DISTRIBUTED RESOURCE ALLOCATION AND PERFORMANCE OPTIMIZATION FOR VIDEO COMMUNICATION OVER MESH NETWORKS BASED ON SWARM INTELLIGENCE.

Transcription

1 DISTRIBUTED RESOURCE ALLOCATION AND PERFORMANCE OPTIMIZATION FOR VIDEO COMMUNICATION OVER MESH NETWORKS BASED ON SWARM INTELLIGENCE A Dissertation presented to the Faculty of the Graduate School University of Missouri-Columbia In Partial Fulfillment Of the Requirements for the Degree Doctor of Philosophy by BO WANG Dr. Zhihai He, Dissertation Supervisor DECEMBER 2007

2 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled DISTRIBUTED RESOURCE ALLOCATION AND PERFORMANCE OPTIMIZATION FOR VIDEO COMMUNICATION OVER MESH NETWORKS BASED ON SWARM INTELLIGENCE presented by Bo Wang, a candidate for the degree of Doctor of Philosophy, and hereby certify that in their opinion it is worth acceptance. Dr. Zhihai He Dr. Curt H. Davis Dr. Guilherme DeSouza Dr. Justin Legarsky Dr. Wenjun Zeng

3 To my loving parents Pingxin Wang and Xianju Cheng, my caring sister Li Wang, and my precious wife Xiaoning Lu

4 ACKNOWLEDGMENTS On completion of my dissertation, I wish to acknowledge the following persons who helped me during my long path: First, I would like to express my sincere gratitude to my dissertation advisor, Dr. Zhihai He. His intellectual guidance, keen insight, motivation and encouragement have been the great support for me throughout the whole work during my research. I would also like to express my gratitude to Dr. Curt H. Davis, Dr. Guilherme DeSouza, Dr. Justin Legarsky, and Dr. Wenjun Zeng, for agreeing to serve as members of my guidance and doctoral committees, and for their time, consideration and suggestion to help improve the quality of my dissertation. I would like to extend my appreciation to my current labmates at Video Processing and Networking Lab, Xi Chen, York Chung, Xiwen Zhao, and Jay Eggert for their technical assistance and wonderful friendship. I would also like to thank all my friends at Columbia, Missouri, who have made my life at University of Missouri-Columbia such an unforgettable experience. I am also thankful to the following former and current staff at University of Missouri- Columbia, for their various forms of support during my graduate study: Jim Fischer, Shirley Holdmeier, Betty Barfield, and Kelly Scott. Last, I would like to acknowledge my family for all their love and support. I especially would like to acknowledge my parents Pingxin Wang, Xianju Cheng and my sister Li Wang for their love, support and encouragement and understanding in dealing with all the challenges I have faced in my life. I would specially acknowledge my wife, Xiaoning Lu, for her ii

5 support, encouragement, understanding and unwavering love in the past ten years of my life. Without their love and support, I could not go through so far. Nothing in a simple paragraph can express the love I have for the four of you. iii

6 TABLE OF CONTENTS ACKNOWLEDGMENTS ii LIST OF TABLES viii LIST OF FIGURES ix ABSTRACT xiii CHAPTER Introduction Overview This Work Major Contributions of the Research Convex Mapping of High-dimensional Resource Constraints Distributed Optimization over Wireless Sensor Networks Distributed Rate Allocation for Video Mesh Networks Distributed Resource Allocation for Wireless Video Sensor Networks Evaluation of PSO Algorithm Dissertation Organization Background and Related Work Network Resource Allocation Overview Resource Allocation Over Mesh Networks Previous Optimization Techniques Optimization Algorithms iv

7 2.2.2 Applications Particle Swarm Optimization PSO Algorithm Social Behavior Applications Summary Particle Swarm Optimization over Wireless Sensor Networks Convex Mapping of High-dimensional Resource Constraints WVSN Operation Models WVSN Performance Optimization Transform of the Solution Space Performance Optimization Using PSO Experimental Results Summary Distributed Optimization over Wireless Sensor Networks Optimization Problems Using Decentralized PSO Source Localization Application Experimental Results Comparison with Gradient Search Algorithms Summary Distributed Rate Allocation for Video Mesh Networks Introduction Related Work Major Contribution Resource Allocation for Video Mesh Networks Formulation of Generic Resource Allocation Problems Basic Framework for Distributed Resource Allocation v

8 4.3 Distributed Rate Allocation for Video Mesh Network Problem Formulation Decomposition Distributed and Asynchronous Particle Swarm Optimization Original Optimization Problem Decomposition In-Network Fusion and Particle Migration Handling Network Bottleneck Issue Using Collaborative Resource Control Algorithm Description of DAPSO Experimental Results Simulation Setup Convex Distributed Rate Allocation and Performance Optimization Application Nonconvex Distributed Rate Allocation and Performance Optimization Application Comparison with Gradient-Based Lagrangian Dual Algorithms Discussion and Conclusion Distributed Resource Allocation for Wireless Video Sensor Networks Introduction Related Work Major Contribution Energy Efficient Resource Allocation and Performance Optimization Channel Module Power-Rate-Distortion Module and Transmission Behavior Analysis Resource Allocation and Performance Optimization Using Particle Swarm Optimization Optimization Problem Formulation vi

9 5.3.2 Experimental Results Energy Efficient Distributed and Asynchronous Particle Swarm Optimization Decomposition of Original Optimization Problem Algorithm Design of EEDAPSO Experimental Results Discussion and Conclusion Evaluation of PSO Algorithm Convergence Analysis of PSO Algorithm Random Search Techniques PSO Convergence Analysis Optimization Algorithms Evaluation and Comparison Optimization Algorithms Introduction Test Problem Analysis Experimental Results Summary Conclusion and Future Work Conclusion Future Work Bibliography VITA vii

10 LIST OF TABLES Table 4.1 Total distortion and communication cost for DAPSO and centralized optimization algorithm under different topologies Configuration of the channel model parameters PSO algorithm parameters used in the test Simulation Results for PSO, GA, and BFGS Algorithms viii

11 LIST OF FIGURES Figure 1.1 Illustration of mesh networks Video Streaming over Ad hoc networks Decomposition of the original optimization problem Illustration of particle swarm optimization Pseudo code for the basic PSO algorithm Performance optimization with swarm intelligence and convex projection The performance metric decrease as the particles update their positions of PSO with convex mapping The traces of the all particles moving in the solution space of PSO with convex mapping A WSN topology with 20 sensors and 24 links The performance function value decrease as the particles update their positions of decentralized PSO The traces of the particles moving in the sensor field of decentralized PSO Optimization convergence of WSN with 20 sensors wake up and different link number in decentralized PSO Comparison of decentralized PSO and subgradient search algorithm on optimization convergence Illustration of video communication over mesh networks Illustration of distributed asynchronous optimization Distributed and asynchronous PSO algorithm An example of bottleneck link in a multi-hop mesh network ix

12 4.5 Illustration of particle migration A randomly generated video mesh network with 16 nodes, 15 links and 6 video sessions Convergence of network utility functions of DAPSO to the global optima of convex function The traces of the particles moving in critical link A of convex function The traces of the particles moving in critical link B of convex function The traces of the particles moving in critical link C of convex function Convergence of network utility function of DAPSO to the global optima of nonconvex function: w1=1 and w2= The traces of the particles moving in critical link A of nonconvex function: w1=1 and w2= The traces of the particles moving in critical link B of nonconvex function: w1=1 and w2= The traces of the particles moving in critical link C of nonconvex function: w1=1 and w2= Convergence of network utility function of DAPSO to the global optima of nonconvex function: w1=1 and w2= The traces of the particles moving in critical link A of nonconvex function: w1=1 and w2= The traces of the particles moving in critical link B of nonconvex function: w1=1 and w2= The traces of the particles moving in critical link C of nonconvex function: w1=1 and w2= Comparison between DAPSO and distributed gradient search with different starting price setting The total source rate beyond link capacity during iteration Response of DAPSO to video content change x

13 4.22 Different wireless sensor network topologies with different number of sessions (a) An example video mesh network; and (b) convergence of network utility function with DAPSO (a) An example video mesh network; and (b) convergence of network utility function with DAPSO (a) An example video mesh network; and (b) convergence of network utility function with DAPSO A WVSN topology with 9 sensor nodes, 8 links and 4 video sessions The average overall video distortion decrease as the particles update their positions under different PSO size The average source rate for each video stream updates during PSO iteration The average encoding power for each video stream updates during PSO iteration The average packet loss probability for each video stream decreases during PSO iteration The average video distortion decreases during EEDAPSO search process The average video distortion decreases during EEDAPSO search process The source rate for every video stream on critical link A updates during EEDAPSO search process The source rate for every video stream on critical link B updates during EEDAPSO search process The encoding power for every video stream updates during EEDAPSO search process The transmission power on every link updates during EEDAPSO search process The packet loss probability for each video stream decreases during EEDAPSO search process (a) An example WVSN; and (b) convergence of network utility function with EEDAPSO xi

14 5.14 (a) An example WVSN; and (b) convergence of network utility function with EEDAPSO Plot for a nonconvex objective function Convergence probability under different initial random solution (a) Convergence of function value for problem H1; and (b) convergence of function value for problem H (a) Convergence of function value for problem H3 (n=4); and (b) convergence of function value for problem H3 (n=8) xii

15 ABSTRACT Mesh networking technologies allow a system of communication devices to communicate with each other over a dynamic and self-organizing wired or wireless network from everywhere at anytime. Important examples of mesh networking include wireless sensor networks, multimedia communication over community networks and Internet, peer-to-peer video streaming, etc. Large-scale mesh communication networks involve a large number of heterogeneous devices, each with different on-board computation speeds, energy supplies, and communication capabilities, communicating over the dynamic and unreliable networks. How to coordinate the resource utilization behaviors of these devices in a large-scale mesh network such that each of them operates in a contributive fashion to maximize the overall performance of the system as a whole remains a challenging task. Network resource allocation and performance optimization can be formulated as a network utility maximization (NUM) problem under resource constraints. In many mesh networking applications, especially in video communication over mesh networks, network utility maximization is often a high-dimensional, nonlinear, constrained optimization problem. An effective solution to this type of problems needs to meet the following three requirements: distributed, asynchronous, and non-convex. In this work, based on swarm intelligence principles, we develop a set of distributed and asynchronous schemes for resource allocation and performance optimization for a wide range of mesh networking-based applications. To successfully apply the swarm intelligence principle in distributed resource allocation and performance optimization in large-scale mesh networks, there are three important issues that need to be carefully investigated. First, existing PSO schemes are not able to efficiently handle constraints, especially xiii

16 constraints in a high-dimensional space. However, in large-scale video mesh networks, the resource constraints are often represented by a convex region embedded in a very high dimensional space. To address this issue, we propose to transform the solution space defined by resource constraints into a convex region in a low-dimensional space. We then merge the convex condition with the swarm intelligence principle to guide the movement of each particle to efficiently search for the optimum solution. Second, distributed optimization requires decomposition of centralized network utility function and resource constraints into local ones. However, in video mesh networking, the resource utilization behavior of neighboring network nodes are highly coupled and interwound. In this work, we propose various methods and approaches for decomposition of network utility function and interwound resource constraints. Third, one of the key challenges in resource allocation and performance optimization is to handle critical / bottleneck links which have very limited resource however are shared by multiple video communication sessions. We observe that the resource allocation results at these critical links have direct impact on the overall system performance. To address this issue, we propose various schemes to fuse the resource allocation information of neighboring optimization modules, propagate and share the resource allocation results at critical links, use this external information to guide the of movements of particles in each local optimization module to efficiently search for the optimum solution. Our extensive experimental results in distributed resource allocation and performance optimization demonstrate that the proposed schemes work efficiently and robustly. Compared to existing algorithms, including gradient search and Lagrange optimization, the proposed approach had the advantage of faster convergence and the ability to handle generic network utility functions. Compared to centralized performance optimization schemes, the proposed approach significantly reduces communication overhead while achieving similar performance. The distributed algorithms for resource allocation and performance optimization provide analytical insights and important guidelines for practical design of large-scale video mesh networks. xiv

