论文研究Nash Bargaining Game Based Joint Resource Allocation and Hungarian Method Based Partner Selection for WirelessCooperative Networks.pdf

无线协作网络中基于Nash讨价还价博弈的资源联合分配和基于Hungarian方法的协作伙伴选择，黄双林，魏蛟龙，研究了无线协作通信网络中自私节点之间基于Nash讨价还价博弈的资源分配和协作伙伴选择的问题。系统每一个参与协作的节点既是源节��
国酗技论文在线 http://www.paper.edu.cn lated problems and their solutions focus on efficiency. The fairness issue was mostly ignored However, in many practical scenarios, nodes'selfishness raises doubts on whether a relay node would like to spend its valuable resource in forwarding packets for other users. For example in a commercial wireless network. all mobile nodes are assumed to be selfish rational and energyconstrained. Cooperation may cause significant costs and the users bearing the greatest immediate cost may not achieve the greatest immediate benefits. In this case, a node may exhaust all of its valuable resource to relay other nodes'data and does not obtain any imme diate profits, which hurts the interests of the selfish node and is unsustainable. Therefore, it is necessary for a network to adopt a fair strategy of distributing cooperation gains so that the individual nodes are satisfied immediately In our daily life, the market often serves as a central plat form where buyers and sellers gather together, negotiate transactions and exchange goods so that they can be satisfied im mediately through bargaining and buying or selling. Similarly the cooperation game theory just provides a flexible and natural tool to explore how the selfish nodes bargain with each other and mutual aid. The pioneering work can be found in the following references. In 8 and g], a CDMAbased twouser cooperative modulation scheme was proposed. The main dea is to a llow each user to retransmit received in formation estimates of their partner such that information of each user is received at the receiver with the higher rate. This work is further extended in [1, in which the ergodic capacity and the outage behavior were analyzed for a threeuser case under quasistatic fading channels, when relay nodes followed various co operative protocols, e.g. detectandforward and amplifyandforward cooperative protocols Ilowever, the cooperative behavior based on cooperation game is not considered. Recently, the NBS approach was employed to solve the fairness problem. The resource allocation problem in 10 was formulated as a cooperative gane using NBS, which allowed mobile nodes to fairly share a downlink bandwidth among themselves. In order to fairly allocate subcarrier, rate and power for multiuser orthogonal frequencydivision multipleaccess systems, a scheme based or NBS was proposed in [11]. However, the node cooperation was not considered. In order to promote cooperation,the authors in [ 12) presented a price pair incentive mechanism to arb: trate resource allocation. In [13, based on the NBs, the authors proposed a novel twotier quality of service(QoS)framework and a scheduling scheme for QoS provisioning in worldwide interoperability for microwave access networks. A resource allocating scheme based on the NBS for downlink orthogonal frequencydivision multiple access(OFDMA) wireless networks was proposed i [14]. The authors in[15] proposed a cooperation bandwidth allocation strat egy for the throughput per unit power increase. In 16, the authors considered an bandwidth exchange incentive mechanism as a means of providing incentive for forwarding data. However only bandwidth allocation problem was considered to encourage cooperation in [1416. In [1719 the power allocation problem was considered to encourage cooperation. The authors i: 国酗技论文在线 http://www.paper.edu.cn D 2 1: The system model for cooperative transmission with terminals N and N, transmitting information to destination D; and D; respectively [17 considered fair power sharing between a user and its partner for an optimal signaltonoise ratiO(SNR)increase. From an energyefficiency perspective based no NBS, the authors in [18 studied a cellular framework including two mobile users desiring to communicate with a com mon base station. In order to obtain both user fairness and network efficiency, a cooperative powercontrol game model based on Nash bargaining was formulated in 19 However, the bandwidth only or power only allocation problem was studied in previous work, ignoring the JBPA in wireless network communication. Furthermore, the cooperative partner selection is a key problem 20\ and also ignored when the number of mobile users is no less than three. Motivated by the aforementioned works we constructed a symmetric wire less system model consisting of two user nodes and two destination nodes, which is shown in Fig 1. In the model, it is assumed that each user acts as a, source as well as a potential relay Furthermore, the proposed model represents a more general scenario, comparing to previous work. By bandwidth and power exchanging, each user has the opportunity to share the others resources(e.g, bandwidth and power) and seek other users help to relay its data to obtain the cooperative diversity, and vice versa. The cooperation degree between partner depends on two factors: one is the bandwidth and power contribution of each node on the cooperative rate increment; the another is their channel condition of each node on its cooperation benefit in terms of cooperative rate increment. Later, a multiuser bargaining algorithm is proposed based on optimal coalition pairs among users. With the llungarian method and JBPA scheme obt ained by the algorithm the overall rate for the network