /F8 11.955 Tf 11.95 0 Td[(E Xn,ccQcn(t)+XccYc(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE Xc(r,c(t)+)^p,c(t)!. (2) 37 PAGE 38 Notethat( 2 )issatisedforalltimeslots.Therefore,wetakeasumontimeslots=0,,T)]TJ /F3 11.955 Tf 11.96 0 Td[(1andhave E(PF(ZZZ(T))))]TJ /F6 11.955 Tf 11.95 0 Td[(E(PF(ZZZ(0)))TK+T)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0JO())]TJ /F9 7.97 Tf 11.29 14.94 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E Xn,ccQcn()+XccYc()!)]TJ /F9 7.97 Tf 11.29 14.95 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JE Xc(r,c()+)^p,c()!. (2) Divide( 2 )byTandrearrangetermstoobtain 1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E Xn,ccQcn()+XccYc()!+1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JE Xc(r,c()+)^p,c()!K+1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JO()+E(PF(ZZZ(0))) T (2) wherethenonnegativityofthepotentialfunctionisutilized.WersttakeliminfT!1onbothsidesof( 2 )andhave8 liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E Xn,ccQcn()+XccYc()!+liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JE Xc(r,c()+)^p,c()!K+liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0JO(). (2) 8Weassumethattheinitialqueuesizesofrealdataqueuesandvirtualqueuesarebounded. 38 PAGE 39 Notethatthersttermof( 2 )isnonnegativeandthuswehave liminfT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0O()liminfT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0E Xc(r,c()+)^p,c()!)]TJ /F6 11.955 Tf 13.15 8.08 Td[(K J. Recallthattheanalysisaboveappliestoarbitrarilysmall0 PAGE 40 wheremaxisthemaximumpossibledatarateonalink.Moreover,by( 2 )and( 2 ),wehaveOmax=CwhereCisthenumberofowsinthenetwork.Therefore,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0E Xn,ccQcn()+XccYc()!K+JC max. Usingthefactthat X(t)Y(t)8t)limsupX(t)limsupY(t),(2) wehave,foreveryowc,theaveragequeuelengthonitsroutingpathisboundedby limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0E cXj=0Qcc(j)()!K+JC cmax. (2) Finally,byapplyingMarkovInequality,weconcludethatalldataqueuesinthenetworkarestable,basedonthefactthattheRHSof( 2 )isbounded. 2.5.3ProofofQoSProvisioning Similarto( 2 ),foreveryvirtualqueueYc(t),wecanobtain limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1X=0E(Yc())K+JC cmax. (2) Therefore,bysimilaranalysisandthedenitionofvirtualqueues,weconcludethatthevirtualqueuesarestableandthustheminimumdataraterequirementsimposedbymultimediaowsareachieved. Next,weshowthatQADPindeedprovidesaservicedifferentiationsolutionontheguaranteedmaximumendtoenddelaysforallmultimediaows.Denotetheactualexperiencedaveragedelayofowcas!c.ByLittle'sLaw,!cisapproximated9by 9Notethatweconsideraheavyloadednetwork.Propagationdelaysareassumedtobenegligiblecomparedtoqueueingdelaysandthusareomitted. 40 PAGE 41 !c=averageoverallqueuelengthonthepathofowc averageincomingrateofowc=limsupT!11 TPT)]TJ /F5 7.97 Tf 6.59 0 Td[(1=0EPcj=0Qcc(j)() limsupT!11 TPT)]TJ /F5 7.97 Tf 6.58 0 Td[(1=0E(Rc()). (2) InlightofthestabilityofvirtualqueueYc(t),wehavelimsupT!11 TPT)]TJ /F5 7.97 Tf 6.59 0 Td[(1=0E(Rc())c.Hence,wecanobtain !cK+JC cmaxc.(2) Equivalentlyspeaking,todifferentiatetheguaranteedmaximumdelaybound,weneedtondasetofweightssuchthat cc`c=dd`d(2) issatisedforanypairofmultimediaowscandd.Therefore,itisstraightforwardtoverifythattheweightassignmentalgorithminQADPindeedprovidesaservicedifferentiationsolutionwheretheguaranteedmaximumendtoenddelaysofmultimediaowsaredistinguishedaccordingtoapplicationdependentservicelevelrequests,whichcompletestheproofofTheorem 2.1 .TheperformanceofQADPalgorithmwillbeevaluatednumericallyinSection 2.6 2.6Simulations 2.6.1SingleHopWirelessCellularNetworks Werstconsiderasinglehopwirelesscellularnetworkwithdownlinkmultimediatransmissions,asshowninFigure 23 .Thebasestation(BS)isassociatedwiththreeuserswithinnitebackloggedtrafc.Aseparatequeueismaintainedbythebasestationforeveryuser.Inaddition,ateachtimeslot,BScanonlytransmittooneparticularuser.Awirelesslinkisassumedtohavethreeequallypossiblechannelstates,i.e.,Good,Medium,Bad.Thecorrespondingtransmissionratesforthreechannelstatesare20,15and10bitsperslot,respectively.Therefore,thebasestationencountersacomplex 41 PAGE 42 opportunisticschedulingproblemwheretherevenuemaximizationproblem,networkstabilityaswellastheQoSrequirementsneedtobeaddressedsimultaneously. Figure23. Asinglehopwirelesscellularnetworkwiththreeusers Withoutlossofgenerality,weassumethattheminimumaverageraterequirementsforuser1,2,3are=[1,2,3]bitsperslot.Inaddition,toprovideservicedifferentiation,thenetworkoffersthreeprioritizedservicelevels,e.g.,Platinum,GoldandSilver,wherelevelPlatinumpossessesthehighestpriorityintermsofendtoenddelayupperbound.Weassumethatuser1hasaservicelevelrequestforPlatinumwhileuser2and3demandforlevelGoldandSilver,respectively,accordingtotheupperlayerapplications.OthersystemparametersareassumedtobeRmaxc=20bitsperslotforallowsandmax=100.WenextimplementQADPalgorithmfordifferentvaluesofJwhereJ=[50,100,500,1000,5000,10000,20000,50000,100000].Everyexperimentissimulatedfor500000timeslots. Figure 24 depictsthesystemrevenue,i.e.,thesolutionof( 2 )byQADPalgorithm,withrespecttodifferentvaluesofJ.Asshownin( 2 )anddemonstratedpictoriallyinFigure 24 ,theachievedsystemrevenuebyQADPconvergesgraduallytotheoptimumsolutionasJgrows.NotethatthevaluesofsystemrevenuesarealmostindistinguishablewhenJ50000.Figure 25 illustratestheactualexperiencedaveragedelaysofallthreeowswithdifferentvaluesofJ,wherethedelaysofuser1touser3arecomparedfromlefttoright.Itisworthnotingthat,notonlythemaximumguaranteedendtoenddelaysaredistinguishedanalyticallybyQADP,butalsothe 42 PAGE 43 Figure24. ImpactofdifferentvaluesofJontheperformanceofQADP Figure25. ImpactofdifferentvaluesofJontheaverageexperienceddelays actualexperiencedaveragedelaysareprioritizedforallthreeowswithdistinctservicelevelrequests.Morespecically,user1,i.e.,thePlatinumuser,enjoysadelaywhichislessthanhalfofthatofuser2andonethirdofthatofuser3,forallvaluesofJ,asdemonstratedinFigure 25 .However,asshownbyFigure 24 andFigure 25 jointly,whilealargervalueofJyieldsanimprovementontheperformanceofQADP,theendtoenddelaysofallthreeowsareaugmentedconcurrently.Therefore,bytuningJ,atradeoffbetweenoptimalityandaveragedelayscanbeachieved.ThetimeaverageadmittedratesofmultimediaowsareshowninTable 21 .Weobservethatinallcases,theaverageratesofowsexceedtheminimumdataraterequirementsspeciedby. ThesamplepathsofpriceadaptationsandqueuebacklogevolutionsareillustratedinFigure 27 andFigure 26 ,respectively,fortherst100timeslotswithJ=50000. 43 PAGE 44 Table21. Averageadmittedratesformultimediaows Flow1 Flow2 Flow3 J=50 4.01 2.71 3.07 J=100 4.04 2.72 3.09 J=500 4.05 2.91 3.11 J=1000 4.31 3.07 3.16 J=5000 4.51 3.57 3.42 J=10000 4.63 3.94 3.62 J=20000 4.75 4.28 4.09 J=50000 5.09 4.71 4.61 J=100000 5.13 4.88 4.85 Notethat,inFigure 27 ,thepricesimposedforthreeusersaredynamicallyadjustedateverytimeslot.ItisworthnotingthatwheneveradataqueueinFigure 26 hasatendencytobuildup,thepriceimposedbyQADPalgorithm,asshowninFigure 27 ,risescorrespondingly,whichinturndiscouragestheexcessiveadmittedrateandthusallqueuesinthenetworkremainbounded.Asaresult,thestabilityofthenetworkisachieved. 2.6.2MultiHopWirelessNetworks WenextconsideramultihopwirelessnetworkwithatopologyshowninFigure 21 .Therearethreemultimediaowsexistinthenetwork,denotedbyFlow1,2,3.TheroutingpathsofowsarespeciedbyP1=fA,B,C,Dg,P2=fF,G,C,DgandP3=fE,F,G,Hg.Withoutlossofgenerality,weassumeatwohopinterferencemodelwhichrepresentsthegeneralIEEE802.11MACprotocols[ 36 37 ].Othercongurationsarethesameasthesinglehopscenariodescribedaboveexceptthatthepossiblelinkratesareassumedtobe40,30,20bitsperslotforthreechannelconditions.Weobservethatinthisnetworktopology,linkC!DandlinkF!Garesharedbytwodifferentows.Therefore,intheschedulingpartofQADP,theparticularowwithalargerweightedqueuebacklogdifferenceshouldbeselected. InFigure 28 ,wespecicallydepictthedynamicsofthreevirtualqueuesfortherst400timeslotswithJ=50000.UnlikethesinglehopcaseinFigure 26 ,thevirtual 44 PAGE 45 Figure26. Queuebacklogdynamicsforallusers Figure27. PricedynamicsinQADPforallusers queuesbehaveremarkablydifferentinthismultihopscenario.Itisworthnotingthatwhilethevirtualqueuesofuser1and3haverelativelylowoccupancies,thevirtualqueueassociatedwithuser2suffersalargeraveragebacklog.Intuitively,duetotheunderlyingtwohopinterferencemodel,linkG!Cneedstobescheduledexclusivelyinthenetworkforsuccessfultransmissions.Inotherwords,linkG!Cisthebottleneckofthenetwork.Therefore,toensurenetworkwidestability,amuchmorestringentregulationisenforcedontheadmittedrateofow2.Asaconsequence,althoughremainsbounded,thevirtualqueueofow2accumulatesmorebacklogscomparedtoothercompetitiveows.Inaddition,wecomparethetimeaveragequeuebacklogsofalldataqueuesinthenetwork,fromlefttoright,inFigure 29 .Wecanobservethatthedataqueuesonthepathofow1havefeweraveragebacklogsduetothehighest 45 PAGE 46 Figure28. VirtualqueuebacklogupdatesinQADPforJ=50000 Figure29. AveragequeuebacklogsinthenetworkforJ=50000 prioritywithrespecttotheservicelevel,i.e.,Platinum.Onthecontrary,thequeuesonthepathofow3havelargerbacklogscomparedtoothertwoows.ItisnoticeablethatQ3FandQ3Ghaveconsiderablylargeraveragequeuesizes.ThisisbecausethatQ3FhastosharethelinkrateofF!GwithQ2F.Nevertheless,Q3FpossessesasmallershareofbandwidththanQ2Fduetothelowerprioritizedservicelevelassociatedwithow3.Evenworseyet,bothlinkF!GandG!Hhavelessopportunitytobescheduledduetotheirlocationsandtheunderlyinginterferencemodel.Therefore,theaveragebacklogsonthepathofuser3hashigheroccupanciescomparedtoothertwocompetitiveows.TheaveragedataratesareprovidedinTable 22 .Notethattheminimumaverageraterequirementsofallmultimediaowsaresatisedsimultaneously,asexpected.Thetradeoffbetweenoptimalityandaveragedelay,whichiscontrolledbydifferentvaluesof 46 PAGE 47 J,aswellasthenetworkservicedifferentiationintermsofdelaysareanalogoustothesinglehopscenariodiscussedabove.Duplicatedsimulationguresareomitted. Table22. Averageadmittedratesformultimediaows Flow1 Flow2 Flow3 J=50 3.04 2.08 3.88 J=100 3.12 2.03 3.98 J=500 3.17 2.04 4.60 J=1000 3.30 2.03 5.04 J=5000 4.13 2.01 6.45 J=10000 4.36 2.02 7.32 J=20000 5.08 2.03 8.12 J=50000 6.43 2.01 9.18 J=100000 7.63 2.01 9.89 2.7Conclusions WeconsideramultihopwirelessnetworkwheremultipleQoSspecicmultimediaowssharethenetworkresourcejointly.Tomaximizetheoverallnetworkrevenue,weproposeadynamicpricingbasedalgorithm,namely,QADP,whichachievesasolutionthatisarbitrarilyclosetotheoptimum,subjecttonetworkstability.Moreover,aweightassignmentmechanismisintroducedtoaddresstheservicedifferentiationissueformultimediaowswithdifferentdelaypriorities.Intandemwiththevirtualqueuetechniquewhichprovidesminimumaveragedatarateguarantees,theQoSrequirementsofmultiplemultimediastreamsareaddressedeffectively. Inthiswork,theapplicationdependentservicelevelrequestsandtheusers'utilityfunctionsareassumedtobeattainedtruthfully.Themechanismdesignforstrategyproofuserinformationacquisitionseemsinterestingandneedsfurtherinvestigation.Moreover,inthiswork,weassumethattheinformationofchannelstates,i.e.,S(t),isavailableforQADPalgorithm,whichiseitheracquiredbythecentralbasestationorapproximatedbythedistributedlocalschedulingalgorithms.Asafuturework,anonlinechannelprobingmechanismneedstobeincorporatedintoQADP,assuggestedin[ 39 ]. 47 PAGE 48 CHAPTER3ENERGYCONSERVINGSCHEDULINGINWIRELESSNETWORKS 3.1Introduction Therehasbeenalotofinterestoverthepastfewyearsincharacterizingthenetworkcapacityregionaswellasdesigningefcientschedulingalgorithmsinmultihopwirelessnetworks.Duetothestochastictrafcarrivalsandtimevaryingchannelconditions,supportinghighthroughputandhighqualitycommunicationsinmultihopwirelessnetworksisinherentlychallenging.Toutilizethescarcewirelessbandwidthresourceeffectively,schedulingalgorithmswhichcandynamicallyallocatethenetworkresource,i.e.,selectactivelinks,areinvestigatedintensivelyinthecommunity.Forexample,MaxWeightalgorithm,a.k.a.,backpressurealgorithm,hasbeenextensivelystudiedintheliterature,e.g.,[ 5 14 34 37 40 ],followingtheseminalworkof[ 12 ].TheMaxWeightalgorithmenjoysthemeritofselfadaptabilityduetoitsonlinenature.Inaddition,MaxWeightalgorithmisknowntobethroughputoptimal[ 17 ].Thatistosay,theMaxWeightalgorithmcanstabilizethenetworkunderarbitrarytrafcloadthatcanbestabilizedbyanyotherpossibleschedulingalgorithms.Therefore,theMaxWeightalgorithmattractssignicantattentionandbecomesanindispensablecomponentforlinkschedulinginnetworkprotocoldesigns,e.g.,[ 41 43 ]. WhilethethroughputoptimalityoftheMaxWeightalgorithmiswellunderstood,theenergyconsumptioninducedbytheMaxWeightalgorithmislessstudiedintheliterature.However,duetothescarcityofenergysuppliesinwirelessnodes,itisimperativetostudytheenergyconsumptionoftheschedulingalgorithmwhichisofspecialinterestinenergyconstrainedwirelessnetworkssuchaswirelesssensornetworks.IsthethroughputoptimalMaxWeightschedulingalgorithmalsoenergyoptimal?Inthiswork,weshowthattheanswertothisquestionisno.ThereasonisthatthevastenergyconsumptionsduringpacketretransmissionsarecompletelyneglectedbytheMaxWeightalgorithm.Forexample,in[ 16 ],anenergyoptimalcontrolscheme 48 PAGE 49 isproposedwhereaminimumpowerexpenditureisachieved.However,asinotherrelatedworks,e.g.,[ 17 ],thewirelesschannelsin[ 16 ]areassumedtobeerrorfree,i.e.,allthetransmissionsareassumedtobesuccessful.Nevertheless,inpractice,wirelesschannelsareerrorproneanddatatransmissionsaresubjecttorandomfailuresduetothehostilechannelconditions.Therefore,beforeapacketcanbesuccessfullyremovedfromthetransmitter'squeue,severaltransmissionsmayhaveoccurred,includingtheoriginalattemptandtheposteriorretransmissions,whichdepleteasignicantamountofenergyforthetransmitter.However,inthetraditionalMaxWeightalgorithm,suchenergyconsumingretransmissionsinducedbychannelerrorsareoverlooked.Intuitively,fromenergysavingperspective,thepossibilitythataparticularlinkisselectedfortransmissionsshouldrelyonnotonlythequeuedifferencebetweenthetransmitterandthereceiver,whichisthedesignrationaleofthetraditionalMaxWeightalgorithm,butalsothepotentialenergyconsumptionsofretransmissionsinducedbyerroneouschannels.Wewillmakethisintuitionpreciseandrigorousinthefollowingsections. Inthiswork,weproposeaminimumenergyscheduling(MES)algorithmwhichconsumesanamountofenergythatcanbepushedarbitrarilyclosetotheglobalminimumsolution.Inaddition,theenergyefciencyattainedbytheMESalgorithmincursnolossofthroughputoptimality.TheproposedMESalgorithmsignicantlyreducestheoverallenergyconsumptioncomparedtothetraditionalMaxWeightalgorithmandremainsthroughputoptimal.Therefore,ourproposedMESalgorithmismorefavorablefordynamiclinkschedulingandnetworkprotocoldesignsinenergyconstrainedwirelessnetworkssuchaswirelesssensornetworks. Therestofthischapterisorganizedasfollows.Section 3.2 describesthesystemmodelusedinthiswork.TheproposedMESalgorithmisintroducedinSection 3.3 wheretheperformanceanalysisisprovided.SimulationresultsareshowninSection 3.4 andSection 3.5 concludesthischapter. 49 PAGE 50 3.2SystemModel Weconsiderastaticmultihopwirelessnetworkdenotedbyadirectedgraph(N,L)whereNisthesetofnodesandLdenotesthesetoflinksinthenetwork.WeusejXjtorepresentthecardinalityofsetX.Timeisslottedbyt=0,1,2,andineverytimeslot,theinstantaneouschannelstateofalink(a,b)2LisdenotedbySa,b(t)whereaandbarethetransmitterandthereceiverofthelink.Inthiswork,weuseS(t)todenotethechannelstatevectorofthewholenetwork.NotethatS(t)remainsconstantwithinonetimeslot,however,itissubjecttochangesontimeslotboundaries.WeassumethatS(t)hasanitebutpotentiallylargenumberofpossiblevaluesandevolvesfollowinganirreducibleMarkovianchainwithwelldenedsteadystatedistributions.Nevertheless,thesteadystatedistributionandthetransitionprobabilitiesareunknowntothenetwork.GivenaninstantaneouschannelstateSa,b(t),thetransmissionofapacketonlink(a,b)issuccessfulwithaprobabilityofpa,b(t),iflink(a,b)isactiveandsuffersnointerferencefromconcurrenttransmissions.Fromthenetwork'sperspective,ateachtimeslott,aninterferencefreelinkschedule,denotedbyI(t),isselectedfromafeasibleset(t),whichisconstrainedbytheunderlyinginterferencemodel,e.g.,Khopinterferencemodel[ 44 ],aswellasotherlimitationssuchasduplexconstraintsandpeakpowerlimitations.Denotegua,b(t)asthenominallinkrateiflink(a,b)isselectedbythenetworkandthechanneliserrorfree,i.e.,pa,b(t)=1.Theactualdatarateoflink(a,b)ishencerepresentedbyua,b(t)=gua,b(t)pa,b(t).Weassumethatua,b(t)isupperboundedbyaconstantumaxforall(a,b)2L.Inpractice,umaxcanbedeterminedbythenumberofantennasequippedinasinglenodeaswellasthecoding/modulationschemesavailabletothenetwork.Wedenoteu(t)asthelinkratevectorofthenetworkattimet.Apparently,u(t)isafunctionofbothI(t)andS(t). ThenetworkconsistsofjFjowsindexedbyf=1,2,,jFj.EachowfisassociatedwitharoutingpathRf=[nf0,nf1,,nff]wherenfj,j=0,,fdenotesthenodesonthepathofowf.Ateachtimeslott,thestochasticexogenousarrivals 50 PAGE 51 ofowf,i.e.,thenumberofnewpacketsthatareinitiatedbynodenf0,isdenotedbyAf(t).Fortheeaseofexposition,weassumethatAf(t)isi.i.d.foreverytimeslotwithanaveragerateoff.Inaddition,thearrivalprocessesofallowsareassumedtobeindependent.Wefurtherassumethatthemaximumnumberofnewpacketsgeneratedbyaowduringonetimeslotisupperbounded,i.e.,Af(t)Amax,8f,t.Weemphasizethatthei.i.d.assumptionincursnolossofgeneralityandourmodelcanbeextendedtocaseswhereAf(t)isnonstationaryinastraightforwardfashion,asin[ 14 ]. Everynodeinthenetworkmaintainsaseparatequeueforeachowthatpassesthroughit.DenoteQfn(t)asthequeuebacklogattimet,fornoden,wherefisoneoftheowsthattraversesthroughn,i.e.,n2Rf.WeassumethatQfn(t)=0,8tifn=nff.Thatistosay,ifapacketreachesthedestination,weconsiderthepacketasleavingthenetworkimmediately.DenetheoverowfunctionofQfnas O(M)=limsupt!11 tt)]TJ /F5 7.97 Tf 6.58 0 Td[(1X=0Pr(Qfn()>M).(3) Thequeueisstableif[ 14 ] limM!1O(M)!0(3) andthenetworkisstableifallindividualqueuesarestableconcurrently. Fornotationbrevity,wedenethenetworkadmissionratevectoras=fn,f,8n,fgwheren,fistheaverageexogenousarrivalrateofowfonqueue1Qfn.Wehaven,f=fifn=nf0andn,f=0otherwise.Denoteasthenetworkcapacityregion[ 12 ],namely,thesetofallfeasibleadmissionratevectors,i.e.,,thatthenetworkcansupport,inthesensethatthereexistsaschedulingalgorithmwhichstabilizesthenetworkundertrafcload.Itisshownin[ 12 ]and[ 14 ]thatisconvex,closedandbounded. 1Notethatwithaslightabuseofnotation,weuseQfntodenoteboththequeueitselfandthenumberofpacketsinthequeue. 51 PAGE 52 Withoutlossofgenerality,weconsideranenergyconsumptionmodelasfollows.Recallthatthesuccessfultransmissionprobabilityoflink(a,b)ispa,b(t).Therefore,foraparticularpackettobetransmittedfromatob,onaverage,anumberof1 pa,b(t)transmissionsareneededinordertoerasethispacketfromthequeueofnodea.Fromthetransmitter'sperspective,however,everytransmission,eithertheoriginalattemptorretransmissions,coststhesameamountofenergy.Denotea,bastheenergyneededtotransmitapacketonlink(a,b).Forexample,a,bcanbeproportionaltothedistancebetweennodeaandb.Therefore,inordertotransmitapacketsuccessfullyon(a,b),nodeaneedstospendatotalenergyofa,b pa,b(t)onaverage.Forthereceiverb,weassumethattheenergydepletiononsuccessfulpacketsreceptionsaredominant,i.e.,theenergyspentforoverhearingandshortACKmessagesareneglected.Denotea,bastheenergyconsumedforasuccessfulpacketreceptionindemodulationanddecodingonnodeb.Withadatarate2ofua,b(t)onlink(a,b),theoverallenergyspentduringtimeslottisgivenby Ga,b(t)=ua,b(t)a,b pa,b(t)+a,b=gua,b(t)(a,b+a,bpa,b(t)). (3) Notethatduetothestochasticnatureofwirelesschannels,Ga,bisarandomvariable.Westressthatthesimpleenergyconsumptionmodelaboveisnotessentialandouranalysiscanbeextendedtoothermorecomplexformsofenergymodelsstraightforwardly,aswillbeshowninthenextsection. 2Theunitofdatarateinthisworkisdenedaspackets/slot.Itisworthnotingthatotherunitssuchasbits/slotarealsoapplicable. 52 PAGE 53 Inaddition,weassumethattheenergyconsumptionofthewholenetworkduringonesingletimeslotisupperbounded,i.e., X(a,b)2LGa,b(t)Gmax,8t.(3) Therefore,tominimizetheenergyconsumption,theobjectiveofthenetworkistondaschedulingalgorithmwhichsolves MinimumEnergySchedulingProblem: minimizeC=limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 G(t) (3) s.t. thenetworkremainsstable,and G(t)=E0@X(a,b)2Lha,b(t)a,b pa,b(t)+a,b1A(3) istheexpectedoverallnetworkenergyconsumptionduringtimeslott,withrespecttotherandomnessofarrivalprocessesandchannelvariations.Notethatha,b(t)istheactualnumberofsuccessfullytransmittedpacketsonlink(a,b)duringtimeslottand3ha,b(t)ua,b(t). Inthenextsection,wewillproposeaminimumenergyscheduling(MES)algorithmwhichminimizestheaveragenetworkenergyconsumptionasymptoticallysubjecttonetworkstability.Inaddition,theproposedMESalgorithmisthroughputoptimal,inthesensethattheMESalgorithmcanensurethenetworkstabilityforallfeasiblenetworkadmissionratevectorsinthenetworkcapacityregion.Restated,thesetoffeasiblearrivalratessupportedbytheMESalgorithmisthesupersetofallotherpossible 3Theinequalityholdswhennodeahaslesspacketstotransmitthantheallocateddatarateua,b(t). 53 PAGE 54 schedulingalgorithms,includingthosewithaprioriknowledgeonthefuturisticarrivalsandchannelconditions. 3.3MinimumEnergySchedulingAlgorithm 3.3.1AlgorithmDescription Theminimumenergyscheduling(MES)algorithmisgivenasfollows. MESALGORITHM: Ateverytimeslott: Everylink(a,b)2Lndstheowfwhichmaximizes maxf:(a,b)2Rf2Qfa(t))]TJ /F6 11.955 Tf 16.77 8.09 Td[(Ja,b pa,b(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Qfb(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Ja,b(3) whereJisapositiveconstantwhichistunableasasystemparameter. Everylink(a,b)2Lcalculatesthelinkweightas Ha,b(t)=2Qfa(t))]TJ /F6 11.955 Tf 16.77 8.09 Td[(Ja,b pa,b(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Qfb(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Ja,b+(3) where[x]+denotesmax(x,0). Forthenetwork,alinkscheduleI(t)isselectedwhichsolves maxI(t)2(t)X(a,b)2Lua,b(t)Ha,b(t)(3)END NotethattheproposedMESalgorithmisdifferentfromtheMaxWeightalgorithmproposedin[ 12 13 ].IntheoriginalMaxWeightalgorithm,theweightofaparticularlink(a,b)isthequeuedifferencebetweennodeaandb.However,intheMESalgorithm,asindicatedby( 3 ),thelinkweightHa,b(t)isrelatedtothepotentialenergyconsumptionsonthislinkunderthecurrentchannelconditionaswell.Morespecically,thelinkweightintheMESalgorithmisthequeuedifferencesubtractedbyaweightedenergyconsumptionfactor,i.e.,Ja,b pa,b(t)+a,b,whereJrepresentstheweight.Intuitively,ifthecurrentchannelisunfavorable,i.e.,pa,b(t)issmall,theenergyrequiredforasuccessful 54 PAGE 55 transmissionisremarkablylargeduetotheretransmissionsandthusthelinkshouldbeselectedlesslikely.Therefore,thelinkweightintheMESalgorithm,i.e.,Ha,b(t),canbeviewedasabalancebetweenthequeuedifferenceandtheenergyconsumptionsunderthecurrentchannelcondition.Morespecically,intheoriginalMaxWeightalgorithm,ifalinkhasalargerqueuedifference,thelinkismorelikelytobeselectedfortransmissions.However,intheMESalgorithm,boththequeuedifferenceandtheenergyexpenditurearetakenintoconsideration.Inaddition,asindicatedby( 3 ),byreplacingJa,b pa,b(t)+a,bwithothermetrics,ourMESalgorithmcanincorporateotherformsofenergyconsumptionmodelsstraightforwardly. ObservethatifJ=0,theMESalgorithmreducestotheoriginalMaxWeightalgorithm,i.e.,theenergyconsumptionsduringretransmissionsareomitted.Therefore,theMaxWeightalgorithmisaspecialcaseofourMESalgorithm.Ontheotherhand,ifweletJ!1,theperformanceoftheMESalgorithmcanbepushedarbitrarilyclosetotheglobalminimumsolution,aswillbeshownanalyticallyinSection 3.3.3 .However,asillustratedin( 3 )and( 3 ),whenJincreases,thenetworkbecomesmorereluctanttotransmissions,forthesakeofenergyconservation,unlesstheaccumulatedqueuebacklogsaresignicantlylarge.Intuitively,alargeraveragequeuesizeinducesalongerexperienceddelayfortransmissions.Therefore,thesystemparameterJisessentiallyacontrolknobwhichprovidesatradeoffbetweentheenergyoptimalityandtheexperienceddelayinthenetwork. NotethatsimilartothetraditionalMaxWeightalgorithm,thecalculationin( 3 )iscentralized.However,following[ 12 ],muchprogresshasbeenmadeineasingthecomputationalcomplexityandderivingdecentralizedsolutionsforthecentralizedMaxWeightalgorithm,e.g.,[ 35 38 40 42 49 ].Itisworthnotingthatsolving( 3 )isequivalenttondingthemaximumweightindependentsetintheconictgraph,whichiscombinatorialinnatureandthusisintrinsicallydifculttosolve.Anaturalheuristic,denotedbyGreedyMax,isdescribedasfollows.First,thelinkwiththehighestweight 55 PAGE 56 inthenetworkisselected.Next,thelinkwiththesecondhighestweightwhichdoesnotinvolveconictswithanypreviouslyselectedlinks,isselectedandtheiterationcontinuesuntilnolinkcanbeadded.BasedonGreedyMax,anovelprepartitionbasedapproachisintroducedin[ 35 ].Theauthorsprovethatifthetopologyofthenetworksatisescertainconditions,a.k.a.,localpoolingfactorconditions,GreedyMaxcanachievethesamethroughputasMaxWeight.Furthermore,treebasedtopologyareshowntosatisfythelocalpoolingconditions.Inlightofthis,[ 35 ]utilizesgraphalgorithmstopartitionthewholenetworkintotreeswhereeachtreeisallocatedanorthogonalchannel.Asaresult,thewholenetworkcanoperatewithsimpleGreedyMaxalgorithmandachievethesamethroughputasMaxWeight.Thislineofresearchisfurthersimpliedby[ 36 ]wheretheauthorprovesthatalocalgreedymaximalalgorithmcanobtainthesameperformanceastheglobalGreedyMaxalgorithm.Specically,ineachtimeslot,alinkonlyneedstocompareitsweightwithlocalneighboringlinkstodecideafeasibletransmissionschedule.Therefore,intandemwiththetreeprepartitionmethodin[ 35 ],thecomplexschedulingalgorithmin( 3 )canbeimplementedinafullydistributedfashion.Anotherfeasibledirectionistoutilizethedistributedrandomaccessapproximationschemes,e.g.,[ 37 38 42 ].Forexample,in[ 42 ],eachnodeinthenetworkutilizesanIEEE802.11MACprotocolwherethechannelaccessprobabilityisdynamicallyadjustedinaccordancetothelinkweight,i.e.,Ha,b(t)inourscenario.Theeffectivenessofsuchrandomaccessbaseddistributedapproximationsisstudiedextensivelyin[ 43 ]and[ 42 ].Wenotethat,althoughdistributedimplementationisnotthefocusofthiswork,ourproposedMESalgorithmcanbeapproximatedwellbythesolutionssuggestedintheabovepapers. Ifanadmissioncontrolmechanismisimplementedinthenetworktoregulatetheoverwhelmingarrivals,inordertoachieveanaverageminimumrateprovisionforeachow,wecanutilizetheconceptofvirtualqueuesintroducedin[ 17 34 ],whereforeach 56 PAGE 57 owf,wedeneavirtualqueueYf(t)whichisinitiallyempty,i.e., Yf(0)=0,8f(3) andthevirtualqueueupdatingdynamicisdenedas Yf(t+1)=[Yf(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Rf(t)]++f8f(3) whereRf(t)istheadmittedtrafcforowfattimeslottandfisafeasibleminimumraterequirementofowf.Intuitively,ifeachofthevirtualqueuesinthesystemisstable,theaveragearrivalrateshouldbelessthantheaveragedeparturerateofthevirtualqueueandhencewehave limT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Rf(t)f,8f(3) whichisexactlythedesiredminimumraterequirementforowf.Moreover,thevirtualqueuesareeasytoimplement.Forexample,thesourcenodeofowfcanmaintainasoftwarebasedcountertomeasurethebacklogupdatesofvirtualqueueYf(t).Therefore,theminimumraterequirementscanbeincorporatedintotheproposedschemestraightforwardly. 