Noncooperative Carrier Sense Game in Wireless Networks

Transcription

1 5280 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER 2009 Noncooperatve Carrer Sense Game n Wreless Networks Kyung-Joon Park, Jennfer C. Hou, Fellow, IEEE, Tamer Başar, Fellow, IEEE, and Hwangnam Km, Member, IEEE Abstract The performance of carrer sense multple access (CSMA) wreless networks heavly depends on the level of spatal reuse,.e., how many concurrent transmssons are allowed. Spatal reuse s prmarly determned by physcal carrer sense, and a key parameter for physcal carrer sense s the carrer sense threshold. Our focus s on how to control the carrer sense threshold for mprovng network performance. We present a noncooperatve game-theoretc framework, whch leads to a fully dstrbuted algorthm for tunng the carrer sense threshold. We ntroduce a utlty functon of each node, whch s a nondecreasng concave functon of the carrer sense threshold. A prcng functon s further ntroduced to mtgate severe nterference among nodes. The cost functon s defned as the dfference between the prcng and the utlty functons. We prove that the noncooperatve carrer sense game admts a unque Nash equlbrum (NE) under some techncal condtons. We derve suffcent condtons that ensure the convergence of the synchronous and asynchronous update algorthms. Based on the analyss, we propose a fully dstrbuted algorthm, enttled noncooperatve carrer sense update algorthm (NCUA). Our smulaton study ndcates that NCUA outperforms standard CSMA wth respect to the per-node throughput by 10 50%. Index Terms CSMA wreless network, spatal reuse, physcal carrer sense, noncooperatve game. I. INTRODUCTION MULTI-HOP wreless networks, such as wreless mesh networks, have emerged to be a promsng costeffectve paradgm for the next-generaton wreless technology [1]. In these networks, nstead of usng a pre-nstalled central entty for coordnatng the rado channel, a dstrbuted access mechansm s usually deployed to arbtrate access to the shared medum. One of the most wdely used medum access mechansms s carrer sense multple access (CSMA). A crtcal performance metrc n wreless networks s network capacty,.e., the average number of data bts that can be transported smultaneously n the network. In CSMA wreless networks, network capacty heavly depends on the level of spatal reuse,.e., how many concurrent transmssons Manuscrpt receved November 28, 2008; revsed June 30, 2009; accepted July 17, The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was H. