Multi-Channel Assignment in Wireless Sensor Networks: A Game Theoretic Approach

Transcription

1 Multi-Channel Assignment in Wireless Sensor Networks: A Game Theoreti Approah Qing Yu, Jiming Chen, Yanfei Fan, Xuemin (Sherman) Shen and Youxian Sun State Key Lab of Industrial Control Tehnology, Zhejiang University, Hangzhou, 31007, P.R.China tsingyu@zju.edu.n; yxsun@iip.zju.edu.n Department of Eletrial and Computer Engineering, University of Waterloo, Waterloo, ON, NL 3G1, Canada {jmhen, yfan, xshen@bbr.uwaterloo.a Abstrat In this paper, we formulate multi-hannel assignment in Wireless Sensor Networks (WSNs) as an optimization problem and show it is NP-hard. We then propose a distributed Game Based Channel Assignment algorithm () to solve the problem. takes into aount both the network topology information and transmission routing information. We prove that there exists at least one Nash Equilibrium in the hannel assignment game. Furthermore, we analyze the sub-optimality of Nash Equilibrium and the onvergene of the Best Response in the game. Simulation results are given to demonstrate that an redue interferene signifiantly and ahieve satisfatory network performane in terms of delivery ratio, throughput, hannel aess delay and energy onsumption. I. INTRODUCTION Wireless Sensor Networks (WSNs) with low-ost, lowpower and multi-funtionality, make their appliations with large sale and dense deployments possible. While traditional ways to use a single hannel among the whole WSNs in the same geographial region, an not work properly when the WSNs beome larger and denser, sine frequent interferene among the WSNs will degrade the network performane dramatially. The half-duplex radio transeiver in urrent available sensor node is already able to operate on multiple hannels. Thus, using the available multiple hannels effetively to exploit parallel transmission in WSNs beomes very attrative. There have been a onsiderable number of multi-hannel protools for wireless networks [1], [], [3], [4], [5]. However, most of them make strong assumptions that the transeivers either use the frequeny hopping spread spetrum wireless ards or an operate on multiple hannels simultaneously. Unfortunately, these protools are not appliable to WSNs, beause urrent available sensor node eah with only one simple half-duplex radio transeiver an not satisfy these strong assumptions. Besides, the extra overhead of some protools aused by dynami hannel negotiations or RTS/CTS pakets, poses major hallenge on both onstrained energy and limited bandwidth in WSNs. Reently, several multi-hannel protools have been proposed for WSNs. Most of them dynamially assign hannels The researh is supported in part by NSFC Guangdong Provine joint Projet grant no. U ; NSFC grant nos and ; 863 High- Teh Projet no. 007AA04101, China and NSERC, Canada to links aording to the transmitting flows in the network [6], [7], [8], [9]. Although these dynami protools an redue interferene to some degrees, the requirements of frequent negotiations and highly aurate time synhronization inur extra large overhead. Alternatively, stati hannel assignment protools are based on network topology [10], [11]. They work with less overhead than the dynami ones but an not redue interferene effiiently, sine they neither take full advantage of the routing information nor dynamially trak the transmitting flows. For example, [10] uses network topology information to balane the hannel usage among twohop neighbors. Without onsidering routing information, it may make the links not involved in the routing bandwidthexess while the links involved in the routing bandwidthtight. In this paper, we take into aount both Topology Information and Routing Information (TIRI) so as to redue the interferene more effiiently. However, it is diffiult, in general, to fully exploit the TIRI in WSNs. For example, TMCP [11] statially divides the whole network into mutually exlusive single-hannel subtrees, whih takes advantage of the inter-tree routing information but does not exploit the intratree routing information. Hene, in order to fully exploit TIRI, all sensor nodes must be involved in the hannel assignment and share information interatively. Game theory is the study of the interation of autonomous agents. The features of WSNs deentralized operation, self onfiguration, and energy awareness result in that eah sensor node in WSNs behaves as an autonomous agent, and thus make game