17 Chapter 1 Introduction Mesh networking technologies allow a system of communication devices to communicate with each other over a dynamic and self-organizing wired or wireless network from everywhere at anytime. Important examples of mesh networking include wireless sensor networks, multimedia communication over community networks and Internet, peer-to-peer video streaming, etc. Large-scale mesh communication networks involve a large number of heterogeneous devices, each with different on-board computation speeds, energy supplies, and communication capabilities, communicating over the dynamic and unreliable networks. How to coordinate the resource utilization behaviors of these devices in a large-scale mesh network such that each of them operates in a contributive fashion to maximize the overall performance of the system as a whole remains a challenging task. 1.1 Overview Currently there has been considerable research interest in the mesh networks, as illustrated in Fig Mesh network consists of nodes that communicate to each other and are capable of hopping radio messages to a base station where they are passed to a PC or other client. Each network node also acts as a router, forwarding data packets to other nodes. All wireless mesh networking systems share a set of common requirements, include low power consumption, ease of use, scalability, responsiveness, and range. Because of their ad hoc nature, mesh networks can respond to conditions much more quickly and reliably than static 1

18 networks. If a node in a mesh network fails, the other nodes in the mesh will notice the failure and adjust their routing accordingly without human intervention. The resilience, flexibility, and decentralized administration make the mesh networks more attractive than the traditional networking systems. Figure 1.1: Illustration of mesh networks. Network resource allocation and performance optimization can be formulated as a network utility maximization (NUM) problem under resource constraints. In many mesh networking applications, especially in video communication over mesh networks, network utility maximization is often a high-dimensional, nonlinear, constrained optimization problem. An effective solution to this type of problems needs to meet the following three requirements: (1) Distributed. Lack of a centralized powerful node for computation, mesh networks are often not able to support centralized computation. In addition, centralized performance optimization introduces significant communication overhead, becomes extremely costly or even infeasible in large-scale mesh networks. The decomposability structure of network utility maximization leads to the most appropriate distributed algorithm for a given network resource allocation problem. Decomposition theory naturally provides the mathematical method to build the foundation for the distributed control of networks. This helps us obtain the most appropriate distributed algorithm for a given network resource allocation problem, such as distributed routing, scheduling to power control and congestion control. (2)Asynchronous. Network nodes are communicating with each other in an asynchronous and on-demand manner. This requires the distributed performance optimization to also 2

19 operate in an asynchronous fashion. (3)Non-convex. As the network utility functions in many video mesh networking applications are highly nonlinear and non-convex, an effective solution should be able to handle generic non-convex objective functions. Based on Kelly s work [1] in 1998, the framework of NUM has found many applications in network resource allocation algorithms and network congestion control algorithms [2, 3, 4, 5], such as Internat rate allocation and TCP congestion control. These work interpret source rates as primal variables and link congestion prices as dual variables to solve an implicit global utility optimization problem or lagrange dual problem. Previous research use price-based strategy [1, 3, 6, 7, 8], where prices are computed to reflect relations between resource demands and supplies, to coordinate the resource allocations at multiple hops. Their results show that the price is effective as a method to arbitrate resource allocation. Most of the papers in the vast related on network resource allocation use a standard lagrange dual based distributed algorithm [1, 3]. While it is well known that dual based distributed algorithm needs utility function to be convex or concave. This is the major drawback of dual base distributed algorithm when the application is inelastic or utility functions are nonconvex, which leads to divergence of congestion control. Several existing distributed optimization algorithms based on incremental sub-gradient search [9, 10] assume that the objective function to be additive and convex. And the drawbacks of the gradient or subgradient based distributed algorithm are that it is sensitive to local optima or saddle points, and it also suffers from slow convergence speed. 1.2 This Work In this work, based on swarm intelligence principles, we develop a set of distributed and asynchronous schemes for resource allocation and performance optimization for a wide range of mesh networking-based applications. Particle swarm optimization (PSO), developed by Eberhart and Kennedy in 1995[11, 12], is a population based stochastic optimization technique which is inspired by social 3

20 behavior of bird flocking or fish schooling. PSO shares many similarities with evolutionary computation techniques, such as Genetic Algorithms (GAs). The main advantage of PSO over other global optimization strategies is that the large number of random solutions of PSO algorithm make the technique avoid dropping in the local minimum of the optimization problem, so that the PSO algorithm does not need to have requirement for the objective function of the optimization problem. In the past several years, PSO algorithm has been successfully applied in many research and application areas. It is demonstrated that PSO gets better results in a faster, cheaper way compared with many other methods. To successfully apply the swarm intelligence principle in distributed resource allocation and performance optimization in large-scale mesh networks, there are three important issues that need to be carefully investigated. First, existing PSO schemes are not able to efficiently handle constraints. To address this issue, we propose to transform the solution space defined by resource constraints into a convex region in a low-dimensional space. We then merge the convex condition with the swarm intelligence principle to guide the movement of each particle to efficiently search for the optimum solution. Second, distributed optimization requires decomposition of centralized network utility function and resource constraints into local ones. However, in video mesh networking, the resource utilization behavior of neighboring network nodes are highly coupled and interwound. In this work, we propose various methods and approaches for decomposition of network utility function and interwound resource constraints. Third, one of the key challenges in resource allocation and performance optimization is to handle critical / bottleneck links which have very limited resource however are shared by multiple video communication sessions. To address this issue, we propose various schemes to fuse the resource allocation information of neighboring optimization modules, propagate and share the resource allocation results at critical links, use this external information to guide the of movements of particles in each local optimization module to efficiently search for the optimum solution. Our extensive experimental results in distribution resource allocation and performance 4

21 optimization demonstrate that the proposed schemes work efficiently and robustly. Compared to existing algorithms, including gradient search and Lagrange optimization, the proposed approach had the advantage of faster convergence and the ability to handle generic network utility functions. Compared to centralized performance optimization schemes, the proposed approach significantly reduces communication overhead while achieving similar performance. The distributed algorithms for resource allocation and performance optimization provide analytical insights and important guidelines for practical design of large-scale video mesh networks. 1.3 Major Contributions of the Research This section presents several evolutionary schemes based on swarm intelligence which solve the different performance optimization problems over wireless sensor networks, including the nonlinear nonconvex optimization problems which can not be solved by the lagrange dual algorithm, meanwhile the proposed algorithms have fast convergence speed Convex Mapping of High-dimensional Resource Constraints There is a significant body of research work on performance optimization of wireless sensor networks, such as energy minimization, rate allocation and topology control [13, 14]. These results and algorithms are generic in their nature, and could be used to improve the performance of WVSNs. However, they have not considered the unique characteristics of WVSN, such as the complex nonlinear resource utilization behavior of each sensor node function, which will potentially render these analysis and algorithms inefficient or even impractical. In this dissertation, we first consider the unique characteristics of WVSN and develop an evolutionary optimization scheme using a swarm intelligence principle to solve the WVSN performance optimization problem. We transform the solution space set by the flow balance and energy constraints to a convex region in a low-dimensional space. Our analysis shows 5

22 that this transform can reduce the computational complexity and remove the interdependence between the control variables. We then merge the convex property of the new solution space with the original swarm intelligence principle to guide the movement of each particle which automatically satisfies the constraints during the evolutionary optimization process. Finally our experimental results demonstrate that the proposed performance optimization scheme is very efficient Distributed Optimization over Wireless Sensor Networks Recently, several distributed optimization algorithms based on gradient search have been proposed in the literature [9, 15, 10]. Most of existing approaches assume that the objective function to be additive and convex. Otherwise, it will be very difficult to assure convergence of the distributed gradient search algorithm, and will be sensitive to the local optima or saddle points. In addition, existing algorithms also suffer from slow convergence speed problem. In this dissertation, we develop an evolutionary distributed optimization scheme using swarm intelligence principle [16], called decentralized particle swarm optimization (DPSO), to solve the distributed WSN optimization problem. Based on the particle swarm intelligence principle, sensor nodes share information with each other through local information exchange and communication to solve a joint estimation or optimization problem. The proposed DPSO scheme has low communication energy cost and assures fast convergence. In addition, the objective function does not need to be convex. We use source location as an example to demonstrate the efficiency and evaluate the performance of the proposed DPSO algorithm. The proposed DPSO algorithm is a distributed algorithm with the most advantage that it will not be sensitive to the local optima or saddle points, and has very fast convergence speed compared with distributed gradient search algorithm. Our experimental results demonstrate that the proposed optimization scheme is very efficient and outperforms the existing distributed gradient-based optimization schemes. 6

23 1.3.3 Distributed Rate Allocation for Video Mesh Networks To successfully deploy the video mesh networking technology, there are a number of issues that need to be carefully investigated, including packet routing, flow control, Quality of Service guarantee, resource allocation, and performance optimization. Within the context of large-scale mesh networks, a distributed and asynchronous solution to the resource allocation and performance optimization is highly desired. This dissertation presents a distributed and asynchronous particle swarm optimization (DAPSO) evolutionary technique to optimize the network utility maximization problems. Unlike many network rate allocation and performance optimization algorithms which can only handle convex network utility functions, the proposed scheme is able to handle generic nonlinear noncovex network utility functions, and has very fast convergence speed compared with gradient based lagrange dual algorithm. For the DAPSO algorithm, we need to take two steps: (1) decompose the global resource parameters and network utility function; and (2) handle inter-dependent resource constraints, such as bottle neck links. We will use a specific rate allocation and performance optimization problem in wireless video sensor network as an example to demonstrate the efficiency and performance of the proposed scheme and compare its performance with other algorithms, such as gradient based lagrange dual algorithm. Our simulation results demonstrate that the proposed performance optimization scheme is very efficient and can significantly enhance the network utility Distributed Resource Allocation for Wireless Video Sensor Networks In the WVSNs, the two major operations on each video sensor are video compression and wireless transmission. And the wireless transmission is also restricted to the transmission bandwidth. In this dissertation, we will analyze the power-rate-distortion (P-R-D) module of the video encoding and its distortion performance, and will also analyze the transmission power consumption for the wireless video communication and its impact for the video 7

24 quality. Finally, based on these modules, we will approach a distributed and asynchronous algorithm for the energy efficient resource allocation and performance optimization over wireless video sensor networks. In this dissertation, we present an energy efficient distributed and asynchronous PSO (EEDAPSO) algorithm to solve the resource allocation and performance optimization problem over wireless video sensor network. We first decompose the original optimization problem into several sub-optimization problems, and next design the proper algorithm to handle the interdependent constraints and do the performance optimization. Compared with the centralized algorithm, our simulation results demonstrate that this proposed distributed and asynchronous resource allocation and performance optimization scheme is very efficient and it only needs very low communication cost Evaluation of PSO Algorithm Random Search Techniques are convergent algorithms for constrained nonlinear problems. Based on this, we provide the mathematical convergence analysis for PSO algorithm. We proved that PSO algorithm is a local convergence algorithm, which means that after predefined number of function iteration, all the solutions will converge to an optimum solution which is no guaranteed a global optimum. However, the experimental results based on many difficult optimization problems show that the large number of random solutions of the PSO algorithm that make up this technique converges to its global optimum in a good opportunity. Meanwhile, we compare PSO algorithm with genetic algorithm (GA), and Broydon-Fletcher-Goldfarb-Shanno (BFGS) quasi-newton algorithm for their global search capabilities based on a suite of difficult analytical optimization problems, the experimental results show that the PSO algorithm has the better convergence probability to the global optimum. 8

25 1.4 Dissertation Organization Following the Introduction, the dissertation is organized as follows. Chapter 2 first presents a background introduction and related works for network resource allocation optimization problems. Next introduces several previous optimization algorithms mostly used in the resource allocation optimization problems, including convex optimization, lagrange duality, gradient or subgradient methods. Meanwhile a brief discussion about these algorithms is given. An algorithm based on swarm intelligence, called particle swarm optimization (PSO), is also introduced. For PSO algorithm which has many attractive features including ease of implementation and the fact that no gradient information requirement, its social behavior and applications are also discussed. Chapter 3 first presents an evolutionary optimization scheme using swarm intelligence principle to solve the WVSN performance optimization problem. The algorithm transforms the original high-dimensional constrained optimization problem to a problem in the low dimension without constraints which can be solved by PSO algorithm. Next proposes a distributed algorithm, called decentralized PSO (DPSO), to solve the generic parameter estimation or performance optimization problems over WSNs. For the proposed DPSO scheme, there is no requirement for the objective function, and the scheme will not be sensitive to the local optima or saddle points. The source localization application demonstrates the efficiency of the DPSO algorithm. Chapter 4 presents a distributed and asynchronous particle swarm optimization (DAPSO) evolutionary technique to optimize the network utility maximization problems. The proposed DAPSO algorithm can solve the rate allocation and performance optimization problem for video communication over mesh networks very efficiently. The DAPSO algorithm is easy and powerful, has fast convergence speed and can also solve the generic nonconvex optimization problems efficiently which can t be solved by the traditional lagrange dual algorithm. Chapter 5 presents an energy efficient distributed asynchronous PSO (EEDAPSO) 9