3.3.2ThroughputOptimality WerstshowthattheproposedMESalgorithmisthroughputoptimal,asgiveninthefollowingtheorem. Theorem3.1. TheproposedMESalgorithmisthroughputoptimal,i.e.,foranarbitrarynetworkadmissionratevectorwhichisinsideofthenetworkcapacityregion,MESstabilizesthenetworkunder. Proof. WerstprovidethequeueupdatingequationofQfn.Notethatapacketisremovedfromthetransmitter'squeueifandonlyifitisreceivedbythereceiversuccessfully.Therefore,thequeueupdatingdynamicisgivenby Qfn(t+1)[Qfn(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uoutn,f(t)]++uinn,f(t)+An,f(t)(3) 57 PAGE 58 whereuoutn,f(t)anduinn,f(t)aretheallocateddatarateontheoutgoinglinkandtheincominglinkofnoden,withrespecttoowf.Notethatuoutn,f(t)=guoutn,f(t)poutn,c(t)andpoutn,c(t)isthecurrentsuccessfultransmissionprobabilityonthislink.Notethatuoutn,f(t)=0,8tifnodenisthedestinationnodeofowfanduinn,f(t)=0,8tifnodenisthesourcenodeofowf.Furthermore,An,f(t)=Af(t)ifnisthesourcenodeofowfandAn,f(t)=0otherwise. From( 3 ),wehave (Qfn(t+1))2)]TJ /F3 11.955 Tf 11.96 0 Td[((Qfn(t))2)]TJ /F3 11.955 Tf 5.48 9.69 Td[((umax)2+(umax+Amax)2)]TJ /F3 11.955 Tf 9.3 0 Td[(2Qfn(t))]TJ /F6 11.955 Tf 5.48 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(An,f(t). (3) Next,wesum( 3 )overthewholenetworkonalldataqueuesandobtain Xn,f(Qfn(t+1))2)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xn,f(Qfn(t))2B)]TJ /F3 11.955 Tf 11.95 0 Td[(2Xn,fQfn(t))]TJ /F6 11.955 Tf 5.48 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t) (3) where B=jNjjFj)]TJ /F3 11.955 Tf 5.48 9.68 Td[((umax)2+(umax+Amax)2(3) isaconstant. DenoteQQQ(t)=fQfn(t),8f,ngastheinstantaneousqueuebacklogsinthenetwork.WetaketheconditionalexpectationwithrespecttoQQQ(t)on( 3 )andhave E Xn,f(Qfn(t+1))2jQQQ(t)!)]TJ /F6 11.955 Tf 11.95 0 Td[(E Xn,f(Qfn(t))2jQQQ(t)!B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.47 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(An,f(t)jQQQ(t). Dene GMESQ(t)=E Xn,fhoutn,f(t)outn,f poutn,f(t)+hinn,f(t)inn,fjQQQ(t)! 58 PAGE 59 whereoutn,f(inn,f)denotestheenergyneededforapackettransmission(reception)ontheoutgoing(incoming)linkofnoden,withrespecttoowf. Inaddition,wedene GMES(t)=E Xn,fhoutn,f(t)outn,f poutn,f(t)+hinn,f(t)inn,f! astheexpectednetworkwideenergyconsumptionduringtimeslott,byfollowingtheproposedMESalgorithm.Apparently,wehave E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(GMESQ(t)= GMES(t). Next,weaddbothsidesbyJGMESQ(t)whereJisapositiveconstant,andhave E Xn,f(Qfn(t+1))2jQQQ(t)!)]TJ /F6 11.955 Tf 11.96 0 Td[(E Xn,f(Qfn(t))2jQQQ(t)!+JGMESQ(t)B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t)jQQQ(t)+JGMESQ(t)B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 9.69 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t)jQQQ(t)+JE Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,fjQQQ(t)!. (3) DenotetheR.H.S.of( 3 )as.Itisofgreatimportancetoobservethat,fromthealgorithmdescriptionoftheMESalgorithmabove,ateverytimeslott,theMESalgorithmessentiallyminimizestheR.H.S.of( 3 )overallpossibleschedulingalgorithms. Sinceliesintheinteriorofthenetworkcapacityregion,itimmediatelyfollowsthatthereexistsasmallpositiveconstant>0suchthat +=f(1,1+),,(n,f+),g2.(3) 59 PAGE 60 ByinvokingCorollary3.9in[ 17 ],weclaimthatthereexistsarandomizedschedulingpolicy,denotedbyRA,whichstabilizesthenetworkwhileprovidingadatarateof E uoutn,f(t))]TJ ET q .478 w 197.33 60.4 m 215.53 60.4 l S Q BT /F6 11.955 Tf 197.33 71.73 Td[(uinn,f(t)jQQQ(t)=n,f+,8t(3) where uoutn,c(t), uinn,c(t)arethelinkdataratesinducedbytherandomizedpolicyRA.Therefore,wehave RA=B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Pn,fQfn(t)+JEPn,f uoutn,f(t)outn,f poutn,f(t)+ uinn,f(t)inn,fjQQQ(t). (3) Notethatthelasttermin( 3 )istheactualenergyconsumptionbyRAalgorithmduringtimeslott.Following( 3 ),wehave RAB)]TJ /F3 11.955 Tf 11.95 0 Td[(2Xn,fQfn(t)+JGmax.(3) Inlightof( 3 ),wehave E)]TJ 5.48 .72 Td[(Pn,f(Qfn(t+1))2jQQQ(t))]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 .72 Td[(Pn,f(Qfn(t))2jQQQ(t)+JGMESQ(t)MESRAB)]TJ /F3 11.955 Tf 11.95 0 Td[(2Pn,fQfn(t)+JGmax. (3) Next,wetaketheexpectationwithrespecttoQQQ(t)on( 3 )andhave E)]TJ 5.48 .71 Td[(Pn,f(Qfn(t+1))2)]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 .71 Td[(Pn,f(Qfn(t))2+J GMES(t)B)]TJ /F3 11.955 Tf 11.95 0 Td[(2E)]TJ 5.48 .72 Td[(Pn,fQfn(t)+JGmax. (3) 60 PAGE 61 Notethattheaboveinequalityholdsforanytimeslott.Hence,wesumoverfromtimeslot0toT)]TJ /F3 11.955 Tf 11.96 0 Td[(1andobtain E)]TJ 5.48 .72 Td[(Pn,f(Qfn(T))2)]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 .72 Td[(Pn,f(Qfn(0))2+JPT)]TJ /F5 7.97 Tf 6.59 0 Td[(1t=0 GMES(t)TB)]TJ /F3 11.955 Tf 11.95 0 Td[(2PT)]TJ /F5 7.97 Tf 6.59 0 Td[(1t=0Pn,fE)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qfn(t)+TJGmax. (3) Next,werearrangetermsanddivide( 3 )byTandhave 21 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0Xn,fE)]TJ /F6 11.955 Tf 5.48 9.69 Td[(Qfn(t)B+JGmax+E)]TJ 5.47 .71 Td[(Pn,f(Qfn(0))2 T)]TJ /F6 11.955 Tf 9.3 0 Td[(J1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t))]TJ /F6 11.955 Tf 13.15 9.03 Td[(E)]TJ 5.48 .71 Td[(Pn,f(Qfn(T))2 T. (3) Notethatthelasttwotermsof( 3 )arebothnonpositive.BytakinglimsupT!1onbothsidesof( 3 ),weattain limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Xn,fE)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qfn(t)B+JGmax 2<1.(3) Therefore,foreveryindividualqueueQfn,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qfn(t)B+JGmax 2<1.(3) Finally,byinvokingMarkovinequality,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Pr)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qfn(t)>M PAGE 62 3.3.3AsymptoticEnergyOptimality Inthissection,weshowthattheMESalgorithmyieldsanasymptoticoptimalsolutionto( 3 ),i.e.,theaverageenergyconsumptioninducedbytheMESalgorithmcanbearbitrarilyclosetotheglobalminimumsolutionof( 3 ),byselectingasufcientlylargevalueofJ. Theorem3.2. DeneCMESandCastheaverageenergyconsumptioninducedbytheMESalgorithmandtheoptimal(minimum)solutionof( 3 ),respectively.TheperformanceoftheMESalgorithmisgivenby CMESC+B J(3) whereBisdenedin( 3 ).Therefore,bychoosingJ!1,theperformanceoftheMESalgorithmcanbepushedarbitrarilyclosetotheoptimumsolutionC. Proof. First,denotetheoptimalsequenceoflinkrates,whichgeneratestheoptimumsolutionC,asuuu(0),uuu(1),,uuu(t),.Next,letusconsideradeterministicpolicy,denotedbyDE,whichallocatesexactlytheoptimumlinkdataratesoneverytimeslott.Similarto( 3 ),wehave E)]TJ 5.48 .71 Td[(Pn,f(Qfn(t+1))2jQQQ(t))]TJ /F6 11.955 Tf 11.95 0 Td[(E)]TJ 5.48 .71 Td[(Pn,f(Qfn(t))2jQQQ(t)+JGMESQ(t)MESDE (3) where DE=B)]TJ /F3 11.955 Tf 11.95 0 Td[(2Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(An,f(t)jQQQ(t)+JE Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,fjQQQ(t)!. (3) Itisworthnotingin( 3 ),onlythedataratesarereplacedbytheonesgeneratedbytheDEalgorithmwhereasallothervaluesremainthesame.Notethatinthiscase,the 62 PAGE 63 optimaldataratesuoutn,f(t)anduinn,f(t)areknownasaprioribytheDEalgorithmandthusareconstantswithrespecttoQfn(t). Wetaketheexpectationonbothsidesof( 3 )andhave E Xn,f(Qfn(t+1))2!)]TJ /F6 11.955 Tf 11.95 0 Td[(E Xn,f(Qfn(t))2!+J GMES(t)B)]TJ /F3 11.955 Tf 9.3 0 Td[(2E Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.96 0 Td[(uinn,f(t))]TJ /F8 11.955 Tf 11.95 0 Td[(n,fjQQQ(t)!+JE Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,f!. (3) Weemphasizethat,however,inthisscenario,atarbitrarytimeslott,thevalueof uoutn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uinn,f(t))]TJ /F8 11.955 Tf 11.96 0 Td[(n,f(3) couldbeeithernonpositiveornonnegative.Inotherwords,therelationshipof( 3 )doesnothold.Tocircumventthis,werstsum( 3 )overtimeslotst=0,,T)]TJ /F3 11.955 Tf 12.3 0 Td[(1anddivideitbyT.Thus,weattain J1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t)B+J1 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0E Xn,fuoutn,f(t)outn,f poutn,f(t)+uinn,f(t)inn,f!+E)]TJ 5.48 .72 Td[(Pn,f(Qfn(0))2 T)]TJ /F3 11.955 Tf 11.96 0 Td[(21 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0(t) where (t)=E Xn,fQfn(t)E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(uoutn,f(t))]TJ /F6 11.955 Tf 11.95 0 Td[(uinn,f(t))]TJ /F8 11.955 Tf 11.96 0 Td[(n,fjQQQ(t)!.(3) 63 PAGE 64 BytakinglimsupT!1,wehave JlimsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t)B+JC)]TJ /F3 11.955 Tf 9.3 0 Td[(2limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0(t). (3) Notethat lim(AB)=lim(A)lim(B)(3) iflim(A)andlim(B)existandarebounded.Recallthatintheprevioussection,wehaveshownthat limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Xn,fE)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Qfn(t)(3) existsandisnite.Moreover,sinceuoutn,f(t)anduinn,f(t)aretheoptimumsolutionto( 3 ),thenetworkstabilityisachievedunder.Therefore,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0uoutn,f(t)limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0uinn,f(t)+n,f sinceotherwise,thenetworkstabilitycannotbeensuredbyuuu.Finally,weconcludethat limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0(t)(3) isnonnegativeandthusthelasttermof( 3 )canbeomitted.Consequently,bydividingJonbothsidesof( 3 ),wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 GMES(t)B J+C.(3) Therefore,Theorem 3.2 holds. 3.4Simulations WeconsideramultihopwirelessnetworkillustratedinFigure 31 .Therearethreeowsinthenetwork,denotedbyFlow1,2,3.TheroutingpathsofowsarespeciedbyR1=fA,B,C,Dg,R2=fF,G,C,DgandR3=fE,F,G,Hg.Theexogenous 64 PAGE 65 arrivalprocessesareBernoulliprocesseswithanaveragerateof5packetsperslotforallthreeows.Withoutlossofgenerality,weassumeatwohopinterferencemodelwhichrepresentsthegeneralIEEE802.11MACprotocols[ 36 37 ].Thenominallinkrateofawirelesslinkisassumedtobe20packetsperslot.Foraparticularlink,therearethreeequallypossiblechannelstates,i.e.,Good,Medium,Badwherethecorrespondingsuccessfultransmissionprobabilitiesare0.8,0.6,0.3,respectively. Figure31. Networktopologywithinterconnectedqueues Fortheeaseofexposition,weassumethata,b=a,b=50Jforalllinksinthenetwork[ 50 ].WeinvestigatetheaverageenergyconsumptioninducedbytheMaxWeightalgorithmandtheproposedMESalgorithmwithdifferentvaluesofJ,i.e.,50,100,200,500,800,1000,2000,5000,8000,10000,12000,14000,16000,18000,20000,25000.Everysimulationisexecutedfor10000timeslots. Figure 32 depictstheaverageenergyconsumptionpertimeslotfortheMaxWeightalgorithmandtheMESalgorithmwithdifferentvaluesofJ.AsillustratedinFigure 32 ,whenJ=0,theMESalgorithmreducestotheoriginalMaxWeightalgorithmandyieldsthesameamountofenergyexpenditure.However,asJincreases,theenergyconsumptioninducedbytheMESalgorithmdecreasesremarkably.Inaddition,asshownanalyticallyinTheorem 3.2 ,theMESalgorithmapproachestotheglobalminimumenergyexpendituregraduallyasJincreases.ToachieveabetterunderstandingontheimpactoflargervaluesofJ,wecomparetheaveragequeuebacklogsofalldata 65 PAGE 66 Figure32. ComparisonoftheenergyconsumptionsoftheMaxWeightalgorithmandtheMESalgorithm queues,withJ=50,150,350and500,inFigure 33 andFigure 34 ,wherethequeuesinthenetworkareindexedby1to9intheorderofQ1A,Q1B,Q1C,Q2F,Q2G,Q2C,Q3E,Q3FandQ3G.Itisworthnotingthat,asJgrows,theaveragequeuebacklogsinthenetworkincreasescorrespondingly.FollowingLittle'sLaw,largerqueuebacklogsyieldlongernetworkdelays.Therefore,whilethenetworkwideenergyexpenditureisnoticeablyreduced,alargervalueofJyieldsalongeraveragedelayinthenetwork.Asaconsequence,atradeoffbetweentheenergyoptimalityandtheexperienceddelaycanbeattainedbytuningJproperly. Sofar,throughoutthiswork,wehaveassumedthattheenergyineachnodeisconstrainedyetsufcientlylarge.Next,weinvestigatetheperformanceoftheMESalgorithminmultihopwirelessnetworkswhereeachnodehasalimitedandniteamountofenergy.Morespecically,thesamenetworkinFigure 31 isconsidered,however,weassumethateachnodeinthenetworkhasabatterywithaninitialenergyof1Joule.WecomparetheperformanceoftheMaxWeightalgorithmwiththeMESalgorithmfordifferentvaluesofJ,intermsofthenetworklifetime,whichisdenedasthetimeinstancewhentherstnodeinthenetworkdepletesthebatterycompletely.ThevaluesofJare50,200,500,800,1000,2000,5000,8000,10000,15000and20000. 66 PAGE 67 Figure33. TheaveragequeuebacklogsinthenetworkforJ=50and150 Figure 35 pictoriallycomparesthenetworklifetimeinducedbytheMaxWeightalgorithmandthatoftheMESalgorithmfordifferentvaluesofJ.Similartothepreviousscenario,whenJissmall,theMESalgorithmyieldssimilarperformance,intermsofthenetworklifetime,comparedtotheMaxWeightalgorithm.However,whenJincreases,theMESalgorithmsignicantlyoutperformsthetraditionalMaxWeightalgorithm.ItisworthnotingthatwhenJ=20000,theMESalgorithmprolongsthenetworklifetimebymorethantwiceasmuchasthatoftheMaxWeightalgorithm!Bythesametoken,atradeoffbetweenthenetworklifetimeandtheexperienceddelayofthenetworkcanbecontrolledeffectivelybytuningthevalueofJ. 3.5Conclusions Inthischapter,weinvestigatetheenergyconsumptionissueoftheschedulingalgorithmsinmultihopwirelessnetworks.WeshowthatthetraditionalMaxWeightalgorithm,whichiswellknowntobethroughputoptimal,isnotenergyoptimaldueto 67 PAGE 68 Figure34. TheaveragequeuebacklogsinthenetworkforJ=350and500 Figure35. ComparisonofthelifetimeoftheMaxWeightalgorithmandtheMESalgorithm theoverlookoftheenergyconsumptionsinducedbyinevitablepacketretransmissions.Inlightofthis,weproposeaminimumenergyscheduling(MES)algorithmwhichsignicantlyreducestheenergyconsumptioncomparedtotheoriginalMaxWeightalgorithm.Inaddition,weanalyticallyshowthattheMESalgorithmisessentiallyenergyoptimalinthesensethattheaverageenergyexpenditureoftheMESalgorithmcanbe 68 PAGE 69 pushedarbitrarilyclosetotheglobalminimumsolution.Moreover,theimprovementontheenergyefciencyisachievedwithoutlosingthethroughputoptimality.Therefore,theproposedMESalgorithmisofgreatimportancefornetworkprotocoldesignsinenergyconstrainedmultihopwirelessnetworkssuchaswirelesssensornetworks. 69 PAGE 70 CHAPTER4CHANNELANDPOWERALLOCATIONINWIRELESSMESHNETWORKS 4.1Introduction Metropolitanwirelessmeshnetworksgainenormouspopularityrecently[ 51 ].Thedeploymentofwirelessmeshnetworksnotonlyfacilitatesthedatacommunicationbyremovingcumbrouswiresandcables,butalsoprovidesameansofInternetaccessscheme,whichisafurthersteptowardsthegoalofcommunicatinganywhereanytime.Nomatterwherethelocationisorthepurposethatthewirelessmeshnetworkisdeployed,thesameconceptuallayeredarchitectureisutilized.Figure 41 illustratesthehierarchicalstructureofwirelessmeshnetworks.Theperipheralnodesaretheaccesspoints(AP)whichprovidewirelessaccessfortheendusers,orclients.EachAPisassociatedwithameshrouter.Theycanbemanufacturedinasingledevicewithtwoseparatefunctionalradios[ 52 ][ 53 ],orsimplyconnectedwithEthernetcables[ 54 ].Themeshroutersarecapableofcommunicatingwitheachotherviathewirelessbackbone.ThecentralnodeisagatewaymeshrouterwhichfunctionsasaninformationexchangebetweenthewirelessmeshaccessnetworkandothernetworkssuchastheInternet.Boththeroutingalgorithmicdesignandchannelassignmentforbackbonemeshroutersareinterestingissuesandattracttremendousattention[ 55 57 ]. Figure41. Hierarchicalstructureofwirelessmeshaccessnetworks Inthiswork,weinvestigateanimportantissuewhichneedstobesolvedinwirelessmeshnetworks.AsinFigure 41 ,theAPanditsassociatedclientsformaregularWLANcell,whichoperateswiththedefactoIEEE802.11standards. 70 PAGE 71 Thethroughputofonecelldependsonthesignaltointerferenceplusnoiseratio(SINR)experiencedatthereceiverwheretheinterferencemainlycomesfromtheotheroperatingcells.Forexample,ifeachofthecellsoperateswithIEEE802.11bstandard,wecanutilizeadifferentfrequencybandsuchasIEEE802.11aorWiMAX[ 58 ],fortheintercellcommunicationamongmeshroutersandhencecausesnointerferencetointracelltransmissions.However,thecochannelinterferencefromotheroperatingcellsisinevitableduetothelimitationofavailabletransmissionchannels,e.g.,3nonoverlappingchannelsinourexample.MostcurrentofftheshelfAPsarecapableofadjustingthetransmissionrateaccordingtothemeasuredchannelconditionwhichisindicatedbytransmissionbiterrorrate(BER).Givenaparticularmodulationscheme,BERisuniquelydeterminedbytheSINRexperiencedbythereceiverofthelink.Generallyspeaking,higherSINRvalueyieldslowerBERandhigherdatarate.Therefore,themutualinterferencedramaticallydegradesthetransmissionrateofeachcellandtheaggregatedthroughputofthewholenetwork[ 59 ].EachAPattemptstotunethephysicalparameterssuchasoperatingfrequency1andtransmissionpowerinordertomaximizetheSINRandhencethethroughput.Inourwork,weinvestigatetheissueofmaximizingtheoverallthroughputofthenetwork,denedasthesummationofthroughputofallcells,byndingtheoptimalfrequencyandtransmissionpowerallocationstrategy.Also,duetotheconcernofscalabilityandcomputationalcomplexity,wepreferadecentralizedsolutiontothethroughputmaximizationproblem. Unfortunately,thethroughputmaximizationproblemischallenging.Forexample,thefrequencyandpowerselectedbyoneAPaffectstheSINRofotherAPs,andviceversa.Worseyet,iftheAPsbelongtodifferentregulationentities,thenoncooperativeAPsmayonlywanttomaximizetheirowncell'sthroughputratherthantheoverallone.Therefore,thethroughputmaximizationproblemiscoupledandndingtheoptimum 1Wewillusefrequencyandchannelinterchangeably. 71 PAGE 72 solutionisnotstraightforward.Moreover,traditionalsiteplanningmethodsincellularnetworksarenotfeasibleeither.Forexample,thenetworkadministratormaywanttoaddmoreAPswhenmoreusersarejoiningthenetworkordisablesomeAPswheretheassociatedusersfailtopaythebill.Thenetworktopologyisnotstatic,althoughthechangestakeplaceslowly.Therefore,thedemandforadaptabilityandlightcomputationburdenrequiresadecentralizedsolutionforthethroughputmaximizationproblem. Inthiswork,weanalyzethethroughputmaximizationproblemforbothcooperativeandnoncooperativescenarios.Inthecooperativecase,wemodeltheinteractionamongallAPsasanidenticalinterestgameandpresentadecentralizednegotiationbasedthroughputmaximizingalgorithmforthejointfrequencyandpowerassignment.Weshowthatthisalgorithmconvergestotheoptimalfrequencyandpowerassignmentstrategy,whichmaximizestheoverallthroughputofthewirelessmeshaccessnetwork,witharbitrarilyhighprobability.InthecasesofnoncooperativeAPs,weprovetheexistenceofNashequilibriaandshowthattheoverallthroughputperformanceisusuallyinferiortothecooperativecases.Tobridgetheperformancegap,weproposealinearpricingschemetocombatwiththeselshbehaviorsofnoncooperativeAPs. Therestofthischapterisorganizedasfollows.Section 4.2 outlinesthesystemmodelweconsideredinthework.ThecooperativewirelessmeshaccessnetworksandthenoncooperativecounterpartareinvestigatedinSection 4.3 andSection 4.4 ,respectively.AnextensionofourmodelisdiscussedinSection 4.5 andtheperformanceevaluationisprovidedinSection 4.6 .Finally,Section 4.7 concludesthischapter. 4.2SystemModel Inthiswork,weconsiderawirelessmeshaccessnetworkillustratedinFigure 41 .EachAPandcorrespondingclientsformacell.Withoutlossofgenerality,weassumethatallthecellsoperatewithIEEE802.11bstandardandtheinterferenceexclusivelycomesfromthecellswithsamefrequency.Furthermore,thedistancebetweencellsaresufcientlylargeinthesensethattheaccumulatedinterferenceexperiencedatthe 72 PAGE 73 receiveronlyaffectstheSINRvalueandnotblockthewholetransmission.WeassumethatthechannelsareslowvaryingadditivewhiteGaussiannoise(AWGN)channels.Thechannelgainsofeachpairofnodesareassumedtobeconstantoverthetimeperiodofinterest.Asweareinterestedinthemaximumachievablethroughput,weconsidertheworstcasewhereallAPsaresaturated.Inotherwords,theAPsalwayshavepacketstotransmitandtheycancommunicatewitheachotherviathebackbonemeshrouterswithnegligibledelay.Also,weassumethattheAPsaretransmittersandclientsarereceiversduetothedominanceofdownlinktrafc,asassumed2in[ 61 ][ 62 ]and[ 63 ].Weonlyfocusonthejointfrequencyandpowerallocationwherethecontentionbehaviorislessrelevantandthusomitted.Therefore,wecansimplifyourmodelasthatalltheAPsaretransmittingdatatotheassociatedclientsconsistently.WeassumethateachAPiscapableofadjustingtheoperatingfrequencyandpoweraswellasacquiringtheSINRvaluesmeasuredattheclientbyshortACKmessages. Letusrstconsiderthesimplestcasewherethereisonlyonecellinthewirelessmeshaccessnetwork,i.e.,asingleWLAN.Inthefollowingtwosections,weassumethattheAPshavepredeterminedandxedmodulationandcodingschemes.Inotherwords,uponreceivingtheSINRvalue3measuredbytheclient,denotedby,theAPtunesthephysicalparametersinordertomaximizethethroughput,whichisdenedas R()=maxRiRi(1)]TJ /F6 11.955 Tf 11.96 0 Td[(Pe(,Ri))(4) whereRiistherawdataratespeciedbytheIEEE802.11standardandR,i.e.,thethroughputofthiscell,isanondecreasingfunctionofreceivedSINR.Peistheerror 2Thedominanceofthedownlinktrafcisveriedbytheexperimentalmeasurementsin[ 60 ]aswell.3Althoughthereisnointerferenceinthiscase,weadoptSINRinsteadofSNRfornotationconsistency. 73 PAGE 74 probabilityofthetransmissionchannel,whichisafunctionofSINRvalueprovidingthemodulationandcodingscheme[ 64 ].Apparently,ifthereisonlyonecellinthemeshaccessnetwork,theAPwillboostthepowerasmuchaspossibletoincreasethevalueofandthusthethroughputismaximized. WenowconsiderthecaseswhereNcellscoexistinthewirelessmeshaccessnetwork.LetpiandfidenotethepowerandfrequencyfortheithAP,respectively.Weusep=[p1,p2,,pN]andf=[f1,f2,,fN]torepresentthepowerandfrequencyassignmentvectorforallNAPs.Therefore,foreachcelli,thevalueofSINR4,i.e.,i,isafunctionof(p,f).Thethroughputofonecelldependsnotonlyonthepowerlevelandfrequencyofitself,butalsothoseofotherAPsinthenetwork.Therefore,thethroughputmaximizationproblemiscoupledandbynomeansstraightforward. Inthefollowingtwosections,wewilldiscussthescenarioswheretheAPsarecooperativeandnoncooperative,respectively,undertheassumptionthatthemodulationandcodingschemesofAPsarepreloadedandxed.InSection 4.5 ,wewillextendouranalysisbyconsideringthescenariowheretheAPsarecapableofadaptivecodingandmodulation.TheperformanceevaluationofallscenariosareprovidedbysimulationsinSection 4.6 4.3CooperativeAccessNetworks Inthissection,weconsiderthescenarioswhereallAPsinthewirelessmeshaccessnetworkarecooperative.ThetransmissionpowerofAPsarequantizedintodiscretepowerlevelsforsimplicity.Fromthesystempointofview,wewanttondajointfrequencyandpowerlevelassignmentsuchthattheoverallthroughputinthewhole 4Throughoutthechapter,thetermSINRofthecellrepresentstheaverageSINRamongalltheclientsinthecell,whichcanbeobtainedbyamovingaverageofthereportedSINRvalue. 74 PAGE 75 networkismaximized.Ourobjectivefunctioncanbewrittenas Unetwork(p,f)=NXi=1Ri(i)=NXi=1Ri(p,f)(4) whereRiisdenedin( 4 ). However,ndingtheoptimalfrequencyandpowerassignmentwhichmaximizes( 4 )isnontrivial.Theinterdependencymakestheproblemcoupledanddifculttosolvebytraditionaloptimizationmethods[ 65 ].Acombinationof(p,f)isnamedaproleandanaiveapproachtosolvetheproblemistoinvestigateallprolesexhaustively.However,thisisimpossibleinpractice.Forexample,inamediumsizewirelessmeshaccessnetworkwith20APswhereeachhas3frequencychannelsand10powerlevels,thesearchspaceis(310)20proles!Obviously,thecentralizedalgorithmsarenotfavorableinthewirelessmeshaccessnetworkduetothescalabilityconcern.Next,wewillintroduceadecentralizednegotiationbasedthroughputmaximizationalgorithm,fromagametheoreticalperspective. 4.3.1CooperativeThroughputMaximizationGame TheAPsinthewirelessmeshaccessnetworksareconsideredasplayers,i.e.,decisionmakersofthegame.WemodeltheinteractionamongAPsasaCooperativeThroughputMaximizationGame(CTMG),whereeachplayerhasanidenticalobjectivefunctionUi,as Ui(p,f)=Unetwork(p,f)=NXj=1Rj(p,f),8i.(4) Foreachplayeri,allpossiblefrequencyandpowerlevelpairsformastrategyspaceiwhichhasasizeofcl,wherecisthenumberoffrequencychannelsavailableandlisthenumberoffeasiblepowerlevels.Dene =12N.(4) Then,theNplayersautonomouslynegotiateaboutthejointfrequencypowerproleininordertondtheoptimalprolewhichmaximizes( 4 ).However,duetothe 75 PAGE 76 interdependencyamongNplayerscausedbymutualinterference,onequestionofinterestisthatwhetherthisnegotiationwilleventuallymeetanagreement,a.k.a.,aNashequilibrium.TheimportanceofNashequilibrialiesinthatapossiblesteadystateofthesystemisguaranteed.IfthegamehasnoNashequilibrium,thenegotiationprocessneverstopsandoscillatesinaneverlastingfashion.Inaddition,weareconcerningaboutwhattheperformanceofthesteadystateswouldbe,ifexist,intermsofoverallthroughputofthewholenetwork.Weprovideanswerstothesequestionsinthefollowing. Lemma3. TheCTMGisapotentialgame. ApotentialgameisdenedasagamewherethereexistsapotentialfunctionPsuchthat P(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i))]TJ /F6 11.955 Tf 11.96 0 Td[(P(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i)=Ui(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i))]TJ /F6 11.955 Tf 11.95 0 Td[(Ui(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(i)8i,a0,a00(4) whereUiistheutilityfunctionforplayerianda0,a00aretwoarbitrarystrategiesini.Morespecically,wehavea0=[p0i,f0i]anda00=[p00i,f00i].Thenotationofa)]TJ /F9 7.97 Tf 6.59 0 Td[(idenotesthevectorofchoicesmadebyallplayersotherthani.Potentialgameshavebeenbroadlyappliedinmodelingtheinteractionsincommunicationnetworks[ 66 ].Thepopularityisonaccountofthenicepropertiesofpotentialgames,suchas PotentialgameshaveatleastoneNashequilibrium. AllNashequilibriaarethemaximizersofthepotentialfunction,eitherlocallyorglobally. ThereareseverallearningschemesavailablewhichareguaranteedtoconvergetoaNashequilibrium,suchasbetterresponseandbestresponse[ 67 ][ 68 ]. Fordetaileddescriptionaboutpotentialgames,readersarereferredto[ 67 ]and[ 69 ],whichinvestigatethepotentialgametheoryinengineeringcontext. Weobservethatinthecooperativecase,eachplayerhasthesameutilityfunctionasin( 4 ),whichistheoverallthroughputofthenetwork.Apparently,onepotential 76 PAGE 77 functionofthegameisthecommonutilityfunctionitself,i.e., P=U1=U2==UN.