Resource Allocation in Cooperative Communications

Transcription

1 Resource Allocation in Cooperative Communications B.Tech. Project Report Submitted by: Ankit Singh Rawat (Y6078) Kartik Venkat (Y6216) Under guidance of Dr. Ajit K. Chaturvedi and Dr. Aditya K. Jagannatham Department of Electrical Engineering Indian Institute of Technology, Kanpur April 2010

2 Certificate This is to certify that the work contained in this report entitled Resource allocation in cooperative communications byankit Singh Rawat (Y6078) and Kartik Venkat (Y6216) has been carried out under our supervision and that this work has not been submitted elsewhere for a degree or diploma. Dr. Ajit K. Chaturvedi Department of EE Indian Institute of Technology Kanpur, India Dr. Aditya K. Jagannatham Department of EE Indian Institute of Technology Kanpur, India

3 Acknowledgements This project has been a wonderful learning curve for both of us. With great encouragement from our supervisors, we have been able to explore diverse areas and learn new techniques which could be applied to our problem. We would like to thank Dr. Ajit Kumar Chaturvedi and Dr. Aditya K. Jagannatham for their valuable support and continued guidance throughout this project.

4 Abstract In this project, we have developed a framework for resource allocation in a multi user cooperative wireless network. We have considered centralized and decentralized node-relay assignment with the objective of maximizing the sum throughput. We present an efficient polynomial time centralized assignment algorithm to achieve this objective, and also investigate efficient decentralized schemes that do not require a central authority to perform the allocation. We present a novel message passing based assignment scheme which converges to the optimal value, and present an adapted scheme based on distributed auction theory. We also present a simple sub-optimal greedy assignment strategy and provide empirical data to support its practical significance.

5 Contents 1 Introduction 1 2 Literature Survey 5 3 System Model 7 4 Basic Theory Notation Amplify and Forward (AF) Decode and Forward (DF) Sum Rate Maximization Centralized Scheme Problem Statement A Computationally Inefficient Solution Solution Graph Construction Definitions Bipartite Graph Matching Residual Graph Augmenting Path Algorithm Proof of Correctness Note on Fairness Complexity Analysis iv

6 CONTENTS 6 Decentralized Scheme Distributed Combinatorial Auction Approach Greedy Bidding Approach Distributed Message Passing Approach Simulations and Results 30 8 Conclusions 35 References 35 v

7 List of Figures 3.1 System Model Typical Wireless Ad Hoc Netowork Example Graph Construction Residual Graph for Matching, M = Description of Algorithm Illustration of message passing Polynomial runtime of centralized assignment scheme Sum throughput vs. no. of source-destination pairs (for M = N/2) Sum throughput vs. no. of source-destination pairs (for M = N) Sum throughput vs. no. of source-destination pairs (for M = 2N). 34 vi

8 List of Tables 5.1 Throughput matrix of Example Throughput matrix of Example Improvement in performance of relay assignment with message passing rounds vii

9 Chapter 1 Introduction In the past few decades, wireless communication has been one of the technical areas which has experienced exponential advancement in terms of technology as well as users. This mode of communication alleviates the need of physical connectivity among communicating entities via exploiting the channel as a signal carrying medium. This also makes communication among mobile devices possible. The main objective of any mode of communication is to increase the data rates while also maintaining high reliability (lower probability of error) of the data sent over the communication link. However, the wireless channel suffers from unwanted yet inevitable effects (e.g. shadowing, pathloss, multipath fading etc) which make it hard to communicate reliably over the channel. Various diversities of the wireless channels are used as potential solutions to mitigate some of these channel impairments. Spatial diversity is used to combat the deleterious effects of fading via transmitting the signals from different locations, thereby allowing independently faded versions of the signal at the receiver. The idea of multiple input multiple output (MIMO) system was proposed to generate spatial diversity by equipping the wireless device with multiple antennas. However many wireless devices are limited by size and hardware complexity to one antenna and MIMO is not realizable in these cases. Cooperative communications provides an alternative solution for this problem via enabling single antenna wireless devices in a multi-user environment to share their antennas and generate a virtual multi-antenna transmitter in order to achieve spatial diversity. The broadcast and multi-cast nature of the wireless channel is exploited in cooperative communications. The wireless devices which overhear the 1

10 transmission between two intended entities can forward the overheard information and provide another independently faded version of the information at the receiver. Thus, each device in the network is supposed to transmit its own information as well as cooperate in delivering the information originating from other devices. 2

11 Motivation Cooperative communications presents many new challenges to researchers along with the numerous advantages. Relay functionality at the cooperating node is of foremost concern for the realization of cooperative communication. This problem has received large attention of researchers, and significant progress has been made in this area. Another main concern in cooperative communication is the sharing of network resources (potential relays) among users and to investigate resource allocation schemes. While many key results have been obtained in this field, several issues remain unexplored. The optimal assignment of partners (relays) in the multiuser network is one area that demands attention. Further, which criterion for optimality should be considered is another pertinent issue. Both, centralized and decentralized schemes are required to be developed for the network depending on its topology and demands. Furthermore, most of the cooperative systems produced so far are based on some very strong assumptions (e.g., synchronization among cooperative nodes and presence of perfect channel state information etc) which must be relaxed to come up with systems with practical constraints. 3