26 algorithm to solve the resource allocation and performance optimization problem over wireless video sensor networks, including rate allocation and power management. The proposed EEDAPSO algorithm considers both encoding distortion and transmission distortion for the whole video quality over the WVSN, and solves the original optimization problem in a distributed and asynchronous way. Chapter 6 presents the convergence analysis for PSO algorithm based on random search techniques. PSO algorithm has been proved that it is a local convergence algorithm, but the large number of random solutions of the PSO algorithm make this technique converge to its global optimum in a good opportunity. Meanwhile, PSO algorithm is compared with genetic algorithm (GA), and Broydon-Fletcher-Goldfarb-Shanno (BFGS) quasi-newton algorithm for their global search capabilities based on a suite of difficult analytical optimization problems. Finally, Chapter 7 summarizes the major contributions of the research and provides some directions of possible future works. 10

27 Chapter 2 Background and Related Work This chapter presents a background introduction for network resource allocation optimization problems and discusses several optimization algorithms and their applications. First we present a brief introduction for network utility maximization (NUM), which is used for most of network resource allocation problems, and review the currently applications of NUM and also the resource allocation optimal problems in wireless mesh networks. Next we introduce several optimization algorithms mostly used in the resource allocation optimization problems, such as convex optimization, lagrange duality, and gradient and subgradient methods. And the flow control problem solved by using these kinds of algorithms is also discussed. An algorithm based on swarm intelligence, called particle swarm optimization (PSO), is introduced later. For PSO algorithm which has many attractive features including ease of implementation and the fact that no gradient information requirement, its social behavior and applications are also discussed. 2.1 Network Resource Allocation Most of the network resource allocation problems can be formulated as a constrained optimization problems of some network utility functions. Normally, there are three levels methods to efficiently solve a network resource allocation optimization problem. First is on theoretical properties, such as global optimality. It is well known as a convex optimization. Second is on computational properties, such as centralized algorithms. Third is on 11

28 decomposable properties, such as dual optimization. [17, 18] Overview Network utility maximization (NUM) problems provide an important approach to conduct network resource allocation, such as rate allocation or power management. The decomposability structure of network utility maximization leads to the most appropriate distributed algorithm for a given network resource allocation problem. The distributed solutions are particularly attractive in large scale networks, where the centralized solutions are infeasible, too costly, too fragile, and inapplicable [17]. Decomposition theory naturally provides the mathematical method to build the foundation for the distributed control of networks. This helps us obtain the most appropriate distributed algorithm for a given network resource allocation problem, such as distributed routing, scheduling to power control and congestion control. Based on Kelly s work [1] in 1998, the framework of NUM has found many applications in network resource allocation algorithms and network congestion control algorithms [2, 3, 4, 5], such as Internat rate allocation and TCP congestion control. Traffic from such applications is elastic, which is a typical example in TCP traffic over Internet. Other examples include the available bit rate (ABR) service, which enables maximal link utilization over ATM networks [19, 20, 21]. These work interpret source rates as primal variables and link congestion prices as dual variables to solve an implicit global utility optimization problem or lagrange dual problem. Some network flow control problems have been used in the context of congestion avoidance in multihop networks by using max-flow min cost theorems [22, 23]. Furthermore, allocation of limited network resources, such as network bandwidth, power control can also be formulated as basic NUM problems. Meanwhile, the framework of NUM has also been substantially extended from internet congestion control to a general approach of understanding interactions across layers. An optimization framework of layered structure is introduced in [24]. The cross-layer optimization can be applied to enable a clean-slate design of the protocol stack. The algorithm 12

29 obtains relations to different layers of protocol stack, and couples them through a limited amount of information passed back and forth. Different layers iterate on different subsets at different time scales using local information to achieve individual optimization. These local algorithms collectively achieve a global optimization objective. Many networks resource allocation problems can be formulated as a constrained maximization of some utility function. The key problem here is how the available bandwidth within the network should be shared between each controllable traffic. A distributed and asynchronous optimization scheme is particularly attractive in large-scale broadband networks where a centralized optimization scheme is infeasible, too fragile and too costly. There are many research work on network utility performance optimization on many applications using distributed optimization scheme. Primal and lagrangian dual algorithm [1, 25, 3, 4, 26] based on distributed scheme have been proposed to many applications to solve the utility function optimization problem. Standard textbook [27], [28] and [29] summarize the mathematics distributed computation of optimization techniques. In the dual based distributed algorithm, the lagrange dual variables can be interpreted as shadow prices for network resource allocation. Here, each link l calculates a price p l for a unit of bandwidth at link l based on the local source rate, and the source s is controlled by the total price p s = p l, where the sum is taken over all links that source s pathes. Then based on these, the source s chooses a transmission source rate x s to maximize its own benefit U s (x s ) p s x s. These kinds of algorithm require the utility function to be strictly convex or concave, where the convexity property readily leads to a distributed algorithm that converges to a globally optimal resource allocation problem. Normally, the video data processing utility function is nonconvex, these kinds of optimization problems can not be handled properly by previous distributed optimization algorithm. In addition, existing algorithms based on gradient projection also suffer from slow convergence speed problem. Most of the papers in the vast related on network resource allocation use a standard lagrange dual based distributed algorithm. While it is well known that dual based distributed algorithm needs utility function to be convex or concave. This is the major drawback 13

30 of dual base distributed algorithm when the application is inelastic or utility functions are nonconvex, which leads to divergence of congestion control Resource Allocation Over Mesh Networks Wireless Mesh Networks (WMNs) are formed by dynamically self-organized and selfconfigured wireless nodes that use multi-hop wireless relaying. WMNs have emerged as a key technology for next generation wireless networking. In the recent past, based on many advantages it has: (1) inexpensive network infrastructure, (2) easiness of implementation the network, and (3) broadband data support. WMNs attract significant research and inspiring numerous applications, including community meshes, vehicular platoons, and home entertainment networks [30, 31]. However, there are significant challenges in the deployment and optimization for efficient video streaming transmission over such wireless mesh network due to the network resources and dynamics. A wireless mesh network consists of a large number of wireless nodes spreading across a geographical area without the help of a fixed infrastructure. Each node in the network forwards packets for its peer nodes and each end-to-end flow traverses multiple hops of wireless links from a source to a destination. Because of its ad hoc nature, wireless mesh network can respond to conditions much more quickly and reliably than static networks. Ad hoc networks inherit the traditional problems of communication networks, such as bandwidth optimization, power control, and transmission quality enhancement. In wired networks, the flows only contend at the router that performs the flow scheduling. Compared with this, the multihop wireless networks show that the flow also contend at shared channel if they are within the interference ranges of each other. This presents the problem of optimizing resource allocation respecting to both resource utilization and fair across contending multihop flows [32]. Previous research use price-based strategy [1, 3, 6, 7, 8], where prices are computed to reflect relations between resource demands and supplies, to coordinate the resource allocations at multiple hops, and the objective functions here are needed to be strictly convex. 14

31 Their results show that the price is effective as a method to arbitrate resource allocation. The basic idea is to set prices on mutually contending links based on their congestion, and the goal is to allocate the transmission rates in such a way that whole networks utility is maximized. Here, the shadow price is associated with the links to reflect the relations between the traffic load of individual link and its bandwidth capacity, the utility is associated with an end-to-end flow to reflect its resource requirement. Transmission rates are chosen to respond to the aggregated price along every flow such that the whole benefits of flows are maximized. A distributed algorithm is obtained by using the lagrangian dual of the optimization problem and hence decomposing the problem. Some other research use gradient and subgradient search algorithm to handle the distributed optimization problem in sensor networks [9, 10]. Since deployment of such an ad hoc network is fast and flexible, it is very attractive to support real time media streaming over an ad hoc network. Recently there are many research interesting on video streaming over ad hoc networks [33, 34, 35, 36, 37, 38, 39]. When there are multiple video streaming in the network, these streams share and compete for the common resources, such as bandwidth and transmission power. Rate allocation in this case serves the purpose of resource allocation among these streams (see Fig.2.1). A video stream utilizes more network resources and achieves better video quality when it reaches a higher video source rate. The rate allocation algorithm should be fair and efficient among the streams. A distributed algorithm will be desirable and efficient because the whole computational burden is shared by all participating nodes, and the solution can be easily adjusted when the network conditions are fluctuant. 2.2 Previous Optimization Techniques In this section, we present several optimization algorithms mostly used in the resource allocation optimization problems, such as convex optimization, lagrange duality, gradient and subgradient methods, and a brief discussion is given. 15

32 Figure 2.1: Video Streaming over Ad hoc networks Optimization Algorithms I. Convex Optimization Since 1940s, a large effort has been into developing algorithms for solving various classes of optimization problems, analyzing their properties, and developing implementations. Mathematical optimization is used for many applications, such as an aid to a human decision maker, system designer, or system operator [40]. Convex optimization methods are widely used in the design and analysis of communication networks and signal processing algorithms. A convex optimization problem can be formulated as below [40]: min. f 0 (x) (2.1) s.t. f i (x) 0, i = 1,..., m where the functions f 0,..., f m : R n R are convex, satisfy f i (αx + βy) αf i (x) + βf i (y) (2.2) for all x, y R n and α, β R with α + β = 1, α 0, and β 0. 16

33 Convex optimization techniques are very important in engineering application, because a local optimum is also the global optimum [40]. This is the theoretical foundation for gradient search and lagrange dual algorithms used in distributed optimization. Convex optimization problems are the largest subset of optimization problems which are efficiently solvable, whereas nonconvex optimization problems are generally difficult. Currently there are many softwares which can generate accurate and reliable solutions without the headaches of initialization, step size selection, and the risks of trapping in a local optimum[41]. Once an engineering problem can be formulated into a convex optimization problem, it is reasonable to consider it solvable. II. Lagrange Duality Convex optimization techniques have useful Lagrange duality properties. The original optimization problem in Eq.(2.1) can be formulated to a lagrangian function as follow: m L(x, λ) = f 0 (x) + λ i f i (x) (2.3) where λ i is dual variable. The original optimization problem is referred as primal optimization problem, the socalled dual function g(λ) is defined as the minimum of the Lagrangian function i=1 g(λ) = min L(x, λ) (2.4) n x R Notice that, even if the original optimization problem is not convex, the dual function g(λ) is always concave because it is a pointwise minimum of a family of linear functions[41]. The dual variable λ is dual feasible if λ 0. For any primal feasible x and dual feasible λ, it turns out f 0 (x) g(λ). This means the dual function value g(λ) always is the lower bound of the primal function f 0 (x). The maximization lower bound can be obtained on the optimal value f of the original problem by solving the dual optimization problem: max. g(λ) (2.5) s.t. λ 0 17

34 Here the optimal value f is achieved at an optimal solution x, f = f 0 (x ). The basic idea of decomposability structures which lead to distributed algorithms are to decompose the original large optimization problem into distributively solvable subproblems which are coordinated by a high-level master problem by means of some kinds of signaling [27, 28, 29, 42], as illustrated in Fig Most of the existing decomposition techniques can be classified into primal decomposition and dual decomposition [17]. Figure 2.2: Decomposition of the original optimization problem. A primal decomposition is a approach since the original problem allocates the existing resources by directly giving each subproblem the resources that it can use. Dual decomposition method is that the original problem sets the price for the resources to each subproblem which can decide the amount of resources to be used. When the original problem has a coupling constraints, the dual decomposition can be formulated as follows: max. s.t. f i (x i ) (2.6) i h i (x i ) b, i x i X i where X i is the range x i should be inside. If the constraints in (2.6) are absent, then the problem is the primal decomposition problem. After using lagrangian duality properties, 18