(4) Infact,thegameswhereallplayerssharethesameutilityfunctionarecalledidenticalinterestgames[ 70 ],whichisaspecialcaseofpotentialgamesandhenceallthepropertiesofpotentialgamescanbeapplieddirectly. Intheliterature,bothbestresponseandbetterresponsearepopularlearningmechanismsthathavebeenutilizedinpotentialgames[ 71 73 ].Ateachstepofthebestresponseapproach,oneoftheplayersinvestigatesitsstrategyspaceandchoosestheonewithmaximumutilityvalue.Thisupdatingprocedureiscarriedoutsequentially.Theprimarydrawbackofthebestresponseisthecomputationalcomplexity,whichgrowslinearlywiththecardinalityofthestrategyspace.Animprovementofthebestresponseisthesocalledbetterresponse,whereateachstep,theplayerupdatesaslongastherandomlyselectedstrategyyieldsabetterperformance.Thedramaticallyreducedcomputationisthetradeoffwiththeconvergencespeed.BoththebestresponseandthebetterresponsedynamicsareguaranteedtoconvergetoaNashequilibriuminpotentialgames[ 66 ].However,theremaybemultipleNashequilibriainapotentialgameandtheperformanceofdifferentequilibriamayvarydramatically.Therefore,althoughthebestresponseandthebetterresponsecouldguaranteetheconvergence,theymayreachanundesirableNashequilibriumwithinferiorperformance. Figure42. AnillustrativeexampleofmultipleNashequilibria LetusconsideranillustrativeexampleinFigure 42 .TherearefourlabeledAPsinthenetwork.AandBareclosetoeachother,andsoareCandD.Withoutlossof 77 PAGE 78 generality,weassumethattheAPshavethesamepowerandonlyadjusttheoperatingfrequenciesinanorderofA!B!C!Dtoavoidtheinterference.TheadaptationcontinueswiththebestresponsemechanismuntilaNashequilibriumisreached.Supposethattherearetwofrequencychannelsavailable,say1and2.First,Arandomlyselectsonechannel,say1.Bwillpick2.Next,Chasthechancetoupdate.SinceCisclosertoBthanA,channel1willbeselected.Finally,Dwillchoosechannel2.Byinspection,weclaimthatprole1)]TJ /F3 11.955 Tf 12.45 0 Td[(2)]TJ /F3 11.955 Tf 12.44 0 Td[(2)]TJ /F3 11.955 Tf 12.44 0 Td[(1isaNashequilibriumsincenoplayeriswillingtoupdateitsstrategyunilaterally.Meanwhile,weobservethatanotherprole1)]TJ /F3 11.955 Tf 13 0 Td[(2)]TJ /F3 11.955 Tf 13 0 Td[(1)]TJ /F3 11.955 Tf 13 0 Td[(2isalsoaNashequilibrium.Obviously,thesecondNashequilibriumgeneratesmuchlessinterferencethantherstNashequilibriumandhenceyieldssuperiorperformanceintermsofoverallthroughput.However,thebestresponseonlyleadstothelessdesirableNashequilibrium. Infact,theexistenceofmultipleNashequilibriaisobservedin[ 71 ]bysimulations.However,theauthorsfailtospecifywhichonewouldbethesteadystateoftheirgameduetothelimitationofthebestresponse,eveninastatisticalfashion.RecallthattheNashequilibriaarethemaximizersofthepotentialfunctioninpotentialgames,convergingtoaninferiorNashequilibriumanalogouslyindicatesbeingtrappedatalocaloptimumofthepotentialfunction.However,itistheglobaloptimum,i.e.,theoptimalNashequilibrium,thatisthedesirablesteadystatewhichweareyearningfor. Next,weintroduceanegotiationbasedthroughputmaximizationalgorithm(NETMA)whichcanconvergetotheoptimalNashequilibriumwitharbitrarilyhighprobability. 4.3.2NETMANegotiationBasedThroughputMaximizationAlgorithm WeassumethattheAPsarehomogeneousandeachhasauniqueIDforroutingpurpose.EachAPmaintainstwovariablesDpreandDcur.TheAPhastheknowledgeofitscurrentthroughputandrecordsitinDcur.Wheneverthereisachangeofthroughput 78 PAGE 79 causedbyexteriorinterference5,theAPsetsDpre=DcurandresetsDcurwiththenewlymeasuredthroughput.Whenthewirelessmeshaccessnetworkentersthenegotiationphase6,NETMAisexecuted.ThedetailedprocedureofNETMAisprovidedasfollows. NETMA: Initialization: ForeachAP,apairoffrequencyandpowerlevelisrandomlyselected.SetDpre=Dcur,thecurrentthroughput. Repeat: 1. RandomlychooseoneoftheAP,sayk,astheupdatingone,i.e.,eachAPupdateswithprobability1=N. 2. FortheupdatingAPk: (a) Randomlychoosesapairoffrequencyandpowerlevel,sayf0andp0,fromthestrategyspacek.ThentheAPcomputesthecurrentthroughputwithf0andp0andrecordsitintoDcur. (b) BroadcastsashortnotifyingmessagewhichcontainsitsuniqueIDktoalltheotherAPsinthemeshaccessnetwork. 3. ForeachAPotherthank,sayj: (a) Ifthejvaluechanges,recordsthepreviousthroughputintoDpreandthecurrentthroughputintoDcur.Remainsunchangedotherwise. (b) Uponreceivingthenotifyingmessage,athreevaluevectorof[Dpre,Dcur,IDj]issentbacktothekthAP. 4. AfterreceivingallthethreevaluevectorsbycountingtheidentiersIDj,thekthAPcomputesthesumthroughputbeforeandafterf0andp0areselected,whicharedenotedbyPpreandPcur. 5Weassumethechannelisslowvaryingandthechangeofthroughputforasinglecellisduetothemutualinterferenceonly.6Toreducethenegotiationoverhead,anegotiationphasecanbeinitiatedbythenetworkadministratorafteranewcontracteduserjoinsoracurrentuserterminatestheservice,oronadailybasis. 79 PAGE 80 5. Forasmoothingfactor>0,thekthAPkeepsf0andp0withprobability ePcur= ePcur=+ePpre==1 1+e(Ppre)]TJ /F9 7.97 Tf 6.59 0 Td[(Pcur)=(4) 6. ThekthAPbroadcastsanothershortnotifyingmessage,whichindicatestheendofupdatingprocessandaspecicnumber,toalltheotherAPs. Until: Thestoppingcriteria)]TJ /F1 11.955 Tf 10.09 0 Td[(ismet. Notethatinstep 6 ,thespecicformatofdependsonthepredenedstoppingcriterion)]TJ /F1 11.955 Tf 6.78 0 Td[(.Forexample, Ifthestoppingcriterionisthemaximumnumberofnegotiationsteps,isacounterwhichaddsoneaftereachupdatingprocess. IfthestoppingcriterionisthatnoAPhasupdatedforacertainnumberofsteps,isabinarynumberwhere1meansupdating. Ifthestoppingcriterionisthatthedifferencebetweensumthroughputobtainedinconsecutivestepsarelessthanapredenedthreshold,isthecalculatedsumthroughputaftereachupdatingprocess. Wecanhaveotherstoppingcriteria)]TJ /F1 11.955 Tf 6.77 0 Td[(sandcorrespondingformatsofaswell. TheNETMAalgorithmisinspiredbytheworkin[ 74 ],whereasimilaralgorithmwasrstintroducedinthecontextofstreamcontrolinMIMOinterferencenetworks.Thedistinguishingfeatureofthistypeofnegotiationalgorithms,fromthebetterresponseandthebestresponse,istherandomnessdeliberatelyintroducedonthedecisionmakinginstep 5 .TherationalecanbeillustratedinFigure 42 intuitively.Ifthereisnorandomnessindecisionmaking,i.e.,=0,thefourAPsmaygettrappedatalowefciencyNashequilibrium1)]TJ /F3 11.955 Tf 11.91 0 Td[(2)]TJ /F3 11.955 Tf 11.9 0 Td[(2)]TJ /F3 11.955 Tf 11.91 0 Td[(1.However,withtherandomnesscausedbynonzero,theymayreachanintermediatestate1)]TJ /F3 11.955 Tf 12.42 0 Td[(2)]TJ /F3 11.955 Tf 12.42 0 Td[(2)]TJ /F3 11.955 Tf 12.42 0 Td[(2andarriveattheoptimumNashequilibrium1)]TJ /F3 11.955 Tf 12.15 0 Td[(2)]TJ /F3 11.955 Tf 12.16 0 Td[(1)]TJ /F3 11.955 Tf 12.16 0 Td[(2eventually.Moreover,theupdatingruleinstep 5 alsoimpliesthatiff0andp0yieldabetterperformance,i.e.,Ppre)]TJ /F6 11.955 Tf 12.3 0 Td[(Pcur<0,thekthAPwillkeepthemwithhighprobability.Otherwise,itwillchangewithhighprobability. ThesteadystatebehaviorofNETMAischaracterizedinthefollowingtheorem. 80 PAGE 81 Theorem4.1. NETMAconvergestotheoptimalNashequilibriuminCTMGwitharbitrarilyhighprobability. Proof. TheproofofTheorem 4.1 followssimilarlinesoftheproofin[ 74 ]and[ 75 ]. Figure43. MarkovianchainofNETMAwithtwoplayers First,weobservethatthejointfrequencypowernegotiationgeneratesanNdimensionalMarkovianchain.Figure 43 illustratestheMarkovianchainintroducedbyNETMAwithtwoplayers,sayAandB.Letxandybethechoicesforeachplayer,wherex2Aandy2B.Inotherwords,playerAcanchooseafrequencypowerpairfrom[x1,,xcl]andplayerBcanchoosefrom[y1,,ycl].Notethatatanarbitrarytimeinstant,onlyoneoftheplayerscanupdate.InFigure 43 ,forexample,state(x1,y1)canonlytransittoastateeitherinthesameroworthesamecolumn,notanywhereelse.ThisistrueforeverystateintheMarkovianchain.LetSi,jdenotethestateof(xi,yi).Wehave Pr(Sm,njSi,j)=8>>>>>>><>>>>>>>:eP(Sm,n)= 2cl(eP(Sm,n)=+eP(Si,j)=),ifm=iorn=j0,otherwise.(4) 81 PAGE 82 whereisthesmoothingfactorinstep 5 ofNETMAandP(Si,j)isthevalueofthepotentialfunction,i.e.,( 4 ),atthestateofSi,j. LetusderivethestationarydistributionPrforeachstate.Weexaminethebalancedequations.Writingthebalanceequations[ 76 ]atthedashedline,weobtain clXk=2Pr(S1,1)Pr(S1,kjS1,1)=clXk=2Pr(S1,k)Pr(S1,1jS1,k).(4) Bysubstituting( 4 )with( 4 ),wehave Pclk=2Pr(S1,1)eP(S1,k)= eP(S1,1)=+eP(S1,k)==Pclk=2Pr(S1,k)eP(S1,1)= eP(S1,1)=+eP(S1,k)=. (4) Observingthesymmetryofequation( 4 )aswellastheMarkovianchain,wenotethatthesetofequationsin( 4 )areallbalancedifforarbitrarystate~Sinthestrategyspace,thestationarydistributionis Pr(~S)=KeP(~S)=(4) whereKisaconstant.Byapplyingtheprobabilityconservationlaw[ 77 ][ 76 ],weobtainthestationarydistributionfortheMarkovianchainas Pr(~S)=eP(~S)= PSi2eP(Si)=(4) forarbitrarystate~S2. Inaddition,weobservethattheMarkovianchainisirreducibleandaperiodic.Therefore,thestationarydistributiongivenin( 4 )isvalidandunique. LetSbetheoptimalstatewhichyieldsthemaximumvalueofpotentialfunctionP,i.e., S=argmaxSi2P(Si).(4) 82 PAGE 83 From( 4 ),wehave lim!0Pr(S)=1(4) whichsubstantiatesthatNETMAconvergestotheoptimalstateinprobability. Finally,theanalogousanalysiscanbestraightforwardlyextendedtoanNdimensionalMarkovianchainandthuscompletestheproof. InNETMA,thereisnocentralcomputationalunitrequired.ThejointfrequencypowerassignmentisachievedbynegotiationsamongcooperativeAPsandthemaximumoverallthroughputisachievedwitharbitrarilyhighprobability.TheautonomousbehavioranddecentralizedimplementationmakeNETMAsuitableforlargescalewirelessmeshaccessnetworks.Moreover,NETMAhasfastadaptabilityforthetopologychangeofthewirelessmeshaccessnetworks.NETMAdoesnotdependonanyrateadaptionalgorithms,noronanyunderlyingMACprotocols.InoursimulationinSection 4.6 ,weuseIEEE802.11bastheMAClayerprotocol.However,itcanbeeasilyextendedtoarbitraryMACprotocolwithmultiratemultichannelcapability,suchasIEEE802.11a.Inaddition,evenwiththeexistenceofexteriorinterferencesource,suchascoexistingWLANs,NETMAworksproperlyaswellsincetheobjectiveofNETMAistomaximizetheoverallthroughputofthenetworkinthecurrentwirelessenvironments.Thetradeoffbetweenalgorithmicperformanceandconvergencespeediscontrolledbyparameterinstep 5 ,wherelargerepresentsextensivespacesearchwithslowconvergence.Onthecontrary,smallrepresentslimitedspacesearchwithfastconvergence.Notethatthesmoothingfactorhereisanalogoustotheconceptoftemperatureinsimulatedannealing[ 78 ].Therefore,itisadvisablethatatthebeginningperiodofthenegotiation,thevalueofissetwithalargenumberandkeepsdeceasingasthenegotiationiterates.Wechoose=10=k2inoursimulations,wherekdenotesthenegotiationstep. Instep 1 ,werequirethateachAPupdateswithaprobabilityof1=N.Forexample,wecanutilizearandomtokenmechanismwhereeachupdatingAPrandomlyselectsanAPasthenextupdatingAP,i.e.,passingthetoken.NotethatinNETMA,evenan 83 PAGE 84 erroneousoperationhappens,forexample,twoAPsupdateatthesametimeinourcase,itonlyprolongstheconvergencetimeforNETMAyetdoesnotaffectthenaloutputofNETMA.Thisisbecausethatsuchanerror,asveriedin[ 72 ]viaextensivesimulations,hasnoinuenceonthestatisticallymonotonicincreasingtendencyofthepotentialfunction. 4.4NonCooperativeAccessNetworks Intheprevioussection,wediscussthescenarioswhereallAPsinthewirelessmeshaccessnetworkarecooperative,andtheoverallthroughputismaximizedbynegotiationsamongautonomousAPsusingtheNETMAmechanism.However,cooperationisnotalwaysattainable.Althoughthefunctionalityofrelayingpacketsforeachothercanbeachievedbyincentivemechanismssuchas[ 79 ],theadjustableparametersinsideeachcellcannotbeenforcedandeffectivelycontrolled.TheNAPsmaybelongtodistinctselfinterestedusersandtheycareaboutexclusivelytheirownthroughputratherthantheoverallaggregatedthroughput.Inotherwords,theutilityfunctionofeachselshuseris Ui=Ri(i)(4) whereRiisthethroughputoftheithcell,denedin( 4 ).AnalogoustoCTMG,wecanformulatetheinteractionamongNselshAPsasaNoncooperativeThroughputMaximizingGame(NTMG)whereeachAPisattemptingtondthefrequencypowerpairwhichmaximizesitsownSINRvalueaswellasthecorrespondingthroughput.Asinthecooperativecase,eachplayer'sutilityfunctiondependsonthefrequencyandpowerofitselfaswellasthoseofothers.However,NTMGisnolongeranidenticalinterestgame. Lemma4. InNTMG,alltheAPswilltransmitwiththemaximalpowerattheNashequilibrium,ifexists. 84 PAGE 85 Proof. TheproofofLemma 4 isstraightforward.Forasingleplayer,wehave i=pigii Pk2Fi(fi)pkgki+Ni.(4) wheregijisthechannelgainfromcelli'stransmittertoj'sreceiverandNiistheGaussiannoiseatthei'sreceiver.Fi(fi)denotesthesetofcellswhichoperateatthesamefrequencyfiotherthancelli.Notethatgivenotherplayers'strategies,iisamonotonicincreasingfunctionofpiandsoisUi.AssumeataNashequilibriumofNTMG,thekthAPhasapowerlevelofpksatisfying0pk PAGE 86 Theorem4.2. ThereexistsatleastoneNashequilibriuminNTMG. Proof. Letusrstconsiderthesimpliedgame.Foreachplayer,theutilityfunctionisgivenas Ui=)]TJ /F3 11.955 Tf 9.3 0 Td[((Xk2Fi(fi)pkgki+Ni) (4) =)]TJ /F3 11.955 Tf 9.3 0 Td[((Xk6=i,k2Npkgki(fi)]TJ /F6 11.955 Tf 11.95 0 Td[(fk)+Ni) (4) where (k)=8><>:1,ifk=00,otherwise.(4) Weconjecturethatoneofthefeasiblepotentialfunctionsis P=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(1 2Xi2NXk2Fi(fi)pkgki.(4) Thevericationisasfollows. 2P=)]TJ /F13 11.955 Tf 11.29 11.36 Td[(Xi2NXk2Fi(fi)pkgki=)]TJ /F13 11.955 Tf 11.29 24.51 Td[(8<:Xk2Fi(fi)pkgki+Xj6=i,j2NXk2Fj(fj)pkgkj9=;=)]TJ /F13 11.955 Tf 11.29 24.51 Td[(8<:Xk2Fi(fi)pkgki+Xj6=i,j2Nfpigij(fi)]TJ /F6 11.955 Tf 11.96 0 Td[(fj)+Xk2Fj(fj),k6=ipkgkjg9=;=)]TJ /F13 11.955 Tf 11.29 20.44 Td[((Xk6=i,k2Npkgki(fk)]TJ /F6 11.955 Tf 11.95 0 Td[(fi)+Xj6=i,j2Npigij(fi)]TJ /F6 11.955 Tf 11.96 0 Td[(fj)+Xj6=i,j2NXk2Fj(fj),k6=ipkgkj9=;. (4) 86 PAGE 87 Notethat pkgki(fk)]TJ /F6 11.955 Tf 11.96 0 Td[(fi)=pigik(fi)]TJ /F6 11.955 Tf 11.95 0 Td[(fk)(4) foranypairofi,k.Wehave P=)]TJ /F13 11.955 Tf 11.29 20.44 Td[((Xk6=i,k2Npkgki(fk)]TJ /F6 11.955 Tf 11.95 0 Td[(fi)+1 2Q()]TJ /F6 11.955 Tf 9.3 0 Td[(i))(4) where Q()]TJ /F6 11.955 Tf 9.3 0 Td[(i)=Xj6=i,j2NXk2Fi(fi),k6=ipkgkj(4) andQ()]TJ /F6 11.955 Tf 9.3 0 Td[(i)isindependentoffi.Therefore,forarbitrarytwofrequenciesa0anda00ofplayeri,wehave Qa0()]TJ /F6 11.955 Tf 9.29 0 Td[(i)=Qa00()]TJ /F6 11.955 Tf 9.3 0 Td[(i)(4) and P(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i))]TJ /F6 11.955 Tf 11.96 0 Td[(P(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(i)=nPk2Fi(a0)pkgki+1 2Qa0()]TJ /F6 11.955 Tf 9.3 0 Td[(i)o)]TJ /F13 11.955 Tf 11.29 13.27 Td[(nPk2Fi(a00)pkgki+1 2Qa00()]TJ /F6 11.955 Tf 9.3 0 Td[(i)o=Ui(a0,a)]TJ /F9 7.97 Tf 6.58 0 Td[(i))]TJ /F6 11.955 Tf 11.95 0 Td[(Ui(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(i). (4) Therefore,accordingtothedenitionin( 4 ),thesimpliedgameisapotentialgameandhasatleastoneNashequilibrium.Thus,theexistenceofNashequilibriuminNTMGisobtainedfromtheequivalencederivedfromLemma 4 AsshowninLemma 4 ,attheequilibrium,thenoncooperativeAPswillalwaystransmitatthemaximumpowerlevel.ThisseemtobethebestchoiceforeachoneoftheAPs.However,itisusuallynotafavorablestrategyfromasocialwelfarepointofview.Tobridgetheperformancegap,weproposealinearpricingschemetocombatwiththeselshbehaviors,i.e.,theplayersareforcedtopayataxproportionaltotheutilizedresources.Forexample,wecouldimposeapricetoallselshAPsforthepower 87 PAGE 88 theyutilize.HenceforeachAP,theutilityfunctionbecomes Ui=Ri(i))]TJ /F8 11.955 Tf 11.96 0 Td[(ippi(4) whereiprepresentsthepowerutilizationpricespecicfortheithAPandpiisitstransmissionpower.Therefore,themorepowerAPuses,themoretaxithastopay.Byimposingpowerpricesproperly,amoredesirableequilibriummaybeinduced,fromasocialwelfarepointofview.WedenethecorrespondinggameasaNoncooperativeThroughputMaximizationGamewithPricing(NTMGP). Letusrstinvestigatetheimpactofpricesonthebehaviorsofplayers.Ifip=0,wherenopriceisimposed,theithAPwilltransmitatthemaximumpowerandcausesextrainterferencetootherAPs.However,ifweimposeanunbearablyhighprice,saypi=1,theAPwouldrathernottotransmitatall.Basedontheseobservations,weproposeaheuristiclinearpricingschemetoimprovetheoverallthroughputinnoncooperativewirelessmeshaccessnetworks. Toenforcethescheme,weintroduceapricingdictatorunit(PDU)intothenetworkwhichdeterminesthepricesforallAPsandinformsthemtimely.Inaddition,weassumethatthePDUhasthemonitoringcapabilityandisawareoftheoperatingfrequenciesofeachcell.TherearetwopriceschargedbythePDUforeachnoncooperativeAP.Besidesthepowerutilizingpriceip,afrequencyswitchingpriceifisimposedontheithAPwheneveritchangestheoperatingfrequency.Thepricesettingprocessisdescribedasfollows. Pricesettingprocess: PhaseI: ThePDUsets1f=Nf=0and1p=Np=0andallAPsplayNTMGuntilconverges,i.e.,aNashequilibriumisreached. ThePDUcollectsthecurrentthroughputinformationfromeachcell,denotedbyMi,whereiistheindexofthecell. 88 PAGE 89 PhaseII: ThePDUsets1f=Nf=1. ForeachAPindexedbyi=1,,N: 1. ThePDUsetsip=1fortheithAPandlettheAPsplaytheNTMGP.Uponconvergence,thePDUcollectstheoverallthroughput,sayVi,inthecurrentpricesetting. 2. CalculatethepowerutilizingpricefortheithAPas ~ip=Vi)]TJ /F13 11.955 Tf 11.96 8.97 Td[(PNj=1,j6=iMj pmax(4) 3. Resetip=0. Output: Powerutilizingpricevector~p=[~1p,,~Np] Frequencyswitchingpricevector~f=[1,,1] Inthepricesettingprocessabove,thePDUimposeszeropricesforallAPsinitially.Asaconsequence,allAPswilltransmitwithpmaxattheequilibrium,asshowninLemma 4 .Uponconvergence,thePDUxesthefrequencyswitchingpricetoinnitywhichdiscouragesthenoncooperativeAPsfromswitchingchannelsthereafter.In( 4 ),PNj=1,j6=iMjisthesumthroughputforallcellsotherthani,whentheithAPtransmitswiththemaximalpowerduetothezeropowerprice.Similarly,ViisthesumthroughputofothercellswhentheithAPissilentduetotheunaffordablepowerprice.Therefore,in( 4 ),thepowerutilizationpricechargedfortheithAP,a.k.a.,~ip,canbeviewedasacompensationtotheimpactitcausesontheoverallthroughputofothercells.Themorepoweritutilizes,themoresevereitaffectstheotherplayersandthusthemoreitpays,asillustratedin( 4 ).Hence,byimposingtaxesdeliberately,theselshbehaviorsofnoncooperativeAPsareeffectivelydiscouragedandamoredesirableequilibriumcanbeinduced,intermofoverallthroughputofthewholenetwork.TheproofofexistenceofNashequilibriuminNTMGPisstraightforward.Notethattheutility 89 PAGE 90 functionof( 4 )isquasiconcavewithrespecttopower.TheexistenceofpurestrategyNashequilibriumfollowsdirectlyfromtheresultsof[ 80 ].WewillpresentthedetailedperformanceevaluationofCTMG,NTMGandNTMGPinSection 4.6 4.5AnExtensiontoAdaptiveCodingandModulationCapableDevices Sofar,wehaveassumedthatthethroughputofacellisgivenintheformof( 4 ),whichisnotacontinuousfunction.Tobeprecise,inbothSection 4.3 andSection 4.4 ,weareconnedtothetraditionalIEEE802.11familydeviceswherethemodulationandcodingschemesarepredeterminedandxed.Forexample,Table 41 providesamappingbetweenSINRvaluesandcorrespondingrates[ 81 ],thefeasibledatatransmissionrateisadiscretesetandisusuallymuchlessthanthetheoreticalchannelcapacity.WenamesuchdevicesaslegacyIEEE802.11devices. Table41. Dataratesv.s.SINRthresholdswithmaximumBER=10)]TJ /F5 7.97 Tf 6.59 0 Td[(5 Rate(Mbps) MinimumSINR(dB) 1 2.92 2 1.59 5.5 5.98 11 6.99 However,thankstotheadvanceofcodingtechniques,themaximumdatatransmissionratecanbelargelyclosedtothetheoreticalShannoncapacityinAWGNchannels[ 82 ].NotethatachievingthisrequiresvariableratetransmissionsbymatchingtotheinstantaneousSINR,whichcanbeimplementedinpracticethroughadaptivecodingandmodulation(ACM)techniques[ 83 ].ThismotivatesustoextendourresultstoACMcapabledevices.Morespecically,weareconsideringmoreadvancedandpowerfulAPswhichattempttotunethefrequencyandpowerinordertomaximize Ci(i)=Wilog2(1+i)(4) 90 PAGE 91 whereCiistheShannoncapacityoftheithcell,andWiisthebandwidth.Notethatinthisscenario,themaximumachievabletransmissionrate,asdenotedbytheShannoncapacity,isacontinuousvariablewithrespecttoi,ratherthandiscrete,asexempliedinTable 41 .ItisworthnotingthattheonlydifferenceinthisscenarioisthealternativeobjectivefunctionofeachAP.Therefore,alltheresultswehaveobtainedsofarareextendable8tothisspecialscenariowithmerelyachangeoftheobjectivefunction. Furthermore,duetothecontinuityoftheobjectivefunctionin( 4 ),theaforementionedheuristicpricingschemeinSection 4.4 canbeimproved.Withoutlossofgenerality,weassumethatallthecellshaveunitybandwidth.Restated,weassumethatthePDUisacentralizeddevicewhichknowsthechannelenvironmentsufcientlybygreedyacquiring.Inaddition,thePDUisassumedtohavemonitoringcapabilityandisawareoftheoperatingfrequencyofeachcell.ThetailoredpricingschemeforthisACMcapablescenarioisdescribedasfollows. First,thePDUsets1f=Nf=0and1p=Np=0andallAPsplayselshly,untilaNashequilibriumisachieved. ThePDUsets1f=Nf=1. ForeachAPindexedbyi=1,,N,thePDUsets 8Morespecically,inthecooperativecase,wereplace( 4 )withUnetwork(p,f)=PNi=1Ci(i)whereasinthenoncooperativecase,( 4 )and( 4 )arereplacedbyUi=Ci(i)andUi=Ci(i))]TJ /F8 11.955 Tf 11.95 0 Td[(ippi,respectively. 91 PAGE 92 ip=jXk2Fi@Ck @pij=jXk2Fi)]TJ /F6 11.955 Tf 77.21 8.09 Td[(pkgkkgik ln2(1+k)(Pj2Fkpjgjk+Nk)2j=Xk2Fipkgkkgik ln2(1+i)(pkgkk k)2=Xk2Figik2k ln2(1+k)pkgkk. (4) ThenthePDUinformseachAPthecorrespondingprices,i.e.,if=1andipcalculatedin( 4 ). Notethatthemajordifferenceinthispricingschemeliesin( 4 ),wheretheipchargedfortheithAPisthePigouvianpriceleviedtodiscourageitsrecklesspowerincrease. AfterobtainingthepricesimposedbythePDU,eachAPxestheoperatingfrequencyandcalculatestheoptimumpowerpi,whichmaximizesUi=Ci)]TJ /F6 11.955 Tf 12.87 0 Td[(piip,accordingtothefollowingsteps. @Ui @pi=@Ci @pi)]TJ /F8 11.955 Tf 11.95 0 Td[(ip(4) where @Ci @pi=gii ln2(Pk2Fipkgki+Ni)(1+i).(4) Bysetting( 4 )equaltozero,wehave pi=1 ln2ip)]TJ /F13 11.955 Tf 13.15 18.52 Td[(Pk2Fipkgki+Ni giipmaxpmin.(4) where[x]badenotesmaxfminfb,xg,agandpmax,pminrepresentthemaximumandminimumpowerofthedevice,respectively.NotethatPk2Fipkgki+NicanbeeasilymeasuredwhentheithAPsetsitstransmissionpowertozero.Therefore,ndingthe 92 PAGE 93 optimumpowerofeachAPrequiresonlylocalinformationandcanbeimplementedinadistributedfashion. In( 4 ),weobservethatthepricesimposedonAPsdependonthepowervectorp,andviceversa.Therefore,aftereachAPoptimizesitspoweraccordingto( 4 ),thePDUmeasuresthenewvalueofpower,adjuststhepricefollowing( 4 ),andthenannouncesthenewpriceagain.Thepricingsettingprocessin( 4 )andtheutilitymaximizationprocessin( 4 )areexecutedinaniterativemanner,untilconvergence.Thewholepricingschemeissummarizedasfollows. PricingScheme: 1. ThePDUinitiallysetszeroprices.Asaconsequence,allAPswillconvergeataNashequilibriumwithmaximumpower. 2. ThePDUsetsinnityforthefrequencyswitchingpricewhichpreventsthewholenetworkfromunstablefrequencyoscillations. 3. ThePDUsetsandannouncesthepowerpricesforeachAPaccordingto( 4 ). 4. InformedbythePDU,eachAPoptimizesitstransmissionpowerfollowing( 4 ). 5. Gobacktostep 3 untiltheiterationconverges. Theperformanceevaluationofthistailoredpricingschemeisillustratedinthenextsection. 4.6PerformanceEvaluation 4.6.1LegacyIEEE802.11Devices Inthissubsection,werstinvestigatetheperformanceofNETMA,NTMGandNTMGPwithlegacydevices,i.e.,thescenariosweconsideredinSection 4.3 andSection 4.4 .WeassumethewirelessmeshaccessnetworkhasNhomogeneousAPs.Theothersimulationparametersaresummarizedasfollows. EachAPhasamaximumpowerpmax=100mWandaminimumpowerpmin=10mWand10differentpowerlevelsas[10mW,20mW,,100mW]. Thenoiseexperiencedateachreceiverisassumedidenticalandhasapowerof2mW. 93 PAGE 94 AllAPsuseIEEE802.11bstandardastheMACprotocol.Inotherwords,eachAPhasfourfeasibledatarate,1,2,5.5,11Mbpsand3nonoverlappingchannels,i.e.,c=3. Withoutlossofgenerality,weassumethatthereceivedpowerisinverselyproportionaltothesquareoftheEuclidiandistance. Thesmoothingfactordecreasesas=10=k2,wherekisthenegotiationstep. ThestoppingcriteriaforNETMAandNTMGarethemaximumnumberofiterations,denotedby!. Forthesakeofsimplicity,weutilizeatabledrivenrateadaptionalgorithmprovidedinTable 41 .Notethatourresultscanalsobeappliedtoarbitrarypropagationmodels,rateadaptationalgorithmsandunderlyingmultichannelmultirateMACprotocols. 4.6.1.1Exampleofsmallnetworks Werstconsiderasmallwirelessmeshaccessnetworkwith5APs,i.e.,N=5.AllAPsarerandomlylocatedinasquareof10by10area.Theglobaloptimumsolutionisobtainedbyenumeratingallfeasiblestrategies,i.e.,(310)5proles,astheperformancebenchmark.WerstinvestigatethecooperativescenariowhereNETMAmechanismisapplied.Next,thenoncooperativescenarioisconsideredandeachAPoperatesatthemaximumpowerandadjuststhefrequencyonly.ThestoppingcriteriaforbothNETMAandNTMGarethemaximumnumberofiterationswhere!=200.TheperformancecomparisonisshowninFigure 44 Figure44. PerformanceevaluationofthewirelessmeshaccessnetworkwithN=5andc=3 94 PAGE 95 Figure45. ThetrajectoryoffrequencynegotiationsinNETMAwhenN=5andc=3 Figure46. ThetrajectoryofpowernegotiationsinNETMAwhenN=5andc=3 AsindicatedbytheOPcurve,theglobaloptimumobtainedbyenumerationapproachfunctionsastheupperboundoftheoverallthroughput.InFigure 44 ,weobservethatNETMAgraduallycatchesupwiththeglobaloptimumasnegotiationsgo.Asexpected,thenoncooperativeAPsyieldremarkablyinferiorperformanceintermsofoverallthroughput,depictedbytheNTMGcurve.TheinefciencyisduetotheselshbehaviorthatAPstransmitatthemaximumpowerandareregardlessoftheinterference.TheexistenceofNashequilibriuminbothCTMGandNTMGaresubstantiatedbytheconvergenceofcurvesinFigure 44 .Figure 45 andFigure 46 depictthetrajectoriesoffrequencynegotiationsandpowerlevelnegotiationsinNETMA,respectively.Attheinitialization,eachAPrandomlypicksafrequencyandapowerlevel 95 PAGE 96 andnegotiateswitheachotherfollowingNETMAmechanism,untiltheoptimumNashequilibriumisachieved.NotethatwhenthefrequencyvectorandpowervectorconvergeinFigure 45 andFigure 46 ,thecorrespondingoverallthroughputobtainedbyNETMAcatchestheglobaloptimuminFigure 44 simultaneously. 4.6.1.2Exampleoflargenetworks Wenowconsideralargewirelessmeshaccessnetworkwith20APs.Theenumerationapproachisnolongerfeasibleinthisscenarioduetotheenormousstrategyspace.The20APsarerandomlyscatteredinadbydsquare,wherethesidelengthdisatunableparameterinsimulations.WeinvestigatebothcooperativeandnoncooperativecasesrepresentedbyNETMAandNTMGcurves,wherethemaximumnumberofiterationsissetto!=1000.Figure 47 pictoriallydepictstheperformanceinefciencyofNTMGcausedbythenoncooperativeAPswhichtransmitatthemaximumpower.TheaveragethroughputperAPiscalculatedbyaveragingtheresultsof50simulations,foreachvalueofthesidelengthd.InFigure 47 ,itisworthnotingthatasthesidelengthdgetsbigger,theperformancegapbetweenNETMAandNTMGreduces.Thereasonisthatwhentheareaislarge,theimpactofmutualinterferenceislesssevereandsoistheperformancedeterioration.However,whenthenetworkiscrowded,i.e.,dissmall,theselshbehaviorsareremarkablydevastating. ToalleviatethethroughputdegradationbythenoncooperativeAPs,weimplementthelinearpricingschemeintroducedinSection 4.4 .ThethroughputimprovementisillustratedasNTMGPinFigure 47 .Itisnoticeablethatbyutilizingtheproposedpricingscheme,theefciencyofNashequilibriumisdramaticallyenhanced,especiallyforcrowdednetworks.Therefore,theselshincentivesofthenoncooperativeAPshavebeeneffectivelysuppressed. 4.6.2ACMCapableDevices Inthissubsection,weinvestigatetheperformanceofourmodelwithACMcapabledevices.Sincethemajordifferenceliesintheimprovedpricingschemetailoredforthis 96 PAGE 97 Figure47. PerformanceevaluationofthewirelessmeshaccessnetworkwithN=20andc=3 specicscenario,wewillonlyprovidetheperformanceevaluationofthetailoredpricingscheme,i.e.,NTMGP,inordertoavoidduplicateresults.Weconsiderapopulatednetworkwhere30APsarerandomlyscatteredinan100mby100msquare,i.e.,N=30.Allothersimulationparametersarethesameasintheprevioussubsectionexceptthatthepowerisacontinuousvariableinthisscenario.Werstinvestigatetheperformanceintermsofoverallachievablerateofthenetworkwhennopricingschemeisapplied,asaperformancebenchmark.Afterwards,thetailoredpricingschemeisimplementedtoimprovetheequilibriumefciency,a.k.a.,theoverallperformanceattheequilibrium.AsshowninFigure 48 ,theoverallachievablerateofthenetworkisdramaticallyimprovedbythepricingscheme.ThePDUadaptstheannouncedpriceforeachAPaccordingtotheoptimumpower,whichiscalculatedbythepreviousannouncedprice,inaniterativefashion.Theaggregatedachievablerateconvergesasthepricesettingiterationgoes,asdepictedbyFigure 48 Figure 49 showsthetrajectoriesofthepowerutilizationpricesforeachAP.ThecounterpartofactualutilizedpowerisillustratedinFigure 410 ,wherethecurvesstartatpmaxandevolvewiththeannouncedpriceinFigure 49 AsobservedinFigure 48 ,toachieveabetterperformance,thepricesettingprocessneedstobeexecutediteratively,comparingwiththeoneshotheuristic 97 PAGE 98 Figure48. Performanceevaluationofthewirelessmeshaccessnetworkwith/withoutthepricingscheme Figure49. ThetrajectoriesofpowerutilizationpricesofeachACMcapableAP pricingschemeproposedforlegacydevices.Therefore,thefurtherimprovementoftheequilibriumefciencyisachievedasatradeoffofcommunicationoverheads.However,itisworthnotingthatevenattherstiteration,wherethepricesaredeterminedbypmax,theinducedequilibriumyieldsremarkablesuperiorperformancethanthecasewherenopricingschemeisapplied. 4.7Conclusions Inthischapter,weinvestigatethethroughputmaximizationprobleminwirelessmeshnetworks.Theproblemiscoupledduetothemutualinterferenceandhencechallenging.WerstconsideracooperativecasewhereallAPscollaboratewitheachotherinordertomaximizetheoverallthroughputofthenetwork.Anegotiationbasedthroughputmaximizationalgorithm,a.k.a.,NETMA,isintroduced.Weprovethat 98 PAGE 99 Figure410. ThetrajectoriesofactualpowerutilizedbyeachACMcapableAP NETMAconvergestotheoptimumsolutionwitharbitrarilyhighprobability.Forthenoncooperativescenarios,weshowtheexistenceandtheinefciencyofNashequilibriaduetotheselshbehaviors.Tobridgetheperformancegap,weproposeapricingschemewhichtremendouslyimprovestheperformanceintermsofoverallthroughput.Theanalyticalresultsareveriedbysimulations.Inaddition,weextendourmodelandanalyticalresultstothescenarioswheremoreadvancedAPsareutilized,i.e.,thedeviceswiththeadaptivecodingandmodulationcapability.Inthisscenario,weproposeatailoredpricingschemewhichremarkablyimprovestheoverallperformanceinaniterativefashion. 99 PAGE 100 CHAPTER5CROSSLAYERINTERACTIONSINWIRELESSSENSORNETWORKS 5.1Introduction Wirelesssensornetworkshaveattractedsignicantattentioninbothindustrialandacademiccommunitiesinthepastfewyears,especiallywiththeadvancesinlowpowercircuitdesignandsmallsizeenergysupplieswhichsignicantlyreducethecostofdeployinglargescalewirelesssensornetworks.Thesensornetworkscansenseandmeasurethephysicalenvironment,e.g.,temperature,speed,sound,radiationandthemovementoftheobjectetc.Inaddition,wirelesssensornetworkshavebecomeanimportantsolutionformilitaryapplicationssuchasinformationgatheringandintrusiondetections.Otherimplementationsofthewirelesssensornetworksincludethehealthcarebodysensornetworks,vehiculartoroadsidecommunicationnetworks,multimediasensornetworksandunderwatercommunicationnetworks.Formorediscussionsonthewirelesssensornetworks,refertothesurveypaperssuchas[ 84 ]and[ 85 ]. Sincethesensornodesinthenetworkareusuallyplacedinhostileenvironments,wirelesstransmissionsamongsensornodesarestronglypreferred.Inaddition,duetotherestrainedsizeofwirelesssensornodes,thecomputationalcapabilityofasinglenodeislimited.Therefore,themeasuredinformationisusuallytransmittedtoaremotedataprocessingcenter(DPC)forfurtherdataanalysis.Furthermore,duetotheunreliablewirelesslinks,multipledatasinksmayexistinthenetworkwhichcollectthemeasureddataandtransmitthepacketstotheDPCnodesecurelyandreliably,possiblythroughtheInternet. Beforethewidedeploymentofwirelesssensornetworks,asystematicunderstandingontheperformanceofthemultihopwirelesssensornetworksisdesired.However,ndingasuitableandaccurateanalyticalmodelforwirelesssensornetworksisparticularlychallenging.First,thetimevaryingchannelconditionsamongwirelesslinks 100 PAGE 101 signicantlycomplicatetheanalysisforthenetworkperformanceintermsofthroughputandexperienceddelay,eveninanaveragesense[ 1 86 87 ].Secondly,duetotheunpredictabilityofthebehaviorofthemonitoredobject,theexogenoustrafcarrivaltothenetwork,i.e.,thenumberofnewlygeneratedpackets,isastochasticprocess.Therefore,toensurethestabilityofthenetwork,i.e.,tokeepthequeuesinthenetworkconstantlynite,theanalyticalmodelofwirelesssensornetworksshouldcomprisearateadmissioncontrolmechanismwhichcandynamicallyadjustthenumberofadmittedpacketsintothenetwork.Thirdly,duetothehostilewirelesscommunicationlinks,adynamicroutingschemeshouldbeincludedintheanalyticalmodel.Moreover,themodelshouldcapturethecomplexissueofwirelesslinkschedulingwhichissignicantlychallengingduetothemutualinterferenceofwirelesstransmissions.Lastly,inordertofullyexplorethenetworkresourceandtomitigatethenetworkcongestion,anappropriateanalyticalnetworkmodelshouldbeabletodynamicallydeliverpacketsthroughmultipledatasinksandthusanautomaticloadbalancingsolutioncanbeachieved. Intheexistingliterature,mostoftheproposedmodelsforwirelesssensornetworksrelyontheuidmodel[ 6 ],whereaowischaracterizedbyasourcenodeandaspecicdestinationnode,e.g.,[ 3 4 6 7 ].However,thismodelisnotapplicabletothecaseswherethegeneratedpacketscanbedeliveredtoanyofthesinknodes,i.e.,thedestinationnodeisoneofthesinksandisselecteddynamically.Moreover,thisuidmodelneglectstheactualqueueinteractionswithinthewirelesssensornetwork.Inthiswork,tostudythecrosslayerinteractionsofthemultihopwirelesssensornetworks,weproposeaconstrainedqueueingmodelwhereapacketneedstowaitforserviceinadataqueue.Morespecically,weinvestigatethejointrateadmissioncontrol,dynamicrouting,adaptivelinkschedulingandautomaticloadbalancingsolutiontothewirelesssensornetworkthroughasetofinterconnectedqueues.Duetothewirelessinterferenceandtheunderlyingschedulingconstraints,ataparticulartimeslot,onlyasubset 101 PAGE 102 ofqueuescanbescheduledfortransmissionssimultaneously.Todemonstratetheeffectivenessoftheproposedconstrainedqueueingmodel,weinvestigatethestochasticnetworkutilitymaximization(SNUM)probleminmultihopwirelesssensornetworks.Basedontheproposedqueueingmodel,wedevelopanadaptivenetworkresourceallocation(ANRA)schemewhichisacrosslayersolutiontotheSNUMproblemandyieldsa(1)]TJ /F8 11.955 Tf 13.17 0 Td[()nearoptimalsolutiontotheglobaloptimumnetworkutilitywhere>0canbearbitrarilysmall.TheproposedANRAschemeconsistsofmultiplelayercomponentssuchasjointrateadmissioncontrol,trafcsplitting,dynamicroutingaswellasadaptivelinkscheduling.Inaddition,theANRAschemeisessentiallyanonlinealgorithmwhichonlyrequirestheinstantaneousinformationofthecurrenttimeslotandhencesignicantlyreducesthecomputationalcomplexity. Therestofthechapterisorganizedasfollows.Section 5.2 brieysummarizestherelatedworkintheliterature.TheconstrainedqueueingmodelforthecrosslayerinteractionsofwirelesssensornetworksisproposedinSection 5.3 .ThestochasticnetworkutilitymaximizationproblemofthewirelesssensornetworkisinvestigatedinSection 5.4 ,whereacrosslayersolution,a.k.a.,theANRAschemeisdeveloped.TheperformanceanalysisoftheANRAschemeisprovidedinSection 5.5 .AnexamplewhichdemonstratestheeffectivenessoftheANRAschemeisgiveninSection 5.6 andSection 5.7 concludesthischapter. 5.2RelatedWork Tocapturethecrosslayerinteractionsofmultihopwirelesssensornetworks,severalanalyticalmodelshavebeenproposedintheliterature.Forexample,in[ 3 7 88 ],themultihopnetworkresourceallocationproblemhasbeenstudiedthroughauidmodel.Eachow,orsession,ischaracterizedbyasourceandadestinationnodewheresinglepathroutingormultipathroutingschemesareimplemented.Mostoftheworkrelyonthedualoptimizationframeworkwhichdecomposesthecomplexcrosslayerinteractionsintoseparatesublayerproblemsbyintroducingdualvariables.Forexample, 102 PAGE 103 theowinjectionrate,controlledbythesourcenodeoftheow,iscalculatedbysolvinganoptimizationproblemwiththeknowledgeofthedualvariables,a.k.a.,shadowprices[ 6 7 ],ofallthelinksthatareutilized.However,thereareseveraldrawbacksfortheuidbasedmodel.First,tocalculatetheoptimumowinjectionrate,theinformationalongallpathsshouldbecollectedinordertoimplementtherateadmissioncontrolmechanism.Inadynamicenvironmentsuchaswirelesssensornetworks,thisprocessofinformationcollectionmaytakeasignicantamountoftimewhichinevitablyprolongsthenetworkdelay.Secondly,theoptimizationbasedsolutionsusuallypursuexedoperatingpointswhicharehardlyoptimalindynamicwirelesssettingswithstochastictrafcarrivalsandtimevaryingchannelconditions.Thirdly,theuidmodelusuallyassumesthatthechangesoftheowinjectionratesareperceivedbyallthenodesalongitspathsinstantaneously.Theactualqueuedynamicsandinteractionsareneglected. Incontrast,followingtheseminalpaperof[ 12 ],manysolutionshavebeenfocusedonthequeueingmodelforstudyingthecomplexinteractionsofcommunicationnetworks.Neelyet.al.extendtheresultsof[ 12 ]intowirelessnetworkswithtimevaryingchannelconditions.Foramorecompletesurveyofthisarea,referto[ 17 ].ThekeycomponentofthequeuebasedsolutionsinthesepapersistheMaxWeightschedulingalgorithm[ 12 14 ].Intuitively,atatimeslot,thenetworkpicksthesetofqueueswhich(1)canbeactivesimultaneouslyand(2)havethemaximumoverallweight.ItiswellknownthattheMaxWeightalgorithmisthroughputoptimalinthesensethatanyarrivalratevectorthatcanbesupportedbythenetworkcanbestabilizedundertheMaxWeightschedulingalgorithm.Inaddition,theMaxWeightalgorithmisanonlinepolicywhichrequiresonlytheinformationaboutcurrentqueuesizesandchannelconditions.However,onenotoriousdrawbackoftheMaxWeightalgorithmisthedelayperformance.Thereasonisthatinordertoachievethethroughputoptimality,theMaxWeightalgorithmexploresadynamicroutingsolutionwherelongpathsareutilizedevenunderalighttrafcload.Thisphenomenonissubstantiatedviasimulations 103 PAGE 104 by[ 89 ].In[ 89 ],theauthorsproposeavariantoftheMaxWeightalgorithmwheretheaveragenumberofhopsoftransmissionsisminimized.Therefore,whenthetrafcislight,theproposedsolutionprovidesamuchlowerdelaythanthetraditionalMaxWeightalgorithm.However,asatradeoff,theinducednetworkcapacityregionin[ 89 ]isnoticeablysmallerthanthatoftheoriginalMaxWeightalgorithm.Consequently,itisdifculttoprovideaminimumrateguaranteeonallthesessionsinthenetwork.Ourworkisinspiredby[ 89 ].Withrespectto[ 89 ],however,ourworkinnovatesinthefollowingways.First,differentfrom[ 89 ],wefocusonaheavyloadedwirelesssensornetwork.Therefore,oursolutionincorporatesarateadmissioncontrolmechanismwhichisnotconsideredin[ 89 ].Secondly,ratherthanminimizingtheoverallnumberofhops,wemaximizetheoverallnetworkutilitywhichcanalsoensurethefairnessamongcompetitivetrafcsessions.Thirdly,wespecicallyprovideaminimumaveragerateguaranteeforeverysessiontoensuretheQoSrequirement.Fourthly,insteadofasingledestinationscenarioasconsideredin[ 89 ],weextendthemodeltocaseswheremultipledatasinknodesareavailable.Eachsourcenodecandeliverthepacketstoanyofthesinks.Moreover,thedynamicroutingandtheissueofautomaticloadbalancingisrealizedbythenetworkonthey.Finally,while[ 89 ]treatsdifferentsessionsequallywhenminimizingtheoverallnumberofhops,ourmodelprioritizesallthesessionswithdifferentQoSrequirements.Therefore,amoreexiblesolutionwithservicedifferentiationscanbeachieved.Wewillpresenttheconstrainedqueueingmodelinthenextsection. 5.3AConstrainedQueueingModelforWirelessSensorNetworks 5.3.1NetworkModel WeconsideramultihopwirelesssensornetworkrepresentedbyadirectedgraphG=fN,LgwhereNandLdenotethesetofverticesandthesetoflinks,respectively.WewillusethenotationofjAjtorepresentthecardinalityofsetA,e.g.,thenumberofnodesinthenetworkisjNjandjLjisthenumberoflinks.Thetimeisslottedas 104 PAGE 105 t=0,1,andataparticulartimeslott,theinstantaneouschanneldatarateoflink(m,n)2Lisdenotedbym,n(t).Inotherwords,link(m,n)cantransmitanumberofm,n(t)packetsduringtimeslott.Notethatthevalueofm,n(t)isarandomvariable.Denote(t)asthenetworklinkratevectorattimeslott.Inthiswork,weassumethat(t)remainsconstantwithinonetimeslotbutissubjecttochangesattimeslotboundaries.Thevalueof(t)isassumedtobeevolvingfollowinganirreducibleandaperiodicMarkovianchainwitharbitrarilylargeyetnitenumberofstates.However,thesteadystatedistributionsareunknowntothenetwork. Attimeslott,thenetworkselectsafeasiblelinkschedule,denotedbyI(t)=fI1(t),I2(t),,IjLj(t)gwhereIl(t)=1iflinklisselectedtobeactiveandIl(t)=0otherwise.Thesetofallfeasiblelinkschedulesisdenotedby(t)whichisdeterminedbytheunderlyingschedulingconstraintssuchasinterferencemodelsandduplexconstraints.Therefore,selectinganinterferencefreelinkscheduleinthenetworkgraphGisequivalenttotheprocessofattaininganindependentsetintheassociatedconictgraph~G,wheretheverticesarethelinksinGandalinkexistsin~GifthetwooriginallinksinGcannottransmitsimultaneously. 5.3.2TrafcModel ThereareanumberofjSjsourcenodesinthewirelesssensornetworkwhichconsistentlymonitorthesurroundingsandinjectexogenoustrafctothenetwork.Thegeneratedpacketsneedtobedeliveredtotheremotedataprocessingcenter(DPC)inamultihopfashion.Tosimplifyanalysis,weassumethateachsourcenodeisassociateduniquelywithasession.ThesetofsourcenodesisdenotedbyS=fn01,n02,,n0jSjgwheren0s,s=1,,jSjisthesourcenodeofsessions.Itisworthnotingthatthefollowinganalysiscanbeextendedstraightforwardlytothescenarioswhereeachsourcenodemaygeneratemultiplesessions. TherearejDjnumberofsinksinthenetworkwhichareconnectedtotheremotedataprocessingcenterviatheInternet.Inotherwords,thesinknodescanbe 105 PAGE 106 viewedasthegatewaysofthewirelesssensornetwork.DenotethesetofsinksasD=fd1,d2,,djDjg.Inthiswork,weconsiderageneralscenariowherethedatapacketsfromasourcenodecanbedeliveredtotheDPCviaanyofthesinknodeinD.Therefore,differentfromtheexistingliteraturesuchas[ 3 88 89 ],thesourcenodesdonotspecifytheparticulardestinationnodeforthegeneratedpackets.Theselectionofthedestinationnodeisachievedbythenetworkviadynamicroutingschemes.ThenetworktopologyconsideredinthisworkisillustratedinFigure 51 Figure51. Topologyofwirelesssensornetworks Foraparticularnodeinthenetwork,saynoden,wedenotednasthenumberofminimumhopsfromnodentothedthdatasinkinsetD.Dene fn=mind(dn),d=1,,jDj(5) astheminimumvalueofdnfornoden,i.e.,theminimumnumberofhopsfromnodentoasinknodeinsetD.Weassumethatnodenisawareofthevalueoffnaswellasthosevaluesoftheneighboringnodes,whichareattainableviaprecalculationsbytraditionalroutingmechanismssuchasDijkstra'salgorithm. 106 PAGE 107 Attimeslott,theexogenousarrivalofsessions,i.e.,thenumberofnewpackets1generatedbythesourcenodeofsessions,isdenotedbyAs(t).Weassumethatthereisanupperboundforthenumberofnewpacketswithinonetimeslot,i.e.,As(t)Amax,8s,t.Foreaseofexposition,weassumethatAs(t)isi.i.d.overtimeslotswithanaveragerateofs.However,thedataratesfrommultiplesourcenodescanbearbitrarilycorrelated.Forexample,ifthewirelesssensornetworkisdeployedformonitoringpurpose,itisverylikelythatamovementoftheobjectwilltriggerseveralconcurrentupdatesofthenearbysensors. Denotethevector=f1,,jSjgasthenetworkarrivalratevector.Thenetworkcapacityregionisthusdenedasallthefeasible2networkarrivalvectorsthatcanbesupportedbythenetworkviacertainpolicies,includingthosewiththeknowledgeoffuturistictrafcarrivalsandchannelrateconditions.Inthiswork,weconsideraheavyloadedtrafcscenariowherethenetworkarrivalvectorisoutsideofthenetworkcapacityregion.Therefore,inordertoachievethenetworkstability,arateadmissioncontrolmechanismisimplementedatthesourcenodes.Morespecically,attimeslott,weonlyadmitanumberofXs(t)packetsintothenetworkfromthesourcenodeofsessions,i.e.,n0s.Apparently,wehave Xs(t)As(t),8s,t.(5) Inaddition,weassumethateachsessionhasacontinuous,concaveanddifferentiableutilityfunction,denotedbyUs(Xs(t)),whichreectsthedegreeofsatisfactionby 1Weassumethatthepacketshaveaxedlength.Forscenarioswithvariablepacketlengths,theunitofdatatransmissionscanbechangedtobitsperslotandthefollowinganalysisstillholds.2Notethatadditionalconstraintsmaybeimposed.Forexample,theconstraintsontheminimumaveragerateandthemaximumaveragepowerexpenditurecanbeenforced.Formorediscussions,pleasereferto[ 17 ]. 107 PAGE 108 transmittingXs(t)numberofpackets.Itisworthnotingthatbyselectingproperutilityfunctions,thefairnessamongcompetitivesessionscanbeachieved.Forexample,ifUs(Xs(t))=log(Xs(t)),aproportionalfairnessamongmultiplesessionscanbeenforced[ 4 10 11 ]. 5.3.3QueueManagement Foreachnodeninthenetwork,therearejNj)]TJ /F13 11.955 Tf 19.91 3.16 Td[(fnnumberofqueuesthataremaintainedandupdated.ThequeuesaredenotedbyQn,h,whereh=fn,,jNj)]TJ /F3 11.955 Tf 18.66 0 Td[(1.NotethatjNj)]TJ /F3 11.955 Tf 19.53 0 Td[(1isthemaximumnumberofhopsforaloopfreeroutingpathinthenetwork.ThepacketsinthequeueofQn,hareguaranteedtoreachoneofthesinknodesinsetDwithinhhops,aswillbeshowninSection 5.4 .Itisinterestingtoobservethatforanewlygeneratedpacketbysessions,thesourcenode,i.e.,n0s,canplaceitinanyofthequeuesofQn0s,h,whereh=fn0s,,jNj)]TJ /F3 11.955 Tf 17.09 0 Td[(1,forfurthertransmission.Thatistosay,consecutivepacketsfromthesourcenoden0smaytraversethroughdifferentnumberofhopsbeforereachingadestinationsinknodeinsetD.Therefore,whenanewpacketisgenerated,thesourcenodeneedstomakeadecisiononwhichqueuethepacketshouldbeplaced,namely,trafcsplittingdecision.Inaddition,thedecisionshouldbemadepromptlyonanonlinebasiswithlowcomputationalcomplexity. Withaslightabuseofnotation,weuseQn,htodenotethequeueitselfandQn,h(t)torepresentthenumberofqueuebacklogs3intimeslott.Forasinglequeue,sayQn,h,itisstableif[ 13 14 ] limB!1g(B)!0(5) where g(B)=limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Pr(Qn,h(t)>B). Thenetworkisstableifalltheindividualqueuesinthenetworkarestable. 3Intheunitofpackets. 108 PAGE 109 Foralink(n,j)2L,werequirethatthepacketsfromQn,hcanbeonlytransmittedtoQj,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1,ifexists.Therefore,thequeueupdatingdynamicforQn,hisgivenby Qn,h(t+1)24Qn,h(t))]TJ /F13 11.955 Tf 16.97 11.36 Td[(X(n,j)2Lun,hn,j(t)35++X(m,n)2Lum,h+1m,n(t)+XsXhs(t)n=n0s(5) where[A]+denotesmax(A,0)andun,hn,j(t)representstheallocateddatarateforthetransmissionsofQn,h!Qj,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1onlink(n,j),attimeslott,and N)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xh=fnun,hn,j(t)=un,j(t) whereun,j(t)=n,j(t)ifIn,j(t)=1,i.e.,link(n,j)isscheduledtobeactiveduringtimeslott,andun,j(t)=0otherwise.ThenotationofXhs(t)denotesthenumberofpacketsthatareadmittedtothenetworkforsessionsandarestoredinqueueQn,hforfuturetransmissions.TheindicatorfunctionA=1ifeventAistrueandA=0otherwise.Notethattheinequalityin( 5 )incorporatesthescenarioswherethetransmitterofaparticularlinkhaslesspacketsinthequeuethantheallocateddatarate.Weassumethatduringonetimeslot,thenumbersofpacketsthatasinglequeuecantransmitandreceiveareupperbounded.Mathematicallyspeaking,wehave X(m,n)2Lum,h+1m,n(t)uin,8n,h,t, (5) and X(n,j)2Lun,hn,j(t)uout,8n,h,t. (5) 5.3.4SessionSpecicRequirements Inthiswork,weconsiderascenariowhereeachsessionhasaspecicraterequirements.Therefore,toensuretheminimumaveragerate,weneedtondapolicy 109 PAGE 110 that limT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0Xs(t)s,8s.(5) Inaddition,weassumethateachsessioninthenetworkhasanaveragehoprequirements.Morespecically,dene Ms(t)=jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1Xh=gn0shXhs(t) (5) where jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1Xh=gn0sXhs(t)=Xs(t),8s,t. Werequirethatforeachsessions, limT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0Ms(t)s,8s.(5) Notethattheaveragehopforaparticularsessionsisrelatedtotheaveragedelayexperiencedandtheaverageenergyconsumedforthepackettransmissionsofsessions.Therefore,byassigningdifferentvaluesofsands,aprioritizedsolutionamongmultiplecompetitivesessionscanbeachievedforthenetworkresourceallocationproblem. 5.4StochasticNetworkUtilityMaximizationinWirelessSensorNetworks Intheprevioussection,weproposeaconstrainedqueueingmodeltoinvestigatetheperformanceofmultihopwirelesssensornetworks.Themodelconsistsofseveralimportantissuesfromdifferentlayers,includingtherateadmissioncontrolproblem,thedynamicroutingproblemaswellasthechallengeofadaptivelinkscheduling.Tobetterunderstandtheproposedconstrainedqueueingmodel,inthissection,wewillexaminethestochasticnetworkutilitymaximizationproblem(SNUM)inmultihopwirelesssensornetworks.Asacrosslayersolution,anAdaptiveNetworkResourceAllocation(ANRA)schemeisproposedtosolvetheSNUMproblemjointly.TheproposedANRAschemeisanonlinealgorithminnaturewhichprovablyachievesanasymptoticallyoptimalaverage 110 PAGE 111 overallnetworkutility.Inotherwords,theaveragenetworkutilityinducedbytheANRAschemeis(1)]TJ /F8 11.955 Tf 12.22 0 Td[()oftheoptimumsolutionwhere>0isapositivenumberthatcanbearbitrarilysmall,withatradeoffwiththeaveragedelayexperiencedinthenetwork. 5.4.1ProblemFormulation RecallthateverysessionspossessesautilityfunctionUs(Xs(t))whichiscontinuous,concaveanddifferentiable.Withoutlossofgenerality,intherestofthischapter,wewillassumethatUs(Xs(t))=log(Xs(t)).Therefore,inlightofthestochastictrafcarrivalaswellasthetimevaryingchannelconditions,ourobjectiveistodevelopapolicywhichmaximizesStochasticNetworkUtilityMaximization(SNUM)Problem XsE(Us(Xs(t))) (5) s.t. Thenetworkremainsstable. TheaverageraterequirementsofalljSjsessions,denotedby=f1,,jSjg,aresatised. TheaveragehoprequirementsofalljSjsessions,denotedby=f1,,jSjg,aresatised. Notethatiftheunderlyingstatisticalcharacteristicsofthestochastictrafcarrivalsandthetimevaryingchannelconditionsareknown,theSNUMproblemisinherentlyastandardoptimizationproblemandthusiseasytosolve.However,duetotheunawarenessofthesteadystatedistributions,theSNUMproblemisremarkablychallenging.Inaddition,inwirelesssensornetworks,dynamicalgorithmicsolutionswithlowcomputationalcomplexityarestronglydesired.Inthefollowing,weproposeanANRAschemetosolvetheSNUMproblemasymptotically.TheANRAschemeisacrosslayersolutionwhichconsistsofjointrateadmissioncontrol,trafcsplitting, 111 PAGE 112 dynamicroutingaswellasadaptivelinkschedulingcomponents.Moreover,theANRAalgorithmcanachieveanautomaticloadbalancingsolutionbyutilizingdifferentsinknodescorrespondingtothevariationsofthenetworkconditions.TheANRAalgorithmisanonlinealgorithminnaturewhichrequiresonlythestateinformationofthecurrenttimeslot.WeshowthattheANRAalgorithmachievesa(1)]TJ /F8 11.955 Tf 12.21 0 Td[()optimalsolutionwherecanbearbitrarilysmall.Therefore,theproposedANRAalgorithmisofparticularinterestfordynamicwirelesssensornetworkswithtimevaryingenvironments. 5.4.2TheANRACrossLayerAlgorithm BeforepresentingtheproposedANRAscheme,weintroducetheconceptofvirtualqueues[ 16 17 34 ]tofacilitateouranalysis.Specically,foreachsessions,wemaintainavirtualqueueYs,whichisinitiallyempty,andthequeueupdatingdynamicisdenedas Ys(t+1)=[Ys(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Xs(t)]++s,8s,t.(5) Similarly,wedeneanothervirtualqueueforeverysessions,denotedbyZs,andthequeuedynamicisgivenby Zs(t+1)=[Zs(t))]TJ /F8 11.955 Tf 11.96 0 Td[(s]++Ms(t),8s,t,(5) whereMs(t)isdenedin( 5 ).Notethatthevirtualqueuesaresoftwarebasedcounterswhichareeasytomaintain.Forexample,thesourcenodeofeachsessioncancalculatethevaluesofvirtualqueuesYs(t)andZs(t)andupdatethevaluesaccordinglyfollowing( 5 )and( 5 ).Inaddition,weintroduceapositiveparameterJwhichistunableasasystemparameter.TheimpactofJonthealgorithmperformancewillbediscussedshortly.TheproposedANRAcrosslayeralgorithmisgivenasfollows.ADAPTIVENETWORKRESOURCEALLOCATION(ANRA)SCHEME: JointRateAdmissionControlandTrafcSplitting(attimet): 112 PAGE 113 Foreachsourcenode,sayn0s,thereareanumberofqueues,i.e.,Qn0s,h,h=fn0s,,jNj)]TJ /F3 11.955 Tf 17.93 0 Td[(1.Findthevalueofhwhichminimizes Zs(t)h+Qn0s,h(t)(5) wheretiesarebrokenarbitrarily.Denotetheoptimumvalueofhash.Thesourcenoden0sadmitsanumberofnewpacketsas Xs(t)=min]Xs(t),As(t)(5) where ]Xs(t)="J 2)]TJ /F6 11.955 Tf 5.48 9.69 Td[(Zs(t)h+Qn0s,h(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Ys(t)#+.(5) Fortrafcsplitting,thesourcenoden0swilldepositallXs(t)packetsinQn0s,h. JointDynamicRoutingandLinkScheduling(attimet): Foreachlink(m,n)2L,denealinkweightdenotedbyWm,n(t),whichiscalculatedas Wm,n(t)="maxh=fm,,jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1(Qm,h(t))]TJ /F6 11.955 Tf 11.96 0 Td[(Qn,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1(t))#+.(5) NotethatifQn,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1doesnotexist,thetransmissionsfromqueueQm,htoQn,h)]TJ /F5 7.97 Tf 6.59 0 Td[(1areprohibited.Attimeslott,thenetworkselectsaninterferencefreelinkscheduleI(t)whichsolves maxI(t)2(t)m,n(t)Wm,n(t).(5) Iflink(m,n)isactive,i.e.,Im,n(t)=1,thequeueofQm,hisselectedfortransmissionswhere h=argmaxh=fm,,jNj)]TJ /F5 7.97 Tf 8.94 0 Td[(1(Qm,h(t))]TJ /F6 11.955 Tf 11.95 0 Td[(Qn,h)]TJ /F5 7.97 Tf 6.58 0 Td[(1(t)).(5) END Notethat( 5 )issimilartotheoriginalMaxWeightalgorithm,introducedin[ 12 ]andgeneralizedin[ 13 14 34 ].Thedynamicroutingandlinkschedulingareaddressedjointlybysolving( 5 ),whichrequirescentralizedcomputation.However,following[ 12 ],manyworkhavebeenfocusedonthedistributedsolutionsof( 5 ).Althoughthedistributedcomputationissueisnotthefocusofthiswork,weemphasizethatourproposedANRAschemecanbeapproximatedwellbyexistingdistributedsolutionssuchas[ 36 38 41 42 45 46 48 ]. 