Optimal Channel Selection for Cooperative Spectrum Sensing Using Coordination Game

Transcription

1 2012 7th International ICST Conference on Communications and Networking in China (CHINACOM) Optimal Channel Selection for Cooperative Spectrum Sensing Using Coordination Game Yuhua Xu, Zhan Gao and Wei Tian Institute of Communications Engineering PLA University of Science and Technology, Nanjing, , China Abstract This paper studies the problem of sensing channel selection in cognitive radio networks, where multichannel cooperative spectrum sensing and fusing are employed. Since the cognitive radio users can not sense all the channels simultaneously, it is important for the users to select sensing channels properly. We first formulate the channel selection problem as a coordination game model and then propose a learning algorithm to achieve the optimal solution. The proposed algorithm is simple, distributed, and optimal; moreover, it only need slight information exchange. Also, the tradeoff between sensing cost and achievable system throughput is investigated. The proposed solution provides a promising approach for distributed optimization problems, especially the one with heavy computation complexity, and hence can be applied to other scenarios. Index Terms cognitive radio, channel selection, cooperative spectrum sensing, coordination game, spatial adaptive play. I. INTRODUCTION Multichannel cooperative spectrum sensing for cognitive radio (CR) networks is an active research topic recently [1], [2]. By using cooperative sensing and fusing technologies, e.g., OR, AND and SOFT combination rules [7], the sensing performance can be improved significantly. Generally, the CR users can not sense all the channels simultaneously; instead, they can sense only a part of the channels. For a dedicated channel, sensing by a large number of users leads to higher throughput. On the other hand, when there is no user to sense it, no throughput is achieved. Thus, it is not desirable that a user senses a highly crowded channel, whereas other channels are not being sensed by any user. As a result, the key concern for the users is to select the sensing channels properly to maximize the system throughput. To the best of our knowledge, the optimal solution of this problem has not been considered yet in the literature. Thus, it is important to investigate the channel selection problem for cooperative spectrum sensing. In this paper, we firstly formulate a general channel selection problem for cooperative spectrum sensing. It is shown that the channel selection problem for multichannel cooperative spectrum sensing is highly coupled among users; specifically, it is a combinatorial optimization problem and hence NPhard. Therefore, the task of achieving the optimal solutions is challenging, as traditional optimization methods always lead to heavy computation complexity. It is seen that the channel selections of the users are interactive, which motivates us to formulate this problem as a game model [3]. Generally, the task of developing a game-theoretic approach for distributed optimization problems consists of two steps [4]: (i) designing the utility function of a user carefully such that the global utility perfectly coincides with the local utility, and (ii) developing updating rules, with which the users adjust their behaviors to a stable and desirable solution. Following this methodology, we formulate the channel selection problem as a coordination game; furthermore, the optimal channel selection profile is achieved rapidly and distributively by using a game-theoretic learning algorithm, which is called spatial adaptive play (SAP). The results obtained in this paper is promising since the proposed algorithm is simple, distributed and optimal. One might argue that game models are only suitable for multi-agent systems with conflicting objectives, where the users make their decisions autonomously and selfishly. This may be caused by the fact most existing work addressing the application of game models in wireless communication systems always emphasized competitive environment and selfishness. However, it should be pointed out that this is just a narrow version of game models. In fact, game models are suitable for any scenario with interactive multi-user decisions. This can bring about more efficient solutions for solving optimization problems, which has began to draw attention. In particular, there are some studies which utilized game models to solve problems in interactive but non-competitive environment, e.g., potential game formulation for cooperative control [5] and vehicle-target assignment [6]. Based on the above consideration, it is emphasized that the algorithm proposed in this paper provides a proposing approach for distributed optimization problems, especially the one with heavy computation complexity. One can apply our algorithm in other scenarios, e.g., spectrum allocation, power control and task scheduling. Related work can be found in [1], [2]. However, it is noted that our work is differentiated from these work in terms of system model, optimization objective and methodology. The rest of this paper is organized as follows. In Section II, we present the system model and the problem formulation. In Section III, we formulate the sensing channel selection problem as a coordination game and propose a game-theoretic learning algorithm called spatial adaptive play to achieve the optimal solution. In Section IV, simulation results and discussion are presented. Finally, we make conclusion in /12/$ IEEE