theory a promising way to solve the ompeting problems in WSNs, suh as overage [1], routing [13], sensor ativation [14], et. To the best of our knowledge, this paper is the first attempt to use game theory to ope with the hannel assignment problem in WSNs. Based on the observations that the interferene suffered by the sender is related to the number of links heard by the reeiver, and that the routing in appliations of WSNs is generally stati and of tree/forest struture, the whole network an be onsidered to be omposed of lots of parent-hildren sets whih are onstruted from the TIRI. In order to fully exploit TIRI, we model the hannel assignment among those sets as a game with the total interferene of the whole network as the soial objetive. Speifially, the parent in eah set is modeled as a player, its payoff is defined as zero minus the sum of the /10/$ IEEE

2 interferene suffered by the set and that aused by the set. The hannel assignment game evolves aording to the Best Response (BR) dynami, i.e., eah player hooses the hannel that maximizes its payoff aording to the hannels hosen by the other players. We prove that BR onverges into a Nash Equilibrium (NE), and analyze the soial effiieny of NE. Furthermore, we propose, based on BR, a Game Based Channel Assignment algorithm () to handle the NPhard hannel assignment problem in WSNs in a distributed way. The main ontributions of this paper are summarized as follows: We analyze the relationship between the number of links heard by the reeiver and the interferene suffered by sender, and propose an interferene metri. We formulate the hannel assignment in WSNs as an optimization problem and prove that it is NP-hard. Furthermore, we model it as a hannel assignment game to fully exploit TIRI. We show that the game is an exat potential game and there exists at least one NE in the game. We analyze the prie of anarhy of the game and bound it by 1 where is the number of available hannels, whih means any NE is suboptimal. We adopt BR as the evolution rule of the hannel assignment game and prove that BR onverges into an NE in at most ( V 1) iterations, where V is the total number of sensor nodes. We propose a distributed algorithm,, based on BR. an provide a suboptimal solution to the hannel assignment problem in polynomial time, and it is ompatible with both sheduling based and ontention based medium aess ontrol shemes. Simulation results show that redues interferene signifiantly and ahieves better network performane in terms of delivery ratio, throughput, hannel aess delay and energy onsumption than does. The remainder of the paper is organized as follows. Setion II analyzes the interferene and formulates the optimization problem for hannel assignment. We propose the game based algorithm to solve this problem and analyze the onvergene and sub-optimality of the algorithm in Setion III. The performane of the algorithm is evaluated by simulations in Setion IV. Finally, we onlude our paper in Setion V. II. PRELIMINARIES AND PROBLEM FORMULATION A. Interferene Analysis In wireless ommuniations, whether a message gets orrupted or not is losely related to whether other links, whih are heard by the reeiver, are transmitting message or not. In general, as long as there is another message transmitting simultaneously through some of these links, there is a ollision at the reeiver and the message may be orrupted. In this subsetion, we study the relationship between the number of links heard by the reeiver and the interferene suffered by a message transferred to the reeiver in a similar way as [15]. Assume that there are I links heard by the reeiver and the probability that a message is transmitting through a link at a given time is q. Therefore, the probability of a suessful transmission with one try is p =(1 q) I 1. (1) If there is no maximum number of retries onstraint, the average number of retries before a suessful transmission is R = + k=1 kp(1 p) k 1 1= 1 1. () p Apparently, Equation () holds beause the MAC layer allows relatively large number of retries. Substituting Equation (1) into Equation (), we have R =(1+ q 1 q )I 1 1. (3) Sine q is very small for the light load in WSNs, an approximate R an be obtained as follows. R =. q (I 1). (4) 1 q Equation (4) reveals that the more links the reeiver hears, the more retries a message needs to be suessfully reeived. Furthermore, the more times the sender retries, the more energy it onsumes and the