113 PAGE 114 ForthepacketsplacedatqueueQn0s,h,atmosthhopsoftransmissionsareneededinordertoreachoneofthesinknodesinsetD.ThiscanbeveriedstraightforwardlyduetotherequirementthatatransmissionfromQm,htoQn,h)]TJ /F5 7.97 Tf 6.58 0 Td[(1canoccurifandonlyifh)]TJ /F3 11.955 Tf 12.4 0 Td[(1fn.Moreover,thejointrateadmissioncontrolandtheoptimumtrafcsplittingcomponentsofANRAcanbeimplementedbythesourcenodeinadistributedfashion.Notethatinordertocalculatetheinstantaneousadmittedrate,thesourcenodeofsessionsneedsonlytoknowthelocalqueuebackloginformation.Moreover,thedecisionoftrafcsplittingrequiresonlylocalqueueinformationaswell.Therefore,ateverytimeslot,thejointrateadmissioncontrolandtrafcsplittingdecisioncanbemadeonanonlinebasisinaccordancetothetimevaryingconditionsoflocalqueues.Furthermore,wewillshowthatthissimpleadaptivestrategydoesnotincuranylossofoptimality.TheachievednetworkutilityinducedbytheANRAschemecanbepushedarbitrarilyclosetotheoptimumsolution.Next,wewillcharacterizetheglobaloptimumutilityinthenetworkandprovidethemainperformanceresultsoftheproposedANRAscheme. 5.4.3PerformanceoftheANRAScheme Inthissection,werstcharacterizetheglobaloptimumsolutionoftheSNUMproblemin( 5 ).DeneUastheglobalmaximumnetworkutilitythatanyschemecanachieve,i.e.,theoptimumsolutionof( 5 ).InordertoachieveU,itisnaturallytoconsidermorecomplicatedpoliciessuchasthosewiththeknowledgeoffuturisticarrivalsandchannelconditions.However,inthefollowingtheorem,weshowthat,somewhatsurprisingly,theglobaloptimumsolutionoftheSNUMproblemcanbeachievedbycertainstationarypolicies,i.e.,theresponsiveactionischosenregardlessofthecurrentqueuesizesinthenetworkandthetimeslotthatthedecisionismade.Recallthatwehaveassumedthatforeachsession,As(t)isi.i.d.overtimeslots.DenoteA(t)asthevectorofinstantaneousarrivalratesofallsessions,attimeslott.LetAbethesetofallpossiblevalueofA(t).NotethatforeveryelementinA,i.e., 114 PAGE 115 Aa,a=1,,jAj,wehave0AaAmaxwheredenotestheelementwisecomparison.WeuseatorepresentthesteadystatedistributionofAa. Theorem5.1. IftheconstraintsintheSNUMproblemaresatised,themaximumnetworkutility,denotedbyU,canbeachievedbyaclassofstationaryrandomizedpolicies.Mathematically,thevalueofUisthesolutionofthefollowingoptimizationproblem,withtheauxiliaryvariablespkaandRka,as maxXaaXsUs(jSj+1Xk=1pkaRka) (5) s.t. Theconstraintsin( 5 )aresatised. 0RkaAa. pka0,8a,k. PjSj+1k=1pka=1,8a. Proof. WeproveTheorem 5.1 byshowingthatforarbitrarypolicywhichsatisestheconstraintsintheSNUMproblem,theoverallnetworkutilityisatmostU,whichistheoptimumutilityattainedbyaclassofstationaryrandomizedpolicies.Inotherwords,weneedtoshowthat UP=limsupT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0 XsUs(Xs(t))!U(5) whereUPistheaveragenetworkutilityunderapolicyP. ForeachstateinA,sayAa,deneRaasthesetofnonnegativeratevectorsthatareelementwisesmallerthanAa.DeneCRaastheconvexhullofsetRa.Therefore,anypointinCRacanbeconsideredasafeasiblenetworkadmittedratevectorgiventhatthecurrentarrivalratevectorisAa.NotethateverypointinCRaisavectorwithadimensionofjSjby1.Therefore,itcanberepresentedbyaconvexcombinationofatmostjSj+1points,denotedbyRka,k=1,,jSj+1,accordingtoCaratheodory'stheorem.Inlightofthis,werstconsideratimeintervalfrom0toT)]TJ /F3 11.955 Tf 12.19 0 Td[(1.DenoteNa(T) 115 PAGE 116 asthesetoftimeslotsthatA(t)=Aa.Therefore,wecanrewrite( 5 )as UP=limsupT!1XajNa(T)j TXsUs0@jSj+1Xk=1pkaRka1A. Duetothestationaryassumption,wehave UP=XaaXslimsupT!1Us0@jSj+1Xk=1pkaRka1A. Notethatwealsoassumethattheutilityfunctioniscontinuousandbounded.Therefore,thecompactnessoftheutilityfunctionsisassured.Next,wefocusonasubsequenceoftimedurations,denotedbyTi,i=1,,1.Denote UPnet(Ti)=XaaXsUs0@jSj+1Xk=1pka(Ti)Rka1A. Itisstraightforwardtoverifythat UP=limsupi!1UPnet(Ti). Duetothecompactnessoftheutilityfunctions,followingBolzanoWeierstrasstheorem[ 90 ],weclaimthatthereexistsasubsequenceofTi,i=1,,1,suchthat limi!1Us0@jSj+1Xk=1pka(Ti)Rka1A!fUas. DenoteepkaasthevalueswhichgeneratefUas,i.e., fUas=Us0@jSj+1Xk=1epkaRka1A. Wehave UP=limsupi!1Unet(Ti)=XaaXsUs0@jSj+1Xk=1epkaRka1A. AccordingtothedenitionofUin( 5 ),weconcludethatUPU. 116 PAGE 117 Intuitively,Theorem 5.1 indicatesthattheglobalmaximumnetworkutilitycanbeachievedbycertainrandomizedstationarypolicies.However,tocalculateU,thestationarypolicyneedstoknowthesteadystatedistributionswhicharedifculttoobtaininpractice.Inlightofthis,weproposeanadaptivenetworkresourceallocationscheme,namely,ANRA,whichisanonlinesolutionanddoesnotrequiresuchstatisticalinformationasapriori.Fornotationsuccinctness,denote UA(t)=XsE(Us(Xs(t))) astheexpectednetworkutilityinducedbytheANRAscheme.TheperformanceoftheANRAalgorithm,withaparameterJ,isgivenasthefollowingtheorem. Theorem5.2. ForagivensystemparameterJ,wehave liminfT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0UA(t)U)]TJ /F3 11.955 Tf 14.6 10.74 Td[(B J(5) whereBisaconstantandisgivenby B=jNj(jNj)]TJ /F3 11.955 Tf 17.94 0 Td[(1))]TJ /F3 11.955 Tf 5.48 9.69 Td[((uout)2+(uin+Amax)2+Xs)]TJ /F3 11.955 Tf 5.48 9.68 Td[((s)2+(s)2+jSj(Amax)2(jNj)]TJ /F3 11.955 Tf 17.94 0 Td[(1)4+1. Inaddition,theconstraintsintheSNUMproblem,i.e.,( 5 ),aresatisedsimultaneously. Proof. TheproofofTheorem 5.2 isdeferredtoSection 5.5 ItisworthnotingthatifweletJ!1,theperformanceinducedbytheANRAalgorithmcanbearbitrarilyclosetotheglobaloptimumsolutionU.However,asatradeoff,alargervalueofJalsoyieldsalongeraveragequeuesizeinthenetwork.AccordingtoLittle'sLaw,alargerqueuesizecorrespondstoalongeraveragedelayexperiencedinthenetwork.Therefore,byselectingthevalueofJproperly,atradeoff 117 PAGE 118 betweenthenetworkoptimalityandtheaveragedelayinthenetworkcanbeachieved.Wewilldiscussmoreaboutthisissueinthenextsection. 5.5PerformanceAnalysis Inthissection,weprovideaprooftoTheorem 5.2 intheprevioussection.Recallthatin( 5 )and( 5 ),weintroducetwovirtualqueues,i.e.,Ys(t)andZs(t)foreachsessions.Therefore,theaveragerateandtheaveragehoprequirementsfromallsessionsareconvertedintothestabilityrequirementsforthevirtualqueues.Forexample,thevirtualqueueupdateofYs(t)isgivenby( 5 ).IfthevirtualqueueYsisstable,theaveragearrivalrateshouldbelessthantheaveragedeparturerateofthequeue,i.e., slimT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0Xs(t), whichisexactlytheminimumaverageraterequirementimposedbysessions.Bythesametoken,theaveragehoprequirementofsessionsisconvertedtothestabilityproblemofthevirtualqueueZs.Dene QQQ(t)=(QQQ(t),YYY(t),ZZZ(t)),namely,allthedataqueuesandthevirtualqueuesinthenetwork.Ourobjectiveistondapolicywhichstabilizesthenetworkwithrespectto QQQwhilemaximizingtheoverallnetworkutility. Wersttakethesquareof( 5 )andhave Qn,h(t+1)2Qn,h(t)2+0@X(n,j)2Lun,hn,j(t)1A2+0@X(m,n)2Lum,h+1m,n(t)+XsXhs(t)n=n0s1A2)]TJ /F3 11.955 Tf 9.3 0 Td[(2Qn,h(t)0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.35 Td[(X(m,n)2Lum,h+1m,n(t))]TJ /F13 11.955 Tf 11.95 11.35 Td[(XsXhs(t)n=n0s1A. Sinceweassumethateachnodegeneratesatmostonesession,wehave XsXhs(t)n=n0sAmax,8t,n. 118 PAGE 119 Notethatifweallowthatanodecaninitiatemultiplesessions,wehave XsXhs(t)n=n0sjSjAmax,8t,n wherejSjisthenumberofsessionsinthenetwork. Inlightof( 5 )and( 5 ),wehave Qn,h(t+1)2)]TJ /F13 11.955 Tf 11.96 13.27 Td[(Qn,h(t)2uout2+uin+Amax2)]TJ /F3 11.955 Tf 19.26 0 Td[(2Qn,h(t)0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXhs(t)n=n0s1A. (5) Wenextsum( 5 )overallthedataqueuesinthenetwork,i.e.,Qn,h,andhave Xn,hQn,h(t+1)2)]TJ /F13 11.955 Tf 11.96 11.35 Td[(Xn,hQn,h(t)2B1)]TJ /F3 11.955 Tf 19.26 0 Td[(2Qn,h(t)0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXhs(t)n=n0s1A (5) where B1=jNj(jNj)]TJ /F3 11.955 Tf 17.93 0 Td[(1)(uout)2+(uin+Amax)2. NotethatMs(t),denedin( 5 ),satises Ms(t)(jNj)]TJ /F3 11.955 Tf 17.93 0 Td[(1)2Amax. Next,wetakethesquareof( 5 )and( 5 )andthushave Ys(t+1)2Ys(t)2+Xs(t)2+(s)2)]TJ /F3 11.955 Tf 11.96 0 Td[(2Ys(t)Xs(t))]TJ /F8 11.955 Tf 11.96 0 Td[(s and Zs(t+1)2Zs(t)2+Ms(t)2+(s)2)]TJ /F3 11.955 Tf 11.96 0 Td[(2Zs(t)s)]TJ /F6 11.955 Tf 11.96 0 Td[(Ms(t). 119 PAGE 120 Similarly,wesumoverallthesessionsandhave XsYs(t+1)2)]TJ /F13 11.955 Tf 11.96 11.36 Td[(XsYs(t)2B2)]TJ /F3 11.955 Tf 11.96 0 Td[(2XsYs(t)Xs(t))]TJ /F8 11.955 Tf 11.95 0 Td[(s where B2=jSj(Amax)2+Xs(s)2. Also,weobtain XsZs(t+1)2)]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsZs(t)2B3)]TJ /F3 11.955 Tf 11.95 0 Td[(2XsZs(t)s)]TJ /F6 11.955 Tf 11.95 0 Td[(Ms(t) where B3=Xs(s)2+jSj(jNj)]TJ /F3 11.955 Tf 17.94 0 Td[(1)4(Amax)2. DenethesystemwideLyapunovfunctionas L( QQQ(t))=Xn,hQn,h(t)2+XsYs(t)2+XsZs(t)2. Next,wedenetheLyapunovdrift[ 14 ]ofthesystemas =E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(L( Q(t+1)))]TJ /F6 11.955 Tf 11.96 0 Td[(L( Q(t))j Q(t).(5) Dene B=B1+B2+B3, wehave B)]TJ /F3 11.955 Tf 11.96 0 Td[(2Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.04 11.36 Td[(X(m,n)2Lum,hm,n(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXhs(t)n=n0S Q(t)1A)]TJ /F3 11.955 Tf 19.26 0 Td[(2E XsYs(t)(Xs(t))]TJ /F8 11.955 Tf 11.96 0 Td[(s) Q(t)!)]TJ /F3 11.955 Tf 11.95 0 Td[(2E XsZs(t)(s)]TJ /F6 11.955 Tf 11.95 0 Td[(Ms(t)) Q(t)!. 120 PAGE 121 Next,wesubtractbothsidesbyJEPsUs(Xs(t))j Q(t)andhave )]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(Xs(t)) Q(t)!B)]TJ /F3 11.955 Tf 19.27 0 Td[(2Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t) Q(t)1A+2E XsXhQn0s,h(t)Xhs(t) Q(t)!)]TJ /F3 11.955 Tf 11.96 0 Td[(2E XsYs(t)Xs(t) Q(t)!+2XsYs(t)s+2E XsZs(t)Ms(t) Q(t)!)]TJ /F3 11.955 Tf 11.96 0 Td[(2XsZs(t)s)]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(Xs(t)) Q(t)!. (5) WerewritetheR.H.S.of( 5 )as R.H.S.=B+2XsYs(t)s)]TJ /F3 11.955 Tf 11.95 0 Td[(2XsZs(t)s)]TJ /F3 11.955 Tf 19.26 0 Td[(2Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t) Q(t)1A)]TJ /F6 11.955 Tf 19.26 0 Td[(E Xs2Ys(t)Xs(t))]TJ /F13 11.955 Tf 11.95 11.35 Td[(Xs2Zs(t)Ms(t))]TJ /F13 11.955 Tf 11.95 11.35 Td[(XsXh2Qn0s,h(t)Xhs(t)+JXsUs(Xs(t)) Q(t)!. WeobservethatthedynamicroutingandschedulingcomponentoftheANRAschemeisactuallymaximizing Xn,hQn,h(t)E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.35 Td[(X(m,n)2Lum,h+1m,n(t) Q(t)1A.(5) Inaddition,thejointrateadmissioncontrolandtrafcsplittingcomponentoftheANRAschemeisessentiallymaximizing E Xs2Ys(t)Xs(t))]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xs2Zs(t)Ms(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(XsXh2Qn0s,h(t)Xhs(t)+JXsUs(Xs(t)) Q(t)!(5) withtheconstraintsof XhXhs(t)=Xs(t),8s,t.(5) 121 PAGE 122 Toseethis,wecandecompose( 5 )toshowthateachsessionsonlymaximizes JUs(Xs(t))+2Ys(t)Xs(t))]TJ /F3 11.955 Tf 11.95 0 Td[(2Zs(t)XhhXhs(t))]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xh2Qn0s,h(t)Xhs(t).(5) Therefore,theproposedANRAalgorithmindeedminimizestheR.H.S.of( 5 )overallpolicies. Considerareducednetworkcapacityregion,denotedby,parameterizedby>0,as fjn,h+2g(5) whereistheoriginalnetworkcapacityregionand n,h=limT!11 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0XsXhs(t)n=n0s.(5) DeneUastheglobaloptimumnetworkutilityachievedinthereducedcapacityregion.Apparently,wehavelim!0U!U.Inaddition,denoteXhs,(0),Xhs,(1),,Xhs,(t),astheoptimumratesequencewhichyieldsU.DeneXsastheaverageoftheoptimumratesequenceofsessions,inthereducedcapacityregion.ItisstraightforwardtoverifythatXs+isintheoriginalnetworkcapacityregion.Therefore,followingasimilaranalysisasin[ 13 15 91 ],weclaimthatthereexistsarandomizedpolicy,denotedbyR,whichgenerates E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.04 11.36 Td[(X(m,n)2Lum,h+1m,n(t)1AXhs,+(5) ifnisoneofthesourcenodesand E0@X(n,j)2Lun,hn,j(t))]TJ /F13 11.955 Tf 19.03 11.36 Td[(X(m,n)2Lum,h+1m,n(t)1A(5) forothernodes.Furthermore,policyRensures E XhXhs,(t)c+! 122 PAGE 123 and E)]TJ /F6 11.955 Tf 5.48 9.68 Td[(Ms,(t)+s whereMs,(t)isgeneratedbyXhs,(t).DuetothefactthattheproposedANRAschememinimizestheR.H.S.of( 5 )overallallpolicies,includingR,wehave )]TJ /F6 11.955 Tf 11.95 0 Td[(JE XsUs(Xs(t)) Q(t)!B)]TJ /F3 11.955 Tf 11.96 0 Td[(2 Xn,hQn,h(t)+XsYs(t)+XsZs(t)!)]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(XhXhs,(t)+) Q(t)!. Wenexttaketheexpectationwithrespectto Q(t)andobtain L( Q(t+1)))]TJ /F6 11.955 Tf 11.95 0 Td[(L( Q(t)))]TJ /F6 11.955 Tf 11.96 0 Td[(JE XsUs(Xs(t))!B)]TJ /F3 11.955 Tf 11.95 0 Td[(2E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!)]TJ /F6 11.955 Tf 9.3 0 Td[(JE XsUs(XhXhs,(t)+)!. (5) Wesumovertimeslots0,,T)]TJ /F3 11.955 Tf 11.96 0 Td[(1andhave L( Q(T)))]TJ /F6 11.955 Tf 11.96 0 Td[(L( Q(0)))]TJ /F9 7.97 Tf 11.96 14.94 Td[(T)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0JE XsUs(Xs(t))!TB)]TJ /F9 7.97 Tf 11.96 14.94 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0JE XsUs(XhXhs,(t)+)!(5) since E Xn,hQn,h(t)+XsYs(t)+XsZs(t)! isalwaysnonnegative.Next,wedividethebothsidesof( 5 )byTandrearrangetermstohave 1 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0JE XsUs(Xs(t))!1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=0JE XsUs(XhXhs,(t)+)!)]TJ /F3 11.955 Tf 13.42 2.65 Td[(B)]TJ /F6 11.955 Tf 13.15 8.08 Td[(L( Q(0)) T wherethenonnegativityoftheLyapunovfunctionisutilized.Sinceweassumethattheinitialqueuebacklogsinthesystemareniteandthevirtualqueuesareinitiallyempty, 123 PAGE 124 taking!0andliminfT!1yieldstheperformanceresultoftheANRAalgorithmstatedinTheorem 5.2 WenextshowthattheconstraintsoftheSNUMproblemarealsosatised.Toillustratethis,weshowthatthequeuesinthenetwork,includingrealdataqueuesandvirtualqueues,arestable.Basedon( 5 ),wesumovertimeslots0,,T)]TJ /F3 11.955 Tf 12.44 0 Td[(1andhave T)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=02E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!L( Q(0))+T)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0JE XsUs(Xs(t))!+TB.(5) DuetoXs(t)Amaxandtheassumptionsontheutilityfunction,weclaimthatUs(t)isupperboundedanddenotethemaximumutilitywithinonetimeslotasUmax,i.e., Us(t)Umax,8s,t.(5) Dividethebothsidesof( 5 )byTandwehave 1 TT)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xt=02E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!L( Q(0)) T+JjSjUmax+B. BytakinglimsupT!1,wehave limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!JjSjUmax+B 2. (5) Notethattheaboveanalysisholdsforanyfeasiblevalueof.Denote'asthemaximumvalueofsuchthat'isnotempty.Finally,weconcludethat limsupT!11 TT)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xt=0E Xn,hQn,h(t)+XsYs(t)+XsZs(t)!JjSjUmax+B 2'<1.(5) ThestabilityofthenetworkfollowsimmediatelyfromMarkov'sInequalityandthuscompletestheproof. Itisworthnotingthatasshownin( 5 ),alargevalueofJinducesalongeraveragequeuesizeinthenetwork.Therefore,atradeoffbetweenthealgorithm 124 PAGE 125 performanceoftheANRAschemeandtheaveragedelayexperiencedinthenetworkcanbecontrolledeffectivelybytuningthevalueofJ. 5.6CaseStudy Inthissection,wedemonstratetheeffectivenessoftheANRAalgorithmnumericallythroughasimplenetworkshowninFigure 52 .Westressthat,however,thisexemplifyingstudycasereproducesallthechallengingproblemsinvolvedinthecomplexcrosslayerinteractionsintimevaryingenvironments,suchasstochastictrafcarrivals,randomchannelconditionsanddynamicroutingandschedulingetc.AsshowninFigure 52 ,thesourcenodesinthenetworkarenodeAandBwhereasthedestinationsinknodesaredenotedbyEandF. Figure52. Examplenetworktopology Therearesixnodesandtwelvelinksinthenetwork.Therefore,nodeAandBeachmaintainsfourqueues,fromhop2tohop5,andnodeCandDeachmaintainsvequeues,fromhop1tohop5,inthebuffer.Ateachtimeslot,awirelesslinkisassumedtohavethreeequallypossibledatarates4,2,8and16.Thetrafcarrivalsarei.i.d.withthreeequallypossiblestates,i.e.,0,10and20.Theminimumraterequirementsofthe 4Notethattheunitofdatatransmissionsispacketperslot. 125 PAGE 126 twosessionsare5and8andtheaveragehoprequirementsofthesessionsare30and10.Withoutlossofgenerality,weassumethatatagiventimeslot,twolinkswithacommonnodecannotbeactivesimultaneously.Forexample,iflinkA!Bisactive,linkB!A,A!C,C!A,B!DandD!Bcannotbeselected. Figure 53 depictstheaveragenetworkutilityachievedbytheANRAschemefordifferentvaluesofJwhereeachexperimentisexecutedfor50000timeslots.WecanobservefromFigure 53 thattheoverallnetworkutilityrisesasthevalueofJincreases.However,thespeedofutilityimprovementdecreasesandtheachievednetworkutilityconvergestotheglobaloptimumutilityUgradually.TodemonstratethetradeoffofdifferentvaluesofJ,inFigure 54 ,weshowtheaveragequeuesizeinthenetworkforJ=20,50,200,500,1000,2000,5000,10000and20000.Wecanseethat,asexpected,theaveragequeuesizeincreasesasthevalueofJgetslarger.Notethattheaveragequeuesizeisrelatedtotheaveragedelayinthenetwork.Therefore,atradeoffbetweenthenetworkoptimalityandtheaverageexperienceddelaycanbeachievedbytuningthevalueofJ. Figure53. AveragenetworkutilityachievedbyANRAfordifferentvaluesofJ InFigure 55 ,weillustratethesampletrajectoriesoftheadmittedratesoftwosessionswithJ=5000,fortherst50timeslots.Wecanobservethateachsessionadmitsdifferentamountofpacketsintothenetworkadaptivelyfollowingthetimevaryingconditionsofthenetwork.Inaddition,wedepictthetrajectoriesofthefourvirtualqueueswiththesamesettings,inFigure 56 ,fortherst100timeslots.Bycomparing 126 PAGE 127 Figure54. AveragenetworkqueuesizebyANRAfordifferentvaluesofJ Figure 55 andFigure 56 jointly,wecanobservethatfortheminimumratevirtualqueue,sayY1,wheneverthereisthetendencythatthevirtualqueueisaccumulating,asdepictedinFigure 56 ,thecorrespondingadmittedratebysession1increasesinFigure 55 .Bythedenitionofthevirtualqueue,alargerbacklogofY1indicatesthattheaveragedeparturerateofthevirtualqueue,i.e.,theaverageadmittedrate,isinsufcient.Therefore,thesourcenodeofsession1willattempttoincreasetheadmittedrateandthusthebacklogofthevirtualqueuewilldecreaseaccordinglywherethestabilityofthevirtualqueuecanbeassured. Figure55. SampletrajectoriesoftheadmittedratesoftwosessionsforJ=5000 127 PAGE 128 Figure56. SampletrajectoriesofthevirtualqueuesforJ=5000 InFigure 57 ,thetrafcsplittingdecisionsofthetwosourcenodes,i.e.,thehopselectionsofthesourcenodes,areillustrated.Wecanobservethatbothsourcenodesinclinetoutilizethequeueswiththesmallernumberofhops.Thequeueswithlongerhops,e.g.,h=3or4,areusedonlywhenthequeuebacklogsinthequeueswithsmallerhopsareoverwhelmed.Inaddition,wecanseethatonaverage,session2utilizesasmallernumberofaveragehopsthansession1.Recallthatsession2hasamuchmorestringentconstraintontheaveragenumberofhopsthansession1,i.e.,10v.s.30.Therefore,thesourcenodeofsession2,i.e.,n02,inclinestodepositmorepacketsonthequeueswithsmallerhopcounts.Asaconsequence,byassigningdifferentvaluesofrateandhoprequirements,aservicedifferentiationsolutioncanbeachievedbytheANRAschemeamongmultiplecompetitivesessionsinthenetwork.Inaddition,anearoptimalnetworkutilitycanbeattainedsimultaneously. 5.7ConclusionsandFutureWork Inthischapter,weproposeaconstrainedqueueingmodeltocapturethecrosslayerinteractionsinmultihopwirelesssensornetworks.Ourmodelconsistsofcomponentsfrommultiplelayerssuchasrateadmissioncontrol,dynamicroutingandwirelesslinkscheduling.Basedontheproposedmodel,weinvestigatethestochasticnetwork 128 PAGE 129 Figure57. SampletrajectoriesofthehopselectionsforJ=5000 utilitymaximizationprobleminwirelesssensornetworks.Asacrosslayersolution,anadaptivenetworkresourceallocationscheme,a.k.a.,ANRAalgorithm,isproposed.TheANRAalgorithmisanonlinemechanismwhichyieldsanoverallnetworkutilitythatcanbepushedarbitrarilyclosetotheglobaloptimumsolution. Asafuturework,energyawaredistributedschedulingalgorithmsaretobestudiedandevaluated.Inaddition,theextensionofourmodeltowirelesssensornetworkswithnetworkcodingseemsinterestingandneedsfurtherinvestigation. 129 PAGE 130 CHAPTER6THRESHOLDOPTIMIZATIONFORRATEADAPTATIONALGORITHMSINIEEE802.11WLANS 6.1Introduction IEEE802.11WLANhasbecomethedominatingtechnologyforindoorwirelessInternetaccess.WhiletheoriginalIEEE802.11DSSSonlyprovidestwophysicaldatarates(1Mbpsand2Mbps),thecurrentIEEEstandardprovidesseveralavailabledataratesbasedondifferentmodulationandcodingschemes.Forexample,IEEE802.11bsupports1Mbps,2Mbps,5.5Mbpsand11MbpsandIEEE802.11gprovides12physicaldataratesupto54Mbps.Inordertomaximizethenetworkthroughput,IEEE802.11devices,i.e.,stations,needtoadaptivelychangethedataratetocombatwiththetimevaryingchannelenvironments.Forinstance,whenthechannelisgood,ahighdataratewhichusuallyrequireshigherSNRcanbeutilized.Onthecontrary,alowdataratewhichiserrorresilientmightbefavorableforabadchannel.TheoperationofdynamicallyselectingdataratesinIEEE802.11WLANsiscalledrateadaptationingeneral. TheimplementationofrateadaptationalgorithmsisnotspeciedbytheIEEE802.11standard.Thisintentionalomissionourishesthestudiesonthisactiveareawhereavarietyofrateadaptationalgorithmshavebeenproposed[ 92 108 ].Thecommonchallengeofrateadaptationalgorithmsishowtomatchtheunknownchannelconditionoptimallysuchthatthenetworkthroughputismaximized.Accordingtothemethodsofestimatingchannelconditions,rateadaptationalgorithmscanbedividedintotwomajorcategories.Therstoneiscalledclosedlooprateadaptation.Mostschemesinthisapproachenablethereceivertomeasurethechannelqualityandsendsbackthisinformationexplicitlytothetransmitterforrateadaptation.Forexample,thereceiverrecordstheSNRorRSSIvalueofthereceivedpacketandsendsthisinformationbacktothetransmitterviaCTSorACKpacket.Consequently,thetransmitterestimatesthechannelconditionbasedonthefeedbacksignalandadjuststhedatarateaccordingly. 130 PAGE 131 Byutilizingadditionalfeedbackmechanisms,thecloselooprateadaptationalgorithmscanachieveabetterperformancethantheopenloopcounterpart.However,inpractice,thecloselooprateadaptationalgorithmsarerarelyusedincommercialIEEE802.11devices.ThisisbecausethattheextrafeedbackinformationneedstobeconveyedreliablyandhenceaninevitablemodicationonthecurrentIEEE802.11standardisneeded.ThisnoncompatibilityhindersthecloselooprateadaptationalgorithmsfrompracticalimplementationsincurrentofftheshelfIEEE802.11products. Thesecondcategoryofrateadaptationalgorithms,whichispredominantlyadoptedbythevendors,islabeledasopenloopalgorithms.ThewidelyutilizedAutoRateFallbackalgorithm,a.k.a.,ARF,fallsintothiscategory.Asmanyotheropenlooprateadaptationalgorithms,ARFadjuststhedateratesolelybasedontheIEEE802.11ACKpackets.Forexample,inEnterasysRoamAboutIEEE802.11card[ 109 ],twoconsecutiveframetransmissionfailures,indicatedbynotreceivingACKspromptly,inducesaratedownshift,whiletenconsecutivesuccessfulframetransmissionstriggersarateupshift[ 110 ].MostcommercializedIEEEproductsfollowthisup/downscheme[ 95 96 ].Inthischapter,wefocusontheopenlooprateadaptationalgorithmsduetothepracticalmerits.Morespecically,weconsiderathresholdbasedrateadaptationalgorithmwhichworksasfollows.Ifthereareuconsecutivesuccessfultransmissions,thedatarateisupgradedtothenextlevel.Ontheotherhand,ifdconsecutivetransmissionsfailed,aratedownshiftistriggered.SinceARFismerelyaspecialcaseofthisthresholdbasedrateadaptationalgorithm,ouranalysiscanbeappliedtoARFanditsvariantsaswell. Asatradeoffwithsimplicity,thereareseveralchallengesexistingforthethresholdbasedrateadaptationalgorithm.First,duetothetrialanderrorbasedup/downmechanism,itinherentlylacksthecapabilityofcapturingshortdynamicsofthechannelvariations[ 96 ].Totacklethisissue,Qiaoet.al.proposeafastresponsiverateadaptationsolutionin[ 92 ].Byintroducingameasureofdelayfactor,theresponsivenessofthe 131 PAGE 132 thresholdbasedrateadaptationalgorithmcanbeguaranteed.Thesecondchallengeofthethresholdbasedrateadaptationalgorithmistheindifferencetocollisioninducedfailuresandnoiseinducedfailures.ItisworthnotingthatinamultiuserWLAN,anACKtimeoutcanbeascribedtoeitheranMAClayercollision,oranerroneouschannel.However,thethresholdbasedrateadaptationalgorithmisunabletodistinguishthemeffectively.Therefore,excessivecollisionsmayintroduceunnecessaryratedegradationswhichsignicantlydeterioratethesystemperformance.Attributingtheactualreasonofatransmissionfailure,orapacketloss,isnamedlossdiagnosis[ 111 ]andhasattractedtremendousattentionfromthecommunity.Forexample,Choiet.al.[ 112 ]proposeanalgorithmicsolution,whichisspecictothethresholdbasedrateadaptationalgorithm,tomitigatethecollisioneffectinmultiuserIEEE802.11WLANs.Therefore,theperformancedeteriorationbytheindifferencetocollisionscanbecompensatedeffectively. Whilethersttwochallengesofthethresholdbasedrateadaptationalgorithmhavebeentackledeffectively,thethirdmajorobstacle,namely,theoptimalselectionoftheup/downthresholds,remainsasanopenproblem.Asystematictreatmentonhowtoselectthevaluesofuanddinthethresholdbasedrateadaptationalgorithmislackingintheliterature,althoughseveralheuristicsolutionsareproposed[ 93 94 ]. Thecontributionofthisworkistwofold.First,weanalyticallyinvestigatethebehaviorofthethresholdbasedrateadaptationalgorithmfromareverseengineeringperspective.Inotherwords,weanswertheessentialyetunresolvedquestion,i.e.,Whatisthethresholdbasedrateadaptationalgorithmactuallyoptimizing?,byunveilingtheimplicitobjectivefunction.Asaresult,severalintuitiveobservationsofthethresholdbasedrateadaptationalgorithmcanbeexplainedstraightforwardlybyinspectingthisobjectivefunction.Therefore,ourreverseengineeringmodelprovidesanalternativemeanstounderstandthethresholdbasedrateadaptationalgorithm.Ourworkisacomplementtotherecenttrendofreverseengineeringstudiesonexisting 132 PAGE 133 heuristicsbasednetworkingprotocols,suchasTCP(transportlayer)[ 4 113 114 ],BGP(networklayer)[ 115 ]andrandomaccessMACprotocol(datalinklayer)[ 116 ].Tothebestofourknowledge,thisistherstworkofstudyingthresholdbasedrateadaptationalgorithmsfromareverseengineeringperspective.Ourresultsexplicitlyshowthatthevaluesofuanddplayanimportantroleintheobjectivefunctionandthusthenetworkperformancehingeslargelyontheselectionoftheup/downthresholds.Inlightofthis,weproposeathresholdoptimizationalgorithmwhichdynamicallytunestheup/downthresholdsofthethresholdbasedrateadaptationalgorithmandprovablyconvergestothestochasticoptimumsolutioninarbitrarystationaryrandomenvironments.Weshowthattheoptimalselectionofthethresholdssignicantlyenhancesthesystem'sperformance. Therestofchapterisorganizedasfollows.Section 6.2 brieyoverviewsthestateoftheartrateadaptationalgorithmsintheliterature.ThereverseengineeringmodelofthethresholdbasedrateadaptationalgorithmisderivedinSection 6.3 .InSection 6.4 ,thethresholdoptimizationalgorithmisproposed.TheperformanceevaluationsareprovidedinSection 6.5 andSection 6.6 concludesthischapter. 6.2RelatedWork RBAR[ 103 ]proposesanSNRbasedcloselooprateadaptationalgorithmwheretheratedecisionreliesonthefeedbacksignalfromthereceiver.Specically,thereceiverestimatesthechannelconditionanddeterminesaproperrateviatheRTS/CTSexchange.WhileconsistentlyoutperformstheopenlooprateadaptationalgorithmssuchasARF,RBARisincompatiblewiththecurrentIEEE802.11standardbyalteringtheCTSframes[ 96 ].In[ 105 ],ahybridrateadaptationalgorithmwithSNRbasedmeasurementsisproposed,wherethemeasuredSNRisutilizedtoboundtherangeoffeasiblesettingsandthusshortenstheresponsetimetochannelvariations.