increment can be maximized and the resource allocation possesses fairness The rest of this paper is organized as follows. In Section 1. the system model is given. In section 2, the utility functions are presented and the JBPA problem is formulated as a Kperson bargaining game. In Section 3, the joint resource allocation algorithm is presented. Meanwhile a twouser algorithm and a multiuser algorithm are proposed. In Section 4, simulation result evaluation is given. Finally, this paper is concluded in Section 5 5 国酗技论文在线 http://www.paper.edu.cn K 2: Direct, transmission M K 3: Timedivision channel for JBPA 1 System Mode There are K source nodes totally in the model. Any two cooperating source nodes, N and i: anld their corresponding destinations, D: and D; (in particular, DiDi), are shown in Fig. 1 They communicate independent information over the orthogonal channels to the destinations The AF cooperation protocol is used in the model. The AF cooperative transmission between the two nodes occurs in two tiIne slots. The systeIll Inodel is based on frequency division multiple access and each user occupies w hertz bandwidth for transmission. The total power consumptions of each user in the two time slots are the same 1.1 Cooperative Transmission Case The details of cooperation between two nodes are illustrated in Fig 3. Specifically, in time slot. 1, node Ni allocates r; fraction (TiE(0, 1)) of its bandwidth and 1 Si fraction (siE(O, 1)) of its power Pi to relay r, fraction of the data from node N,, and it uses the ri fraction(T E(0 1))of the bandwidth and si fraction(s: E(0, 1)) of its power for its own data transmission. Ii time slot 2, node N; uses ri fraction(r E(O, 1 ))of the bandwidth and 1s, fraction(S; E(, 1)) of its power P, to forward the data originating from node Ni; and it uses the r; fraction of the bandwidth and s, fraction of its power for its own data transmission According to the cooperation details described above, a relay can forward no more than the amount of data as that originating from the source itself. There is r:=1ri, which came from the result of [15. Obviously, both ri and r; should be nonnegative for a meaningful 6 国酗技论文在线 http://www.paper.edu.cn cooperation. Then, we have +7,=1,0 Suppose that subscript denotes source node and superscript denotes destination node. Let Gii represents the channel gain between node Ni and node N; (it i). and Let G; denotes the channel gain between source node Ni and destination node Di. We assume that the noise power spectral density at different receivers is i i.d. with the No. The cooperative transmission consists of two stage. In time slot 1, assumed that ai is the message signal from N to N, and destination Di, then. the achieved SNR helped by N; for N; to D, is given byl3 SiP Gi(1 ,)P 02PGy+(18)PG+o2 and the effective rate of node n at the d, is AF log(1A+Ai) where o?=Ti WNo and Ais PiCi is the SNR that results from the direct transmission (DT) from node mi to d i in the first time slot Similarly, the relayed SNR helped by N for N, to D, is given by S, Pi Gi(lSiPiG 03{5;PGn+(18)PG2+ And thie effective rate of node ni at the d is rjRjiryWlog(1+x+xji) sinG where a=riNO and a is the snr that results from the dt from node n to the Di in the first time slot 1.2 Direct Transmission Case Ilowever, Ni and N, may prefer transmitting its own data independently, if it could make up the opportunity cost of cooperative transmission by direct transmission, as illustrated in Fig. 2. Then the dt rate at Di is Pg? Ri=W log(1+ And the dt rate at the d is PiG R;=wlog(1+ where o2=W No is the AWGN received at the destination Di and D, on the condition of no partner for Cooperation 国酗技论文在线 http://www.paper.edu.cn From the above introduction, it's clear that the resource allocation variables r; and reflect the Ni's rational decisions while r, and s, reflect the decisions of ni,i.e, Ni determines r; and N, determines ri, and that the decisions of one user will affect the choices of its partner Their payout and payoff should be traded off and both users expect an optimal tradeoff. The following sections will focus on in particular this problem's solution that can bring about win win results 2 Utility Function and problem Formulation II this sectiOn, the utility functions of the source nodes in the systeIn inodel are given anld then the model is analyzed via the cooperative game theory 2.1 Utility Function For N; and N, their utility functions U and U can be defined as Ui=r, R AF R AF For the sake of notation simplicity. we define a 6. PiGa P: G Pa:G Then. we have biCisi(1si U=Wbg(1+081+b+c(1s)+小 0) b;c;8(1 b5+c1(1s:)+ Since a node would participate in cooperative transmission only if its effective rate obtained from cooperative transmission is higher than that of DT, it is obvious that any selfish node M will give up cooperation when ri is more than its payoff. Therefore, the minimal values vi and l, must be (13 2.2 Problem formulation The bargaining probleIn based onl the cooperative gane is described as follows. Define the set of players K1, 2, K in the game. S is a convex and closed subset of reh to denote the set of feasible payoff allocations that the players can achieve if they cooperate Let Ukin represents the IniniImal payoff that the k:th player can accept; otherwise, it will 国酗技论文在线 http://www.paper.edu.cn ait cooperation. Suppose that UK ES Uk >UK,VkEK is a nonempty bounded set Define Umin=(Umin, Umin, .. UKin), then the pair(S, Umin )is called as a Kperson bargaining problem Within the feasible set s, nbs provides a unique