longer delay the transmission suffers, whih in turn results in derease of both throughout and delivery ratio. Thus, the average number of retries, R, is a key metri to haraterize the interferene suffered by the sender. Furthermore, aording to Equation (4), I is approximately linear with R, so we adopt I as the interferene metri. B. Problem Formulation The links towards a reeiver an be divided into two ategories: interseting links and interfering links [16]. The transmissions in interseting links aim at the reeiver, while the transmissions in interfering links do not aim at the reeiver but an be overheard by the reeiver. If only single hannel is used, then both interseting links and interfering links may interfere with the transmission aiming at the reeiver. Though we an not redue the interferene aused by interseting links beause there is only one half-duplex radio transeiver in a sensor node and thus the reeiver an not reeive multiple messages simultaneously, we an redue the interferene aused by the interfering links by using multiple hannels. In this paper, we assign hannel in a reeiver-entri way, i.e., eah sensor node is assigned a fixed hannel to reeive message, the neighbors whih have messages to deliver to it should use this hannel to send, and obviously all links always use the hannels that their senders use to send. If there are enough non-overlapping hannels, we an make all interfering links use different hannels from the ones their reeivers use. In this ase, the reeivers an not overhear the transmissions in interfering links and the interferene in the network an be remarkably redued. However, the number of non-overlapping

3 TABLE I NOTATION DEFINITIONS Symbol Definition represents the ardinality of a set. C the set of non-overlapping hannels, i.e., C={1,,...,. V the set of sensor nodes inluding both normal nodes whih know their parents and sink nodes whih only ollet messages and have no parent. E the set of links inluding both interseting links and interfering links. Any pair of onneted normal nodes are onneted by two links with different diretions, while any pair of onneted normal and sink nodes are only onneted by one link. And there is no link between sink nodes. G(V,E) the topology graph of sensor network. It is a direted graph and omposed of V and E. s(e),r(e) the sender and reeiver of link e. parent(i) the parent of sensor node i. Child(i) the set of hildren of sensor node i. P the set of all non-leaf sensor nodes whih have hildren in the tree/forest struture, i.e., P ={1,,...,m. f the hannel assignment that assigns eah non-leaf sensor node one hannel to reeive message, i.e., f : P C. E s the set of interseting links, i.e., E s={e : e E and s(e) Child(r(e)). E f the set of interfering links, i.e., E f ={e : e E and s(e) / Child(r(e)). h(e) the hannel of link e, i.e., h(e)=f(parent(s(e))). the potential interferene that link e may yield to the network, i.e., = Child(r(e)). I(i, f) the interferene suffered by sensor node i in assignment f, i.e., I(i, f)= {e : e E,r(e) =parent(i) and h(e) = f(parent(i)). Sine the sink nodes just ollet messages in the network, they suffer no interferene. L r(f), L u(f) for a given f, L r is the set of all the interfering links that an not be heard by its reeiver, and L u is the set of all the interfering links that still an be heard by its reeiver, i.e., L r(f)={e : h(e) f(r(e)),e E f and L u(f)={e : h(e) =f(r(e)),e E f. hannels is usually fixed and limited in pratie [11]. Hene, the problem beomes to optimally assign the limited hannels to minimize interferene. We first define some notations in Table I and make some useful assumptions as follows: 1) The network routing is of tree/forest struture, i.e., hildren always deliver their messages to their parents in the network; ) Channels are all non-overlapping and do not interfere with eah other; and 3) Communiation between two sensor nodes is symmetri, i.e., given that sensor nodes i and j are onneted, if i uses j s hannel to transmit then i an be heard by j, and vie versa. We then use the total interferene suffered by all sensor nodes as the optimization objetive, sine it is orresponding to the average number of retries over the network and in turn reflets the network performane. The primal optimization problem (PP): GivenG(V,E) and C, the primal optimization problem is to find a hannel assignment f to minimize the total interferene I(i, f). For a given G(V,E) and an assignment