[ 97 ]isanotherexampleofthecloselooprateadaptationalgorithmswhichattemptstoimprovethethroughputbypredictingthechannelcoherencetime,whichisneverthelessdifcult 133 PAGE 134 inpractice.Chenet.al.introduceaprobabilisticbasedrateadaptationforIEEE802.11WLANs.In[ 100 ],arateadaptiveacknowledgementbasedrateadaptationalgorithmisintroduced.ThebasicideaisthatbyvaryingtheACKtransmissionrate,theappropriaterateinformationisconveyedtothetransmitter.WangandHelmy[ 108 ]proposeatrafcawarerateadaptationalgorithmwhichexplicitlyrelatesthebackgroundtrafctotherateselectionproblem.However,aforementionedsolutionseitherrelyonalteringtheframestructures,ordonotconformtothedefactoIEEE802.11standardincommercialusage1.CHARM[ 101 ]avoidstheoverheadofRTS/CTSbyleveragingthechannelreciprocity.However,modicationsontheIEEE802.11standardizedframestructures,e.g.,beaconsandprobesignals,arestillneeded. Inlightofthecomplexityandtheincompatibilityofthecloselooprateadaptationalgorithms,alternativeopenlooprateadaptationalgorithmsareproposed.Althoughusuallyprovidinginferiorperformancethanthecloseloopsolutions,openloopalgorithmssoonbecomethepredominanttechniqueincommercialIEEE802.11devicesduetothesimplicityandthecompatibility.ThemostpopularopenlooprateadaptationalgorithmistheARFprotocolproposedbyAdandLeo[ 104 ]wherearateupshiftistriggeredbytenconsecutivesuccessfultransmissionwhiletwoconsecutivefailuresinducearatedownshift.SampleRate[ 102 ]isanotherwidelyadoptedopenlooprateadaptationalgorithmwhichperformsarguablythebestinstaticsettings[ 96 ].However,aspointedoutby[ 96 ],SampleRatesuffersfromsignicantpacketlossesinfastchangingchannels.ONOE[ 106 ]isacreditbasedmechanismincludedinMadWiFidrivers.Thecreditisdeterminedbythenumberofsuccessfultransmissions,erroneoustransmissionsandretransmissionsjointly.However,itispointedin[ 98 ]thatONOEislesssensitivetoindividualpacketfailureandbehavesoverconservatively. 1Forexample,theylargelyrelyontheRTS/CTSsignalingmechanismwhichishardlyusedinpractice. 134 PAGE 135 Asmentionedpreviously,oneofthedrawbacksofthesimpleopenlooprateadaptationalgorithmsistheinabilitytodiscriminatethecollisioninducedlossesandthenoiseinducedlosses.Whileseveralcollisionawarerateadaptationsareavailableintheliterature[ 95 96 99 107 ],theybaseonthecloseloopsolutionswhichutilizeeitherRTS/CTSsignalingormodicationsonthestandard.In[ 110 ],[ 112 ],Choiet.al.proposeacollisionmitigationalgorithmforopenloopARFalgorithmbasedonaMarkovianmodeling.However,thechannelconditionsareassumedtobeconstantintheirwork.Inthiswork,onthecontrary,weparticularlyfocusonthethresholdoptimizationtocombatwithfastchanneluctuationsratherthancollisions.Therefore,intandemwith[ 110 112 ],ourworkprovideasolutiontojointlymitigatethechanneluctuationsandthemultiusercollisions.Forexample,theup/downthresholds,i.e.,uanddcanberstcalculatedbyourthresholdoptimizationalgorithminSection 6.4 .Next,theyaresubjectedtofurtheradjustmentsfollowing[ 112 ]tomitigatethecollisioneffects. 6.3ReverseEngineeringfortheThresholdBasedRateAdaptationAlgorithm WeconsiderastationinamultirateIEEE802.11WLAN.ThereareNstationsintheWLANwhereeachstation,sayi,hasatransmissionprobabilityofpi.NotethattheequivalenceoftheppersistentmodelandtheIEEE802.11binaryexponentialbackoffCSMA/CAmodelhasbeenextensivelystudiedin[ 116 ]and[ 117 ].Throughoutthiswork,weassumethatthetransmissionprobabilityofeachstationisxed.Theinteractionofthedatarateandthetransmissionprobabilityremainsasfutureresearch.Withoutlossofgenerality,weassumethatthestationshaveasametransmissionprobabilityofp. Inthiswork,wefocusonthewidelydeployedopenloopthresholdbasedrateadaptationalgorithm.Notsurprisingly,thespeedofchannelvariationshasagreatimpactontheperformanceoftherateadaptationalgorithm.Whiletheoriginalintentionofthethresholdbasedrateadaptationalgorithmistomaximizetheaveragethroughput,anaturalquestionarisesthatwhetherthisisindeedthecase.Inthissection,tobetterunderstandtheimpactofuanddontheperformanceofthethresholdbasedrate 135 PAGE 136 adaptationalgorithm,weinvestigatethethresholdbasedrateadaptationalgorithmviaareverseengineeringapproach.WeassumethattheRTS/CTSsignalsareturnedoff.Inatimeslot,sayt,wedenotethechannelstate2ass(t)anddenotethesuccessfultransmissionprobability,giventhecurrenttransmissionrater(t)andthechannelconditions(t),as PS(s(t),r(t))=p(1)]TJ /F6 11.955 Tf 11.96 0 Td[(p)N)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1)]TJ /F6 11.955 Tf 11.95 0 Td[(e(s(t),r(t)))(6) whereedenotestheframeerrorrate(FER)andisgivenby e(s(t),r(t))=1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F6 11.955 Tf 11.96 0 Td[(Pe(s(t),r(t)))L(6) andPe(s(t),r(t))isthebiterrorrate(BER)whichisdeterminedbythecurrentdatarate,i.e.,modulationscheme,andthecurrentchannelcondition.Listheframelengthofthepacket.Similarly,wedenePF(s(t),r(t))=1)]TJ /F6 11.955 Tf 12.32 0 Td[(PS(s(t),r(t))asthetransmissionfailureprobabilityattimet.ItisworthnotingthatbothPSandPFarefunctionsofthecurrentdatarater(t)aswellastheinstantaneouschannelconditions(t),whichisrandom.Particularly,weassumethatthethresholdbasedrateadaptationalgorithmwillincreasethedataratebyanamountofifthereareuconsecutivesuccessfultransmissionsanddecreaseitbyiftherearedconsecutivefailures.Denoteu=u)]TJ /F3 11.955 Tf 11.64 0 Td[(1andd=d)]TJ /F3 11.955 Tf 12.66 0 Td[(1fornotationsuccinctness.Wedeneabinaryindicatorfunction(t)where(t)=1meansthatthetransmissionattimeslottissuccessfuland(t)=0otherwise.Mathematically,theupdatingruleofthethresholdbasedrateadaptationalgorithmcanbewrittenas 2Notethatthenumberoffeasiblechannelstatescanbepotentiallyinnite. 136 PAGE 137 r(t+1)=(r(t)+))]TJ /F14 7.97 Tf 11.65 1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 11.66 1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0+r(t))]TJ /F9 7.97 Tf 11.65 1.79 Td[(o.w. (6) where )]TJ /F9 7.97 Tf 6.78 1.8 Td[(x=8>><>>:1,ifeventxistrue0,ifeventxisfalse. (6) Forexample,)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)=1=1ifthetransmissionattimetissuccessfuland)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=1=0otherwise.Thesymbolofo.w.denotestheeventthatneitheruconsecutivesuccessfulnordconsecutivefailedtransmissionshappened.Forsimplicity,weassumethatthemaximumallowabledatarateissufcientlylargeandtheminimumdatarateiszero,i.e.,nottransmitting.Dene h(t)=[r(t),r(t)]TJ /F3 11.955 Tf 11.95 0 Td[(1),,r(1),e(s(t),r(t)),,e(s(1),r(1))](6) asthehistoryvector.Inaddition,wedene Z(t+1)=Efr(t+1)jh(t)g (6) whereEistheexpectationoperator. Condition: (C.1)Thechannelstatesbetweentwoconsecutivesuccessfultransmissionsortwoconsecutivefailedtransmissionsareindependentrandomvariables. Weemphasizethattherestrictivecondition(C.1)isnotourgeneralassumptioninthiswork.If(C.1)issatised,however,thederivationofthereverseengineeringanalysiscanbepresentedinamoreconciseform,aswillbeshownshortly. 137 PAGE 138 First,weobtainEf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1jh(t)g=Prf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=1=1,)]TJ /F14 7.97 Tf 31.15 1.86 Td[((t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=1=1,,)]TJ /F14 7.97 Tf 12.25 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1=1jh(t)g (6) Ifcondition(C.1)issatised,( 6 )canbefurtherdecomposedas Ef)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1jh(t)g=Prf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=1=1jh(t)gPrf)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1=1jh(t)gPrf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1=1jh(t)g=uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)) (6) Similarly,wehaveEf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0jh(t)g=Prf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=0=1,)]TJ /F14 7.97 Tf 31.15 1.86 Td[((t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=0=1,,)]TJ /F14 7.97 Tf 12.25 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=0=1jh(t)g (6) If(C.1)isassumedtobevalid,wecanobtainEf)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F9 7.97 Tf 6.58 0 Td[(d)=0jh(t)g=dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)) (6) Fornotationsuccinctness,wewilltemporarilyassumethat(C.1)issatised.Theconditionwillberelaxedaftertheimplicitobjectivefunctionisrevealed.Therefore,we 138 PAGE 139 canwrite( 6 )as Z(t+1)=Efr(t+1)jh(t)g=(r(t)+)uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[()dYk=0(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))+r(t) 1)]TJ /F9 7.97 Tf 17.5 14.95 Td[(uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)))]TJ /F9 7.97 Tf 16.65 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))!=r(t)+ uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)))]TJ /F9 7.97 Tf 16.65 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))! (6) Letusrevisit( 6 ),whichcanberewrittenas r(t+1)=r(t)+1 (r(t)+))]TJ /F14 7.97 Tf 11.66 1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 11.66 1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0+r(t))]TJ /F9 7.97 Tf 11.65 1.79 Td[(o.w.)]TJ /F6 11.955 Tf 11.96 0 Td[(r(t)=r(t)+(t) (6) where (t)=1 n(r(t)+))]TJ /F14 7.97 Tf 11.66 1.86 Td[((t)=1)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=1)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(u)=1+(r(t))]TJ /F8 11.955 Tf 11.95 0 Td[())]TJ /F14 7.97 Tf 11.65 1.86 Td[((t)=0)]TJ /F14 7.97 Tf 6.78 1.86 Td[((t)]TJ /F5 7.97 Tf 6.58 0 Td[(1)=0)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0+r(t))]TJ /F9 7.97 Tf 11.65 1.8 Td[(o.w.)]TJ /F6 11.955 Tf 11.96 0 Td[(r(t)o. (6) 139 PAGE 140 Itshouldbenotedthat Ef(t)jh(t)g=uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)))]TJ /F9 7.97 Tf 24.62 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.96 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)). (6) Therefore,weobservethat( 6 )isaformofstochasticapproximationwithrespectto( 6 ).Next,wepresentthereverseengineeringtheoremforthethresholdbasedrateadaptationalgorithm. Theorem6.1. Thethresholdbasedrateadaptationalgorithmof( 6 )isastochasticapproximationwhichsolvesanimplicitobjectivefunctionU(t),withaconstantstepsizeof,whereU(t)isintheformof U(t)=r(t)(uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)))]TJ /F9 7.97 Tf 16.65 14.95 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))o+K (6) ifcondition(C.1)issatisedandKisaconstantwithrespecttorater(t). Proof. Theorem1followsdirectlyfromthepreviousanalysis.Notethatthethresholdbasedrateadaptationalgorithmcanbewrittenas r(t+1)=r(t)+(t)(6) where(t)isthestochasticgradientandsatises Ef(t)jh(t)g=@U @r(t)jh(t)(6) HenceTheorem1holds. Ifcondition(C.1)doesnothold,wecanreplace uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)) 140 PAGE 141 and dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k)) in( 6 )with Prf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=1,,)]TJ /F14 7.97 Tf 12.26 1.86 Td[((t)]TJ /F9 7.97 Tf 6.58 0 Td[(u)=1jh(t)g (6) and Prf)]TJ /F14 7.97 Tf 6.77 1.86 Td[((t)=0,,)]TJ /F14 7.97 Tf 12.26 1.86 Td[((t)]TJ /F9 7.97 Tf 6.59 0 Td[(d)=0jh(t)g (6) respectivelyandthetheoremremainsvalid. ItisworthnotingthattheobjectivefunctionU(t)isatimevaryingfunctionwhichisdeterminedbythedatarateaswellasthechannelconditionsofthepasttimeslotswhere=max(u,d).Notethatthedataratewithinthelasttimeslotsalwaysremainsunchanged.However,thechanneluctuationsaffectthesuccessfultransmissionprobabilityPSandthusaltertheobjectivefunctionU(t). ThepartialderivativeoftheobjectiveofU(t)givenh(t),i.e., g(t)=uYk=0PS(s(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k)))]TJ /F9 7.97 Tf 17.31 14.94 Td[(dYk=0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(PS(s(t)]TJ /F6 11.955 Tf 11.95 0 Td[(k),r(t)]TJ /F6 11.955 Tf 11.96 0 Td[(k))(6) isalsotimevarying.Iftheprobabilitythatthelastutransmissionsareallsuccessfulisgreaterthantheprobabilitythatthelastdtransmissionsareallfailures,thestationtendstoincreasethedatarateandviceversa.Thespeedofrateincreasingordeceasingisdeterminedbythedifferenceofthesetwoprobabilities.Inotherwords,thepartialderivativein( 6 )couldbeeitherpositiveornegativewhichcorrespondstoarateupshiftoraratedownshift.Theabsolutevalueoftheinstantaneousderivativedeterminesthespeedofratechanging. Adirectcomputationofthepartialderivativein( 6 )ischallenging,ifnotimpossible,duetotheuncertaintyinducedbytheunpredictablestochasticchannel.Therefore,thethresholdbasedrateadaptationalgorithm,describedin( 6 ),utilizes 141 PAGE 142 analternativestochasticapproximationapproachwiththeunbiasedestimationofg(t),i.e.,(t)in( 6 ),whichsignicantlyreducesthecomputationalcomplexitysinceonlylocalinformationbasedonACKsisrequired.ThisnatureofsimplicityandpracticabilityfacilitatesthepopularityofthethresholdbasedrateadaptationalgorithmanditsvariantssuchasARF. Toachieveabetterunderstandingof( 6 ),letusconsiderthefollowingextremecasesfromareverseengineeringstandpoint. (u=0,d=0))(u=d=1):Inthiscase,thethresholdbasedrateadaptationalgorithmincreasesthedatarateifthecurrenttransmissionissuccessfulanddecreasesotherwise.From( 6 ),wecanobservethatthisaggressivealgorithmmerelycomparesthesuccessfulprobabilitywiththefailureprobabilityofthecurrenttimeslotandexpectsthatthenexttimeslotwillremainthecurrentchannelcondition. (u=+1,d=0))(u=+1,d=1):From( 6 ),wecanseethatthederivativeisalwaysnegativesincetherstpartof( 6 )iszero.Therefore,thethresholdbasedrateadaptationwillkeepdecreasingdatarateuntiltheminimumdatarateisreached,i.e.,zero,whichisconsistentwiththeintuition. (u=0,d=+1))(u=1,d=+1):Inthisscenario,thesecondpartof( 6 )isalwayszeroandthusthealgorithmwillkeepupgradingdatarateuntilthemaximumdatarateisachieved. (u=+1,d=+1))(u=1,d=+1):Thederivativeof( 6 )isalwayszeroandhencethedatarateneverchanges. Hence,byinspecting( 6 )and( 6 )directly,weprovideanalternativemeanstounderstandthebehaviorofthethresholdbasedrateadaptationalgorithm.In( 6 ),wehaveassumedaconstantstepsizewhileincurrentofftheshelfIEEE802.11devices,adiscretesetofdataratesareprovided.However,continuousrateadaptationcanbeachievedbycontrollingthetransmissionpowerjointlyordeployingAdaptiveCodingandModulation(ACM)capabledevices.Therefore,ourreverseengineeringmodel,whiletsincontinuousratescenarios,providesanapproximatemodelfordiscreterateadaptationscenarios.Webelievethatthereverseengineeringresultinthisworkprovidesarststeptowardsacomprehensiveunderstandingonthe 142 PAGE 143 goodyetsimplerateadaptationalgorithmdesigns.Inaddition,basedontheunveiledimplicitobjectivefunctionof( 6 ),theinteractionsofrateadaptationsamongmultipleIEEE802.11stationscanbeinvestigatedinagametheoreticalframework. Itisimmediatetoobservefrom( 6 )and( 6 )thattheselectionofuanddhassignicantimpactontheperformanceoftherateadaptationalgorithm.Ideally,therateadaptationalgorithmattemptstondthedataratewhichmaximizestheexpectedthroughputinthenexttimeslot,i.e., r0=argmaxrrPS(s(t+1),r).(6) Dene V=rPS(s(t+1),r).(6) Therefore,ifwecanestimatethevalueof@V @randrelateitby g(t)!@V @r,(6) thenthethresholdbasedrateadaptationalgorithmisindeedoptimizingtheexpectedthroughputviathestochasticapproximationapproach.However,toachieveaderivativeestimationofageneralnonMarkoviansystemisnontrivial[ 118 ][ 119 ].Moreover,ifthechannelappearstotallyrandom,e.g.,nonstationaryandfastfading,thereexistsnoeffectiveoptimizationbasedsolutionunlesscertainlevelofknowledgeonthechannelrandomnessisavailable.Therefore,inthenextsection,wewillconsiderastationaryrandomchannelenvironment.Nevertheless,thestochasticchannelcanbeslowvaryingorfastvaryingfollowingarbitraryprobabilisticdistributions.Weproposeathresholdoptimizationalgorithmwhichprovablyconvergestothestochasticoptimumvaluesofuanddandtheoverallperformanceofthenetworkisremarkablyenhanced. 6.4ThresholdOptimizationAlgorithm Inthissection,wemodelthestochasticchannelasastationaryrandomprocessdenotedbys(t).Itisworthnotingthatifthechannelisquasistatic,i.e.,s(t)isa 143 PAGE 144 piecewiseconstantfunction,theprobingbasedrateadaptationalgorithms,e.g.,SampleRate[ 102 ],canachievegoodperformance.Itisinterestingtoobservethateveninaquasistaticenvironment,fordifferentchannelconditions,theoptimalvaluesofuanddmaybedifferent.Onefeasiblewaytondtheoptimumvaluesofuanddwithdifferentchannelconditions,inaquasistaticenvironment,isthesamplepathbasedpolicyiterationapproachintroducedin[ 120 ]and[ 121 ],basedontheMarkovianmodelofthethresholdbasedrateadaptationalgorithmproposedin[ 110 ]. However,itisarguablethatthechannelenvironmentisstochasticandtimevaryinginnature.Therefore,itisnotunusualthatthechannelconditionhasalreadychangedbeforetheoptimizationalgorithmhasreachedasteadystatesolution.Thealgorithmwillbetherebyconsistentlychasingafteramovingobjectandthusthethresholdsarealwayschosensuboptimally.Thisismoresevereinafastfadingstochasticenvironment.Inlightofthis,wealternativelypursuethestochasticoptimumvaluesofthethresholdsinatimevaryingandpotentiallyfastchangingchannelenvironment.Thatistosay,weattempttondthesetofvaluesforuanddwhichmaximizetheexpectedsystemperformancewithrespecttotherandomchannel.Beforeelaboratingfurther,webrieyoutlinethelearningautomatatechniquesbasedonwhichourthresholdoptimizationalgorithmisproposed. 6.4.1LearningAutomata Learningautomatatechniquesarerstintroducedinthecontrolcommunitywhereinmanyscenarios,thesystemistimevaryingandstochasticinnature.Therefore,stochasticlearningapproachesaredesiredtoaddressthestochasticcontrolprobleminrandomsystems.Thebasicideaoflearningautomatatechniquescanbedescribedasfollows.Weconsiderastochasticenvironmentandasetofniteactionsavailableforthedecisionmaker.Eachselectedactioninducesanoutputfromtherandomenvironment.However,duetothestochasticnature,theoutputsforagiveninputmaybedifferentatdifferenttimeinstances.Basedonobservations,alearningautomataalgorithmis 144 PAGE 145 expectedtondanactionwhichisthestochasticoptimumsolutioninthesensethattheexpectedsystem'sobjectiveismaximized. Ateachdecisioninstance,thedecisionmakerselectsoneoftheactionsaccordingtoaprobabilityvector.Theselectedactionisfedtothestochasticenvironmentastheinputandarandomoutputisattained.Thegistoflearningautomatatechniquesliesintheprovableconvergencetotheoptimalsolution,aswillbedenedlater,inarbitrarystationaryrandomenvironment.Thankstothepracticalityandapplicability,learningautomatatechniqueshavebeenbroadlystudiedinvariousaspectsofthecommunicationandnetworkingliteraturesuchas[ 122 ][ 123 ][ 124 ][ 125 ].Inthiswork,weproposealearningautomatabasedthresholdoptimizationalgorithmwhichndsthestochasticoptimumvaluesoftheup/downthresholdsefcientlyinanystationaryyetpotentiallyfastvaryingrandomchannelenvironment.Thedetailedimplementationsareintroducednext. 6.4.2AchievingtheStochasticOptimalThresholds WeconsideranIEEE802.11stationasthedecisionmakerwhichadjuststhevaluesofuandd.Withoutlossofgenerality,weassumethatthemaximumvalueofuanddaremuandmd,i.e.,thefeasiblesetofuisAu=f1,,mugandthatofdisgivenbyAd=f1,,mdg.ThestationmaintainstwoprobabilityvectorsPuandPdforAuandAd,respectively.ThekthelementinPu,i.e.,Pku,denotestheprobabilitythatuissettok.Pkdisdenedanalogously.Simplyput,ateachdecisioninstance,thethresholdoptimizationalgorithmrandomlydeterminesthevaluesofuanddaccordingtoPuandPd.Next,theprobabilityvectors,i.e.,PuandPd,areupdatedandtheiterationcontinuesuntilconvergence,i.e.,Pmu=1andPnd=1wheremandndenotethestochasticoptimalvaluesofuandd,respectively. Deneatimeseriesdenotedbyt=[t0,t1,]wheret0denotesthestartingtimeandotherelementsrepresenttheexacttimeinstanceswhenthedatarateischanged. 145 PAGE 146 DenoteT(j)=[tj)]TJ /F5 7.97 Tf 6.59 0 Td[(1,tj],j=1,2, asthejthtimeperiod,ortimeduration.Itisworthnotingthatwithinaparticulartimeperiod,sayj,thevalueofthedatarate,thevaluesofthresholds,i.e.,uandd,areallxednumbers.Wedenotetheseperioddependentparametersas3r(j),u(j)andd(j).Withthisobservation,weutilizethetimeseriestasthetimeseriesofdecisioninstances.Morespecically,forexample,attimetj,aratechangeistriggeredeitherbyu(j)consecutivesuccessfultransmissionsord(j)consecutivefailedtransmissions.Intandemwiththedatarateup/downshift,newvaluesofthethresholds,i.e.,u(j+1)andd(j+1)aredeterminedbythethresholdoptimizationalgorithmaccordingtotheprobabilityvectorsPuandPd. BesidesPuandPd,weintroduceadditionalvectorsforAuandAd,namely,thecountingvectorsCu,Cd,thesurplusvectorsSu,Sd,theestimationvectorsDu,DdandthecomparisonvectorsZu,Zd.Thedenitionsofparametersofthealgorithmareprovidedasfollows.Algorithm: Parameters4: Pu(Pd):Theprobabilityvectorforu(d)overAu(Ad). mu(md):Themaximumvalueofu(d),orequivalently,thecardinalityofAu(Ad). Cu(Cd):Thecountingvectorofu(d)wherethekthelement,i.e.,Cku(Ckd),denotesthetimesthatkhasbeenselectedasthevalueofu(d). Su(Sd):Thesurplusvectorofu(d)wherethekthelement,i.e.,Sku(Skd),denotestheaccumulatedthroughputwithu=k(d=k). 3Notethatwithaslightabuseofnotation,weusejtodenotethejthtimeduration.4Wepresenttheanalogousdenitionsintheparenthesis. 146 PAGE 147 Du(Dd):Theestimationvectorofu(d)wherethekthelement,i.e.,Dku(Dkd),iscalculatedbyDku=Sku Cku(Dkd=Skd Ckd). R:Theresolutionparameterwhichisapositiveintegerandistunablebythestation. u(d):Thestepsizeparameterofu(d)andisgivenbyu=1 muR(d=1 mdR). 'u('d):Theperturbationvectorofu(d)wherethekthelement,i.e.,'ku('kd),isazeromeanrandomvariablewhichisuniformlydistributedin[)]TJ /F14 7.97 Tf 22.49 5.26 Td[( Cku(Ckd),+ Cku(Ckd)]+1)]TJ /F5 7.97 Tf 6.59 0 Td[(1whereisasystemparameterandiscontrollablebythestation.Thenotationof[a,b]xyrepresents[max(a,y),min(b,x)]. Zu(Zd):Thecomparisonvectorofu(d)wherethekthelement,i.e.,Zku(Zkd),isgivenbyZku=Dku+'ku(Zkd=Dkd+'kd). B:Thepredenedconvergencethreshold,e.g.,1,whichisdeterminedbythestation. J:Arunningparameterwhichrecordstheupdatedmaximumachievedthroughputduringonetimeperiod.Attimet0: Initialization: ThestationsetsPu=[p1,,pi,,pmu]wherepi=1 muforall1imu.Similarly,Pdisgivenby[p1,,pi,,pmd]wherepi=1 mdforall1imd. InitializesCu,Cd,Su,Sd,DuandDdtozeros. Randomlyselectsthevaluesofu(1)andd(1),saym,n,accordingtoPu(1)andPd(1). Transmitswithu=mandd=nuntilT(1)ends,i.e.,adataratechangeistriggered. RecordstheaveragethroughputwithinT(1)asJ.Attimetj(j1): Do: RecordstheaveragethroughputduringTjasO(j).IfO(j)>J,setsJ=O(j)andremainsJunchangedotherwise. 147 PAGE 148 UpdatesthemthelementinthesurplusvectorSuandthenthelementinSdbyaddingthemeasurednormalizedthroughputofthelasttimeperiod,asSmu=Smu+O(j) JandSnd=Snd+O(j) J. UpdatesthecountingvectorsbyaddingonetothemthcounterinCuandthenthcounterinCd,asCmu=Cmu+1andCnd=Cnd+1. UpdatesthemthelementinDuandthenthelementinDdbyDmu=Smu CmuandDnd=Snd Cnd. ForeveryelementinZuandZd,updatesZku=Dku+'ku,k=1,,muandZkd=Dkd+'kd,k=1,,md. FindstheelementinZuwiththehighest5valueofZku,k=1,,mu,say,the~mthelementinAu. Similarly,ndstheelementinZdwhichhasthehighestZkd,k=1,,md,say,the~nthelementinAd. UpdatestheprobabilityvectorsofPuandPdas Pku=max)]TJ /F6 11.955 Tf 5.48 9.69 Td[(Pku)]TJ /F8 11.955 Tf 11.96 0 Td[(u,0ifk6=~m,k=1,,muPku=1)]TJ /F13 11.955 Tf 12.55 11.36 Td[(Xk6=~mPkuifk=~m (6) and Pkd=max)]TJ /F6 11.955 Tf 5.48 9.69 Td[(Pkd)]TJ /F8 11.955 Tf 11.95 0 Td[(d,0ifk6=~n,k=1,,mdPkd=1)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xk6=~nPkdifk=~n (6) whereu=1 muRandd=1 mdR. WiththeupdatedprobabilityvectorsPuandPd,newvaluesofuandd,i.e.,u(j+1)andd(j+1),areselected. StartsthetransmissionsinT(j+1)withu(j+1)andd(j+1). Until: max(Pu)Bandmax(Pd)BwhereBisthepredenedconvergencethreshold. 5Notethatatiecanbeeasilybrokenbyarandomselection. 148 PAGE 149 End Theproposedthresholdoptimizationalgorithmissimilartothestochasticestimatorlearningautomataproposedin[ 126 ].Thekeyfeatureofthisgenreoflearningautomataistherandomnessdeliberatelyintroducedby'kuand'kd.Notethatalthough'kuand'kdarezeromeanrandomvariables,theirvariancesaredependentonthevaluesofCkuandCkd,respectively.Specically,thevariancesapproachtozeroswiththeincreaseofthenumberoftimesthatthecorrespondingvaluesofuanddareselected.Asaconsequence,thethresholdoptimizationalgorithminclinestomorereliablestochasticestimatesandthuspossessesafasterconvergencebehaviorthanotherlearningalgorithms[ 126 ].Thevaluesthathavebeenselectedlessfrequentlystillhavethechanceofbeingconsideredasoptimal.However,themissingprobabilitydiminishestozeroalongiterations.Inthealgorithm,theresolutionparameterRcontrolsthestepsizeofprobabilityadjustmentinthealgorithm.AsmallervalueofRproducesanegrainedprobabilityadjustmentyetunavoidablyprolongstheconvergencetime.TheconvergencethresholdBdeterminesthestoppingcriteriaofthealgorithm.Therefore,atradeoffbetweenoptimalityandconvergenceratecanbeadjustedbytuningthevalueofB. Thesteadystatebehavioroftheproposedthresholdoptimizationalgorithmisprovidedinthefollowingtheorem. Theorem6.2. Theproposedthresholdoptimizationalgorithmisoptimalforanystationarychannelenvironmentwitharbitrarydistribution.Mathematically,foranyarbitrarilysmall>0and>0,thereexistsat0satisfying Prfj1)]TJ /F6 11.955 Tf 11.95 0 Td[(Pmuj PAGE 150 TheproofofTheorem 6.2 followssimilarlinesasin[ 126 ]andisomittedforbrevity.Formorediscussionsonthestochasticestimatoralgorithms,referto[ 127 ]and[ 128 ].Inthenextsection,wewilldemonstratetheefcacyofourproposedthresholdoptimizationalgorithmviasimulations. 6.5PerformanceEvaluation Inthissection,weevaluatetheperformanceoftheproposedthresholdoptimizationalgorithmwithsimulations.Forcomparisons,werstimplementtheheuristicsbasedthresholdadjustmentalgorithmsin[ 93 ]and[ 94 ].In[ 93 ],thedownshiftthresholddisxedto1andthedefaultvalueofuis10.Afterusuccessfultransmissions,adataupshiftistriggeredandifthersttransmissionaftertherateupshiftissuccessful,thealgorithmassumesthatthelinkqualityisimprovingrapidly[ 93 ].Consequently,uissettoasmallnumber,e.g.,u=3,inordertocapturethefastimprovingchannel.Otherwise,thealgorithmassumesthatthechannelischangingslowlyandthusalargervalueofuisdesired,e.g.,u=10.WedenotethisthresholdadaptationschemeasDLAinoursimulations.Anotherwellknownthresholdadjustingscheme,namely,AARF,isproposedin[ 94 ].Similarly,theratedownshiftthresholdisxedtod=2empirically.However,foru,abinaryexponentialbackoffschemeisapplied.Ifthersttransmissionafterarateupshiftfailed,thedatarateisswitchedbacktothepreviousrateandthevalueofuisdoubledwithamaximumvalueof50.Thevalueofuisresetto10wheneveraratedownshiftistriggered. TosimulatetheindoorofceenvironmentforIEEE802.11WLANs,wesimulateaRayleighfadingchannelenvironment.Inotherwords,weassumeaatfadingenvironment.However,itcouldbeeitherafastfadingorslowfadingchannel.TheDopplerspread(inHz)correspondstothechannelfadingspeedwherealargeDopplerspreadvaluerepresentsafastfadingchannelandasmallDopplerspreadindicatesaslowvaryingchannel.ItshouldbenotedthattheDopplerspreadvaluedescribesthetimedispersivenatureofthewirelesschannel[ 129 ],whichisinverselyproportionalto 150 PAGE 151 thechannelcoherencetime.Morespecically,werelatethechannelcoherencetimeTcandtheDopplerspreadvaluefmas[ 129 ] Tc=0.423 fm.