2 channel m by CR user n is given by: Fig. 1. An example of the multichannel cooperative spectrum sensing with four CR users and three licensed channels ( Users 1, 2 and 3 can only sense one channel at a time, while user 4 can sense two channels simultaneously). Section V. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a CR network involving N CR users and M licensed channels. Denote the CR user set as N = {1,...,N} and the licensed channel set as M = {1,...,M}. It is assumed that the channel m has transmission rate R m, m M. The activities of the primary users are assumed to be independent from channel to channel and from user to user. Then, we can assume that each channel is not occupied by the primary users with probability θ m,m M. Due to hardware limitation, the CR users can not sense all the channels simultaneously; instead, CR user n can only sense K n channels at a time, 1 K n <M, n N. Note that K n is not necessarily identical for different n. An example of the considered system model involving four CR users and three licensed channels is shown in Fig. 1. Each CR user senses the spectrum and reports the individual sensing result to a fusion center (FC), which combines these results cooperatively to deduce the final spectrum decisions [7]. Denote the individual detection probability 1 of channel m by CR user n as p nm, 0 <p nm < 1. Denote a n A n as users n s action, where A n is the action space of user n, i.e., A n is the combination set of K n channels taken from M channels at a time. For instance, the action space forcruser1infig.1isa 1 = {1, 2, 3}, while that for user 4isA 4 = {(1, 2), (2, 3), (1, 3)}. The number of the available actions for user n is given by C Kn N = N! (N K. n)!(k n)! Denote the joint action profile of all the CR users as a = {a 1,...,a N }. For the cooperative fusing scheme, the AND combination rule [7] is applied at the FC 2. Then, under the action profile a, the individual detection probability of licensed 1 The detection probability here is defined by the probability that the primary user is idle and detected as idle by the CR user. Although this definition is slightly different from that in previous literature, it does not eventually affect the analysis in this article. 2 The used AND rule is just for the purpose of illustration, and the analysis presented in this paper can easily be extended to other combination schemes, e.g., OR and SOFT combination rules. p n (m, a n )= { pnm, m a n 0, m / a n (1) Furthermore, we assume that the detection of the CR users is independent, then the joint detection probability of channel m by all the CR users is given by: P (m, a) =1 ( 1 pn (m, a n ) ) (2) n N It is assumed that the CR users sensing the same channel share that channel equally. Then, the achievable throughput of CR user n is given by: T n (a) = m a n θ m R m P (m, a) k N I(m, a k), (3) where P (m, a) is the joint detection probability of channel m specified by (2), and I() is an indicator function as follows: { 1, m ak I(m, a k )=. (4) 0, m / a k Then, the system throughput is given by: T (a) T n (a) = θ m R m P (m, a). (5) n N m M The system-centric objective is to find the optimal channel selection profile to maximize the system throughput, i.e., (P 1 :) find a opt arg max T (a) (6) However, P 1 is a combinatorial problem and the task of obtaining a opt involves extremely huge searching space. For instance, consider a small-size system involving ten CR users and six licensed channels, and each user can sense two channels simultaneously. Then, the number of possible action profiles is (C6) Notably, it is difficult to solve by traditional methods. Therefore, a solution with lower complexity is desirable. III. COORDINATION GAME BASED OPTIMAL SOLUTION There exists a fusion center (FC), which has sensing information from all the users. Moreover, the CR users can naturally have information about other users through the FC. Therefore, cooperative decision is then feasible among the CR users. Hence, we establish a coordination game [8] to model the channel selection problem P 1, and propose a learning algorithm to obtain the optimal solution a opt. A. Coordination game model The coordination game is defined by G c = [N, {A n } n N, {u n } n N ], where N = {1,...,N} is the set of players (CR users), A n is the set of available actions (channels) for player n, and u n is the utility function of player n. Coordination game is a kind of game models with identical payoff, which is the system throughput [8], i.e., u n (a n,a n )=T (a), n N (7) 110