12 Objective Our main objective in this project is to study various existing schemes for resource allocation in cooperative communications and to develop novel approaches for the same. There are two fundamental paradigms that fall under the rubric of resource allocation - centralized and decentralized. The criterion for optimality we have selected is maximization of sum throughput. We intend to develop a holistic framework to address both these approaches, with the aim of maximizing the sum throughput in a multiuser cooperative wireless network. 4

13 Chapter 2 Literature Survey The cooperative communication paradigm can traced back to pioneering works done by van der Meulen [1]) as well as Cover and El Gamal [2]. However, current research interests in cooperative communication are inspired by work carried out in early 21 st century (e.g., [3]-[5]). In [6], Nosratinia et. al. have provided a nice introduction of this paradigm. Ng et. al. in [16] presented a general analysis of impact of channel state information (CSI) on the performance of cooperative communication. Their analysis is termed general in the sense that they did not limit their attention to transmitter cooperation strategies only but also considered the receiver cooperation. Acrucialchallengeintheimplementationofcooperativecommunicationishow to assign source-relay pairs before cooperation begins. In general, relay selection can be classified into two categories, either multiple-relay selection or single-relay selection. The performance of multiple-relay participation is fundamentally limited by the orthogonal partitioning of system resources, inefficient usage of power, or difficult synchronization among cooperative nodes. In [13], Zhao et. al. showed that assigning the best relay to a source-destination pair is sufficient to achieve full diversity order. This result has motivated researchers to explore solutions where a single-relay is assigned to a wireless node for cooperation. Resource allocation in cooperative networks has attracted attention from the research community only recently. Related work in this area can be divided into two categories: centralized (e.g., [10], [7]) and distributed (e.g., [9], [8]). In [7], Ng and Yu devised a centralized resource allocation algorithm for power control, bandwidth allocation, relay selection and relay strategy choices in an Orthogonal 5

14 Frequency-Division Multiple Access (OFDMA) based relay network. In [10], Shi et al. have proposed a centralized optimal relay assignment scheme, for maximizing the minimum link capacity in a wireless network over orthogonal relay channels. Bletsas et al. in [9] proposedadistributedrelayselectionscheme,whereoneuser chooses the best end-to-end path among many relays based on instantaneous channel measurements. Wang et al. in [8] haveinvestigatedresourceallocation for cooperative transmission using Stackelberg games. Some researchers have also focused their attention to distributed relay beamforming schemes to realize cooperative communication (e.g., [17]). A significant progress in finding a distributed scheme to perform distributed resource allocation in cooperative communications has been shown by Huang et. al. in [18]. They modeled the entire resource allocation process as an distributed auction, and showed the existence of Nash equilibrium when the network under consideration contains certain properties. Recently, a new cooperative strategy has been proposed for ad-hoc networks in [15], where two transmitters simultaneously transmit their own messages and assist the transmission of another transmitter via relaying its message. Similar to assigning source-relay pairs in previous cooperative strategies, assigning partner pairs is an important problem in multiuser network. 6

15 Chapter 3 System Model We have considered a wireless ad-hoc network with N s source-destination pairs, and N r potential candidates for relaying. Each source-destination pair has its own dedicated channel. Every candidate relay also operates at a different frequency band. If a relay node is cooperating with a given user, it overhears the source information and communicates it to the receiver. Simultaneously, the source also communicates with the destination through its dedicated channel. The destination, by virtue of receiving data from multiple channels, applies maximum ratio combining (MRC) technique. Fig. 3.1 represents the diagrammatic system model. In this scheme - h sr is the channel gain between the source and the relay h sd is the channel gain between the source and the destination h rd is the channel gain between the relay and the destination An illustration of a typical ad-hoc network is shown in Fig. 3.2 Relay (R) h_sr h_rd MRC h_sd Source (S) Destination (D) Figure 3.1: System Model 7

16 Source Receiver (destination) Potential relay nodes Figure 3.2: Typical Wireless Ad Hoc Netowork 8

17 Chapter 4 Basic Theory The selection of relay functionality is a fundamental aspect of cooperative communication. Various relay functionalities have been proposed and analyzed in the literature of cooperative communication. Amplify and forward (AF) and Decode and forward (DF) are widely used among these. We have considered both of them within the framework of our model and a brief description of them is given in the following part of this chapter: 4.1 Notation In Fig. 3.1 :Letthechannelgainsforthesource-relay,relay-destinationandsourcedestination be denoted by h sr, h rd and h sd respectively. Let the SNR for each of these links be denoted by γ sr, γ rd and γ sd respectively. The signal transmitted by the source is x s. The gaussian noise on the direct channel is n d and on the relay channel is n r 4.2 Amplify and Forward (AF) In this relaying scheme, each user forwards the noisy version of the partner node simply amplifying it while satisfying the transmission power constraint. In this case, received signal at the receiver (for an amplification factor A) forwardedvia relay (y rd )canberepresentedas: 9