f, wehave I(i, f) = +. (5) e E s e L u(f) Sine both the total potential interferene that all the interseting links may generate and the one that all the interfering links may generate are onstants in a given G(V,E), wehave =A, + =B, (6) e E s e L u(f) e L r(f) where A and B are onstants regardless of f. Substituting Equation (6) into Equation (5), we have I(i, f) =A + B. (7) e L r(f) Hene PP an be equivalently transformed into a dual optimization problem. The dual optimization problem (DP): GivenG(V,E) and C, the interferene redution problem is to find a hannel assignment f to maximize the total removed interferene U(f) =. e L r(f) III. GAME BASED CHANNEL ASSIGNMENT In this setion, we first analyze the hardness of the hannel assignment problem in WSNs, then model the problem as a hannel assignment game, and finally provide a distributed algorithm based on the game to handle the problem. A. Problem Assessment The minimum same-olor edges oloring problem (MSCP): Given undireted graph G(V, E) and integers 0< K V, 0 M< E, find a oloring (i.e., assign eah vertex one of the K olors) suh that the number of same-olor edges (i.e., the olors of its two vertexes are the same) is not more than M. Lemma 1: MSCP is NP-omplete. Proof: we set M=0 in MSCP, thus MSCP is restrited to GRAPH K-COLORABILITY problem [17] whih is a typial NP-omplete problem. Hene, MSCP is NP-omplete. The relative deision problem (RP): Given G(V,E), C and a non-negative integer W, determine whether there exists a hannel assignment f suh that I(i, f) W. Theorem 1: PP is NP-hard. Proof: To prove this theorem, it is suffiient to prove RP is NP-omplete. It is easy to see that RP is in NP. Then, we transform MSCP into RP. For an arbitrary G, wereateag in the following way. For eah node i in G, we reate two normal sensor nodes in G, i.e., p i and s i, and p i is the parent of s i. Two direted edges are added: one is from s i to p i and the other is from p i to s i. For eah edge e ij in G, we reate two direted edges in G: one is from s i to p j and the other is from p j to s i. Finally, we reate a sink sensor node r and V edges eah of whih is from p i to r. Thus, we get a new graph G(V,E) where V = V +1 and E =3 V + E. Apparently, this transformation is polynomial time. Let the hannel assigned to p i in G orrespond to the olor assigned to i in G and the interfering links heard by the non-leaf sensor nodes to the same-olor edges. Thus, MSCP with K, M and G is transformed, in polynomial time, into RP with C =K, W = V + V +M and G. Aording to Lemma 1, MSCP is

4 NP-omplete, and hene RP is also NP-omplete. Therefore, PP is NP-hard. Obviously, DP is also NP-hard as DP is ompletely equivalent to PP. B. Channel Assignment Game Sine DP is NP-hard, it is hard to find a solution with both polynomial exeution time and optimal result. Instead, we model a hannel assignment game to onstrut distributed algorithm to solve DP with polynomial exeution time and suboptimal result in this subsetion. Realling the assumption of tree/forest struture, we onstrut parent-hildren sets from the network to exploit its TIRI, i.e., eah non-leaf sensor node in the network and its hildren onstitute a Parent-Children Set (PCS). We define interfering PCSs as a pair of PCSs suh that the message sent by a hild in one PCS may be heard by the parent in the other, e.g., PCS3 and PCS4, and PCS1 and PCS4 in Fig. 1. Correspondingly, the parents in the pair of PCSs are alled interfering parents, e.g., player3 and player4 in Fig. 1. All the interfering parents of parent i is denoted by a set Θ(i). In fat, the interferene suffered by any hild in a PCS is determined by the hannel usage among the PCS and its interfering PCSs. In other words, the more parents in Θ(i) use the same hannel as parent i uses, the more interferene the hildren of parent i suffer. Thus, in order to minimize the interferene suffered by their hildren, the PCSs should autonomously ompete and interat with eah other for the hannel usage. In this ase, the hannel assignment among the PCSs an be naturally modeled as a hannel assignment game. In the hannel assignment game, the players are all parents of the PCSs, i.e., P. Eah player i hooses a hannel s i C as its strategy. The strategies of all players make a hannel assignment s=(s 1,s,..., s m ), and we denote the strategies of all players exept player i by s i. The payoff of player i is a funtion of s and denoted by u i (s). To minimize the total