35 the problem in (2.6) is changed to max. s.t. f i (x i ) λ T ( i i x i X i h i (x i ) b) (2.7) Then the dual problem is solved by updating the dual variable λ min. g(λ) = i g i (λ) + λ T b (2.8) s.t. λ > 0 where g i (λ) is the maximization value obtained from dual function by using lagrangian for a given λ. If the dual function g(λ) is differentiable, the dual problem in (2.8) can be solved by using gradient methods mentioned in the following section. III. Gradient and Subgradient Methods Lagrange duality properties can lead to decomposability structures [17]. After performing a decomposition, the objective function of the optimization problem may or may not be differentiable. Gradient and subgradient methods are a popular technique for iteratively optimization problems of differentiable and nondifferentiable functions. These methods are distinguished and very convenient because of their simplicity, little requirements of memory usage, and amenability for parallel implementation [43, 28]. Below is a general concave maximization problem over a convex constraint set: max. f 0 (x) (2.9) s.t. x S Both the gradient and subgradient projection methods generate a sequence feasible points {x(t)} as follow: x(t + 1) = [x(t) + α(t)p(t)] S (2.10) where p(t) is a gradient of subgradient of f 0 at x(t), α(t) is a positive step size, and [ ] S denotes the projection onto a feasible convex set S. However, there is a difference between 19

36 gradient and subgradient methods. Each iteration of the subgradient method may not improve the objective value as happens with a gradient method. For sufficiently small step size α(t), the distance of the current point x(t) to the optimal solution x decrease makes the subgradient method converge. There are many results on convergence of the gradient and subgradient methods with different choices of step size. For a constant step size α, more convenient for distributed algorithms, the gradient method converges to the optimal value provided that step size is sufficiently small, whereas for the subgradient method, it is guaranteed to converge to within some range of the optimal value, in other words, the subgradient method finds an ɛ suboptimal point within a finite number of steps. The major drawbacks for the gradient and subgradient methods are that they have slow convergence speed and are sensitive to the local optimum and saddle points if the objective function is not strictly convex or concave Applications I. Basic Model of Flow Control Let us consider a generic flow control optimization problem over a large-scale communication network with a set L = {1,..., L} logical links, each link with a capacity of c l, l L. The communication network can be wired or wireless. The whole network is shared by a set S = {1,..., S} sources. Each source s emits one flow and transmits at a source rate x s which satisfies m s x s M s (m s 0 and M s < are the minimum and maximum transmission rates), uses a set L(s) L of links in its path, and has a utility function U s (x s ). Each link l is shared by a set S(l) = {s S l L(s)} of sources. The flow control optimization problem can be formulated as the problem of maximizing the network utility s U s(x s ), over the S sources, subject to linear network flow constraints s S(l) x s c l. Using centralized scheme to solve this optimization problem would require to know all utility functions, which is not reality in many applications. The existing distributed optimization algorithms based on incremental subgradient method and traditional 20

37 lagrange dual distributed algorithm requires the utility function U(x) to be strictly concave or convex, which can not be satisfied to many reality applications. These kinds of algorithms need critical convexity properties to prove convergence to global optimum. The NUM problem optimization model can be formulated as max. s.t. U s (x s ) (2.11) s S(l) s S(l) x s c l, l = 1,..., L where U( ) is the utility functions. In general, it is a nonlinear, nonconvex function. Existing distributed optimization algorithms based on incremental sub-gradient search [9, 10] assume that the objective function U(x) is additive and convex, and the algorithms based on pricebased lagrangian dual flow control optimization [1, 25, 3] assume that the objective function is increasing and strictly concave. II. Dual Optimization A dual optimization is appropriate when the problem has a coupling constraint such that the optimization problem can be decomposed into several subproblems. Normally, dual optimization uses Lagrange duality properties. The basic idea in Lagrange duality is to relax the original optimization problem in(2.11) by transferring the constraints to the objective in the form of a weighted sum. Define the Lagrangian as: L(x, p) = U s (x s ) λ l ( x s c l ) s l s S(l) = (U s (x s ) x s λ l ) + (2.12) λ l c l s l L(S) l The optimization problem of the lagrangian dual problem is max L(x, p) = (U s (x s ) x s λ s ) + x s l s.t. x s M, s = 1,..., S λ l c l (2.13) where M = [m s, M s ] denotes the range which x s must lie in, and λ s = λ l. l L(s) 21

38 The dual problem is solved by updating the dual variable λ s : min λ D(λ) = s s.t. λ 0 D s (λ s ) + l λ l c l (2.14) where D s (λ s ) is the dual function obtained as the maximum value of the lagrangian solved in (2.13) for a given λ. This approach will have appropriate results only if strong duality holds, this means the original optimization problem should be convex. Here we can interpret λ l as the price per unit bandwidth at link l, then the λ s is the total price per unit bandwidth for all links in the path of source s, x s λ s represents the bandwidth cost to source s, and D s (λ s ) represents the maximum benefit for source s can achieve at the given price λ s [3]. The total price λ s for source s summarizes all the congestion information that s needs to know. For a given λ, a unique maximizer exists since original objective function is strictly convex. The important point here is for a given λ, individual source s can solve problem in (2.14) separately without the need to know the information of other sources. If the dual function is differentiable, then the dual problem in (2.14) can be solved by using the gradient projection method. The link price λ l is adjusted as follows: λ l (t + 1) = [λ l (t) α D λ l (x(t), λ(t))] (2.15) where α is a parameter for step size, [x] =max{x, β}, and β is the link price lower bound. D is continuously differentiable D (x(t), λ(t)) = λ l s S(l) x s (t) c l = x l (t) c l (2.16) where x l is the aggregate source rate at link l. Then substituting (2.16) to (2.15), we will get the price adjustment equation for link l L. λ l (t + 1) = [λ l (t) + α(x l (t) c l )] (2.17) From the demand and supply, if the demand for aggregate source rate at link l is less than the supply c l, then reduce price λ l, otherwise increase the price λ l. The adjustment for 22

39 price is completely distributed and can be implemented by only using local information. In each iteration, source s solves problem in (2.14) individually and propagates its result x s (λ) to all the links on its path. Meanwhile link l will update its price λ l based on current source rate demand and propagates the new price to sources path this link. The dual variable (price) λ will converge to the dual optimal after finite iterations and the primal variable (source rate) x will also converge to the optimal solution. The basic dual optimization algorithm can be summarized as follows: Step 1: Initializes λ l (0) to some positive value for all links. Step 2: Each source s locally solves problem when receives the total price λ s in its path and propagates this new source rate x s to the whole network. Step 3: Each link l updates its price λ l when receives all rates x s that go through this link and propagates the new price to all sources that path this link. Step 4: Go through Step 2 to Step 3 until the termination criterion is satisfied. In Low s work, he has already proof that when the following condition are satisfied: the original utility function is strictly convex, twice continuously differentiable, the curvatures of utility function are bounded away from zero and the time between consecutive updates is also bounded, then starting from any initial rates inside the range M = [m s, M s ] and prices λ(0) 0, every accumulation point (x, λ ) of the sequence (x(t), λ(t)) generated by the basic dual optimization algorithm mentioned above are primal optimal. Moreover, for all sources s, the error in price estimation and rate calculation both converge to zero, and the gradient estimation error is also converges to zero [3]. 2.3 Particle Swarm Optimization Particle swarm optimization (PSO), developed by Kennedy and Eberhart in 1995 [11, 12, 16], is a promising population-based new optimization technique which models the set of 23

40 potential problem solutions as a swarm of particles moving about in a virtual search space. Some of the attractive features of the PSO include the ease of implementation and the fact that no gradient information is required. It can be widely used to solve many different optimization problems, including most of the problems that can be solved by using Genetic Algorithms (GAs). Many popular optimization algorithms are deterministic, like the gradient-based algorithms. Compared with gradient-based algorithms, the PSO algorithm is a stochastic algorithm that does not need any gradient information. This allows the PSO algorithm to be used on many functions where the gradient algorithm is either unavailable or computationally to obtain. In the past several years, PSO algorithm has been successfully applied in many research and application areas. PSO is also attractive because that there are few parameters to adjust. It is demonstrated that PSO gets better results in a faster, cheaper way compared with many other methods PSO Algorithm The original PSO algorithm was based on the sociological behavior associated with birds flock. The method of PSO is inspired by the movement of flocking birds and their interactions with their neighbors in the group. Instead of using evolutionary operators to manipulate the individuals, like in other evolutionary computational algorithms, each individual in PSO in the search space with a velocity which is dynamically adjusted according to its own flying experience and its companions flying experience [44]. The high-level idea of PSO can be summarized as follows. To find the minimum of an objective function f(x) (x is a vector) within a solution space, the PSO algorithm starts with a set of candidate solutions called particles, {x p 1 p P } distributed in the solution space. A typical value of P is between 20 and 50 [16]. During the optimization process, each particle x p moves within the solution space in search for the minimum of f(x), and the corresponding movement path is denoted by x p (t), where t represents time. At each 24

41 time step, the movement of particle x p is given by x p (t + 1) = x p (t) + v (2.18) where v = w v + c 1 Θ 1 [x s p x p (t)] + c 2 Θ 2 [x g x p (t)] (2.19) Here, w, c 1 and c 2 are weighting factors, Θ 1 and Θ 2 are two random numbers, these parameters can influence the maximum step size that a particle can reach in a single iteration. x s p is the best solution that the particle itself has found so far, and x g is the best solution that all particles have found so far, the value of v should be clamped in the range [ v max, v max ] to reduce the likelihood that the particle might leave the search space. Fig. 2.3 gives the illustration of PSO algorithm. Each particle, when determining its next move in search for the global optimum, always balances the behaviors of its own and the group [16]. Figure 2.3: Illustration of particle swarm optimization. The PSO algorithm consists of repeated applications of the particle updated equation represented in Eq. (2.18). Fig. 2.4 gives a pseudo code example for the basic PSO algorithm. Note that the two if statements are the equivalent of applying Eq. (2.18) and Eq. (2.19) respectively. The initialization mentioned in the PSO algorithm includes the following: (1). Initialization of each particle x m (x m is a n-dimensional vector). Each coordinate particle x m 25

42 should be initialized to a value drawn from the uniform random distribution on the search space. This distributes the initial position of all the particles randomly throughput the search space. (2). Initialization of each v m (v m is a n-dimensional vector) of each particle x m. Initializes each v m to a value drown from the uniform random distribution on the interval [ v max, v max ]. Alternatively, the v m of each particle could be initialized to 0, since the starting positions of each particle are already randomize. The stop condition mentioned in the PSO algorithm depends on the type of problems being solved. Normally, the PSO algorithm runs for a fixed number of iterations or until a specified error bound is reached. Figure 2.4: Pseudo code for the basic PSO algorithm Social Behavior A number of scientists have created computer simulations of various interpretations of the movement of a bird flock or fish school. It does not seem a large leap of logic to suppose that some same rules underlie animal social behavior, including herds, school, flocks and that of humans[12]. Sociobiologist E. O. Wilson states that individual members of the school can 26

43 profit from the discoveries and previous experience of all other members of the school during the search for food, this advantage can become decisive, outweighing the disadvantages of competition for food item, whenever the resource is unpredictably distributed in patches[45]. We can achieve that social sharing of information offers an evolutionary advantage from this statement. This hypothesis is the fundamental to the development of particle swarm optimization[12]. The psychological assumptions of PSO theory are general and noncontroversial. In the search of consistent cognitions, individuals will tend to retain their own best beliefs, and will also consider the beliefs of other colleagues. Adaptive change happens when individuals perceive that others beliefs are better than their own. This concept is not new, what is new is that these simple concepts taken together create an evolutionary information processing technique which is powerful enough to manage the huge amount of information comprising human knowledge[46] Applications Currently, PSO has been applied to solve many problems. One of the first applications of PSO is in neural network introduced by J. Kennedy and R. Eberhart in One of the authors first experiments involved training weights for a three layers neural network solving the XOR problem [12]. Their results show that the PSO algorithm performs very well on this problem. Most of PSO applications are involved in neural networks. Currently there are many modified PSO approaches used on many aspects in neural network[47, 48, 49], including fuzzy neural networks, B-Spline for nonlinear system identification, feedforward neural networks with weight decay, and etc. Furthermore, paper[50] combines PSO algorithm and fuzzy neural network together to handle the financial risk early warning problem, which is the foundation of the effective risk management. Genovesi [51] uses PSO to handle the electromagnetic optimization problems. In this literature, the problem of synthesis of frequency selective surfaces is handled by simultaneously 27