3 where a n is the action profile of all the players except n. Theorem 1. G c is a potential game which has at least one pure strategy Nash equilibrium (NE), and the optimal action profile a opt constitutes a pure strategy NE of G c. Proof: We construct the potential function as follows: Φ(a n,a n )=T (a n,a n ) (8) where T (a n,a n ) is the system throughput specified by (5). Then, according to (6) and (8), we have: a opt arg max Φ(a n,a n ) (9) Suppose that an arbitrary player n unilaterally changes its channel selection from a n to a n, then the change in individual utility function caused by this unilateral change is given by: u n (a n,a n ) u n (a n,a n )=T (a n,a n ) T (a n,a n ) (10) Meanwhile, the change in the potential function caused by this unilateral change is given by: Φ(a n,a n ) Φ(a n,a n )=T (a n,a n ) T (a n,a n ) (11) From (10) and (11), it is seen that the change in individual utility function caused by any player s unilateral deviation is the same as the change in the potential function. Thus, according to the definition given in [9], G c is a potential game with system throughput serving as the potential function. There are several attractive properties of potential games and the following are the most important two: (i) every potential game has at least one pure strategy NE, and (ii) any global or local maxima of the potential function constitutes a pure strategy NE. Based on the above properties, Theorem 1 follows. According to Theorem 1, the optimal joint action profile can emerge as the results of interactions of individual decisions. However, normally multiple NE points exist in G c, and some are suboptimal [3]. Thus, we seek for a learning algorithm that achieves the optimal NE in the following. B. Achieving optimal channel selection profile via spatial adaptive play With the problem now formulated as a potential game, there are large number of learning algorithms available in the literature to achieve pure strategy NE, e.g., the well-known best response dynamic [9]. However, a main drawback of the above learning algorithms is that they may converge to the undesirable NEs. Thus, we resort to spatial adaptive play (SAP) [10] to achieve the optimal NE. To characterize SAP, we extend the game to a mixed strategy form. Let the mixed strategy for player n at iteration k be denoted by probability distribution q n (k) Δ(A n ), where Δ(A n ) denotes the set of probability distributions over action set A n. In SAP, exactly one player is randomly selected to update its action according to the mixed strategy while all other players repeat their actions. Formally, SAP is described at the top of this page. It is noted the procedure of the proposed algorithm is similar to that proposed in our recent work [4]. Spatial adaptive play for sensing channel selection Initialization: Set k =0and each CR user n N randomly selects an available action a n (0) from its available action set A n with equal probability. All the CR users report their current action a n (0) and their detection probability p nm, n N,m M, to FC. Then, FC broadcasts the detection probabilities p nm, n N,m M, to all the CR users. Step 1: A CR user, say i, is randomly selected with probability 1/N. Then, FC sends a message including the current action profile a(k) to CR user i. Step 2: All the CR users other than i keep their actions unchanged, i.e., a i (k +1) = a i (k). Meanwhile, with the message received from FC, user i calculates the utility functions over its all available actions, i.e., u i ( a i,a i (k)), a i A i. Then, it randomly chooses an action according to the mixed strategy q i (k +1) Δ(A i ), where the a i th component q ai i (k +1) of the mixed strategy is given as: q ai i (k +1)= exp{βu i (a i,a i (k))} (12) a i A i exp{βu i ( a i,a i (k)) for some learning parameter β > 0. Finally, user i report the new action a i (k +1) to FC. Step 3: If the predefined maximum number of iteration steps is reached, stop; else go to Step 1. Note that the SAP is implemented distributively, and the message exchange only occurs between the selected user and FC in each iteration. Furthermore, the asymptotic behavior of the mixed strategy is characterized by the following theorem. Theorem 2. With a sufficiently large β, SAP converges to the optimal joint action a opt with arbitrary high probability 3. Proof: According to [4], [5], [10], the stationary distribution μ(a) Δ(A) of the joint action profiles for a potential game, β >0, is given as: exp{βφ(a)} μ(a) = a A exp{βφ( a)} (13) where Φ() is the potential function specified in (8). From (9), the maximum potential function is then determined by Φ max =Φ(a opt ).Now,setβ,wehave: exp ( βφ(a opt ) ) exp ( βφ( a) ), a {A a opt } (14) Based on (13) and (14), the following can be obtained: lim μ(a opt) =1 (15) β Hence, Theorem 2 follows. Remark: The most proposing features of the proposed solution are: (i) the updating procedure, as specified by (12), is simple and distributed, (iii) the message exchange is slight, and (iii) the optimal solution is achieved. 3 Here, arbitrary probability means that the probability is sufficiently approaching one, say