interferene of the network, we onsider both the interferene suffered by the hildren of player i and that aused by its hildren when onstruting u i (s). Upon hoosing a hannel, player i must try its best to hoose a hannel different from its interfering players to minimize the interferene suffered by its hildren; On the other hand, it also should bring as less interferene as possible to the hildren of its interfering players beause exessive selfishness may result in extra interferene to eah other s hildren. Therefore, we define u i (s) in two parts as follows: u i (s) =, (8) e X(i,s) e Y (i,s) X(i, s) = {e : e L u (s),s(e) Child(i), Y (i, s) = {e : e L u (s),r(e) =i. For example, in Fig. 1, the payoffs of player1, player, player3 and player4 are -3, -4, -4 and -3, respetively. The hannel assignment game is designed to be a repeated game, and players negotiate the hannel usage aording to the player1 player player3 1 1 PCS1 PCS PCS3 player4 PCS4 Fig. 1. An example of hannel assignment among interfering PCSs, where player and player3 use hannel 1 to reeive message, and player1 and player4 use hannel to reeive message. The solid arrows represent interseting links, while the dashed arrows represent interfering links. Best Response (BR) dynami. Speifially, in eah iteration of the game, eah player hooses the hannel to maximize its payoff based on the hannel assignment in last iteration, and the hannel hosen in last iteration is preferred if it is among those hannels that maximize the payoff, but the interfering players an not hange their hannels simultaneously. However, BR does not guarantee onvergene in all ases and the stable state is not always with optimal soial utility. Hene, we study the harateristis of our hannel assignment game in terms of onvergene and sub-optimality in the following. We begin with the definition of the famous stable state in game theory Nash Equilibrium (NE), and then disuss the onvergene of BR to NE and the sub-optimality of NE. Nash Equilibrium: In a game, a strategies profile s is alled NE if and only if, for eah player i and an arbitrary strategy s i in its strategy spae, the following inequality is always satisfied, u i (s ) u i (s i,s i). (9) To demonstrate the onvergene of BR, we have the following theorem. Theorem : The hannel assignment game is an exat potential game [18]. Proof: We onstrut the potential funtion as follows: ϕ(s) = 1 u i (s). (10) Consider player i unilaterally hanges its strategy from s 1 i to s i,wehave ϕ(s i,s i ) ϕ(s 1 i,s i ) = 1 [u j (s i,s i ) u j (s 1 i,s i )] = 1 j P { j {s j=s i Θ(i) j {s j=s 1 i Θ(i) [u j (s i,s i ) u j (s 1 i,s i )] [u j (s 1 i,s i ) u j (s i,s i )] +u i (s i,s i ) u i (s 1 i,s i ). (11)

5 Let Z(a, b)={e : s(e) Child(b),r(e) =a, wehave [u j (s i,s i ) u j (s 1 i,s i )] j {s j=s i Θ(i) = = j {s j=s i Θ(i) { e Z(j,i) e X(i,(s i,s i)) + e Z(i,j) e Y (i,(s i,s i)) = u i (s i,s i ). (1) Similarly, the seond sum in Equation (11) is equal to u i (s 1 i,s i). Hene, Equation (11) beomes ϕ(s i,s i ) ϕ(s 1 i,s i )=u i (s i,s i ) u i (s 1 i,s i ). (13) Aordingly, we prove that the hannel assignment game is an exat potential game and one of its potential funtions is ϕ(s). Sine ϕ(s) is bounded and of integral value, Equation (13) indiates that ϕ(s) will ontinue inreasing in BR until it reahes a loal maximum point. Hene, the number of iterations to onverge is finite. From this finite improvement property [18], we have the following orollary. Corollary 1: The existene of NE in the hannel assignment game is guaranteed, and the hannel assignment maximizing ϕ(s) must be an NE. Theorem 3: In the hannel assignment game, BR always onverges to an NE, and the number of iterations to onverge is less than ( V 1). Proof: The potential funtion ϕ(s) = 1 { e X(i,s) e Y (i,s) = = (14) e L u(s) e Y (i,s) For eah i P, Y (i, s) V 1 Child(i) and Child(i) V 1. Thus, ϕ(s) Child(i) ( V 1 Child(i) ) (15) = ( V 1) Child(i) + Child(i) > ( V 1) (16) Hene, we have 0 ϕ(s) > ( V 1). Aording to BR, eah player tries to hoose the hannel that maximizes its own payoff in eah iteration. Thus, in iteration k, the situation is always one of the following two ases: Case 1: none of the players hanges hannel, i.e., for eah player i and an arbitrary s i, u i (s k ) u i (s i,s k i ). Aording to the definition of NE, s k is an NE. Case : at least one player hanges hannel. Suppose that there are m 1 suh players and denote them by a set Ω(k). Aording to BR, these