44 optimizing both the real and binary parameters. PSO algorithm is used for design optimizations of electromagnetic devices in [52]. A modified PSO (Quantum PSO) algorithm[53] is applied to linear array antenna synthesis, which is one of the standard problems used by antenna engineers. PSO algorithm is also applied to the parameter extraction of an equivalent circuit model [54] recently. And PSO combined with adaptive simulated annealing with tunneling (ASAT) are applied to the RF circuit synthesis techniques in [55]. Another application for PSO algorithm is on sensor networks [56, 57, 58, 59, 60]. A sink node placement optimization problem is discussed in [56]. Target location estimation problem is discussed in [57]. Paper [58] uses PSO algorithm for cluster formation in wireless sensor networks, and paper [60] uses PSO algorithm for clustering node problems in ad hoc sensor networks. An optimization problem of multicast routing in sensor networks in discussed in [59]. And in [61], PSO algorithm is used to solve programming problems in MIMO-based cross layer design for sensor networks. 2.4 Summary Network resource allocation optimization has been widely used in Internet, TCP traffic and ATM performance optimizations. Most of the network resource allocation problems can be formulated as a constrained optimization problems of some network utility functions. Network utility maximization (NUM) has been used for most of network resource allocation problems, and there are many applications currently, such as resource allocation optimization problems over wireless mesh networks. Meanwhile, the framework of NUM has also been substantially extended from internet congestion control to a general approach of understanding interactions across layers. There are several basic optimization concepts most used in the resource allocation optimization problems, such as convex optimization, lagrange duality, gradient and subgradient methods. The basic network flow control problem has already been successfully solved by 28

45 lagrangian dual algorithm. The mainly drawback for this algorithm is that it needs the objective function to be strictly convex or concave, otherwise the optimization problem can not be handled properly. And for the gradient search algorithm, it also suffers from the slow convergence speed. For the PSO algorithm, it has many attractive features including ease of implementation and the fact of no gradient information requirement. The main advantage of PSO over other global optimization strategies is that the large number of random solutions that make up the PSO make the technique avoid dropping in the local optima or saddle points of the optimization problem. Meanwhile, unlike Genetic Algorithm (GA), PSO has no evolution operators such as crossover and mutation and also has fast convergence speed. 29

46 Chapter 3 Particle Swarm Optimization over Wireless Sensor Networks A wireless sensor network (WSN) is a system of spatially distributed sensor nodes to collect important information about the target environment. In this Chapter, we first consider the unique characteristics of WVSN and develop an evolutionary optimization scheme using a swarm intelligence principle to solve the WVSN performance optimization problem [62]. We transform the solution space set by the flow balance and energy constraints to a convex region in a low-dimensional space without constraints. We then merge the convex condition with the swarm intelligence principle to guide the movement of each particle during the evolutionary optimization process. Our experimental results demonstrate that the proposed performance optimization scheme is very efficient. Next, we develop an evolutionary distributed optimization scheme using swarm intelligence principle, called decentralized particle swarm optimization (DPSO), to solve the generic distributed WSN parameter estimation or optimization problem [63]. Based on the particle swarm intelligence principle, sensor nodes share information with each other through local information exchange and communication to solve a joint estimation or optimization problem. We use the swarm intelligence principle to guide the movement of each particle during the evolutionary optimization process. The proposed distributed optimization scheme has low communication energy cost and assures fast convergence. In addition, the objective function is not required to be convex. 30

47 We use source location as an example to demonstrate the efficiency of the proposed DPSO scheme. Our experimental results demonstrate that the proposed optimization scheme is very efficient and outperforms the existing distributed gradient-based optimization schemes. 3.1 Convex Mapping of High-dimensional Resource Constraints WSNs have been envisioned for a wide range of applications, such as battlefield intelligence, environmental tracking, and emergency response. Each sensor node has limited computational capacity, battery supply and communication capability [13]. A wireless video sensor network (WVSN) is a system of spatially distributed video sensors which capture, process and transmit video information over a wireless ad hoc network. In WVSN, each sensor, equipped with a video camera, is able to capture, process and transmit visual information about the circumstance activities. Compared to other conventional sensor networks, GPS and temperature sensor networks, the WVSN has the following unique characteristics: (1) The video sensor data is voluminous. (2) Video data compression is computationally intensive and energy consuming, which consumes about 60-80% of the total energy [64]. Therefore, the WVSN operates under severe bit and energy constraints. Performance optimization of a large-scale WVSN under resource constraints is a nonlinear, high-dimension, constrained optimization problem. There is a significant body of research work on performance optimization of wireless sensor networks, such as energy minimization, rate allocation and topology control [13, 14]. These results and algorithms are generic in their nature, and could be used to improve the performance of WVSNs. However, they have not considered the unique characteristics of WVSN, such as the complex nonlinear resource utilization behavior of each sensor node function, which will potentially render these analysis and algorithms inefficient or even impractical. 31

48 3.1.1 WVSN Operation Models We assume that the WVSN has K video sensor nodes {V i }, and has L communication links. Each node compresses the video sensor data and let the output bit rate be R i. Let P i be the energy consumption used in video compression. According to our previous work on energy consumption modeling and power-rate-distortion analysis for video compression [64], the coding distortion of the compressed video data is give by D(R i, P i ). Our experimental studies [64] suggest that the P-R-D behavior can be well captured by the following model D(R i, P i ) = σ 2 i 2 λr i g(p i ) (3.1) Here, σ 2 i represents the picture variance at node V i and λ represents the resource utilization efficiency of the video encoder. g(p ) is the power consumption model of the microprocessor. The analysis in [64] suggests that g(p ) = P 1 γ, 1 γ 3. We assume that the performance of the whole WVSN is measured by the overall video distortion D = K D(R i, P i ). (3.2) i=1 The power consumption in wireless transmission between nodes can be modeled as P t (i, j) = c i,j r i,j, where P t (i, j) is the power dissipated at node i when it is transmitting to node j, r i,j is the bit rate transmitted from node i to node j, and c i,j is the power consumption cost of radio link from node i to node j. The cost can be computed by c i,j = α + β d m ij, (3.3) where α is the distance-independent constant term, β is coefficient term associated with the distance-independent term, d ij is the distance between node i and j, and m is the path loss index. The power dissipation at a receiver can be modeled as [14]: P r (i) = ρ j i r ji (3.4) where j i r ji is the total bit rate of the received video data at node i and ρ is a constant. 32

49 3.1.2 WVSN Performance Optimization The WVSN performance optimization has to satisfy the flow balance and energy constraints. Let E i be the energy supply at node V i, and T the operational lifetime of the sensor network. Mathematically, the optimization problem can be formulated as min D = K D(R i, P i ) {R i,p i },{r ij } i=1 s.t. r ib + r ij r ji = R i, j i j i P i T + ρ r ki T + c ij r ij T + c ib r ib T E i, k i j i 0 r ib, r ij r max, 0 R i R max, 0 P i P max (3.5) where r ij and r ib are data rates transmitted from node i to node j and from node i to base-station B, r max, respectively [14]. R max and P max are the maximum possible values of rates and power. The first constraint is the flow balance constraint, stating that the total data transmitted from node i should be equal to total data received from other nodes plus the data generated by itself. The second constraint is the power consumption constraint, representing that the total energy used for data processing should not exceed the energy constraint. In the following, we develop a matrix representation of the performance optimization problem in (3.5). We assemble all the link rate variables {r ij } into a link rate vector r, and all the node bit rates {R i } into vector R, all the power consumption in video compression {P i } at each node into vector P. Let X = [r, R, P], which represents all the control variables that need to be determined by the performance optimization. Note that the flow balance and energy constraints are both linear. Therefore, the performance optimization problem in (3.5) can be written in the following matrix form min X J(X) s.t. M X = 0 N X E 0 0 X X max, (3.6) 33

50 where M represents the topology of the network, and N represents the energy cost for video transmission over the network. E 0 represents the initial power supply. We can see that X is a (2K +L)-dimension vector, M and N are both K (2K +L) matrices. This is a nonlinear, very high-dimensional, constrained optimization problem for a large-scale WVSN Transform of the Solution Space Handling the constraints is one of the central challenges in population-based evolutionary optimization, such as PSO. One of the existing approaches to handle constraints is as follows: move the particle (or compute the next generation) according to the swarm intelligence principle; check if the new position of the particle satisfies the constraints; if not, this movement of the particle is canceled and the particle stays in its current position. We observe that handling the constraints in the original space of vector X = [r, R, P] is very inefficient. First, in the original space, variables are interdependent. For example, if we change the coding bit rate of node i, then the transmission rates of its associated links, as well as the bit rates of its neighboring nodes will be affected, because they have to satisfy the flow balance constraints. Second, according to our preliminary studies, the probability of a solution vector [r, R, P] generated by swarm intelligence principle satisfying the constraints is extremely small. This will render the optimization algorithm very inefficient. In this paper, we use a linear representation of the solution space to define a linear transform, which is able to map the high-dimensional constrained original solution space into a convex region in a low-dimensional space without constraints. We then develop a new evolutionary optimization scheme using the swarm intelligence principle and the convex properties to solve the performance optimization in this low-dimensional convex space. I. Linear Representation of the Solution Space First, we consider the flow balance constraint M X = 0. This constraint implies that 34

51 the vector X must be in the null space of matrix M. Therefore, X can be represented by n 1 X = y i g i = GY, (3.7) i=1 where G = [g 1, g 2,..., g n1 ] are the orthonormal basis vectors of the null space and Y = [y 1, y 2,..., y n1 ] t are coefficients. We can see that n 1 = (2K + L) rank(m). Second, we consider the energy constraint. We assume that the optimum performance is achieved when all energy is used, i.e, N X = E 0. According to (3.7), we have N G Y = E 0. (3.8) We suppose X 0 is one specific solution (not necessary the minimum solution) which satisfies the flow balance and energy constraints. Therefore, X 0 must be in the null space of M. Let X 0 = GY 0 Since N G Y 0 = E 0, Therefore, N G (Y Y 0 ) = 0. (3.9) This implies that Y Y 0 is in the null space of matrix N G. Let H = [h 1, h 2,..., h n2 ] be the orthonomal basis vectors of the null space. Therefore, we have n 2 Y Y 0 = z i h i = HZ. (3.10) II. Transform of the Solution Space i=1 Eqs. (3.7) and (3.10) tell us that if a vector X satisfies the flow balance and energy constraints, then it can be represented in a low-dimensional space by vector Z. Since G and H are both orthonormal matrices, from Eqs. (3.7) and (3.10), we can see that Z = H t G t [X X 0 ]. (3.11) This defines a transform from the original solution space S = {X X satisfies all the con straints in (3.6)} to a new space S : F : X Z = H t G t [X X 0 ]. (3.12) 35

52 The new solution space is given by S = {Z 0 F 1 (Z) X max }. (3.13) This transform approach is very important and has the following advantages. Lemma 1 The dimension of the new space S is much lower than the original space S. The transform is able to reduce number of dimensions by up to 2K, where K is the number of sensor nodes. Proof The matrices G and H can be obtained from the singular value decomposition of matrices M and NG. Therefore, n 1 = (2K +L) rank(m) and n 2 = (2K +L) rank(m) rank(ng). Therefore, the number of reduced dimensions is rank(m) + rank(ng). If the sensor network is not ill-conditioned, such as there is no orphan nodes, rank(m)+rank(ng) should be close or equal to 2K. A detailed proof of these arguments is omitted here due to page limitation. Lemma 2 The new space S is convex. Proof Notice that the transform F is linear, and the original space S is also convex. Therefore, S is also convex. Figure 3.1: Performance optimization with swarm intelligence and convex projection Performance Optimization Using PSO 36

53 Based on the swarm intelligence principle and the unique properties given by Lemmas 1 and 2, we are going to develop a new evolutionary optimization scheme for WVSN performance optimization. The new swarm optimization will operate in the new solution space S. According to Lemma 1, this will significantly reduce the computational complexity. In addition, in the new space, since the flow balance and energy constraints are implicitly satisfied, the particles can move freely in S. To make sure that each particle move within the convex region of S, we can use the property that S is convex. This implies that if two particles X 1 and X 2 are in S, then λx 1 + (1 λ)x 2 S, 0 λ 1. (3.14) In PSO, the movement of each particle is defined by (2.18) and (2.19). If we choose the weights such that w + c 1 Θ 1 + c 2 Θ 2 = 1, (3.15) then (2.19) can be written as x m (t + 1) = w x m + c 1 Θ 1 x m + c 2 Θ 2 x. (3.16) Here, x m = x m (t) + [x m (t) x m (t 1)], (3.17) is the predicted position of x m (t) according to the inertial velocity. In (3.21), we know that x m and x are in S because they are the best positions that have been found so far by the particle itself and the whole group. However, x m is not necessarily in S. If not, we find the convex projection for x m as follows. Let {x i 1 i I} be the set of all particles, including the initial particles and their moving histories. We know that these particles are all in S. Therefore, according to the convex condition, their weighted sum is also in S. Let I [p 1, p 2,...p I] = arg min p i x i x m 2, (3.18) {p i } where 0 p i 1 and I p i = 1. Then, x m = I p ix i S which is the convex projection of i x m. In (3.21), we replace x m by x m, then the new position of the m-th particle x m (t + 1) i i 37