and fair Pareto optimal operation point According to Nash's game theory on bargaining problem, in the analysis of the Kperson bar gaining problem, the cooperative solution should satisfy six axioms. Suppose that the cooper ative solution is U*=(Ui, U) and the contextualized formulations of those axioms are given as follows 1 )Individual Rationality: Ui>Umn, Vi 2 Feasibility:Ux∈S. 3Symmetry: The cooperative solution is not affected when the positions of those two players are exchanged 4 Pareto Optimality: For every (Ui, U,). if (Ui, U>U,Vi, then (U,U=U, Vi 5 Independence of Irrelevant Alternatives: If U*(S Umn)E SCS, then U*(S, Umin US.U 6 Independence of Linear Transformations: For any monotone incremental linear function F, we always have U*[F(S), (Umim)= FU(S, Uman) Nash proved the following theorem, showing that there is exactly one Nbs satisfying the above axioms 21 Theorem 1: Existence and Uniqueness of NBS: There is a unique solution function that satisfies all above six axioms, and this solution satisfies arg max Ui>>U II(UU For the twopersoll bargaining probleIn, the NBS function is expressed as U=arg max(UiU7)(UjUmm (15 As discussed above, eacl user has Vi as its objective function, which is bounded above anld has a convex, nonempty and closed support. The objective is to maximize all Ui simultaneously a d keep fair. Umin denotes the minimal performance( direct transmit performance), and Umir is the initial agreement point. For the problem(19), it is a JBPA problem and its objective function is not concave. Therefore there might be an infinite number of local maximal points The problem. then, is to find a simple way to choose the operating point in S for all users, such that this point is optinal and fair. For this hard problein, we will discuss it in the next section 9 国酗技论文在线 http://www.paper.edu.cn 3 Joint Resource Allocation Algorithm Firstly, the twouser case is studied in this section and a fast twouser bargaining algorithm is proposed. TheIl, a lllultipleuser algorithiIl using coalitiOns is developed 3.1 Bargaining algorithm for TwoUser Case Since it 's impossible to reach the closedform solution of(15 ), we developed a numerical search algorithm, by which the global maximum of(15)rather than a local maximum can be obtained According to the decomposition optimization theory 22 123, the optimization problem of (15 can be equivalently decomposed into the following two problems. Firstly, the bandwidth allocation ratio can be obtained by solving U*(x,,s;,S)=g U( (16 (0,1 Where U*(ri, Fi, si, Si is the maximal solution for given s; and s,, not the optimal solution r: and r: are the corresponding ba ratios Secondly, the optimal Pa ratios is obtained by solving )=arg U(示 17 ∈(0,1) Then, we compare aIU*(Fi,Fi, si: Si) and choose the maximal one for all s, and s;. This way, we can obtain the optimal solution, U(ri The following gives the proof of Theorem 2 which indicate that(16)and(17) both have a unique Nash equilibrium solution Theorem 2:(Existence of Unique Nash Equilibrium) For given Si and s,, Vsi, S,E(0, 1) the twouser bargaining game admits a unique Nash equilibrium solution r=(ri, Ti). For given ri and ri, Vri,T; E(0, 1), the twouser bargaining game admits a unique Nash equilibrium solution s=(si, si) Proof: See Appendix A n what follows, we firstly put forward a fast iterative algorit hm for searching the maximal BA rat For given s: and s,, there exist the corresponding Ba ratios r; and r:. Substituting Ti and r* into R AF and R AP respectively we have A w log(1+.+ bc2:2(1) (18 个b2+c2(16)+产 RAF=Wlog(1+ b;G;s(1s) b+c;(1s:)+ 10 国酗技论文在线 http://www.paper.edu.cn This way, Rii and rii will not include variables Ti and Ti. So we have U(T,. TK, Si, si)arg Imax P:iap R2)(Ti RARn ys2,s;∈(0,1) For probleIn(20), by taking the derivative to r'i and r; respectively, anld equating then to zero, we get =(:(t,(t)=21 RAF R (21) D 1R D 7=1(:(,()2 AF (22) It is obvious that r*+r; =l, which means that there is one variable only between T: and p;. So the iterations of the Ba ratio updating can be expressed as follows 7;(+1)=L;(r2() (23) We show next the convergence of the iterations in(23) by proving that the Ba ratio updating function I(r: (t)is a standard function(24) Definition 1. A function L;(F) is standard if for all ri>0, the following properties are satisfied(24 Positivity. Ii(r:)>0 Monotonicity. If r:>rl, then Ii (ri>IG! Scalability. For all c> 1. (li(T:)>li(ari) Proposition 1. The function I; (;)is standard Proof. See Appendix b In 24, a proof has been given. Starting from any initial feasible BA ratios ri and Ti, the BA ratios produced after several iterations via the standard ba function always converges to a unique fixed point The problen(17)is a combinatorial probleInl involving two continuous variables, si: and Si. Ilowever, continuous power adaptation is infeasible in practical networks because there are only several discrete power levels for each node. Suppose that the number of power levels for each nlode is M. The coMputation coMplexity of searching the PA ratios is O(M ). In order to explain the path to obtain the optimal allocation ratios ri, si and si, the iteration searching algorithm is shown in table 1 3.2 MultipleUser Algorithm A twostep algorithm is proposed for solving the JBPA problem for all users of a Knode network(K> 2) in a centralized way. First, all users are grouped into pairs, in which the cooperating nodes form a coalition. Then, the Hungarian method 25 is employed to solve the
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