18 4.3 Decode and Forward (DF) y rd = h rd (Ah sr x s + n r )+n r. (4.1) is: Similarly the signal received by the receiver through direct communication (y sd ) y sd = h sd x s + n r. (4.2) The information at the receiver is combined using MRC (maximal ratio combining) which results into the effective SNR : γ AF = γ sd + which translates into the throughput γ sr γ rd 1+γ sr + γ rd (4.3) 4.3 Decode and Forward (DF) In this scheme, the relaying node tries to decodes the data received form the partner node before forwarding it to the destination. After successfully, decoding the received signal relaying node again encode it and forwards to the destination. The received information at the receiver via relay can be expressed as: y rd = h rd x r + n r (4.4) Receiver combined this signal with the signal directly received from the source using MRC which give the effective SNR: γ DF = γ sd + min (γ sr,γ rd ). (4.5) which translates into expression for throughput 10

19 4.4 Sum Rate Maximization Note: With knowledge of the channel states and SNR, it is possible for us to compute the link throughputs in the network for consideration in the optimal relay assignment problem 4.4 Sum Rate Maximization We are given N s source-destination pairs and N r relays. We are given the capacity C iφ for the direct channel between the source and destination, as well as the capacity C ij between the i th source-destination communicating through the j th relay channel, for all j and i. Mathematically, the sum rate maximizing assignment problem can be formulated as follows - We are interested in determining α = {α 1,α 2...α Ns }, i =1, 2...N s,whereα j {φ 1,φ 2,...,φ Ns, 1, 2...N r } such that α =argmax α C iαi ) (4.6) ( Ns i=1 with the following constraints α i = α j when i = j : i, j {1, 2,...N s } (4.7) and, α i = φ j when j = i (4.8) It is clear from the above formulation that the problem we are intending to address is a combinatorial optimization problem. 11

20 Chapter 5 Centralized Scheme Introduction There has been much research on selecting an optimal relay node from a set of candidate nodes for a single source-destination pair. In a wireless ad hoc network, there are multiple source-destination pairs competing for the same pool of relays. Our focus is therefore, to develop a criterion to perform an optimal relay assignment for all users in a network from amongst a set of available relays. We will consider assignments that will maximize the sum throughput of the network, while assigning at most one relay to a source-destination pair. A centralized scheme assumes complete information about all participants in the network. In our case, this translates to channel state information and SNR for all participating links, thereby enabling us to compute the requisite link capacities. In [10] Shi et al. have developed an efficient centralized node relay assignment scheme that maximizes the minimum capacity over all possible assignments. We have studied and understood their approach to develop or own model for analysis. We have also simulated the algorithm proposed by them We shall now proceed to develop a centralized relay assignment algorithm to achieve sum rate maximization in the given network setting. 12

21 5.1 Problem Statement 5.1 Problem Statement We are given N s source-destination pairs and N r relays. We are given the capacity C iφ for the direct channel between the source and destination, as well as the capacities C ij between the i th source-destination communicating through the j th relay channel, for all j and i. Our aim is to assign each source-destination pair to at most one unique relay, such that this assignment maximizes the sum throughput of the network. In other words, there exists no other assignment which can have greater sum throughput A Computationally Inefficient Solution Given all the link throughput values, we are required to come up with the optimal assignment as stated in equation 4.5. A brute force solution to the above stated problem would be to compute the sum rate for all possible assignments, and to select the one which maximizes the required sum metric. As can be easily seen, this problem has a running time that is exponential in the input size. More specifically it will take O((N r + N s )(N r + N s 1)...(N r )), which is computationally intractable. Therefore, there is a need to develop an efficient solution to the above problem which is computationally efficient (polynomial running time). 5.2 Solution We have formulated a graph theoretic solution to the above problem based on network flows. We model the given problem as a weighted bipartite graph matching problem. Graph matching problems are well studied in algorithms literature. Bipartite Matching has been discussed in [14]. We use the augmenting path approach to generate an optimal assignment. The problem model, along with the algorithm, proof of correctness and complexity analysis are detailed herewith Graph Construction Let G be a bipartite graph with vertex set V = X Y, wherex,y are disjoint sets. 13

22 5.2 Solution S1 R1 S2 R2 R3 Figure 5.1: Example Graph Construction Each vertex in X represents a unique source-destination pair. X = N s Each vertex in mathcaly represents a unique relay or a direct channel between some sourcedestination pair. Y = N r + N s We construct the edge set E as follows - For every node i X there exists an edge e ij between i X and j Y, ifit is possible for the said source-destination pair to use the j th channel Each such edge e ij will have an associated non-negative real number c ij Note :TheN s direct channels represented by nodes in Y will have one edge each corresponding to their source-destination. The relays will have edges corresponding to multiple nodes in X. The Fig. 5.1 gives an example construction of such a bipartite graph for N s =2,N r = Definitions Bipartite Graph AbipartitegraphG can be partitioned into two disjoint vertex sets V = X Y, such that every edge e E has one end in X and the other in Y 14