(6) Therefore,alargerDopplerspreadvalueindicatesasmallchannelcoherencetimewhichrepresentsafastfadingscenario.WeconductthesimulationsundervariouschannelfadingconditionsviadifferentDopplerspreadvalues.WeconsidertheIEEE802.11bPHYspecication,i.e.,theavailabledataratesare1Mbps,2Mbps,5.5Mbpsand11MbpsandtheRTS/CTSsignallingschemeisturnedoff.Sincetheobjectiveofourproposedthresholdoptimizationalgorithmistocombatwiththechannelvariation,thecollisioneffectisomitted.Therefore,weconsideraWLANwithonestationconsistentlytransmittingpacketstotheAP.However,notethat[ 112 ]providesacomplementarysolutiontomitigatethecollisioneffectandthusouralgorithmscanworkcollectivelyasajointsolution.Weemphasizethatthissimplicationdoesnotinduceanylossofgeneralitysincealthoughseeminglysimple,itproducesallthechallengingproblemsinvolvedinrateadaptationalgorithmsinatimevaryingstochasticchannelenvironment.ThedatatrafcisgeneratedusingconstantbitrateUDPtrafcsourcesandtheframesizeissetto1024octets.Thepowerofthetransmitterandthepowerofthermalnoisearesetto50mWand1mW,respectively.TheSNRBERrelationisgivenbyTable 61 whichisderivedfrom[ 130 ]. WevarytheDopplerspreadvaluetosimulatethevariouschannelfadingspeeds.ForeachvalueofDopplerspreadvalue,thesimulationisexecutedfor1000secondsandtheaveragethroughputofthefollowingalgorithms6,whicharecommonlybasedonthe 6TheperformancecomparisonofARFandotheropenlooprateadaptationalgorithmssuchasSampleRateandONOEhasbeenstudiedextensivelyin[ 96 ]and[ 98 ]forvariouschannelconditions. 151 PAGE 152 thresholdbasedup/downmechanism,arecompared:(1),OPtheoptimumthroughputattainedbyassuminganoraclewhichforeseesthevariationofchannelandadaptsthedatarateoptimally.Thiscurveisattainedasaperformancecomparisonbenchmark;(2),ARFthethresholdbasedrateadaptationalgorithmwithu=10andd=2;(3),DLAthedynamicthresholdadjustmentalgorithmproposedin[ 93 ];(4),AARFthethresholdadjustmentalgorithmproposedin[ 94 ];(5),TOAthethresholdoptimizationalgorithmproposedinthiswork.Thealgorithmsarecomparedwitheachotherintermsoftheaveragesystemthroughput(inMbps).ForTOA,withoutlossofgenerality,weassumethatmu=md=10andtheresolutionparameterRis1.TheconvergencethresholdBis0.999andis1.TheperformancecurvesoftheaforementionedalgorithmsareplottedinFigure 61 InFigure 61 ,weobservethatexcepttheOPcurve,allotherrateadaptationalgorithmssufferfromperformancedegradationswithlargeDopplerspreadvalues.RecallthatalargeDopplerspreadvalueindicatesafastfadingchannelenvironmentandhencetheaveragethroughputdeterioratesduetotheincompetencyofcapturingshortchanneluctuations.Amongwhich,ARFandAARFprovideworstperformancewithaslightdifference.DLAperformsbetterduetothecapabilityofswitchingbetween Table61. SNRv.s.BERforIEEE802.11bdatarates SNR BPSK QPSK CCK CCK (dB) (1Mbps) (2Mbps) (5.5Mbps) (11Mbps) 1 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 5E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 8E)]TJ /F5 7.97 Tf 6.59 0 Td[(2 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(1 2 1E)]TJ /F5 7.97 Tf 6.58 0 Td[(6 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 4E)]TJ /F5 7.97 Tf 6.59 0 Td[(2 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(1 3 6E)]TJ /F5 7.97 Tf 6.58 0 Td[(8 2.1E)]TJ /F5 7.97 Tf 6.59 0 Td[(4 1.8E)]TJ /F5 7.97 Tf 6.58 0 Td[(2 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(1 4 7E)]TJ /F5 7.97 Tf 6.58 0 Td[(9 3E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 7E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 5E)]TJ /F5 7.97 Tf 6.59 0 Td[(2 5 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 2.1E)]TJ /F5 7.97 Tf 6.59 0 Td[(6 1.2E)]TJ /F5 7.97 Tf 6.58 0 Td[(3 1.3E)]TJ /F5 7.97 Tf 6.58 0 Td[(2 6 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.5E)]TJ /F5 7.97 Tf 6.59 0 Td[(7 3E)]TJ /F5 7.97 Tf 6.59 0 Td[(4 5.2E)]TJ /F5 7.97 Tf 6.58 0 Td[(3 7 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1E)]TJ /F5 7.97 Tf 6.59 0 Td[(8 6E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 2E)]TJ /F5 7.97 Tf 6.59 0 Td[(3 8 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(9 1.3E)]TJ /F5 7.97 Tf 6.58 0 Td[(5 7E)]TJ /F5 7.97 Tf 6.59 0 Td[(4 9 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(9 2.7E)]TJ /F5 7.97 Tf 6.58 0 Td[(6 2.1E)]TJ /F5 7.97 Tf 6.58 0 Td[(4 10 2.3E)]TJ /F5 7.97 Tf 6.59 0 Td[(10 1.2E)]TJ /F5 7.97 Tf 6.59 0 Td[(9 5E)]TJ /F5 7.97 Tf 6.59 0 Td[(7 6E)]TJ /F5 7.97 Tf 6.59 0 Td[(5 ... ... ... ... ... 152 PAGE 153 Figure61. Averagethroughputv.s.Dopplerspreadvalues thelargevalueandthesmallvalueofufordifferentchannelconditions.Ourproposedthresholdoptimizationalgorithm,asdemonstratedinFigure 61 ,consistentlyoutperformsotheropenlooprateadaptationalgorithmsandbridgestheperformancegapwiththeOPcurve.Thissuperioritybecomesremarkablysignicantinfastfadingstochasticchannelenvironments,i.e.,scenarioswithlargeDopplerspreadvalues.Whileotheralgorithmsarejeopardizedbytheunmatureratechangesduetotheuncertaintycausedbytherandomandfastvaryingchannelenvironment,ourproposedthresholdoptimizationalgorithm,basedonthelearningautomatatechniques,isabletondthestochasticoptimumvaluesofuanddwhichmaximizetheexpectedsystemperformance.Therefore,ourschemeisparticularlysuitableforfastchangingyetstatisticallystationaryrandomchannelenvironmentswhereotheropenlooprateadaptationsolutionsusuallyprovideunsatisfactoryperformance. Toillustratetheprocessofndingthestochasticoptimumvaluesofuandd,inFigure 62 ,weprovidethetrajectoriesofuanddinasamplesimulationrunwithaxedDopplerspreadvalueof10.Itisobservablethatstartingfromtheinitialpoint,thethresholdoptimizationalgorithmadaptsthevaluesofuanddontheyalongwiththeratechanges.Thealgorithmsoonndsthestochasticoptimumvaluesanduanddconvergetotheoptimumsolutionseffectively.Theevolutionsoftheprobabilityvectors, 153 PAGE 154 Figure62. Thetrajectoriesofuanddinachievingthestochasticoptimumvalues i.e.,PuandPd,aredemonstratedinFigure 63 andFigure 64 ,respectively.Notethat,inFigure 63 ,thesixthcurvewhichrepresentsthevalueofP6usoonexcelsothersandapproachesto1.Correspondingly,thevalueofuconvergesto6rapidlyasdepictedinFigure 62 .InFigure 64 ,thesecondcurveapproachestounitygraduallywhileothersdiminishtozeros.Asaconsequence,inFigure 62 ,thevalueofdconvergestothestochasticvalue,i.e.,2.Inthissamplerunofsimulation,thestochasticoptimumvaluesofuandd,whichmaximizestheexpectedthroughputofthesystem,isgivenbyu=6andd=2.Theproposedthresholdoptimizationalgorithmndsthesevalueseffectivelyandefciently,whileprovidingsuperiorperformancethanothernonadaptivethresholdbasedrateadaptationalgorithms,asdemonstratedinFigure 61 Figure63. EvolutionoftheprobabilityvectorPu 154 PAGE 155 Figure64. EvolutionoftheprobabilityvectorPd 6.6ConclusionsandFutureWork Inthischapter,weinvestigatethethresholdbasedrateadaptationalgorithmwhichispredominantlyutilizedinpracticalIEEE802.11WLANs.Althoughwidelydeployed,theobscureobjectivefunctionofthistypeofrateadaptationalgorithms,commonlybasedontheheuristicup/downmechanism,islesscomprehended.Inthiswork,westudythethresholdbasedrateadaptationalgorithmfromareverseengineeringperspective.Theimplicitobjectivefunction,whichtherateadaptationalgorithmismaximizing,isunveiled.Ourresultsprovide,albeitapproximate,ananalyticalmodelfromwhichthethresholdbasedrateadaptationalgorithm,suchasARF,canbebetterunderstood. Inaddition,weproposeathresholdoptimizationalgorithmwhichdynamicallyadaptstheup/downthresholds,basedonthelearningautomatatechniques.Ouralgorithmprovablyconvergestothestochasticoptimumsolutionsofthethresholdsinarbitrarystationaryyetpotentiallyfastchangingrandomchannelenvironment.Therefore,bycombiningourworkwiththethresholdadaptationschemein[ 112 ],wherethethresholdsareadjustedtomitigatethecollisioneffects,ajointcollisionandrandomchannelfadingresilientsolutioncanbeattained. Inthiswork,toemphasizeontheimpactofrandomchannelvariations,werestrictourselvestothescenariowhereallstationshaveaxedandknowntransmissionprobabilityp.Theinteractionofthetransmissionprobabilitiesandthedataratesseems 155 PAGE 156 interestingandneedsfurtherinvestigation.Duetothecompetitivenatureofchannelaccess,astochasticgameformulationmaybeutilized. 156 PAGE 157 CHAPTER7STOCHASTICTRAFFICENGINEERINGINMULTIHOPCOGNITIVEWIRELESSMESHNETWORKS 7.1Introduction Thepastdecadehaswitnessedtheemergenceofnewwirelessservicesindailylife.Oneofthepromisingtechniquesisthemetropolitanwirelessmeshnetworks(WMN),whichareenvisionedasatechnologywhichadvancestowardsthegoalofubiquitousnetworkconnection.Figure 71 illustratesanexampleofwirelessmeshnetwork.Thewirelessmeshnetworkconsistsofedgerouters,intermediaterelayroutersaswellasthegatewaynode.Edgeroutersaretheaccesspointswhichprovidethenetworkaccessfortheclients.Therelayroutersdeliverthetrafcaggregatedattheedgerouterstothegatewaynode,whichisconnectedtotheInternet,inamultihopfashion. Figure71. Architectureofwirelessmeshnetworks Whilethecurrentdeployedwirelessmeshnetworksprovideexibleandconvenientservicestotheclients,theperformanceofameshnetworkisstillconstrainedbyseverallimitations.Therstbarrierisduetothemultihopnatureofthewirelessmeshnetwork,wherethenodesingeographicproximitygenerateseveremutualinterferenceamongeachotherandthusthenetworkperformanceisdevastated.Toaddressthisproblem,severalschedulingschemeshavebeenproposedintheliterature[ 131 ].Recently,anovelcodingbasedschemewhichmayproduceaninterferencefreewirelessmesh 157 PAGE 158 network,isproposed[ 132 ].AnotherexampleoftheinterferencefreenetworkistheCDMAbasedwirelessmeshnetworks[ 133 ]wherebyassigningorthogonalcodesforeachlink,thenetworkthroughputisremarkablyimproved. Thesecondhinderanceforthenetworkperformanceisthelimitedusablefrequencyresource.Incurrentwirelessmeshnetworks,theunlicensedISMbandsaremostcommonlyadoptedforbackbonecommunications.Notsurprisingly,thewirelessmeshnetworkislargelyaffectedbyallotherdevicesinthisISMband,e.g.,nearbyWLANsandBluetoothdevices.Moreover,thelimitedbandwidthoftheunlicensedbandcannotsatisfytheincreasingdemandforthebandwidthduetotheevolvingnetworkapplications.Ironically,asshownbyavarietyofempiricalstudies[ 134 ],thecurrentallocatedspectrumisdrasticallyunderutilized.Asaconsequence,theurgetoexploretheunusedwhitespaceofthespectrum,whichcansignicantlyenhancetheperformanceofthewirelessmeshnetworks,attractstremendousattentioninthecommunity[ 135 139 ]. Cognitiveradiosareproposedasaviablesolutiontothefrequencyreuseproblem[ 131 ].Thecognitivedevicesarecapableofsensingtheenvironmentandadjustingthecongurationparametersautomatically.Iftheprimaryuser,i.e.,thelegitimateuser,isnotusingtheprimarybandcurrently,thecognitivedevices,namely,secondaryusers,willutilizethiswhitespaceofthespectrum.Incorporatingwiththeestablishedinterferencefreetechniquessuchas[ 132 ]and[ 18 ],thethroughputofthewirelessmeshnetworkcanbedramaticallyenhanced.Theprotocoldesignforcognitivewirelessmeshnetworks(CWMN),ormoregenerally,multihopcognitiveradionetworks,isaninnovativeandpromisingtopicinthecommunity[ 140 ]andhasbeenlessstudiedintheliterature.Inthiswork,weconsideracognitivewirelessmeshnetworkwheretheunlicensedband,e.g.,ISMband,isutilizedbythemeshroutersforthebackbonetransmission.Moreover,eachrouterisacognitivedeviceandhenceiscapableof 158 PAGE 159 sensingandexploitingtheunusedprimarybandsfortransmissionswhenevertheprimaryusersareabsent. Inthiswork,weinvestigateanimportantyetunexploredissueinthecognitivewirelessmeshnetworks,namely,thestochastictrafcengineeringproblem.Morespecically,weareparticularlyinterestedinhowthetrafcinthemultihopcognitiveradionetworksshouldbesteered,undertheinuenceofrandombehaviorsofprimaryusers.Itisworthnotingthatgivenaroutingstrategy,thecorrespondingnetwork'sperformance,e.g.,theaveragequeueingdelayencountered,isarandomvariable.Thereasonisthattheavailablebandwidthforaparticularlinkdependsontheappearanceofalltheaffectingprimaryusers.Ifalltheprimaryusersarevacant,alinkcanutilizeallavailablefrequencytrunkscollectivelybyutilizingadvancedphysicallayertechniques,e.g.,OFDMA.However,ifalltheprimaryusersarepresent,theonlyavailablefrequencyspaceistheunlicensedISMbandandthusthetrafconthislinkwillexperiencelongerdelaythanthepreviouscase.Inotherwords,theperformanceofatrafcengineeringsolutionhingesintenselyontheunpredictablerandombehaviorsoftheprimaryusers.Weemphasizethatinmultihopcognitiveradionetworks,thisdistinguishingfeatureofrandomness,inducedbytherandombehaviorsofprimaryusers,mustbetakenintoaccountinprotocoldesigns.Duetothelocationdiscrepancy,itispossiblethatsomenodeisaffectedbymanyprimaryuserswhileothersarenot.Asaconsequence,ifweroutethetrafcviathisparticularnode,thetransmissionsaremorelikelytobecorruptedbythereturnsoftheprimaryusers.Apparently,afavorablesolutionismoreinclinedtosteerthetrafcfromthoseseverelyaffectedarea,tothepathswhicharelessaffectedbytheprimaryusers.Wewillmakethisintuitiveapproachpreciseandrigorousinthiswork.Toourbestknowledge,thisworkistherstworkonthetrafcengineeringprobleminmultihopcognitiveradionetworks,withaspecialfocusontheimpactofrandombehaviorsfromtheprimaryusers. 159 PAGE 160 Therestofthischapterisorganizedasfollows.TherelatedworkisreviewedinSection 7.2 andSection 7.3 providesthesystemmodelofourwork.ThestochastictrafcengineeringproblemwithconvexityisinvestigatedinSection 7.4 .InSection 7.5 ,weextendourframeworktothenonconvexstochastictrafcengineeringproblem.PerformanceevaluationisprovidedinSection 7.6 ,followedbyconcludingremarksinSection 7.7 7.2RelatedWork Traditionaltrafcengineering(TE)algorithmsareproposedasthesolutiontothetrafcmanagementofthenetworkinacostefcientmanner.DifferentfromthetraditionalQoSrouting,thetrafcengineeringsolutionnotonlyguaranteesacertainQoSlevelforeachow,butalsooptimizesaglobalperformancemetricoverthewholenetwork,bysplittingtheingresstrafcoptimallyamongseveralavailablepaths.ThemultipathroutingisusuallysupportedbytheMultiProtocolLabelSwitching(MPLS)techniqueswheretheexplicitroutingpathforapacketispredeterminedratherthanbeingcomputedinahopbyhopfashion.Forapairofsourceanddestinationnodes,thesetofavailablepaths,a.k.a.,labelswitchedpaths(LSP),areestablishedandmanagedbysignallingprotocolssuchasRSVPTE[ 141 ]andCRLDP[ 142 ]ormanuallyconguration.Thetraditionaltrafcengineeringsolutionevolvestothestochastictrafcengineering(STE)solutionwhenuncertaintyexistsinthenetwork,e.g.,therandomreturnsoftheprimaryusersinourscenario.TEsolutionsrequireconsistentroutechangeswhichareunfavorableinthatthenetworkwillbeoverwhelmedbytheoscillationsinducedbytheunpredictablebehaviorsoftheprimaryusers.Inlightofthisstabilityconcern,STEsolutionalternativelypursuesanoptimummultipathroutingstrategysuchthattheexpectedutilityofthenetworkismaximized.Thestochastictrafcengineeringwithuncertaintiesarediscussedintheliteraturesuchasin[ 143 ]and[ 144 ].However,thepreviousworksusuallyassumeaprobabilitydistributionoftheuncertainty,whileinourscenario,thebehaviorsoftheprimaryusersarecompletelyunpredictable 160 PAGE 161 fromthemeshnetwork'spointofview.Distinguishingfromthepreviousworks,weproposeanalgorithmicsolutionwhichrequiresnopriorknowledgeaboutthedistributionoftheuncertainty,inastochasticnetworkutilitymaximizationframework.Itisworthnotingthatourworkdifferfromthetraditionalstatedependenttrafcengineeringsolutionaswell.Forexample,in[ 145 ],astatedependenttrafcengineeringsolutionisproposed.However,theauthorsassumethatthesystemstate,i.e.,thecurrentvalueofuncertainty,isfullyobservable.Inourapproach,wedonotassumethattheingressnodehastheperfectknowledgeofthecurrentappearanceofthenetwork.WewilldiscussthisissueindetailinSection 7.4 Recently,cognitivewirelessmeshnetworks(CWMN)haveattractedgreatattentionintheliterature.In[ 135 ],thechannelassignmentisdiscussedinaCWMN.In[ 136 ],aclusterbasedcognitivewirelessmeshnetworkframeworkisproposed.Theinfrastructurebasedcognitivenetworkisdiscussedin[ 137 ]withafocusonthecooperativemobilityandthechannelselectionschemes.Thespectrumsensingandchannelselectionarejointlyconsideredinauniedframeworkin[ 138 ].Inaddition,theIEEE802.16hisintheprocessofincorporatingthecognitiveradiosintotheWiMAXmeshnetworks[ 139 ].However,noneofthepreviousworksconsidersthestochastictrafcengineeringproblem.Therefore,asystematicstudyoftheimpactsoftherandomreturnsfromtheprimaryusers,onthenetworkroutingperformanceislackingintheexistingliterature.In[ 146 ],thejointcongestioncontrolandtrafcengineeringproblemisconsidered.Heet.al.proposeadistributedalgorithmtobalancetheuser'sutilityandthesystem'sobjective.However,theauthorsassumetheenvironmentisxedanddoesnotconsidertherandomnesswhichisthedistinguishingyetusuallyoverlookedfeatureincognitiveradionetworks.[ 147 ]and[ 148 ]discusstheroutingissueincognitiveradionetworksyettheimpactofrandomreturnsofprimaryusersisnotinvestigated.Houet.al.[ 149 ][ 150 ][ 151 ][ 152 ]formulatethejointrouting,powerandsubbandallocationproblemincognitiveradionetworksasamixedintegerprogramming.However,the 161 PAGE 162 channels'bandwidthsareassumedtobexed,i.e.,therandombehaviorsofprimaryusersarestillneglected.Ourworkispartiallyinspiredby[ 153 ].However,ourworkdiffersfromtheirsinthreecrucialaspects.First,bytargetingthestochastictrafcengineeringproblem,ourmodeldiffersfromthejointpowerschedulingandratecontrolworkin[ 154 ].Secondly,in[ 153 154 ],theauthorsonlyconsiderasinglepathscenariowhileourworkextendstoamultipathroutingnetworkwherethenetworktrafccanbesteered.Thirdlyandmostimportantly,[ 153 154 ]requirethatthecurrentsystemstateisfullyobservableatthedecisionmaker.Toachievethis,theauthorsassumeacentralizedmechanismwhichknowsallthechannelstatesofallthelinksoverthenetwork.However,ourworkdiffersfrom[ 153 154 ]signicantlyinthatwedonotrequirethatthecurrentsystem'sstateisknown,whichisofgreatpracticalinterestsinceinmultihopcognitivewirelessmeshnetworks,thedecisionmakers,i.e.,theedgeroutersinourscenario,cannotbeawareoftheappearanceofallprimaryusersinthewholenetworkasapriori.Moreover,ourschemesenjoyadecentralizedimplementation,incontrasttocentralizedmechanismsin[ 153 154 ],byutilizingthefeedbacksignalsandlocalinformationonly.Inourpreviousworkof[ 20 ],weproposedaroutingoptimizationschemetocombatwiththerandomnessofinstantaneoustrafcinnoncognitivewirelessmeshnetworks.Withrespectto[ 20 ],thisworkdiffersinthefollowingways.First,inthewirelessmeshnetworksconsideredin[ 20 ],thecapacityofeachwirelesslinkisassumedtobexed,i.e.,timeinvariant.However,incognitivewirelessmeshnetworks,duetotheunpredictableappearanceofprimaryusers,thebandwidthofeachwirelesslinkisrandom.Secondly,in[ 20 ],thequalityofservice(QoS)requirementisnotconsidered.Nevertheless,inthischapter,weparticularlyaddresstheQoSconcernofeachuser,e.g.,theexpectedaccumulateddelayonthepathscannotexceedauserspecicdelaytolerance,aswillbeelaboratedinSection 7.4 .Thirdlyandmostimportantly,theanalysisin[ 20 ]wasbasedupontheassumptionthattheusersallhaveconvexutilityfunctions.Inthiswork,weextendthetechniquestoaddressthescenarios 162 PAGE 163 withnonconvexutilityfunctions.Wewilldiscusstheaforementionedissuesfurtherinthefollowingsections. 7.3SystemModel WeconsideramultihopwirelessmeshnetworkillustratedinFigure 71 whereanuplinktrafcmodelisconsidered,i.e.,alledgeroutersaggregatethetrafcfromclientsanddelivertothegatewaynodeviatheintermediaterelayrouters.Toensureconnectivity,weutilizetheISM2.4Gbandastheunderlyingcommonchannelforthewirelessmeshnetwork.Inaddition,eachlinkcanutilizetheopportunisticchannels,i.e.,secondarybandstoincreasethelink'sachievabledataratewhenevertheprimaryuserisvacant.Weassumethatthereexists1jMjprimaryusers.Eachprimaryuserpossessesalicensedfrequencychannelandeachmeshrouterisacognitivenodewhichhasthecapabilityofsensingthecurrentwirelessenvironment.WemodelthemultihopcognitivewirelessmeshnetworkasadirectionalgraphGwheretheverticesarethenodes.Wealsodenotelink(i,j)aslinke,e2Ewheret(e)=iandr(e)=jrepresentthetransmitterandthereceiveroflinke. Werstconsideraparticularlinkdenotedby(m,n).Theinstantaneousavailablefrequencybands,attimet,foranodeiisdenotedbyIi(t),whichisdeterminedbythecurrentpresenceoftheprimaryusers.BesidestheunderlyingISMband,thecommunicationbetweenmandncanfurtherutilizeallsecondarybandswithinIm(t)TIn(t),ifavailable.Thecurrentcognitiveradiodevicesbenetlargelyfromthesoftwaredenedradio(SDR)techniqueswithadvancedcoding/modulationcapabilities.Forexample,byutilizingthemulticarriermodulation,e.g.,OFDMA,acognitiveradiodevicecanutilizeallthedisjointavailablefrequencybandsimultaneously[ 149 150 155 157 ].Atthetransmitter,asoftwarebasedradiocombineswaveformsfordifferentsubbandsandthustransmitsignalatthesesubbandssimultaneously. 1ThesymbolofjXjrepresentsthecardinalityofthesetX. 163 PAGE 164 Whileatthereceiver,asoftwarebasedradiodecomposesthecombinedwaveformsandthusreceivessignalatthesesubbandssimultaneously[ 151 152 156 157 ].Inthiswork,weassumeaspectrumsensingschemeavailablethateachnodecansensethepresenceoftheprimaryusersinrange,suchas[ 131 158 ],althoughthetimeofrandomreturnscannotbepredicted.Alinkwillutilizealltheavailablevacantbandsandthatthecognitiveradiosarefullduplexandcantransmitatdifferentbandsconcurrently[ 151 152 156 157 ].Wefurtherassumethatsomeschedulingmechanismisinplaceorsomephysicallayermechanismsareutilizedsuchthatthenodescannotinterferewitheachotherduringthetransmissions.Forexample,inamultichannelmultiradiowirelessmeshnetwork,thechannelscanbeassignedproperlythatthetransmissionsdonotinterferewiththeneighboringnodes[ 132 159 ].OtherexamplesaretheOFDMA/CDMAbasedwirelessmeshnetworks[ 133 160 ]wheretheinterferenceamongnodescanbeeliminatedbyassigningorthogonalsubcarriers/codes.Weemphasizethatthisassumptionisonlyforthesakeofmodelingsimplicityanddoesnotincuranylossofgenerality,aswillbeclariedshortly. Itisworthnotingthattheavailablebandwidthofeachlinkinthecognitivewirelessmeshnetworkisarandomvariable.Forexample,attimeinstancet1,nodemhasthreesecondarybandsavailable,i.e.,Im(t1)=fI0,I1,I2,I3gandIn(t1)=fI0,I2,I3,I4,I5gduetothelocationdiscrepancy,whereband0istheunderlyingunlicensedISMbandand1,2,3,4,5arethelicensedbandsofprimaryusers.Thecurrentbandwidthoflink(m,n)isrepresentedbyWm,n(t1)=BW0+BW2+BW3whereBWiisthebandwidthofbandi.Atanothertimeinstancet2,theprimaryuser2returnsandthebandwidthoflink(m,n)becomesWm,n(t2)=BW0+BW3.Inotherwords,thebandwidthoflinksarerandomvariableswhicharedeterminedbytheunpredictableappearanceoftheprimaryusers.Wemodelthisrandomnessinducedbytheprimaryusersasastationaryrandomprocesswitharbitrarydistribution.Thesystemisassumedtobetimeslotted.Ineachtimeslotn,thesystemstateisassumedtobeindependentandisdenotedbyastate 164 PAGE 165 vectors=f1,,jMjg,s2S,wherei=1denotestheabsenceoftheithprimaryuserand0otherwise.Wedenotethestationaryprobabilitydistributionofstatesass.Fortheeaseofexposition,weassumethattheprimaryusersarestatic.However,weemphasizethatourmodelcanbeextendedtomobileprimaryusersscenariosstraightforwardly.Forexample,ifaprimaryuserismovingfollowingaMarkovianwalkmodelwithwelldenedsteadystatedistribution,thefollowinganalysisstillapplies.Withoutlossofgenerality,weexpressthelinkcapacityintheformofCDMAbasednetworks,i.e.,thecapacityofawirelesslinke2E,giventhesystemstates,isdenotedbycse,whichisgivenby[ 4 161 ]cse=Wse1 Tlog2(1+Kse),whereWseisthebandwidthoflinkeinstatesandseisthecurrentSINRvalueoflinke.TheconstantTisthesymbolperiodandwillbeassumedtobeoneunitwithoutlossofgenerality[ 4 ].TheconstantK=)]TJ /F5 7.97 Tf 6.59 0 Td[(1 log(2BER)where1and2areconstantsdependingonthemodulationschemeandBERdenotesthebiterrorrate.WewillassumeK=1inthisworkforsimplicity[ 161 ].NotethatournetworkmodelcanbeincorporatedintoothertypesofnetworkssuchasMIMO,OFDMwithTDMAorCSMA/CAbasedMACprotocolsbymodifyingtheformofthecapacityaccordingly,whichrepresentstheachievabledatarateingeneral.Forexample,ifweconsideraschedulingbasedMACprotocolwhereeachlinkobtainsatimeshareofthechannelaccess,theachievabledatarateisgivenbycse=ecse ewhere eisthefractionoftimethatthelinkisactivefollowingtheschedulingschemeandecseisthenominalShannoncapacityofthelink. TherearejLjunicastsessionsinthenetwork,denotedbysetL,whereeachsessionlhasatrafcdemanddl.Weassociateeachsessionwithauniqueuser.Therefore,wewillusesessionlanduserlinterchangeably.Foreachsessionl2L,wedenotethesourcenodeanddestinationnodeasS(l)andD(l),respectively.Recallthatweassumeanuplinktrafcmodelandthusallthesourcenodesareedgeroutersandthedestinationnodeisthegateway.Furthermore,toimprovethereliabilityand 165 PAGE 166 dependability,weallowmultipathroutingschemes.Wedenotetheavailable2setofacyclicpathsfromS(l)toD(l)byPlandthekthpathisrepresentedbyPkl.Weintroduceaparameterrklastheowallocatedinthekthpathofsessionl.Theoverallowofuserl,representedbyxl,isgivenas xl=24jPljXk=1rkl35dl0(7) where[x]badenotesmaxfminfb,xg,ag.DeneanjEjbyjPljmatrixHlwheretheelementHle,k=1iflinkeisonthekthpathofPland0otherwise.Hence,H=fH1,,HjLjgrepresentsthenetworktopology.NotethatthetrafcsplittingandthesourceroutingareexecutedonthesourcenodeS(l). Foreachlinke2E,thereisanassociatedcostfunction,denotedbylse(fe,cse)wherefeistheaccumulatedowonlinke.Weassumethefunctionlseisanincreasing,differentiableandconvexfunctionoffeforaxedcse.Forexample,ifweassumelse(fe,cse)=1 cse)]TJ /F9 7.97 Tf 6.58 0 Td[(fewhencsefe,thecostessentiallyrepresentsthedelayforaunitowonlinkeundertheM=M=1assumption.Notethatinourscenario,eventheaccumulatedowfeisxed,thevalueofcostfunctionisrandomduetothestatedependentvariablecse.Fromthenetwork'sperspective,thestochastictrafcengineeringsolutionwilldistributetheaggregatedowamongmultiplepathsoptimally,inthesensethattheoverallnetworkutilityismaximized.Innextsection,wewillformulatethestochastictrafcengineeringprobleminastochasticnetworkutilitymaximizationframework[ 4 ]andprovideadistributedsolutionwhichrequiresnoprioriinformationabouttheunderlyingprobabilitydistribution,i.e.,s,ofthesystemstates. 2TheavailablesetofmultiplepathscanbeobtainedbysignallingmechanismssuchasRSVPTE[ 141 ]orpreconguredmanually.Inthiswork,weassumeapredeterminedsetofacyclicpaths.Theprotocoldesignforacquiringsuchpathsisbeyondthescopeofthiswork. 