4 3 3 System throughput (T) Optimal system throughput Proposed game based solution Iterations (k) Achievable system throughput (T) Individual detection probability increases p = 0.9 p = p = 0.7 p = 0.6 p = Number of CR users (N) Fig. 2. The convergence behavior of the proposed game based solution. Fig. 3. The achievable system throughput versus the number of CR users. IV. SIMULATION RESULTS AND DISCUSSION For presentation, we simulate a small-size CR system, which involves N =9CR users and M =4licensed channels. Each CR user can only sense one channel at a time. Furthermore, all the channel are assumed to have unit transmission rate, i.e., R m = 1, and the channel idle probability is set to θ m =0.7, m M; moreover, the individual detection probabilities are set to p nm =0.1, n N,m M.The convergence behavior of the proposed game based solution is shown in Fig. 2. The maximal throughput, T max =2.778, is obtained by exhaustive search. On the other hand, SAP is applied for the game based solution. Specifically, the learning parameter is set to β = k, where k is the number of iterations. Such a linearly increasing form of β is to balance the tradeoff between exploration and exploitation [4]. It is noted that the game based solution catches up with the optimal system throughput in about 55 iterations, whereas the searching space for the exhaustive search is about profiles. The obtained results validate the optimality of the proposed game based solution. Secondly, the achievable system throughput obtained using SAP channel selection algorithm versus the number of CR users N for different individual detection probability p is shown in Fig. 3. It is noted that for a given individual detection probability, e.g., p =0.8, the achievable system throughput increases as N increases but becomes moderate when N becomes sufficiently large. This result can be expected in any cooperative sensing systems. In addition, the system throughput is piecewise linear with N, which is inherently due to the structure of the multichannel cooperative sensing problem. Furthermore, larger p leads to higher system throughput as expected. It is noted that there is a fundamental tradeoff between the sensing cost (larger N and p result in higher sensing cost) and achievable system throughput. Then, the choice of them should be application-dependent in practice. V. CONCLUSION We proposed a coordination game to find the optimal solution for the problem of channel selection in cognitive radio systems with multichannel cooperative spectrum sensing. It is analytically shown that the proposed solution achieves the optimal solution rapidly and distributively. The proposed solution provides a promising approach for distributed optimization problems, especially the one with heavy computation complexity. Again, it is emphasized here that game models are suitable for any interactive scenarios, no matter the environment is competitive or non-cooperative. The key concerns are designing utility function carefully and developing efficient updating rules, which may vary from scenario to scenario. ACKNOWLEDGMENT This work was supported by the National Basic Research Program of China under Grant no. 2009CB320400, the National Science Foundation of China under Grant nos and , and in part by the Jiangsu Province Natural Science Foundation under Grant no. BK REFERENCES [1] R. Fan and H Jiang, Optimal multi-channel cooperative sensing in cognitive radio networks, IEEE Trans. Wireless Commun., vol. 9, no. 3, pp , [2] R. Fan and H Jiang, Q. Guo, and Z. Zhang, Joint optimal cooperative sensing resource allocation in multichannel cognitive radio networks, IEEE Trans. Veh. Tech., vol. 60, no. 2, pp , [3] R. Myerson, Game Theory: Analysis of Conflict. Cambridge, MA: Harvard Univ. Press, [4] Y. Xu, J. Wang, Q. Wu, et al. Opportunistic spectrum access in cognitive radio networks: Global optimization using local interaction games, IEEE J. Sel.Topics Signal Process., vol. 6, no. 2, pp , [5] J. Marden, G. Arslan, and J. Shamma, Cooperative control and potential games, IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, vol. 39, pp , [6] G. Arslan, J. Marden, and J. Shamma, Autonomous vehicle-target assignment: a game theoretical formulation, ASME Journal of Dynamic Systems, Measurement and Control, vol. 129, pp ,

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