23 5.2 Solution s t Figure 5.2: Residual Graph for Matching, M = Matching AmatchingM in G is a subset of edges M E such that each vertex appears in at most one edge in M. The size of a matching is M The cost of a matching is C M = e ij M c ij In our construction, ˆM, suchthat ˆM = N s is the minimum cost perfect matching if C ˆM C M, M : M = N s Residual Graph Let M be a matching. We add two new nodes s and t to the graph. We add edges (s, x) forallnodesx X that are unmatched and edges (y, t) forallnodesy Y that are unmatched. An edge e =(x, y) isdirectedfromxto y if e/ M, andfrom y to x if e M. For all edges oriented from y to x, wereplacetheedgeweightc xy with the negative value c xy. All edges of the form (s, x) and(y, t) areassignedweight0. The weighted, directed, bipartite graph so obtained for a given matching M is referred to as the residual graph G M 15

24 5.2 Solution Augmenting Path In the residual graph G M,letP be a directed path from s to t. Ifsuchapathexists, then we can construct a matching M,s.t. M = M + 1. This can be achieved in the following manner- if edge e in path P is directed from X to Y, thenitisaddedtom if edge e in path P is directed from Y to X, thenitisdeletedfromm We call P,theaugmentingpathanduseittoaugmentthematchingM Algorithm We now present the optimal node relay assignment algorithm to maximize sum capacity. Main Algorithm 1: N s number of source-destination pairs 2: N r number of relay channels 3: Construct Graph G 4: S =max(capacity(i, j)) over all possible e(i, j) E 5: for every edge e ij E, c ij S capacity(i, j) 6: m 0 7: M φ 8: while m = N s do 9: construct G M,theresidual graph for matching M 10: compute the shortest path P from s to t in G M 11: call Augment(P, M, G M ) 12: m m +1 13: end while 14: M is the minimum cost perfect matching of size N s 15: e(x, y) M assign x th source destination pair to y th relay channel Augment(Path P, Matching M, Graph G M ) 1: for every edge e P do 2: if e is directed from X to Y in G M then 16

25 5.2 Solution BEGIN Graph Construction Source nodes = N Relay nodes = R M = 0 M = N? Yes return Matching No Construct Residual Graph END Find shortest path from s to t Augment Matching with shortest path M = M + 1 Figure 5.3: Description of Algorithm 3: e =(x, y) 4: Add e to M 5: else if e is directed from Y to X in G M then 6: e =(y, x) 7: Delete (x, y) fromm 8: else continue to next edge in P 9: end if 10: end if 17

26 5.2 Solution Proof of Correctness In the above algorithm, we are iteratively constructing a matching of size k +1 from amatchingofsizek. We claim that the final matching of size N s obtained, is the minimum cost matching. Observation: Given a matching M and a path P from s to t in G M,letM be the matching obtained by augmenting path P to M. Then M = M +1 and C M = C M + cost(p ) Our aim therefore is to iteratively augment along paths of minimum cost to obtain amatchingofsizen s A Negative Cycle is a collection of vertices forming a closed loop path such that the sum of edge weights < 0 Lemma: Let M be a matching of size N s. If there exists a negative cost cycle C in G M,thenMis not minimum cost Proof: Note that s and t cannot be part of a cycle in G M, because there are no edges emerging from s or t. If we were to use the edges of cycle C to augment the matching M, wewillgetanothermatchingm of size N s. However, C M = C M + cost(c) where cost(c) < 0. Thus, M is not of minimum cost. Lemma: Let M be a matching of size N s. If there are no negative cost directed cycles in G M,thenMis a minimum cost perfect matching Proof: Let M be a matching of size N s of smaller cost than M. Considerthesetof edges γ = M M. This edge set corresponds to a set of vertex disjoint directed cycles in G M.ThecostofthissetisC M C M < 0. Thus, it must be that at least one of the cycles has negative cost. With the above result, it is clear that we want to generate matchings of larger and larger sizes while ensuring that in no iteration do we have a negative cycle. This is the motivation behind choosing the smallest weight path from s to t as our augmenting path. When we terminate with a matching of size N s in the above algorithm, it is the one that has minimum cost Maximum Sum Throughput 18

27 5.2 Solution Note on Fairness We can trivially extend our solution model to address an objective function that is a weighted sum of link throughputs in the multi-user environment. This can be used to account for fairness criteria, and associating a scalar importance with the participating nodes in the network Complexity Analysis The algorithm that we have proposed above is computationally efficient. In this section we shall prove that the running time of the algorithm is polynomial in the size of the input. It is important to appreciate that the brute force solution to this problem would be exponential in the input size thereby making it intractable. G is the graph that we have used to model the problem. The number of vertices V =2N s + N r = O(N s + N r, the number of edges E = N s N r + N s = O(N s N r ). The algorithm can be divided into three main steps from the time complexity perspective Residual Graph Computing the residual graph takes O( V + E ) time Computing Shortest Path We are using the bellman ford algorithm to compute the shortest path from s to t in the residual graph. This algorithm has a worst case running time of O( V E ) Augmenting the Path This method will take O( P ) time,wherep is the augmenting path. The algorithm is iterative in nature, the number of iterations being N s in our case. Thus the worst case performance of our algorithm is O(N s V E ) =O(Ns 2 N r (N s + N r )). Example 5.1: We presented the algorithm with the throughput values given in Table 5.1 (for a multiuser network with N s =5andN r =4). For the above data, our algorithm presented us with the optimal assignment to maximize sum throughput. The returned assignment has been indicated in the 19