54 must be in S according to the convex condition. Using this convex projection method, we can make sure that each particle moves within the solution space Experimental Results We test the proposed performance optimization scheme with different WVSN topology. In the following experiment, the network has K = 10 nodes and L = 15 links. The particle population size is 20, and the maximum generations is The other parameters are set as follows: λ = and γ = 0.5, α = 50, β = , m = 2, ρ = 5, λ k = and σ 2 k = 10. Fig. 3.2 shows that the overall video distortion metric quickly reaches to its minimum after several generations. Fig. 3.6 shows the movement path of each particle in the solution space (projected onto a 2-D plane for illustration purpose). After all the particles converge, an optimum solution is found. We can see that the proposed performance optimization works very efficiently. Our experimental results with other settings of WVSN yield similar results Minimum Funcation Value Iteration Number Figure 3.2: The performance metric decrease as the particles update their positions of PSO with convex mapping. 38

55 Dimension Dimension 1 Figure 3.3: The traces of the all particles moving in the solution space of PSO with convex mapping Summary In this section, based on the unique properties of WVSN and the swarm intelligence principle, we have developed an evolutionary optimization scheme to solve the nonlinear constrained performance optimization problem in WVSN. This section focuses on the analytic development of the performance optimization method. The most challenge for PSO algorithm is handling constraints. Our analysis shows that we can transform the solution space into a convex region in a low-dimension space to reduce the computational complexity and remove the interdependence between the control variables. The convex property of the new solution space can be incorporated in the original swarm intelligence principle to guide the movement of particles such that each particle in the new generation automatically satisfies the constraints. 39

56 3.2 Distributed Optimization over Wireless Sensor Networks A major challenge in WSN system design is data collection and analysis. Since each node has very limited resources for communication and computation, it is inefficient or even impractical to transmit all sensor data to a central location for information processing, such as estimation and optimization. Therefore, distributed in-network signal processing and optimization are highly desired, especially in large-scale WSN systems [9]. Recently, several distributed optimization algorithms based on gradient search have been proposed in the literature [9, 10]. Most of existing approaches assume that the objective function to be additive and convex. Otherwise, it will be very difficult to assure convergence of the distributed gradient search algorithm. In addition, existing algorithms also suffer from slow convergence speed problem. In this section we present a new distributed scheme, called decentralized PSO [63], which will not be sensitive to local optimum or saddle points and has fast convergence speed Optimization Problems Using Decentralized PSO Let us consider a generic parameter estimation or optimization problem over a WSN with N sensors and each sensor taking M measurements about the target phenomenon. The problem is to estimate a p 1 vector parameter θ R p from M independent measurements x collected by n distributed selected sensors. The model is min f(x, θ) s.t. θ R p (3.19) where f( ) is a cost or utility functions. In general, it is a nonlinear function. Existing distributed optimization algorithms based on incremental sub-gradient search assume that the objective function f(x, θ) is additive and convex. However, in this work, there are no such requirements and f(x, θ) can be a generic nonlinear function. 40

57 Our major idea in DPSO can be summarized as follows. Suppose there are K sensor nodes which are involved in the distributed optimization or estimation task. We partition the swarm particles (typically, in the range of particles) into K sub-groups with each node managing a sub-group of particles. More specifically, it is tasked to evaluate the objective function for each particle in its sub-group and manage their movements based on the swarm intelligence principle described in Eq. (2.19). After each iteration, the sensor node finds the minimum solution within its sub-group and shares this sub-group minimum with its neighboring nodes. Upon receiving the external information from its neighbors about their sub-group minimum, each sensor node uses these external information to guide the particles inside its sub-group so as to facilitate the search and optimization process. The decentralized PSO algorithm can be stated as follows: For each particle cluster { For each particle { Repeat initialized particle until it satisfies all the constraints}} Do { Compare the received other cluster best solution with local cluster best solution, update the local solution with the best one. For each particle { Use normal PSO algorithm to find particle itself best solution and find new cluster best solution among all cluster particles.} Transmit new cluster best solution to other particle clusters. }While max iterations or convergence criteria is met. 41

58 In each particle cluster, the movement of each particle is defined by (2.18) and (2.19). Here we choose the weights such that w + c 1 Θ 1 + c 2 Θ 2 = 1 (3.20) then (2.19) can be written as x m (t + 1) = w x m + c 1 Θ 1 x p + c 2 Θ 2 x g (3.21) where, x m = x m (t) + [x m (t) x m (t 1)] (3.22) is the predicted position of x m (t) according to the inertial velocity Source Localization Application In this section, we will use source location as an example to evaluate the performance of the proposed DPSO algorithm. Location estimation of an acoustic source is an important problem in both environmental and military applications [65]. Source localization problem has been traditionally solved through nonlinear least-squares estimation, which is equivalent to the maximum likelihood estimation when the observation noise is modeled as a white gaussian noise [66]. In this source localization scenario, sensors near the source or target are waken up and each sensor collects M measurements about the source location using received signal strength. In the sensor network field, the acoustic source is located at an unknown position θ. We assume that each sensor knows its own location r i, i=1,...,n. And the received signal strength measurement model for the j-th measurement, j=1,...,m, at node i is given by the following signal propagation model [65] x i,j = A θ r i β + ω i,j (3.23) where A is a constant, β is the signal energy decay exponent and ω i,j are independent Gaussian noise at each sensor i for every measurement j. 42

59 A maximum likelihood estimate ˆθ is the global minimum formulated as ˆθ = arg min θ 1 n M n i=1 M A (x i,j θ r i β )2 (3.24) j=1 Sheng s work uses Maximum likelihood estimator to solve the acoustic source localization problem in sensor network [67], but this approach requires sensors to transmit all their data to a fusion center for processing, which is impractical because of requiring massive amount energy and bandwidth for a large network. In Rabbat and Nowak s work [9, 15], they propose gradient based distributed algorithm to solve the problem. The advantage of this distributed algorithm is that it dose not need all the data transmitted to a central point for processing compared with algorithm provided in [67]. But the drawback of this distributed algorithm is that it is sensitive to local optimum and saddle points. In the decentralized PSO algorithm, we assume each sensor knows all other sensors measurements and positions information after initial communication. In each particle cluster, it finds its own cluster best estimation parameter. And during the estimation process, each sensor node only transmits local best estimation parameter through communication links. After communicating with other particle clusters and updating each cluster s best estimation parameter, all particles will find the best solution when all the particles converge. Compared with gradient based distributed algorithm, the decentralized PSO algorithm will not suffer from local optimum or saddle points Experimental Results We test the proposed distributed optimization scheme using decentralized PSO with different WSN topologies. In the following experiment, a sensor network was constructed by placing 100 sensors uniformly distributed in a sensor field. Here in our experiment, one sensor field has a size of unit square. Sensor location is denoted by r i = (x, y). We randomly select n sensors to wake up to collect measurements and estimate source location. In total, there are K links between these sensors. These links should let all the selected sensors to join the communication. The other parameters are set as follows: M = 10, 43

60 A = 100, and β = 2. The particle cluster size selects 2, we only assign 2 particles for each sensor. All sensors make measurements at a signal-to-noise ratio (SNR) of 3 db. Fig. 3.4 depicts an example network topology using 20 sensors and 22 links. Fig. 3.5 shows that the overall minimum objective function quickly reaches to its minimum after several iterations with DPSO. Fig. 3.6 shows the movement path of each particle in the sensor field. Here in each particle sub-group, we only trace one particle s moving path. From these experimental results, we can see that the proposed scheme works very efficiently. Our experimental results with other settings of WSN with different wake up sensor number and link number yield similar results. Due to page limitation, they are omitted here Sleep Sensor Wake Up Sensor Target Particle Y Position X Position Figure 3.4: A WSN topology with 20 sensors and 24 links. In the sensor network which has fixed number of wake up sensors, when we increase the number of communication links in the network, we will reach faster convergence. Fig. 3.7 shows the experimental results for one sensor network with 20 sensors waken up. As we can see that as the number of communication links increases, the sensor nodes are able to share information more efficiently, therefore speed up the DPSO convergence process. As a 44

61 Minimum Function Value Iteration Number Figure 3.5: The performance function value decrease as the particles update their positions of decentralized PSO. result, the number of iterations that are needed for reaching the minimum reduces Comparison with Gradient Search Algorithms In this section, we compare the proposed DPSO with distributed gradient search algorithms proposed in the literature [9, 15]. We will first give a short review about distributed incremental gradient algorithm [9, 15]. This distributed incremental algorithm is a cycled parameter estimation. On the k-th cycle, sensor i (i = 1,..., n) receives an estimation ψ k i 1 from its neighbour and updates as following: ψ k i = ψ k i 1 α f i (ψ k i 1) (3.25) where α is a positive step size, and f i (ψ k i 1) represents the gradient of f i at ψ k i 1 f i (ψ) = 1 m m A (x i,j ψ r i β )2 (3.26) j=1 At the beginning, sets arbitrary initial condition ψ 0 0 = ˆθ 0, and we will have ˆθ k = ψ k n after a 45

62 Y Position X Position Figure 3.6: The traces of the particles moving in the sensor field of decentralized PSO. complete cycle through the network. Each subiteration only focuses on optimizing a single function f i, which only depends on local data at sensor i. The minimum objective function values found by the decentralized PSO and gradient search algorithms were recorded as a function of the iteration number of the search process. The comparisons are based on the same sensor network topology with the same number of waken-up sensors and communication links. The experimental results are based on the average of 100 experiments Fig 3.8 shows the comparison results of DPSO and gradient search algorithms on optimization convergence. From the experimental results, we can see that the decentralized PSO scheme is more efficient than the sub-gradient search algorithm Summary In this section, we have developed an evolutionary optimization scheme, called decentralized PSO (DPSO), to solve the WSN distributed generic parameter estimation or optimization problem based on swarm intelligence principles. The basic operation involves 46

63 Iteration Number Communication Link Number Figure 3.7: Optimization convergence of WSN with 20 sensors wake up and different link number in decentralized PSO. parameter estimation in each sensor and shares its own local results with its neighbors. The proposed DPSO algorithm does not have requirement for the objective function, is not sensitive to the local optimum or saddle points, and has very fast convergence speed compared with gradient search method. Simulation results show that our evolutionary optimization scheme is very efficient for different network topology. 47

64 Decentralized PSO Subgradient Search Minimum Function Value Iteration Number Figure 3.8: Comparison of decentralized PSO and subgradient search algorithm on optimization convergence. 48

65 Chapter 4 Distributed Rate Allocation for Video Mesh Networks Rate allocation and performance optimization problem is the most important aspect in multi-hop video mesh networks. Video communication over mesh networks has found many important applications, including webcam video communication over Internet, video surveillance, and wireless vision sensor networks. We consider a large number of video sessions sharing a mesh network. Since video streaming is bandwidth-demanding while the network has a limited bandwidth resource, it is important for us to optimally allocate this critical network resource among video sessions to maximize the overall system performance or network utility. This optimization is often a high-dimensional nonlinear constrained optimization problem. For large-scale networks, a centralized and synchronous solution to this rate allocation and performance optimization problem is too costly, non-scalable, fragile, and even infeasible in many cases. In this chapter, based on a swarm intelligence principle, we develop a distributed and asynchronous particle swarm optimization (DAPSO) scheme for distributed rate allocation and performance optimization [68]. Specifically, we study the problem of decomposition of a network utility function with global resource parameters and inter-dependent resource constraints into local optimization problems, each being solved by particle swarm optimization. We develop in-network fusion and particle migration schemes to exchange information between neighboring PSO modules and coordinate 49