players must not be interfering players, and hene, ϕ(s k ) ϕ(s k 1 )= [u i (s k ) u i (s k 1 )]. (17) i Ω(k) Moreover, eah term in the sum of Equation (17) is no less than 1. Thus, ϕ(s k ) ϕ(s k 1 ) m 1. (18) Therefore, before satisfying Case 1, eah iteration will make ϕ inrease at least one. Sine 0 ϕ> ( V 1), BR will take at most ( V 1) iterations to onverge to an NE. Remark 1: Aording to Theorem 3, the number of iterations to onverge to an NE is O( V ).InSetionIII-C,itwill be shown that eah iteration an be exeuted in polynomial time. Thus, the hannel assignment problem an be handled in polynomial time. Remark : In Inequalities (15) and (16), the lower bound of ϕ(s) is very relaxed. The number of links towards a sensor node is usually bounded by a onstant K, whih is related to the node density. The hildren of a non-leaf sensor node are usually omparable in number to its interfering links. Let 0 < δ < 1 be the upper bound of the ratio of interfering links to all links towards a non-leaf sensor node. Then ϕ(s) Kδ( V 1). Moreover, ϕ(s) usually inreases more than one in one iteration beause many players ould hange their hannels simultaneously if they are not interfering players. Hene, the number of iterations to onverge is usually muh less than ( V 1). From Equation (14), ϕ(s) is equal to the total interferene aused by the unremoved interfering links. Thus, from Equation (6), ϕ(s) also reflets the soial objetive in DP. Aording to Corollary 1, the optimal assignment must be an NE. Based on these observations, we further study the soial effiieny of NE as follows. Corollary : Any NE of the hannel assignment game is loally optimal, i.e., in an NE, all non-leaf sensor nodes an not redue interferene unilaterally and thus DP reahes a loal maximum point. Theorem 4: The Prie of Anarhy [19]: let s o be the optimal solution of DP and s be an arbitrary NE, then we have 1 U(s o ) 1. Proof: Aording to the definition of NE, under hannel assignment s, the payoff of any player in the hosen hannel is no less than the potential payoffs in the other hannels in C, and obviously no less than the average potential payoff over all the hannels in C. Thus, we have the following inequality. U(s ) { u i (s ) = e X(i,s ) + e Y (i,s ) 1 { e {s(e) Child(i) E f +. (19) e {r(e)=i E f

6 residual interferene ratio (a) bound number of nodes residual interferene ratio (b) bound 0 iterations to onverge () number of nodes iterations to onverge (d) 15 Fig.. Numerial results of in terms of onvergene and sub-optimality delivery ratio (a) number of flows throughput(kbps) (b) number of flows hannel aess delay(s) () number of flows energy onsumption per byte(mwhr).8 x (d) number of flows Fig. 3. Performane omparison between and with number of flows inreasing Therefore, the potential funtion an be bounded as: ϕ(s ) 1 { e {s(e) Child(i) E f + e {r(e)=i E f = 1 e E f = B (0) (1) Thus, aording to Equations (6) and (14), we have U(s ) = B =B + ϕ(s ) () e L u(s ) B B 1 U(s o ) (3) Apparently, U(s ) U(s o ), and thus, 1 U(s ) U(s o 1. (4) ) Remark 3: Aording to Theorem 4, any NE is a suboptimal solution of DP. Taking Miaz a typial sensor node for example, there are 8 non-overlapping hannels available, whih is verified in [11]. Aording to Inequality (3), in an NE, at least 87.5% of the interferene aused by interfering links is redued. Aording to Inequality (4), the NE is at most 1.5% worse than the optimal assignment. Remark 4: Sine ϕ(s) is of integral value, and it is not always that all the interfering players of a player use different hannels and have the same potential interferene to it, then the relaxations in Inequalities (19) and (0) are usually a little exessive. Thus, NE is usually muh loser to the optimal assignment than the given bound implies. Sine the sub-optimality of NE is losely related to, we have an enouraging orollary as follows when is large enough. Corollary 3: When >max Θ(i), any NE is optimal and removes all the interferene aused by interfering links. Proof: Sine > max Θ(i), inanarbitraryne,any player i an and have to hoose a hannel s i whih is different from all its interfering players s. Hene, u i (s i,s i ) = 0. Furthermore, ϕ(s i,s i )=0. Aording to Equation (), we have U(s )=B. Moreover, U(s ) U(s o ) B. Therefore, U(s )=U(s o )=B. In summary, in the hannel assignment game, BR will onverge to an NE after at most ( V 1) iterations, and the sub-optimality of NE is guaranteed by 1 U(s ) U(s o ) 1.