28 5.2 Solution Source-Destination R1 R2 R3 R4 φ Table 5.1: Throughput matrix of Example 5.1 table itself. The total sum throughput in the optimal assignment =

29 Chapter 6 Decentralized Scheme The centralized schemes assume the existence of a central unit (CU) which has the global knowledge of the channel state information and heterogeneous power constraints at various nodes in the network. CU solves the optimization problem of allocating the relays among the users maximizing certain objective function. Centralized information exchange and coordination, however does not scale well with the size of the network and the scheme loses its charm. Moreover, in the case of ad hoc networks, assumption of a central authority seems implausible as the whole motivation behind ad hoc networks is to be able to communicate without any major infrastructure in place. In this chapter we discuss the idea of distributed relay assignment. The performance of distributed resource allocation schemes gets affected by the fact that each user does not have global channel knowledge, and hence is unable to make a decision that is optimal for the overall network. Schemes following game theoretic approach similar to the channel allocation in cognitive radio network are given in [11], [12]. The system model is the same as described in the previous chapter with N s source-destination (s-d) pairs and N r relays, all operating on orthogonal frequency channels. In a given multi-user cooperative network, we define the concept of network connectivity. Here, we apply the constraint that a given source destination pair can be served by specific relays only. Thus, for each source-destination pair s i we have a set of potential relays R i = {r1,r i 2..r i k(i) i } which constitute a subset of all available relays in the network. We can model this communication network as a graph G =(V, E). In this graph, an edge exists between a source-destination node 21

30 and a relay node, if they can communicate (i.e. they are connected). This notion of connectivity is useful in sharing local information amongst users to arrive at a better overall decision for the network. In this model, we assume that each user (source) has information pertaining to the channel gains of the direct channel between the source and destination, as well as CSI pertaining to the source-relay and relay-destination channels for each relay R i (for source s i ). Each of the N r relays has CSI of the source-relay and relay-destination links for those sources that are connected to the same. Using the channel state information, it becomes possible for a communicating entity to compute the capacity of that particular link. The assumptions made in this regard are valid in light of [19], where Guo et. al. have presented a medium access control (MAC) protocol for multi-user cooperative networks. This protocol explains the higher layer issues pertaining to acquiring such information in a network scenario as we have considered. Game Theory has traditionally been used to model resource allocation problems in wireless networks. Modeling the channel allocation problem as a non-cooperative game is an example where game theory is used to come up with a distributed channel allocation scheme (e.g., [11],[12]). In these papers, M channels get distributed among N sources in a decentralized fashion with the intention to maximize a certain network performance metric. Each user attempts to acquire a resource while aiming to maximize a suitably chosen utility function. This allows for decision making in the absence of a central authority. Usually, these resource allocation schemes are optimal in the game theoretic sense, i.e. the assignments converge to a pareto optimal or Nash equilibrium. Game theory therefore, seems a natural approach towards the distributed relay allocation problem that we are addressing. The works presented in [8] and[18] are also motivated with the intention to apply game theory for allocating resources in cooperative networks, using Stackelberg games and Auction theory respectively. In their work in [18], Huang et. al. have considered a network, where each relay serves multiple source-destination pairs via allocating its power among them. This power allocation problem has been modeled as an auction, where each source submits a bid to the candidate relay which would allow the relay to allocate an optimum amount of power for that particular source. Sources calculate their bids to maximize their utility function based on their local information. 22

31 This paper deals with fundamental questions : 1) When is it beneficial to relay? and 2) How does each relay allocate its power amongst contending sources? with an interesting approach, while providing analytical expressions for bid calculation. The authors have shown the existence of Nash equilibrium under the satisfaction of certain initial constraints, and have also presented a generalization of their scheme for networks with multiple relays. There is a crucial difference between our system model and the system model considered in [18]. Instead of distributing relay power among multiple sourcedestination pairs, our objective is to assign an entire relay to serve a particular source-destination pair. Therefore, even as the problem is fundamentally one of resource allocation in a distributed network setting, the actual nature of the resources are different. But we derived a motivation for applying auction theory to our problem from their work. Moreover, our problem can be modeled as a combinatorial assignment problem. A rich reference for the distributed combinatorial auctions is [20]. Since each user in our network has only local information about channel gains, in order to arrive at a globally optimal decision, one requires information sharing amongst nodes. This may be achieved via message passing amongst connected nodes in the cooperative network. This forms the basis of optimal decisions in a decentralized framework. This message passing in the network is similar to the concept invoked in decoding of LDPC codes using tanner graphs. In this chapter, we have presented a novel relay allocation scheme based on distributed message passing, which converges in terms of sum throughput to the global optimum. Rest of the chapter is divided into three sections. In each section, we propose different distributed relay allocation schemes as solutions to our problem. We discuss three solutions that we have developed in the context of our problem, namely the distributed combinatorial auction approach, the greedy bidding strategy, and the distributed message passing algorithm. We then present simulations, analysis and comparison of the schemes under various criteria of performance. For our comparisons, we shall invoke the centralized assignment developed in chapter 5 as the benchmark for performance, since it is the global optimal assignment. 23