66 their search behaviors and optimization processes. We also develop a collaborative resource control scheme to efficiently handle network bottleneck issues. Our extensive simulation results demonstrate that the proposed DAPSO algorithm is very effective for distributed rate allocation and performance optimization. The proposed optimization framework is generic and can be extended to other network resource allocation and performance optimization problems with slight modification. 4.1 Introduction A wireless mesh network (WMN) is a set of fixed and/or mobile nodes that self-assemble into a dynamic multi-hop ad hoc network [69]. In this work, we study collaborative video communication over a large-scale mesh network where a large number of sender devices transmit compressed video data, either storage (pre-compressed) or live video data, to a large number of receivers through multi-hop transmission with packet relay. The communication link between two neighboring network nodes can be either wired or wireless. This type of video mesh networking technology is found in many important applications, such as webcam video communication over Internet / enterprise / community networks, video surveillance, and image/video sensor networks [69, 70, 30]. To successfully deploy the video mesh networking technology, there are a number of issues that need to be carefully investigated, including packet routing, flow control, Quality of Service guarantee, resource allocation, and performance optimization [69]. In this work, we focus on rate allocation and performance optimization. Video streaming is bandwidthdemanding while the network has a limited bandwidth resource, it is important for us to optimally allocate this critical network resource among different video sessions so as to maximize the overall system performance or network utility. This optimization is often a high-dimensional nonlinear constrained optimization problem. We propose to develop a distributed asynchronous optimization scheme to solve the rate allocation and performance optimization problem. The proposed optimization framework is generic and can be extended 50

67 to other network resource allocation and performance optimization problems with slight modification Related Work Rate allocation has been extensively studied in rate-distortion analysis and quality optimization for point-to-point video communication [71, 72]. It has also been considered in the broad context of resource allocation [14]. Although a number of cross-layer resource allocation and performance optimization schemes have been developed, they mostly focus on centralized performance optimization for single-stream video communication over networks with relatively simple topologies (e.g. a chain topology) [70, 73]. Within the context of large-scale mesh networks, especially video mesh networks, the network resource allocation and performance optimization is often a high-dimensional constrained nonlinear optimization problem with a large number of resource parameters (optimization variables) [17]. In this case, a distributed and asynchronous solution to the resource allocation and performance optimization is highly desired [17]. This is because, first, in large-scale mesh networks, communicating the information that is required by each step of the optimization algorithm from every network node to a central location often involves a significant communication overhead and a large communication delay. However, a distributed solution only requires local information exchanges. Second, because of the large communication delay in gathering global information, a centralized rate allocation algorithm is often not able to quickly respond to local changes in network conditions and time-varying video data characteristics. However, a distributed approach has the advantage of quick response to local changes. Third, a distributed solution is scalable. The resource allocation and performance optimization procedure can be easily extended when new nodes are added into the mesh network. Therefore, a distributed and asynchronous optimization scheme is particularly attractive in large-scale networks [74, 17]. A number of distributed network utility optimization schemes based on prime and Lagrangian dual and gradient search have been developed in the literature [1, 9, 10, 25, 3, 26, 51

68 38, 75]. For those dual-based distributed algorithms, the Lagrange dual variables can be considered as prices for network resource allocation [1, 25, 3]. Most of existing approaches, especially those based on incremental sub-gradient search [9], assume that the objective utility function to be additive and convex. Convex functions have unique properties. A local minimum is also a global minimum and the duality gap between the prime and dual optimization problems is zero [17]. For nonconvex objective functions, it will be very difficult to assure convergence of the distributed optimization process. However, in many scenarios of resource allocation and performance, especially in video communication over mesh networks, the utility function is often non-convex. In some cases, the utility function even has no explicit expression due to the inherent difficulty in its mathematical modeling and we can only obtain its function value for a given set of independent variables. How to develop a distributed and asynchronous scheme to optimize a generic nonlinear objective function over large-scale networks remains an open and challenging problem. In addition, with the context of rate allocation and performance optimization for large-scale video mesh networks, some important issues, such as network bottleneck links and their impact to the overall video streaming and fast response to time-varying video characteristics, have not been sufficiently addressed. However, within the context of video communication over networks, the relationship between the video quality of service metric (or system performance metric) and resource utilization parameters are often nonlinear and complex. Therefore, there is a need to develop a distributed asynchronous optimization algorithm which is able to handle generic nonlinear network utility functions. In general, the rate allocation and performance optimization for video communication over network problem is a nonlinear, nonconvex optimization problem. Existing distributed optimization algorithms based on incremental sub-gradient search [9] [10] assume that the objective function U(x) is additive and convex, and the algorithms based on price-based lagrangian dual flow control optimization [1] [25] [3] assume that the objective function is increasing and strictly concave. Distributed optimization scheme proposed in [38] for multiple video streams in networks can still only handle the convex 52

69 optimization problem Major Contributions In this paper, based on a swarm intelligence principle [16], we develop a distributed asynchronous particle swarm optimization (DAPSO) scheme to solve the rate allocation and performance optimization problem for large-scale video mesh networks. The major contributions of this work include: (1) we explore the idea of particle swarm optimization which provides a natural and ideal platform for distributed resource allocation, collaborative video communication, and network performance optimization. (2) We develop simple yet efficient schemes for local DAPSO modules to exchange information which enable the fast convergence of DAPSO. (3) We develop a simple yet efficient scheme to address the network bottleneck issue in distributed rate allocation and performance optimization. (4) Unlike many network resource allocation performance optimization algorithms in the literatures which are able to handle convex network utility functions, the proposed optimization framework is generic, is able to handle nonconvex network utility functions, even does not require their specific expressions, and can be extended to other network resource allocation and performance optimization problems with slight modification. 4.2 Resource Allocation for Video Mesh Networks In this section, we formulate the resource allocation problem for video mesh networks and then discuss a generic distributed solution for this type of problems Formulation of Generic Resource Allocation Problems In this work, we model the mesh network as a graph with V network nodes V = {1, 2, V } and L logical links L = {1, 2, L}. The mesh network is shared by a set of video transmission sessions (or streams), denoted by S = {1, 2,, S}, as illustrated in Fig In a large-scale mesh network, for example, a community or enterprize network 53

70 supporting online video chatting or conference services, there could be a large number of simultaneous video sessions crossing over the mesh network and sharing communication links. Figure 4.1: Illustration of video communication over mesh networks. In performance optimization of video mesh networks under resource constraints, all network nodes need to collaborate in video streaming and network resource utilization so as to maximize the overall system performance under resource constraints. Let X = {x 1, x 2,, x N } be the set of resource parameters. Example resource parameters include encoding bit rate of a video stream, link transmission rate, transmission power, and delay bound 1 [70, 76]. There are two types of resource parameters, local and global resource parameters. A local parameter is associated with a specific network node or link, or a local neighborhood of nodes or links. For example, data processing and transmission power are two local resource parameters. A global parameter involves a set of networks nodes or links that spatially distribute over the network. For example, the encoding bit rate of a video streaming is a global parameter since a video stream involve a series of network nodes and link along its transmission path. The network operates under resource constraints. There are two types of constraints: 1 In video communication over multi-hop networks, delay is also an important resource since, with a large delay bound, the network has more flexibility in scheduling so as to improve its data communication efficiency. 54

71 independent constraints, such as the energy constraint, which are uniquely associated with one specific node or link; and inter-dependent constraints, such as flow balance constraints, which are often associated with a neighborhood of network nodes or links and different constraints are inter-dependent of each other. As in the literature, a network utility function, denoted by U, is used to describe the overall system performance. Since the overall system performance depends on the configuration of resource parameters, therefore, U is a function of {x 1, x 2,, x N }, denoted by U(x 1, x 2,, x N ). Now, the problem performance optimization under resource constraints can be formulated as max U(x 1, x 2,, x N ) (4.1) s.t. independent constraints interdependent constraints. In the following, we will use rate allocation as an example to show how a practical resource allocation problem could be fitted into the formulation in (4.1) Basic Framework for Distributed Resource Allocation Distribution of computation implies decomposition. More specifically, we need to decompose the global optimization problem with a global network utility function (objective function) and all resource constraints into a set of local optimization problems with a local objective function and local resource constraints, as illustrated in Fig There are two basic decomposition methods, primal and dual decomposition. The former is based on decomposing the original primal problem, whereas the latter is based on decomposing the Lagrangian dual problem. In dual decomposition, a pricing method is often used to coordinate the resource utilization between different optimization modules [17]. After decomposition, neighboring optimization modules are allowed to exchange status 55

72 information through local communication. In optimization over networks, it is highly desirable that the optimization modules operate in an asynchronous manner. More specifically, each local optimization module immediately moves to its next optimization step / iteration using the available status information received so far from neighboring nodes instead of waiting for synchronous status information exchange with other nodes. If a distributed asynchronous optimization scheme is effective, after a number of rounds of local optimization and information exchange, the overall system performance should approach the optimum obtained by global optimization. If not possible, at least a sub-optimum should be achieved. From (4.1), we can see that the major task in decomposing a global constrained optimization problem into a set of local optimization modules is to decompose the network utility function with global resource parameters and handle those inter-dependent resource constraints. To decompose the global utility function, our basic idea is to introduce a set of local resource parameters to replace the global ones and decompose the network utility function into local ones which only have local resource parameters. In doing so, we need to make sure theoretically and/or experimentally that the local network utility functions converge to the global optimum during distributed optimization. To handle those inter-dependent resource constraints, our basic idea is to use local communication between optimization modules to exchange information and let them negotiate with each other to make sure that the inter-dependent resource constraints are satisfied during distributed optimization. In the following sections, based on this observation and a swarm intelligence principle [16], we will develop a distributed asynchronous scheme called DAPSO for rate allocation and performance optimization for video mesh networks. 4.3 Distributed Rate Allocation for Video Mesh Network In this section, we first formulate the rate allocation problem within the basic framework of (4.1). We discuss how this problem can be decomposed into local optimization problems 56

73 Figure 4.2: Illustration of distributed asynchronous optimization. to be solved using DAPSO Problem Formulation In rate allocation, the major resource constraint is in the form of link capacity and the resource parameters are the bit rate of each video stream. While determining the bit rates of video streams in order to maximize the overall video communication performance, we need to make sure that the aggregated bit rate of all video streams on every link of the network does not exceed its link capacity [2, 75]. More specifically, suppose we have N video streams (sessions) that are sharing communication links and crossing over the network. Let x n X = {x 1, x 2,, x N } be the bit rate of video stream n. Note that x n is a global resource parameter. Let X L [l] = {x n X video stream n uses link l}. (4.2) which is the subset of video sessions that use link l. Let C l be the link capacity. Here, the link capacity can be considered as the average number of information bits per second that can be successfully transmitted over the link. A number of factors, including SIR (signal-interference ratio), transmission distance, communication protocols, forward error correction (FEC) schemes, and packet re-transmission, 57

74 contribute to this link capacity [77]. According the link capacity constraint, we have x n C l, 1 l L. (4.3) x n X L [l] In the following, we define the network utility function for video communication over mesh networks. The ultimate goal in video communication system design is to provide users with the best-quality videos. Therefore, the system performance should be measured by the overall video quality. A commonly used metric for measuring the quality of a single video stream is the encoding distortion. As suggested by the literature on rate-distortion (R-D) modeling for video coding [72], we use the following R-D model D(x n ) = σ 2 n 2 λx n, (4.4) to describe the relationship between video coding distortion D and source rate x n for video stream n. Here, σn 2 represents the picture variance. More specifically, within the context of motion prediction based video coding, it is the variance of difference picture after motion compensation. λ is an encoder-related parameter. It should be noted that in this work we just use the R-D model in (4.4) as an example to demonstrate the proposed DAPSO algorithm. Certainly, this model can be replaced by any other more accurate R-D model developed in the literature [72, 78]. As we can see from the following sections, the optimization procedure of the proposed DAPSO algorithm does not depend on the specific expression of the R-D model. Within the context of video communication over mesh networks with a large number of simultaneous video streams, we need establish a performance metric function, or a network utility function, to characterize the overall system performance. One commonly used measure to describe the overall video quality of multiple video streams is the aggregated video distortion, i.e., N N U(x 1, x 2,, x N ) = D(x n ) = σn 2 2 λx n. (4.5) n=1 n=1 At this moment, we consider stationary video sources and assume that σn 2 is constant. As noted by a number of researchers, when characterizing the overall quality over multiple 58