166 PAGE 167 7.4StochasticTrafcEngineeringwithConvexity 7.4.1Formulation Inthestandardnetworkutilitymaximizationframework,eachuserhasautilityfunctionUl(xl),whichreectsthedegreeofsatisfactionofuserlbytransmittingatarateofxl,e.g.,Ul(xl)=log(xl).Inthissection,weassumetheutilityfunctionstobeconcaveanddifferentiable.ThenonconvexutilityfunctionsareconsideredinSection 7.5 .Notethatthefairnessissuecanbeembodiedintheutilityfunctions[ 4 ].Forexample,intheseminalpaper[ 113 ],thelogutilityfunctionsareadoptedtoachievetheproportionalfairnessamongdifferentows. Deneafeasiblestochastictrafcengineeringsolutionasr=[r1,,rjLj]whererl,[r1l,,rjPljl].WecanformulatethestochastictrafcengineeringproblemasP1: maxr0Xl2LUl(Xk2Plrkl)s.t.Xk2Plrkldl8l2L (7)Xs2SsXk2Plrkl0@Xe2Pkllse(fe,cse)1Abl8l2L (7)feXs2Sscse8e2E (7)fe=Xl2LXk2PlHle,krkl8e2E (7)cse=Wse1 Tlog2(1+Kse)8e2E (7) wheree2Pklrepresentsthelinksalongthekthpathofuserl.ThevariableinP1isthevectorofr.Therstsetofconstraintsreectthattheoveralldataratesofallpathscannotexceedthetrafcdemanddl.Thesecondsetofconstraintsindicatethatfor 167 PAGE 168 eachuserl,theexpectedcosthastobenomorethanapredenedconstraintbl.Thethirdsetofconstraintsrepresentthattheaggregatedowonlinkecannotexceedtheaveragelinkcapacity.Apparently,iftheunderlyingprobabilitydistributionofeachstatesisknownasapriori,P1isadeterministicconvexoptimizationproblemandthuseasytosolve.However,inpractice,theaccuratemeasurementofprobabilitydistributionisanontrivialtask.In[ 20 ],weutilizedastochasticapproximationbasedapproachtocircumventthedifcultyofestimatingtheprobabilitydistribution.Inthefollowing,wewillextendthistechniqueanddevelopatailoreddistributedalgorithmtoaddresstheissuesoftimevaryinglinkcapacitiesaswellastheuserspecicQoSrequirements,whichareofparticularinterestinmultihopcognitivewirelessmeshnetworks. First,denetheLagrangianfunctionofP1asL(r,,,v)=Xl2LUl(Xk2Plrkl)+Xl2Ll(dl)]TJ /F13 11.955 Tf 12.24 11.35 Td[(Xk2Plrkl)+Xl2Lvl0@bl)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xs2SsXk2Plrkl(Xe2Pkllse(fe,cse))1A)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xe2Ee(fe)]TJ /F13 11.955 Tf 11.95 11.36 Td[(Xs2Sscse)=Xl2L(Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.24 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F6 11.955 Tf 9.3 0 Td[(vlXs2SsXk2Plrkl(Xe2Pkllse(fe,cse))9=;)]TJ /F13 11.955 Tf 11.96 11.35 Td[(Xe2Ee(fe)]TJ /F13 11.955 Tf 11.95 11.35 Td[(Xs2Sscse)=Xs2Ss(Xl2L Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.25 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F13 11.955 Tf 11.58 11.35 Td[(Xk2Plrkl(Xe2Pkl(vllse(fe,cse)+e))1A+Xe2Eecse) 168 PAGE 169 DeneMs(,,v)=supr0(Xl2L Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.24 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F13 11.955 Tf 11.58 11.35 Td[(Xk2Plrkl(Xe2Pkl(vllse(fe,cse)+e))1A+Xe2Eecse) (7) Let~rbetheoptimumsolutionof( 7 ).Wewilldiscusshowtoobtain~rshortly.ThedualfunctionofP1isobtainedby g(,,v)=Xs2SsMs(,,v).(7) Thus,thedualproblemofP1isgivenbyP2: min,,v0g(,,v).(7) 7.4.2DistributedAlgorithmicSolutionwiththeStochasticPrimalDualApproach Inthissubsection,weproposeadistributedalgorithmicsolutionofP1,orequivalentlyP2,basedonthestochasticprimaldualmethod.Inordertoreachthestochasticoptimumsolution,thedualvariables,andvareupdatedaccordingtothefollowingdynamics l(n+1)=[l(n))]TJ /F8 11.955 Tf 11.95 0 Td[(l(n)l(n)]+8l2L (7) e(n+1)=[e(n))]TJ /F8 11.955 Tf 11.95 0 Td[(e(n)e(n)]+8e2E (7) vl(n+1)=[vl(n))]TJ /F8 11.955 Tf 11.95 0 Td[(b(n)l(n)]+8l2L (7) where[x]+denotesmax(0,x)andnistheiterationnumber.l(n),e(n)andb(n)arethecurrentstepsizeswhilel(n),e(n)andl(n)arerandomvariables.Moreprecisely,theyarenamedthestochasticsubgradientofthedualfunctiong(,)andthefollowing 169 PAGE 170 requirementsneedtobesatisedEfl(n)j(1),,(n)g=@lg(,,v)8l2L (7)Efe(n)j(1),,(n)g=@eg(,,v)8e2E (7)Efl(n)jv(1),,v(n)g=@vlg(,,v)8l2L (7) whereE(.)istheexpectationoperatorand(1),,(n),(1),,(n)andv(1),,v(n)denotethesequencesofsolutionsgeneratedby( 7 ),( 7 )and( 7 ),respectively.ByDanskin'sTheorem[ 162 ],wecanobtainthesubgradientsasl(n)=dl)]TJ /F13 11.955 Tf 12.25 11.35 Td[(Xk2Pl~rkl(n)8l2L (7)e(n)=cse(n))]TJ /F3 11.955 Tf 12.05 2.66 Td[(~fe(n)8e2E (7)l(n)=bl)]TJ /F13 11.955 Tf 12.24 11.35 Td[(Xk2Pl~rkl(n)Xe2Pkllse(~fe(n),cse(n))8l2L (7) where~rklistheoptimumsolutionof( 7 ).Notethatcse(n)denotestheinstantaneouschannelcapacityonlinkeatiterationn. WenextshowhowtocalculateMs(,,v)in( 7 ),i.e.,ndingtheoptimumsolution,denotedby~r,whichmaximizesXl2L Ul(Xk2Plrkl)+l(dl)]TJ /F13 11.955 Tf 12.24 11.36 Td[(Xk2Plrkl)+vlbl)]TJ /F13 11.955 Tf 11.58 11.35 Td[(Xk2Plrkl(Xe2Pkl(vllse(fe,cse)+e))1A+Xe2Eecse (7) Notethatwhenupdatingtheprimalvariable,i.e.,r,thelinkcostsaredeterministicwhichareobtainedviathefeedbacksignal,e.g.,ACKmessages.Therefore,byutilizingthesamestochasticsubgradientapproach,wehave rkl(n+1)=rkl(n)+r(n)(n)dl0 (7) 170 PAGE 171 where (n)=@Ul @Pk2Plrkl(n))]TJ /F8 11.955 Tf 11.95 0 Td[(l)]TJ /F13 11.955 Tf 13.55 11.36 Td[(Xe2Pkl(e+vlle(fe,cse)) (7) isthestochasticsubgradientmeasuredattimen. Theorem7.1. TheproposedalgorithmconvergestotheglobaloptimumofP1withprobabilityone,ifthefollowingconstraintsofstepsizesaresatised:(1)(n)>0,(2)P1n=0(n)=1,and(3)P1n=0((n))2<1,8l2Lande2E,whererepresentse,l,bandrgenerally. Proof. First,letusrevisittheupdatingequationsof( 7 ),( 7 )and( 7 ).Notethatinthestochasticsubgradientapproach,themeasuredvaluesofl(n),e(n)andl(n)areconsideredastheinstantaneousobservationoftherealgradients,denotedby l(n), e(n)and l(n),respectively.Weconsidertherelationshipofl(n)and l(n)forinstance.Theobservationvalue,i.e.,l(n),canberewrittenas l(n)=l(n))]TJ /F25 11.955 Tf 11.95 0 Td[(E(l(n))+E(l(n)))]TJ ET q .478 w 305.43 335.28 m 329.96 335.28 l S Q BT /F8 11.955 Tf 305.43 345.92 Td[(l(n)+ l(n)= l(n)+E(l(n)))]TJ ET q .478 w 248.61 362.17 m 273.13 362.17 l S Q BT /F8 11.955 Tf 248.61 372.82 Td[(l(n)+l(n))]TJ /F25 11.955 Tf 11.95 0 Td[(E(l(n))= l(n)+]l(n)+[l(n) (7) whereEistheexpectationoperatorand ]l(n)=E(l(n)))]TJ ET q .478 w 275.28 460.8 m 299.8 460.8 l S Q BT /F8 11.955 Tf 275.28 471.44 Td[(l(n) (7) [l(n)=l(n))]TJ /F25 11.955 Tf 11.95 0 Td[(E(l(n)). (7) Notethat]l(n)isthedifferencebetweentheexpectationoftheobservationsandtherealgradient.Hence,itisthebiasedestimationerrorterm.Next,weexaminethat E([l(n)j\l(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1),,[l(0))=08n.(7) 171 PAGE 172 Therefore,theseriesof[l(n)isamartingaledifferencesequence[ 163 ].Therelationshipof( 7 )indicatesthattheobservationvalueistherealgradientdisturbedbyabiasedestimationerroraswellasamartingaledifferencenoise.Wenextinvestigatetheconvergenceconditionsofthestochasticprimaldualapproach.For]l(n),thefollowingrequirement 1Xn=0e(n)jE(l(n)))]TJ ET q .478 w 261.26 122.19 m 285.78 122.19 l S Q BT /F8 11.955 Tf 261.26 132.83 Td[(l(n)j<1(7) issatisedduetothestationaryassumption.Similarly, E([l(n)2)=E)]TJ /F3 11.955 Tf 5.48 9.68 Td[((l(n))]TJ /F25 11.955 Tf 11.96 0 Td[(E(l(n)))2(7) isboundedaswell.Thesimilaranalysiscanbeextendedtoe(n),l(n)and(n)in( 7 ),( 7 )and( 7 )straightforwardly.Therefore,thestandardconditionsaresatisedandtheconvergenceresultofTheorem1follows[ 8 ]. Itisworthnotingthattheaforementioneddistributedalgorithmenjoysthemeritofdistributedimplementationfromanengineeringperspective.Withthecurrentvaluesofdualvariables,eachsourcenodeS(l)optimizes( 7 )accordingto( 7 )and( 7 ).Theinformationneedediseitherlocallyattainableoracquirablebythefeedbackalongthepaths.Forexample,thechannelstatesoftheintermediatenodesalongpathscanbepiggybackedbytheendtoendacknowledgementmessagesfromthedestinationnode,i.e.,thegatewaynodeinourscenario.Thesourcenodeupdatesthelandvlaccordingto( 7 )and( 7 )wheretheneededinformationiscalculatedby( 7 )and( 7 ),respectively.Foreachlinke,thecurrentstatusof( 7 )ismeasured.Next,thevalueofeisupdatedfollowing( 7 ).Theiterationcontinuesuntilanequilibriumpointisreached.Notethatourframeworkcanincorporatethewirelesslossynetworkscenariosbyreplacingtheowratewiththeeffectiveowrateintheleakypipeowmodel[ 164 ]. 172 PAGE 173 7.5StochasticTrafcEngineeringwithoutConvexity Thusfar,wehaveconsideredthescenarioswherealltheusershaveconcaveutilityfunctions.However,inpractice,severalnetworkapplicationsmaypossessanonconcaveutilityfunction.Forexample,inadatastreamingapplication,theuserissatisediftheachieveddatarateexceedsathreshold,wheretheutilityfunctionisastepfunctionandthustheconvexitydoesnotpreserve.Therefore,theproposedstochasticprimaldualapproachinSection 7.4 cannotbeappliedhere.ItisworthnotingthatwecanstillformulatethestochastictrafcengineeringproblemasinP1exceptthattheoptimizationproblemisastochasticnonconvexprogramming,whichisNPhardingeneral,andcomputationallyprohibitivetosolveeveninacentralizedfashion[ 165 ].Inthefollowingsection,wewillproposeanalgorithmicsolutiontothenonconvexstochastictrafcengineeringproblem,basedonthelearningautomatatechniques.Moreover,weanalyticallyshowthattheproposedalgorithmwillconvergetotheglobaloptimumsolutionasymptotically,inadecentralizedfashion. WerstconvertthecompactstrategyspaceofeachuserintoadiscretizedsetdenotedbyR.Morespecically,eachuser,sayl,maintainsaprobabilityvectorpl,kforeachpathk2Pl.Thesegmentof[0,rm]isquantizedintoQsectionswherermisthemaximumallowedtransmissionrateonanypath.Inotherwords,thecontinuousvariablerklistransformedintoadiscreterandomvariable,rql,k,withinadiscretizedsetRwithQ+1elements.ThedatarateisrandomlyselectedfromRaccordingtotheprobabilityvectorofpl,kwheretheqthelement,pql,k,q=0,,Q,denotestheprobabilitythatthelthusertransmitswitharateofrql,k=qrm QonthekthpathofPl.Associatedwitheachprobabilityvectorpl,k,thereisaweightingvectorwl,kwiththesamedimensionof1(Q+1).Theprobabilityvectorpl,kisuniquelydeterminedbytheweightingvectorwl,kbythesoftmaxfunction[ 166 ], pql,k=ewql,k PQq=0ewql,k8l,k,q(7) 173 PAGE 174 wherewql,kistheqthelementofwl,k,q=0,,Q. Next,weformulateanidenticalinterestgamewheretheplayersarethejLjsourcenodesandthecommonobjectivefunctionistheoverallnetworkutility,i.e.,thesummationoftheutilityfunctions.Inaddition,foreachsourcenode,ateamoflearningautomata[ 167 ]isconstructed.Ateachtimestep,everysourcenodepicksthedataratesonitsownpathsaccordingtotheprobabilityvectors,whicharedeterminedbytheweightingvectors.Basedonthefeedbacksignal,whichwillbedenedshortly,eachsourcenodeadjuststheweightingvectorsandtheiterationcontinues.Theexecutedalgorithmoneverysourcenode,sayl,isprovidedasfollows.Algorithm: Repeat: Foreverypath,sayk,randomlyselectsatransmissionraterjl,kfromR,accordingtothecurrentprobabilityvectorpl,k(n)wherendenotesthecurrenttimeslot. Afterreceivingthefeedbacksignal(n)fromthegatewaynode,ifthecostconstraintissatised,theweightingvectorwl,kisupdatedaswql,k(n+1)="wql,k(n)+(n)(n)(1)]TJ /F6 11.955 Tf 28.6 8.09 Td[(ewql,k PQq=0ewql,k)+p (n)&ql,k(n)#L0forq=j;wql,k(n+1)=hwql,k(n)+p (n)&ql,k(n)iL0forq6=j; (7) Otherwise,theweightingvectorremainsthesame. Theprobabilityvectorpl,kisthenupdated,followingequation( 7 ). Until: max(pl,k(n+1))>BwhereBisapredenedconvergencethreshold. Inthealgorithm,(n)isthelearningparameterofthealgorithmsatisfying0<(n)<1.Lisasufcientlylargeyetnitenumberwhichkeepstheweightingvectorbounded.Thesequenceof&ql,k(n)isasetofi.i.d.randomvariableswithzeromeanandavarianceof2(n).Theglobalfeedbacksignal(n)iscalculatedbythegatewaynode 174 PAGE 175 andsentbacktoallsourcenodes,as (n)=Pl2L(Ul(Pk2Plrkl)) J(7) whereJisanumbertonormalizethefeedbacksignal.Forexample,wecansetJtothemaximumvalueofoverallutilitytillnandupdatethisvalueonthey.Therefore,thevalueof(n)lieswithin[0,1].Ulisthenonconvexutilityfunctionforthelthuser.Notethattheutilityfunctionsofallusersareassumedtobetrulyacquiredbythegatewaynode[ 168 ].Inpractice,thevalueof(n)canbecirculatedefcientlybyestablishedmulticastalgorithmssuchas[ 169 ].Basedonthefeedback,thelearningautomatateamadjuststheweightingvectorinadecentralizedfashion.Inaddition,notethatBisthepredenedconvergencethreshold,e.g.,B=0.999,whichprovidesatradeoffbetweentheperformanceofthealgorithmanditsconvergencespeed. Beforeanalyzingthesteadystatebehavioroftheproposedalgorithm,werstdiscussthefollowingconcepts. Denotethemaximumnetworkutilityoftheoriginaltrafcengineeringproblem,i.e.,P1,asO.Next,wedenethenaloutcomeoftheproposedalgorithmasO0.Wesaythatthealgorithmprovidesanaccuratesolution,ifforanyarbitrarilysmall>0,thereexistsaQ0suchthat jO)]TJ /F6 11.955 Tf 11.95 0 Td[(O0j<8Q>Q0(7) Apotentialgame[ 67 ]isdenedasagamewherethereexistsapotentialfunctionVsuchthat V(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))]TJ /F6 11.955 Tf 11.96 0 Td[(V(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(l)=Ul(a0,a)]TJ /F9 7.97 Tf 6.58 0 Td[(l))]TJ /F6 11.955 Tf 11.95 0 Td[(Ul(a00,a)]TJ /F9 7.97 Tf 6.58 0 Td[(l)8l,a0,a00(7) whereUlistheutilityfunctionforplayerlanda0,a00aretwoarbitrarystrategiesinitsstrategyspace.Thenotationofa)]TJ /F9 7.97 Tf 6.59 0 Td[(ldenotesthevectorofchoicesmadebyallplayersotherthanl. 175 PAGE 176 Aweightedpotentialgame[ 67 ]isdenedasagamewherethereexistsapotentialfunctionVsuchthat (V(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))]TJ /F6 11.955 Tf 11.96 0 Td[(V(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))hi=Ul(a0,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l))]TJ /F6 11.955 Tf 11.96 0 Td[(Ul(a00,a)]TJ /F9 7.97 Tf 6.59 0 Td[(l) (7) foralll,a0,a00wherehi>0. Accordingtothedenitions,itisapparentthattheformulatedidenticalinterestgameisaspecialcaseofweightedpotentialgames.Inthefollowingtheorem,wewillprovidetheconvergencebehaviorofamoregeneralsettingforweightedpotentialgames,andhencetheresultappliestoourspecicscenarionaturally. Theorem7.2. ForanNpersonweightedpotentialgamewhereeachpersonrepresentsateamoflearningautomata,theproposedalgorithmcanconvergetotheglobaloptimumsolutionasymptotically,whichisanaccuratesolutiontotheoriginalstochastictrafcengineeringproblem,forsufcientsmallvalueofand. Proof. Theprooffollowssimilarlinesasin[ 167 ].However,weextendtheresulttoamoregeneralsettingwheretheunderlyingNpersonstochasticgameisaweightedpotentialgame.Therefore,theproofin[ 167 ]canbeviewedasaspecialcase.DeneVasthepotentialfunctionofthegame.Notethattheselectedrateisdeterminedbytheprobabilityvectorwhichisgenerateduniquelybytheweightingvector.Therefore,wecanviewtheweightingvectorasthevariableinthiscaseandtheobjectivefunctionisgivenbyz=E(Vjw).Intheupdatingprocedureof( 7 ),signalisreplacedbyV.Werstverifythat @z @wql,k=@E(Vjw) @wql,k=@Pqpql,kE(Vjw,rql,k) @wql,k=Xqewql,k PQq=0ewql,k(1)]TJ /F6 11.955 Tf 28.6 8.09 Td[(ewql,k PQq=0ewql,k)E(Vjw(n),rql,k). 176 PAGE 177 NotethatE(V(1)]TJ /F6 11.955 Tf 28.59 8.09 Td[(ewql,k PQq=0ewql,k)jw(n))=Xqpql,k(1)]TJ /F6 11.955 Tf 28.59 8.09 Td[(ewql,k PQq=0ewql,k)E(Vjw(n),rql,k)=Xqewql,k PQq=0ewql,k(1)]TJ /F6 11.955 Tf 28.6 8.08 Td[(ewql,k PQq=0ewql,k)E(Vjw(n),rql,k)=@z @wql,k. Next,itisstraightforwardtoverifythatthestandardconditionsin[ 170 ](Chap.6,Theorem7)aresatised.Weomitthevericationssincetheyarethesimilarproceduresasin[ 167 ].Thus,weconcludethattheabovedynamicweaklyconvergestothefollowingSDE[ 170 171 ] dw=rz+dW(7) forasufcientlysmall!0whereisthestandarddeviationofthei.i.d.randomvariables&ql,kandWisastandardWienerProcess.NotethattheSDE( 7 )fallsintothecategoryofLangevinequation[ 172 ]whichiswellknownthattheprobabilitymeasureconcentratesontheglobalmaximumsolutionofzforasufcientlysmall[ 167 172 ].Therefore,weconcludethatintheweightedpotentialgamescenario,theproposedalgorithmwillconvergetotheglobaloptimumoftheobjectivefunction,forthequantizeddataratesetting.TheassociationoftheaccuratesolutiontotheoriginalstochastictrafcengineeringproblemofP1followstheresultof[ 173 ]. NotethatTheorem2establishestheconvergencebehavioroftheaforementionedalgorithmwithnoadditionalrequirementfortheproblemstructure.Incontrast,weproposeastochasticprimaldualapproachinSection 7.4 ,whichrequirestheunderlyingproblemtobeastochasticconvexprogramming.Thestochasticprimaldualapproachcannotbeappliedefcientlyotherwise.However,theaforementionedlearningbasedalgorithmissuitableforalmosteveryaspectsuchasstochastic 177 PAGE 178 nonconvexprogrammingandstochasticmixedintegerprogramming.Theasymptoticallyconvergenceresultstillholds.Therefore,inthiswork,theproposedalgorithmsprovidetwodifferentexemplifyingmethodsforprotocoldesignsunderthestochasticenvironment.However,itisworthnotingthatforthelatterapproach,thetradeoffforgeneralapplicabilityistheconvergencespeed.Inotherwords,inordertoachieveanaccurateresult,i.e.,whenissmall,theconvergencespeedmaybeslow.Theactualconvergencespeeddependsonthevaluesof,Q,,aswellastheinherentstructureoftheproblemandhenceisdifculttoquantify.Fortunately,inpracticalapplications,achievingagoodenoughresultissometimessatisfactory.Thistradeoffcanbeachievedbyutilizingdiminishingvaluesofand,asdemonstratedinthesimulatedannealingliterature[ 78 ]and[ 174 ].Tosumup,ifthestochastictrafcengineeringproblempossessesanicepropertyofconvexity,thealgorithmbasedonthestochasticprimaldualapproachinSection 7.4 isrecommendedduetoitsnicedecomposedstructureandcomputationallyefcientsolution.However,iftheproblemisnonconvexinnature,thelearningautomatabasedalgorithmcanbeutilizedtoachieveanapproximatesolution.Thetradeoffbetweentheaccuracyandconvergencespeedcanbetunedbyadjustingthevaluesofand. 7.6PerformanceEvaluation Inthissection,wepresentasimpleyetillustrativeexampletodemonstratethetheoreticalresults.Weconsideracognitivewirelessmeshnetwork3depictedinFigure 72 .Therearethreeedgeroutersasthesourcenodes,denotedbyA,B,C,whichtransmittothegatewaynodeGWviatherelayroutersX,YandZ.Amongallfeasible 3Figure 72 onlyshowsthelinksontheavailablepathsobtainedbythesignallingmechanismsormanuallycongurations.Theactualphysicaltopologyofthenetworkcanbepotentiallylarger. 178 PAGE 179 paths,weselectthefollowingavailablepathsforedgerouters,assummarizedinTable 71 Figure72. Exampleofcognitivewirelessmeshnetwork Table71. Availablepathsforedgerouters A P1A: fA!X!GWg P2A: fA!X!Y!GWg P3A: fA!X!Y!Z!GWg B P1B: fB!X!GWg P2B: fB!Y!GWg P3B: fB!Y!X!GWg P4B: fB!Y!Z!GWg P5B: fB!Z!GWg C P1C: fC!Z!GWg P2C: fC!Z!Y!GWg P3C: fC!Z!Y!X!GWg Thereareveprimaryusersinthearea,denotedby1,2,3,4and5whereeachonehasaprimarybandof10MHz.ThecommonISMbandisassumedtobe10MHz.Thereturnprobabilityoftheprimaryusersisgivenas$=[0.2,0.3,0.4,0.3,0.3].Thetransmittingpowerofeachnodeisxedas100mWandthenoisepowerisassumedtobe3mW.Weconsideramodelwherethereceivedpowerisinverselyproportionaltothesquareofthedistance.Notethatthetransmittingpowerisuniformlyspreadonallavailablebands.Inaddition,weexplicitlyspecifytheaffectingprimaryusersforaparticularnode.Weusefi,j,k,gtorepresentthataparticularnodeisaffectedbyprimaryuseri,j,k,.Forexample,nodeXislabeledwithf1,2gwhichindicatesthat 179 PAGE 180 thetransmissionofnodeXwilldevastatethetransmissionsofprimaryuser1and2ifthecorrespondingprimarybandisutilized.Notethatthecentralnode,namely,Y,ismostseverelyaffectedbyallprimaryusers.Intuitively,toachieveanexpectedoptimumsolution,thestochastictrafcengineeringalgorithmsareinclinedtosteerthetrafcawayfromY.Wewilldemonstratethisdetoureffectnext. Werstconsiderthecognitivewirelessmeshnetworkwithconvexity,e.g.,Ul(xl)=logxltoachieveaproportionalfairnessamongtheows[ 161 ].Thelinkcostisassumedtobeintheformoflse(fe,cse)=1 cse)]TJ /F9 7.97 Tf 6.58 0 Td[(fe,whichreectsthedelayexperiencedforaunitowonlinkeundertheM=M=1assumption[ 76 ].Notethatiffece,thecostis+1.Wesetthetrafcdemandofalledgeroutersasdl=30Mbpswhilethecostbudgetisbl=5.Thestepsizesarechosenas=1=nwherenisthecurrentiterationstep.Figure 73A illustratesthetrajectoriesoftheratevariablesandFigure 73B showstheconvergenceofthenetworkoverallutilityaswellastheindividualutilityfunctions4.Weobservethatwhiletheratevariablesconvergeastheiterationsgo,theoverallobjective,i.e.,thesumoftheindividualutilities,approachestotheglobaloptimumindicatedbythedashedline,whichisattainedbycalculatingthesteadystatedistributionfollowingthereturnprobability$. Inaddition,Table 72 providestherateoneachpathafterconvergenceforasamplerunofthealgorithm.Forcomparison,weprovidetheconvergencerateswhennodeYisswitchedfromthemostaffectednodetotheleastaffectednode,i.e.,nodeYcanutilizeallveprimarybandsallthetime,inTable 73 .FromTable 72 and 73 ,itisinterestingtonotethat,intherstscenario,eachuserallocatesarelativelysmallamountofowonthepathswhichtraversethroughnodeY.Onthecontrary,whennodeYislessaffected,alltheowsallocatenoticeablylargerdataratesonpathsthattraversethrough 4NotethatFigure 73B alsoreectstheevolutionofthethroughputofeachedgerouterlogarithmically. 180 PAGE 181 ATrajectoriesofRateVariables BTrajectoriesofUtilityFunctions Figure73. Cognitivewirelessmeshnetworkswithconvexity YdespitethefactthatnodeYisthecentralnodewhichisleastfavorablebytraditionaltrafcengineeringsolutions.Therefore,ourproposedstochastictrafcengineeringalgorithmisofparticularinterestformultihopcognitivewirelessmeshnetworksduetothecapabilityofsteeringthetrafcawayfromtheseverelyaffectedareasautomatically,withoutapriorknowledgeoftheunderlyingprobabilisticstructure,inadistributedfashion. 181 PAGE 182 Table72. ConvergencerateswhenYisaffectedbyallveprimaryusers A P1A: 15.3035 P2A: 3.2458 P3A: 2.7362 B P1B: 7.3725 P2B: 1.9584 P3B: 2.1752 P4B: 2.6458 P5B: 5.3489 C P1C: 17.3051 P2C: 2.3792 P3C: 2.5675 Table73. ConvergencerateswhenYisnotaffectedbyanyoftheprimaryusers A P1A: 9.6413 P2A: 11.1335 P3A: 9.1861 B P1B: 5.0101 P2B: 6.2625 P3B: 4.9711 P4B: 4.9921 P5B: 8.7878 C P1C: 15.3118 P2C: 9.1042 P3C: 5.6121 Wenextconsideracognitivewirelessmeshnetworkwithnonconvexutilityfunctions.Specically,weconsidertheutilityfunctionas Ul(xl)=8>><>>:1,ifxl2Mbps0,ifxl<2Mbps.(7) whileothersettingsarethesameasinthepreviousscenario.Additionally,weutilizediminishingvaluesofandas=1=nand=1=n,wherenistheiterationstep5.Withoutlossofgenerality,wesetL=100andthequantizationlevelQ=20.The 5Byutilizingdiminishingparameters,atradeoffbetweentheperformanceandtheconvergencespeedcanbeachievedbytuningthedecreasingspeed[ 78 ]. 182 PAGE 183 maximumallowedratermisassumedtobe10.Therefore,thediscretizeddataratesetisgivenbyR=[0.5,1.0,1.5,,9.0,9.5,10.0].Figure4illustratestheevolutionoftheprobabilityvectorofpA,1.Notethatastheiterationsevolve,theprobabilityofp20A,1,i.e.,theprobabilitythatrouterAchoosesthetwentiethdatarate(rminthiscase),excelsothersandapproachesto1asymptotically.WeplottheevolutionsoftheprobabilityvectorsofotherpathsinFigure5collectively.Foreachpath,theprobabilityofselectingoneparticulardataratesoonexcelsothers.WeobservethatrouterAselectsthetwentieth,therstandthefourthdaterateonitsthreepathsasymptotically.Meanwhile,routerBinclinestochoosetheeighth,thesecond,therst,thefourthandthetwentiethdatarateonitspaths.ThesteadystatedataratesforrouterCisthetwentieth,therstandtherstelementinR,asdepictedinFigure5.ItisinterestingtonoticethatalltheroutersautomaticallydetourthetrafcfromtheseverelyaffectednodeYbyallocatingmoredatarateonotherpaths. Figure74. TrajectoryoftheprobabilityvectorofrouterA'srstpath 7.7Conclusions Inthischapter,weinvestigatethestochastictrafcengineering(STE)problemincognitivewirelessmeshnetworks.Toharnesstherandomnessinducedbytheunpredictablebehaviorsofprimaryusers,weformulatetheSTEprobleminastochasticnetworkutilitymaximizationframework.Forthecaseswhereconvexityholds,wederiveadistributedalgorithmicsolutionviathestochasticprimaldualapproach,whichprovablyconvergestotheglobaloptimumsolution.Forthescenarioswhereconvexityisnot 183 PAGE 184 Figure75. Trajectoriesoftheprobabilityvectorsonotherpaths attainable,weproposeanalternativedecentralizedalgorithmicsolutionbasedonthelearningautomatatechniques.Weshowthatthealgorithmconvergestotheglobaloptimumsolutionasymptotically,undercertainconditions. Inourwork,werestrictourselfinasinglegatewayscenario.Theextensiontothemultiplegatewayscenarioseemsinterestingandneedsfurtherinvestigation.Inaddition,inthiswork,weconsideracooperativecasewherealltheedgeroutersattempttomaximizetheoverallnetworkperformance.Inthecaseswheretheedgeroutersarenoncooperative,eachplayerisinterestedinitsownutilityratherthanthesocialwelfare.Stochasticgametheoryprovidesafeasibletooltoaddressthenoncooperativecase,whichremainsasfutureresearch.Wealsoassumeanegligibledelayforthefeedbacksignalwhileinamoregeneralcase,theimpactoffeedbackdelayneedsfurtherinvestigation.Onefeasiblesolutionistoutilizethedistributedrobustoptimizationframework[ 175 ]wheretheworstcaseperformanceismaximizedgiventhatthefeedbackdelay/erroriswithinareasonablerange.Ourworkinitiatesarststeptoinvestigatetheimpactofunpredictablereturnsofprimaryusers,onthestochastictrafcengineeringproblemincognitivewirelessmeshnetworks. 184 PAGE 185 CHAPTER8CONCLUSIONS Inthisdissertation,weprovideefcientandeffectivesolutionstoanumberofdynamicresourceallocationandoptimizationproblemsinwirelessnetworkswithtimevaryingchannelconditions. Forwirelessadhocnetworks,werstdevelopadynamicpricingschemewhichcanachieveanetworkwidestabilitywhilemaximizingtheaccumulatedrevenuewithacontrollabletradeoffwiththeaveragedelay.Inaddition,aservicedifferentiationsolutioncanbeachievedbyprioritizingtheowsaccordingtotheirQoSrequirements.Wealsoproposeaminimumenergyschedulingalgorithmwhichminimizestheoverallenergyconsumptioninthenetworkwithoutdecreasingthethroughputregion.Forwirelessmeshnetworks,weinvestigatethejointfrequencyandpowerallocationproblemamongmultipleaccesspointsinagametheoreticalframework.Theproposednegotiationbasedalgorithmcanachievetheoptimalfrequencyandpowerallocationwitharbitrarilyhighprobability.Thepriceofanarchyinnoncooperativewirelessmeshnetworksisalsostudied.Forwirelesssensornetworks,weproposeaconstrainedqueueingmodeltocapturethecomplexcrosslayerinteractionsamongmultiplesensornodes.Wealsostudythestochasticnetworkutilitymaximizationproblemandproposeacrosslayersolutionwhichsolvestheproblemsofadmissioncontrol,loadbalancing,dynamicschedulingandroutinginacollectivefashion.Forwirelesslocalareanetworks,westudythethresholdsbasedrateadaptationalgorithmsfromareverseengineeringperspective,wheretheimplicitobjectivefunctionisrevealed.Wealsoinvestigatethestochastictrafcengineeringprobleminmultihopcognitiveradionetworks,wheredistributedalgorithmsaredevelopedtomaximizetheoverallnetworkutility. 185 PAGE 186 REFERENCES [1] G.R.GuptaandN.Shroff,Delayanalysisformultihopwirelessnetworks,IEEEINFOCOM,2009. 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