32 6.1 Distributed Combinatorial Auction Approach 6.1 Distributed Combinatorial Auction Approach In this approach we consider the source-destination pairs as agents who are competing for a set of favorable tasks (relays). Our aim is to develop an optimal assignment from agents to tasks in a distributed fashion that maximizes the sum utility. Let the network be modeled as a graph G =(V, E). Here V = {1, 2,...,N s } represents the set of source-destination pairs, and E = {(i, j) i, j V} represents the edge adjacency matrix of G. An edge denotes that information exchange between these agents is possible. Neighbors of the i th agent can be represented as belonging to the set N i = {j V (i, j) E}. As far as the relay allocation is concerned, our aim is to achieve a map α : {1, 2,...,N s } {1, 2,...,N r } in a decentralized fashion that maximizes the sum capacity ( N s i=1 C iα(i)). Notation: C ij denotes the capacity when the j th relay is assigned to the i th source. p ij (t) denotes the price bid by user i for the resource j at time t. b ij (t) stores the highest bidding agent for resource j at time t, whichhasthehighestindex. is a predefined system parameter which represents the minimum increase in bid for all agents in the network. Distributed Combinatorial Auction Algorithm [23]: Auction iteration for i th (sd pair) i Require: An assignment α i (t) {1,...,m} and a set of prices p ij (t) 0and highest bidders b ij (t) N for all relays 1 j N r. 1: Update the prices of all tasks and the corresponding highest bidders by p ij (t +1)=max k Ni (t){p ij (t),p k j(t)} and b ij (t +1)=max k arg maxz Ni (t){p ij (t),p kj (t)}{b kj (t)}; 2: if p ii (t) p iαi (t)(t + 1) and b iαi (t)(t +1)= i then 3: Update the assignment by α i (t +1) arg max 1 k Nr {C ik p ik (t +1)}; 4: Set b iαi (t+1) = i and increase the price for task α i (t +1)byp iαi (t+1)(t +1)= p iαi (t+1)(t)+γ i,whereγ i is according to equations (3)-(5). 5: else 6: Remain assigned to task α i (t), i.e., α i (t +1)=α i (t); 7: end if 24

33 6.2 Greedy Bidding Approach Discussion The above scheme develops an auction based method to achieve sum rate optimal assignment within a performance gap. It is important to note that this scheme is applicable only when the number of relays are equal to, or exceeds the number of users. Zavlanos et.al. have proposed a task assignment scheme, in which they have proved the following results pertaining to the performance and convergence of this algorithm. The detailed proofs can be found in [23]. Let C ij denote the capacity matrix, and n be the total number of agents in the network. Results The distributed auction algorithm converges to the final assignment in O( n 2 max(c ij ) min(c ij ) ), where n 1. The final assignment α : {1,...,n} {1,...,m} that is obtained is within n of maximizing the total sum rate. This implies that : n i=1 C iα i S n, where S = max( n 1 C iβ(i)) β : {1,...,n} {1,...,m}. The above results highlight two important aspects of this scheme. Firstly, we are able to pinpoint the performance gap that this scheme will achieve when compared to the centralized scheme. We can further reduce this gap arbitrarily by choosing a suitable. Thetradeoffhoweveristhatasmallerepsilonwillresultinagreater number of iterations for the algorithm to terminate to the required assignment. 6.2 Greedy Bidding Approach We have proposed a new bidding scheme based on a greedy strategy that is aimed at maximizing the increase in throughput of the system at each iteration. This scheme is very simple in terms of computational resources required at nodes, as well as iterations required for achieving convergence. We shall now describe this greedy bidding strategy employed by agents in the network. The performance of this scheme and its comparison with other schemes shall also be outlined. 25

34 6.2 Greedy Bidding Approach Greedy bidding approach 1: Each source will place bid to relay that maximizes its increase in rate: b i =argmax k N r C ij,wherenr is the set of relays that are still unassigned. 2: Each Relay that has been bid for by at least one agent, will choose that source which shall have the maximum capacity upon making this assignment: c j =argmax k Bj (C kj ), where B j represents the set of those sources who bid for j th relay in this round. 3: Update the set of unassigned relays (Nr )andsetofthosesourceswhohavenot been mapped to a relay (Ns ). 4: if, Nr = φ and Ns = φ, then 5: Remaining sources bid for remaining relays in the same fashion. 6: else 7: terminate the allocation process. 8: end if The greedy bidding approach that we have described in this subsection require sources and relays to use their local CSI only. Since a source is not aware of the current state of the entire network, it may take a decision which will subsequently be suboptimal for the cooperative network under consideration. In other words, the presented greedy scheme does not guarantee to converge to the optimal assignment. The following example illustrates this fact: Example 6.1: Consider a cooperative network with 3 source-destination pairs and 2 potential relays. Table 6.1 shows throughputs of all possible communication links in this network. Source-Destination φ R1 R Table 6.1: Throughput matrix of Example 6.1 In the first round of bidding, source 1 and 2 both will bid for relay 1 as this relay provides them the maximum throughput. Source 3 will place bid for relay 2. 26