75 video streams, besides minimizing the aggregated video distortion, we also need to minimize the quality variation between different video streams. From a user s perspective, minimizing the quality variation is also a very important part of maintaining the fairness among users and different video services. By incorporating the quality variation into the network utility function, we have U(x 1, x 2,, x N ) = N D(x n ) + n=1 N n=1 D(x n ) D, D = 1 N Now, the rate allocation and performance optimization can be formulated as [ ] N min U(x 1, x 2,, x N ) = D(x n ) + D(x n ) 1 N D(x n ) N s.t. x n X L [l] n=1 n=1 N D(x n ). (4.6) n=1 (4.7) x n C l, 1 l L. (4.8) x n 0, 1 n N. (4.9) Here, the resource constraints in (4.8) are inter-dependent since they involve global resource parameters in X, while those in (4.9) are independent. From this rate allocation example we can see that the network utility function in video communication over networks U(x 1, x 2,, x N ) is often a nonlinear nonconvex (or nonconcave) function. For large-scale networks, the performance optimization problems in (4.1) and (4.7) are often a high-dimensional nonlinear nonconvex constrained optimization problems. Existing methods developed for convex optimization and existing algorithms for distributed rate allocation, flow control, and resource allocation, such as distributed gradient based lagrange dual algorithm, cannot be applied. In this work, based on a swarm intelligence principle, we develop a distributed and asynchronous particle swarm optimization (DAPSO) algorithm to solve the constrained nonlinear rate allocation problem in (4.7) Decomposition We propose to decompose the global optimization problem in (4.7) into a set of local optimization modules, each of which is associated with a communication link. More specifically, let L = {1, 2, L} be the set of links that are involved in the multiple-session video 59

76 communication, as illustrated in Fig In total, we have L local optimization modules. For each link, which hosts a local optimization module, we define a set of local resource parameters, X l = {x nl }, where x nl represents the bit rate of video session n at local optimization module l. We define the l-th local optimization module to be max X l U l (X l ) = s.t. n X L [l] x nl 0, n X L [l] D(x nl ) + n X L [l] D(x nl ) D, D = 1 N n X L [l] D(x nl ). x nl C l. (4.10) n X L [l]. Note that, for a video session, its bit rates at all links along the transmission path should be the same. In other words, x nl = x nk, l, k L S [n], (4.11) where L S [n] is the set of links used by video session n. This is an inter-dependent flow balance constraint [14]. We can see that the original global optimization problem has been decomposed into L local optimization modules in (4.12) plus a set of inter-dependent resource constraints in (4.13), which will be solved by the proposed DAPSO algorithm to be presented in the following section. It should be noted that the specific decomposition procedure depends on the actual problem formulation. However, we observe that the rate allocation problem in (4.7) is quite representative and many other network resource allocation problems share similar formulation, often having a network utility function as an optimization objective plus inter-dependent resource constraints (e.g. flow balance constraints) and several local resource constraints (e.g. link capacity and energy constraints) [17]. 4.4 Distributed and Asynchronous Particle Swarm Optimization In the proposed DAPSO scheme, each local optimization problem is solved by particle swarm optimization (PSO). Through local communication, neighboring PSO modules 60

77 share important information in a distributed and asynchronous manner to expedite the search process and make sure that the inter-dependent flow balance constraints in (4.13) are satisfied. To do this, we propose to explore two major ideas: (1) in-network fusion and particle migration to handle inter-dependent resource constraints and (2) collaborative resource management to handle network bottleneck issues. In the following sections, we discuss these design issues in more detail Original Optimization Problem Decomposition In rate allocation, we propose to decompose the global optimization problem in (4.7) into a set of local optimization modules, each of which is associated with a communication link. More specifically, let L = {1, 2, L} be the set of links that are involved in the multiple-session video communication, as illustrated in Fig In total, we have L local optimization modules. For each link, which corresponds to a local optimization module, we introduce a set of local resource parameters, X l = {x nl }, where x nl represents the bit rate of video stream n at local optimization module l. We define the l-th local optimization module to be min s.t. U l (X l ) = w 1 x nl C l. n X L [l] D(x nl ) + w 2 n X L [l] D(x nl ) D, D = 1 N n X L [l] D(x nl ). (4.12) n X L [l] x nl 0, n X L [l]. Here w1 and w2 are weight parameters for overall video distortion and distortion difference, and N is the total number of the video streams path link l. Note that, for a video stream, its bit rate on each link along the transmission path should be the same. In other words, x nl = x nk, l, k L S [n], (4.13) where L S [n] is the set of links used by video stream n. This is an inter-dependent flow balance constraint. We can see that the original global optimization problem has been decomposed into L local optimization modules in (4.12) plus a set of inter-dependent resource 61

78 constraints in (4.13), which will be solved by the following DAPSO algorithm In-Network Fusion and Particle Migration In the proposed DAPSO algorithm, each local constrained optimization in (4.12) is solved by the PSO algorithm discussed in Section Each local PSO module has a set of local particles moving around in the solution space defined by local constraints in (4.12) and searching for optimum solution for the local optimization module. Neighboring PSO modules share status information about their group-best, denoted by X g l, and then use this external information to guide the movement of their internal particles so as to meet the inter-dependent resource constraints in (4.13) in a collaborative manner. This is achieved by two major operations: in-network fusion and particle migration, as illustrated in Fig Figure 4.3: Distributed and asynchronous PSO algorithm. More specifically, during in-network fusion, neighboring local PSO modules exchange and fuse information about their group-best particles so as to generate new group-best particles which has the following two properties: (1) they still satisfy the independent constraints in each local PSO module. In other words, they are still in the solution space. (2) They should satisfy the inter-dependent resource constraints better than each group-best particle. For example, consider two neighboring PSO modules l and k, 1 l, k L. Suppose video stream n is optimized by both modules x g nl and xg nk which are the bit rates of video stream 62

79 n in the group-best particles found by local PSO modules l and k, respectively. During in-network fusion, these two modules need to negotiate with each other to determine what is a better bit rate for each, denoted by x g nl and xg nk, such that the inter-dependent resource constraint is better satisfied. The specific fusion rule depends on the actual formulation of the constraint. For example, in the rate allocation case, the following fusion rule can be used x nk = x nl = min(x g nk, xg nl ), (4.14) which are the new group-best particles in local PSO modules l and k. The major reason that we choose the min operation in (4.14) is that those new group-best particles still satisfy the link capacity constraints in both DAPSO modules. Certainly, with this fusion rule applied to all local PSO modules and for all video sessions, the new generation of group-best particles of both modules will satisfy the inter-dependent flow balance constraint better. Once the group-best particle in the local DAPSO module is updated, guided by the swarm intelligence principle in (2.19), it will attract the rest particles towards this new location in the subsequent local search. We call this operation as particle migration because these local particles move towards a new location due to external factors Handling Network Bottleneck Issues Using Collaborative Resource Control One of the major challenges in network resource allocation is to deal with bottleneck links which have very limited bandwidth resources however are shared by a large number of video sessions. This network bottleneck issue has not be adequately addressed in the literature. This problem becomes even more difficult in distributed resource allocation since local optimization modules may not be aware of the bottleneck links in some remote places of the network. Let us consider the example in Fig The network has four video sessions. Video sessions s 1 and s 2 share link l 1 whose link capacity is 300 kbps (kilo bits per second). Sessions s 1, s 3, and s 4 share link l 2 whose capacity is only 180 kbps. Let x n,li, 63

80 1 n 3 and i = 1, 2, be the local bit rate of session s n at local optimization module l i. We have the following two link capacity constraints: Constraint l 1 : x 1,l1 + x 2,l1 300, (4.15) Constraint l 2 : x 1,l2 + x 3,l2 + x 4,l (4.16) Suppose all video sessions have similar scene statistics. Therefore, in the local optimization module l 1, video session s 1 is allocated for 300 / 2 = 150 kbps. However, in module l 2, it is only allocated for 180 / 3 = 60 kbps due to resource competition from video sessions s 3 and s 4. s 1 has to choose 60 kbps since constraints l 1 and l 2 have to be both satisfied. In this case, link l 2 becomes the bottleneck link. From this simple example, we can see that bottleneck links play a critical role in network resource allocation; the overall system performance is controlled by the resource constraints at bottleneck links; and resource constraints at non-bottleneck links, such as that in (4.15), become less critical or even useless. Figure 4.4: An example of bottleneck link in a multi-hop mesh network. To effectively handle bottleneck link in distributed rate allocation, there are two major issues that need to be carefully addressed. First, the information of bottleneck links needs to be properly propagated and shared with other local optimization modules. Second, we need to develop a collaborative resource management scheme to coordinate the resource utilization between video sessions. Video sessions should yield their resources at non-bottleneck 64

81 links to those with bottleneck links. For example, in the example of Fig. 4.4, at link l 1, the local DAPSO module allocates 150 kbps for session s 1. However, s 1 can not use this much bandwidth resource because of its bottleneck at another link l 2. In this case, within the local DAPSO module of l 1, s 1 should yield part of its unused bandwidth resource to s 2. To this end, we need to develop a scheme to intelligently control and manage the resource utilization at each link in a collaborative manner. Otherwise, the video sessions at non-bottleneck links will keep pushing their own resource allocation toward their capacity limits, which might significantly slow down the convergence process. It should be noted that the bottleneck is session-specific. In other words, a link is a bottleneck for one video session but may not be the bottleneck for another one. For example, session s 3 may have a bottleneck at link l 3 if the link capacity of l 3 is very small. On the other hand, a non-bottleneck link for one video session might be a bottleneck for another. For example, link l 1 is a non-bottleneck link for session s 1 but it can be a bottleneck link for session s 2, as illustrated in Fig In addition, since many video sessions are competing for the network resources, as the network resource allocation are gradually adjusted during distributed resource allocation, the bottleneck for a video session might shift from one link to another. This session-specific and dynamic bottleneck issue presents a significant challenge in distributed resource allocation. To address this bottleneck issue, we propose a method called collaborative resource control. Our basic idea is to introduce a resource budget window for each resource parameter at each local DAPSO module. For example, the resource parameter x nl is the bit rate of video session n in the l-th PSO module. We impose the following resource budget window on x nl : C nl x nl C + nl, (4.17) where C nl and C + nl will be adjusted by the following particle migration procedure. The window size is denoted by L = C + nl C nl. Initially, we can set C nl to be 0 and C+ nl to the link capacity so that the resource budget window does not affect the original link capacity 65

82 constraints in (4.12). During distributed rate allocation, we propose to use the linearlyincrease and multiplicative-decrease rule which is used in TCP network control [79] to adjust the resource budget window. More specifically, in the proposed scheme, a bottleneck link is determined based on the following condition: if x g n, = min k L S [n] xg n,k = xg n,l, (4.18) where x g n,k is the bit rate of video session s n in the group-best particle at PSO module k, we claim link l is now the bottleneck for s n. In this case, we linearly increase the window size L by a fixed amount L. (In this work, L is empirically chosen.) This is because, by increasing the resource budget window size L for video session s n on this bottleneck link, the constraint on its bit rate x n,l is further relaxed, thus video session s n has more flexibility in competing for more system resource with other video sessions on this link during the local optimization process. On the other hand, if link l is not the bottleneck link for session s n according to the criteria in (4.18), we reduce the budget window size L by half and center the resource window around x g n,. Note that link l is not the bottleneck link for video session s n, which implies that the resource constraint on the bit rate x n,l is too loose as that in (4.15). In this case, we need to reduce its resource budget window so that it can yield part of its resource to other competing video sessions during the local optimization process. Once the resource budget window is modified, it will serve as a constraint to regulate the particle movement during PSO at link l. It is well known that the linearly-increase and multiplicative-decrease rule in TCP is able to converge to fair bandwidth allocation between different network communication sessions [79]. During our experiments, we also observe that it has a similar behavior in DAPSO, However, at this moment, we are not able to theoretically justify this convergence behavior since the whole DAPSO process is much more complicated than network transmission control Algorithm Description of DAPSO 66

83 Figure 4.5: Illustration of particle migration. In this section, we summarize the DAPSO algorithm. As discussed in Section 4.3.2, we decompose the centralized rate allocation problem in (4.7) into a set of local optimization problems in (4.12). Each local optimization problem is solved with PSO. The proposed DAPSO operates at each link, performing the following three major tasks: (A) allocating the link bandwidth among video sessions that share this link; (B) transmitting its groupbest to its neighboring DAPSO modules; (C) receiving the group-best information from its neighboring DAPSO modules and using them to guide its own particle movement and adjusting the resource budget window, as explained in Sections and In the following, we outline the major actions performed by each DAPSO module. Step 1 Initialization. Initialize the local PSO module by randomly generating a set of particles in the solution space [16]. The total number of particles in each PSO module can be adjusted. In our simulations, we set it to be 10. Also initialize the resource budget window as explained in Section Step 2 Local optimization. Each local DAPSO module moves its own particles in the solution space based on the swarm intelligence principle in (2.19) and records its group-best particle. Step 3 Sharing group-best information. If the group-best particle has been updated, the 67