7 Algorithm 1 Game Based Channel Assignment Algorithm 1: Input: the initial hannel assignment s 0 : Output: the stable hannel assignment NE s 3: for eah iteration k=1,,...,( V 1) do 4: for any non-leaf sensor node i P do 5: //window RTC: 6: ompute the hannel to maximize the non-leaf sensor node i s payoff: h =argmax s i C ui(si,sk 1 i ); 7: if h is not equal to s k 1 i then 8: broadast a REQ message with i s id; 9: else 10: 11: s k i = h; end if 1: ollet messages in window RTC; 13: //window PTC: 14: if node i has reeived REQs in window RTC then 15: find the max id in REQs and reply a PER message to it; 16: end if 17: ollet messages in window PTC; 18: //window STC: 19: if node i has broadasted REQ in window RTC then 0: if node i has reeived PERs from all its neighbors in window PTC then 1: set both its rv h and s k i to h; : broadast a CHA message with h and its id; 3: else 4: set both its rv h and s k i to s k 1 i ; 5: end if 6: end if 7: ollet messages in window STC; 8: //window RCC: 9: if node i has reeived CHAs in window STC then 30: if the id of one CHA is its parent then 31: set its snd h to the h in this CHA; 3: end if 33: rebroadast the CHAs with i s id added; 34: end if 35: ollet messages in window RCC; 36: if node i reeives CHAs in window RCC and the CHAs are not from itself then 37: update hannel information in its RNNT aording to the h and id of the CHAs; 38: end if 39: end for 40: end for 41: //the final assignment s must be an NE. C. Game Based Channel Assignment Algorithm The payoff of a player depends only upon the hannels hosen by its interfering players, so players only need to exhange information with their interfering players to implement BR. Based on BR, we propose a distributed Game Based Channel Assignment algorithm () to ope with DP. For the hannel assignment game, the most important elements are the payoff funtions of players, whih reflet the benefit of players and further determine the NE of the game and its performane, and the BR dynami of players, whih determines the dynami evolution of the game. The payoff funtions and BR also onstitute. Thus, aording to Setion III-B, the existene of stable state, the onvergene of BR to stable state and the sub-optimality of stable state are guaranteed in. The details of are stated as follows: Assume that eah sensor node knows its parent, its hildren and the sensor nodes whih have interfering links towards it. Eah sensor node has a struture (rv h, snd h, id) (i.e., the hannel it uses to reeive, the hannel it uses to send and its ID number, respetively) and a Related Non-leaf Nodes Table (RNNT) whih is used to reord information about its interfering parents: their hannels, their number of hildren and the orresponding intermediate sensor nodes. is onduted in two phases: the interfering parents disovery phase and the hannel negotiation phase. In the first phase, eah sensor node broadasts twie. In the first broadast, eah sensor node broadasts a message with its id and its number of hildren; In the seond broadast, eah sensor node rebroadasts the messages, whih it reeived in the first broadast, with its id added. After these broadasts, eah non-leaf sensor node knows its interfering parents, their number of hildren and the intermediate sensor nodes. All these information an be used to alulate payoff in the next phase. In the seond phase, BR is implemented and the hannel negotiation is onduted iteratively. Eah iteration oupies a time slot and is divided into four time windows: the RTC window, the PTC window, the STC window and the RCC window. This phase is the ore of, and we present its pseudo-ode in Algorithm 1. Apparently, eah iteration in the seond phase ould be ompleted in polynomial time. Additionally, ( V 1) iterations are suffiient to onverge to an NE. Therefore, this algorithm will onverge in polynomial time. In addition, the ommuniations in these two phases use a designated hannel. After the two phases, a fixed hannel is assigned to eah non-leaf node, and then the network starts to use this hannel assignment to ommuniate in the remaining data transmissions. And time synhronization is only needed in the seond phase but not required when transmitting data. Besides, onduting data transmission task, is ompatible with lots of medium aess ontrol shemes, inluding both sheduling based ones and ontention based ones. So we do not intend to design a medium aess ontrol sheme speially for. Currently, IEEE is very popular in WSNs, so we use its medium aess ontrol sheme (i.e., CSMA/CA) in the simulations of this paper with just one modifiation: when performing the CCA, the sensor node should hek both its hannel to send and its parent s hannel to send to avoid olliding with its siblings and parent. IV. PERFORMANCE EVALUATION A. Convergene and Sub-optimality Evaluation In Setion III-B, we analyze the onvergene and suboptimality of theoretially. In this subsetion, we evaluate these harateristis of numerially. We set the field to 00m 00m, and ommuniation radius to 30m. The number of non-overlapping hannels is 8 and the number