35 6.3 Distributed Message Passing Approach Figure 6.1: Illustration of message passing Clearly, after the 1 st round of allocation, relay 1 will be assigned to source 1, and source 3 will get to relay its information through relay 2. In this source 2 will have to rely on its direct channel. The total throughput of this assignment is = This assignment is not optimal as the optimal assignment would map relay 1 to user 1 and relay 2 to user 2 resulting into sum throughput of = Distributed Message Passing Approach As shown in the example described in the previous section, greedy bidding approach does not guarantee the assignment that corresponds to maximum sum throughput of the network. The obvious reason for this shortcoming is the availability of only local CSI at sources. Clearly, increasing the level of CSI at each user would enable them to make selections which are beneficial for the entire network. For instance - in the above example, providing the knowledge of source 1 s throughput via relay 1 and source 3 s throughput via relay 2 to source 2 will ensure that the source 2 bids for relay 2 instead of relay 1 which will eventually lead to an overall better sum throughput. This idea motivated us to devise a distributed scheme which focuses on expanding the view of each source from local to global. Our literature survey also pointed 27

36 6.3 Distributed Message Passing Approach out some recent research work (e.g., [21], [22]), where information is being shared among the users in the form of message passing in order to evolve an optimal resource allocation scheme in a distributed fashion. In the lines of work presented in [21], we incorporated message passing with our greedy bidding approach. Each message passing round can be divided into two half rounds, where in the first half round each source forwards its complete CSI to all the relays it is connected to, and in the second half round all the relays forwards their CSI to all the sources they are connected to. After each complete round of message passing, available information at each node increases and its view expands towards global view. Fig. 6.1 illustrates the message passing mechanism. We may view the graph representing connections among sources and relays through which message passing takes place, as a tanner graph. This idea appears to be very much similar to the belief propagation scheme used in decoding of Low Density Parity Check codes. Distributed message passing 1: Each source performs first half of message passing round. 2: Each Relay performs second half of message passing round. 3: Each source run bipartite graph matching algorithm based on the information available to it and bids for the relay that is mapped to it by algorithm. 4: if, No two or more than two sources bid for the same relay, then 5: Perform another round of message passing before final relay allocation. 6: else 7: Terminate the allocation process. 8: end if Convergence The above algorithm converges to the optimal assignment. After each iteration, the channel state information at each user increases, moving towards global CSI required to make the optimal relay selection. Let, in any given connected component of the network graph, there exist n source-destination pairs and m potential relays. A path from agent i to j will exist (due to connectivity) and include at most m 1 hops. Thus, in k m 1iterationsoftheabovescheme,userj will achieve global CSI for the connected component, and will therefore be able to make the optimal 28

37 6.3 Distributed Message Passing Approach assignment. Since all nodes perform the computation in parallel, it is clear that the algorithm will converge to the global optimum in at most m 1 iterations,wherem is the maximum number of relays in any connected component. Note that a trivial bound on m is N r In this chapter, we have described novel relay selection schemes which operate in a distributed fashion in a multi user cooperative network environment. We have employed ideas and results from graph theory, auction theory, and combinatorial optimization in developing these schemes. In the following chapter, we shall proceed to analyze and compare each of the above designed schemes in a manner that is representative of their practical performances. 29

38 Chapter 7 Simulations and Results Simulations have been performed by randomly placing communicating nodes in asquaregridofdimensions500mx500m. The channel coefficients are calculated using random samples of the Rayleigh distribution. Path loss has been incorporated with a path loss exponent α =2. Thenoisepowerhasbeenfixedat10 11 W. The transmitting power P T =1WandtherelaypowerP R = 1 W. Capacity has been calculated using the AF Relay model, though the results we have established are valid for DF relaying strategy as well. We observed the running time of the centralized algorithm for different input sizes (network size). We varied the number of source destination pairs, and the number of relays in the same ratio. Fig. 7.1 illustrates that the running time of our algorithm is upper bounded by a polynomial in the network size, as proved in the theory. Complexity: The distributed message passing scheme is a polynomial time algorithm. As we have established that the centralized scheme is polynomial time in chapter 5, and we are invoking the same scheme at nodes here after each iteration; a bound on the number iterations has also been specified, rendering this scheme polynomial in running time complexity. The greedy scheme while being sub-optimal, is the most efficient scheme as far as complexity and running time is concerned. As we have analyzed in the previous chapter, it is a selection based on only the local information at each available node, and thus no extraneous computation is required. The running time of distributed auction algorithm has been mentioned 30

39 !1 10!2 10!3 runtime of centralized scheme "(N 4 ) No. of source!destination pairs! Figure 7.1: Polynomial runtime of centralized assignment scheme in the corresponding section in chapter. This scheme highlights the running time vs. optimality gap trade off. In order to obtain a representative platform upon which we shall be comparing our schemes, we have performed an average case analysis of the throughput over several instances of the random topology described above. In all analysis that follows, the centralized scheme is used as a benchmark to compare the distributed schemes with. Fig. 7.2 shows the variation of average throughput with network size for each of the four schemes when the number of relays is half that of the number of source destination pairs. Fig. 7.3 illustrates the same parameter for the situation where number of available relays in the network is equal to that of source-destination pairs. Fig. 7.4 shows the performance of various relay selection schemes with 2N relays. In all these figures, curves named as message passing shows the performance 31

40 of distributed message passing scheme after two rounds of message passing. It is clear from these figures that the greedy bidding scheme is sub-optimal. However, the performance gap with optimal scheme (centralized scheme) is not significant for small size cooperative network. The performance gap is noticeable only when the network size grows larger. Another important observation is that performance gap with large number of relays is less than that with small number of relays. This implies that taking optimal decision is crucial in a resource limited cooperative network Sum throughput Centralized Scheme Greedy Bidding Message Passing No. of source destination pairs Figure 7.2: Sum throughput vs. no. of source-destination pairs (for M = N/2) Table 7.1 intends to illustrate the fact that increasing the level of information at sources leads them to distributively take decisions which are beneficial for the entire network. We can see that the performance of the greedy scheme exceeds significantly just after one round of message passing. In almost all the cases, distributed approach 32

41 Sum throughput Centralized Scheme Greedy Bidding Message Passing (2 rounds) Distributed Auction No. of source destination pairs Figure 7.3: Sum throughput vs. no. of source-destination pairs (for M = N) achieves the optimal performance after only two rounds of message passing. But, the improvement over the distributed scheme after one message passing round is not that significant. This leads us to conclude that in many practical ad hoc networks, distributed scheme with single round of message passing can give assignments which perform very close to the performance of optimal relay assignment. 33

42 Sum throughput Centralized Scheme Greedy Bidding Message Passing Distributed Auction No of source destination pairs Figure 7.4: Sum throughput vs. no. of source-destination pairs (for M = 2N) Relay assignment scheme N= Centralized scheme Greedy bidding scheme roundMessagePassing roundMessagePassing Table 7.1: Improvement in performance of relay assignment with message passing rounds. 34

43 Chapter 8 Conclusions In this work we address the important problem of resource allocation in cooperative wireless networks. We have investigated both centralized and decentralized schemes towards relay allocation with the aim of maximizing sum throughput in the network. We have proposed an efficient optimal centralized relay allocation scheme. We have also developed multiple distributed relay allocation strategies and have performed acomparativeanalysisofthesameundervariousimportantcriteria. The schemes we have developed address the issue of trade-off between performance gap and complexity of the algorithms. We observe that a greedy selection strategy based on local information performs very close to the optimal for small scale networks. The distributed message passing scheme converges closer to the optimal assignment with each round of message passing. We find that message passing with one round of information exchange gives very good results in the sum throughput sense. A scheme based on distributed auction theory is also proposed, which can be applied when relays exceed the number of sources. This scheme clearly highlights the trade-off between time taken for convergence and optimality gap. We have provided a holistic picture of the relay allocation problem in the multiuser cooperative network scenario and solutions to the same. Moreover, the solutions we have proposed are general and can be applied to any relay functionality. Future work in this area can address issues such as lack of synchronization among cooperative nodes or imperfect channel state information at the sources. 35

44 References [1] E. C. van der Meulen, Three terminal communication channels, Advances in Applied Probability, vol.3,pp ,1971. [2] T. M. Cover and A. A. G. Gamal, Capacity theorems for the relay channel,ieee Trans. Info Theory, vol.25,no.5,pp ,1979. [3] A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity - part I, II, IEEE Transactions on Communications, vol.51,pp ,2003. [4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: Efficient protocols and outage behavior, IEEE Transactions on Information Theory, vol.50,pp ,2004. [5] J. N. Laneman and G. W. Wornell, Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks, IEEE Transactions on Information Theory, vol.49,pp ,2003. [6] A. Nosratinia, T. E. Hunter and A. Hedayat, Cooperative communication in wireless networks, IEEE Communication Magazine, October2004. [7] T. Ng and W. Yu, Joint optimization of relay strategies and resource allocations in cooperative cellular networks, IEEE Journal on Selected Areas in Communications, vol25,no.2,pp ,2007. [8] B. Wang, Z. Han, and K. J. R. Liu, Distributed Relay Selection and Power Control for Multiuser Cooperative Communication Networks Using Buyer / Seller Game, in Proc. IEEE INFOCOM, Anchorage, AK, May