8 delivery ratio (a) throughput(kbps) (b) hannel aess delay(s) () 0.01 energy onsumption per byte(mwhr) 4.5 x (d) Fig. 4. Performane omparison between and with inreasing delivery ratio (a) ommuniation radius(m) throughput(kbps) (b) ommuniation radius(m) hannel aess delay(s) () ommuniation radius(m) energy onsumption per byte(mwhr) x 10 6 (d) ommuniation radius(m) Fig. 5. Performane omparison between and with ommuniation radius inreasing of sensor nodes is Aording to Inequality (3), we use 1/ as the upper bound of residual interferene ratio, i.e., [ e L u(f) ]/B. For omparison purpose, is also used as a baseline, and this paper implements the even-seletion version of, whih an be used when non-overlapping hannels are not suffiient and leads to less interferene [10]. We depit the numerial results in Fig., where eah point is the average result of 50 independent omputations. From Fig.. (a) and (b), we an observe: 1) always redues more interferene than, and its residual interferene is far less than the upper bound, whih means NE is usually muh better than its lower bound and loser to the optimal solution in Inequality (4); ) With the number of sensor nodes inreasing, the residual interferene ratio of inreases. That is beause the inreasing number of sensor nodes in the same area exaerbates interferene; and 3) With the inreasing, the residual interferene ratio of dereases quikly to zero, sine inreasing gives more hoies to eah nonleaf sensor node and removes more interferene. From Fig.. () and (d), we an see: 1) onverges very fast and usually less than 50 iterations, whih is far less than the bound ( V 1) ; ) Approximately, the number of iterations needed for onvergene inreases linearly with the number of sensor nodes, sine more nodes in networks ause more hannel hanges; and 3) With the inreasing, the number of iterations first inreases and then dereases, sine the number of sensor nodes to hange hannel inreases with the inrease of the, when it is relatively small, and the probability that sensor nodes beome stable at early iteration inreases with the inrease of the number of hannels, when it is relatively large. B. Network Performane Evaluation uses game theory as a tool to exploit TIRI as muh as possible to redue the interferene suffered by the network. In this subsetion, we evaluate its network performane and ompare it with by simulations with OMNeT++. We ondut three groups of simulations. In all these simulations, the field is set to 00m 00m, in whih 30 sensor nodes are uniformly distributed. They form a forest with 16 sink sensor nodes. The energy and time parameters of sensor nodes are in aordane with CC430 [0], and the modified CSMA mentioned in Setion III-C is used to ontrol medium aess. In the first group of simulations, we set the number of non-overlapping hannels to 6 and ommuniation radius to 30m, and vary the number of flows from 5 to 55. The rate of eah flow is 5 pakets per seond. These flows

9 are randomly generated in spae but the number of flows transmitting simultaneously is fixed aording to the setting. The simulation results are shown in Fig. 3 where eah point is the average result of 45 independent simulations and so is for the remaining two groups. From Fig. 3, we an see: 1) outperforms in all these metris, i.e., it has larger delivery ratio and higher throughput but yields smaller hannel aess delay (i.e., the time a sensor node spends ompeting for medium aess and transmitting a paket) and lower energy onsumption; ) These advantages get more remarkable with the inrease of the number of flows sine takes full advantage of TIRI but just tries to balane the hannel usage in two-hop neighborhood. And when the load of networks gets heavier, the effet of TIRI beomes more obvious. In the seond group of simulations, we set the number of flows to 40 and ommuniation radius to 30m, and vary the number of non-overlapping hannels from to 8. The way to generate flows is the same as the first group and so is for the third group. The simulations results are shown in Fig. 4. We an see: 1) outperforms in all these metris. ) ahieves the near-best performane with fewer non-overlapping hannels than does, e.g., it almost yields the largest delivery ratio and highest throughput when the is 5, while may ahieve this when the is 7 or more. That is beause an ombine TIRI with the limited hannels rationally to redue interferene in the network. This harateristi of makes it more pratial than, as the nonoverlapping hannels are very limited in pratie [11]. In the third group of simulations, we set the number of non-overlapping hannels to 6 and the number of flows to 40, and vary the ommuniation radius of sensor nodes from 0m to 50m, whih means the number of neighbors of sensor node inreases and generates more interferene. We depit the simulation results in Fig. 5, and i an be seen that : 1) outperforms in all these metris; ) These advantages beome more signifiant with the ommuniation radius inreasing; 3)When the ommuniation radius is small, the delivery ratio and throughput of inrease faster than as it inreases, and when the radius is large, the delivery ratio and throughput of derease slower than as it inreases. The inrement in delivery ratio and throughput with the inrease of radius when the radius is relatively small results from the following reason: the sensor nodes get shorter path to transmit message when the simulation builds the forest. V. CONCLUSIONS In this paper, we have studied how to assign hannels in WSNs to redue interferene and improve network performane. Unlike previous stati assignment protools, we take full advantage of both network topology information and routing information to assign hannels in WSNs. 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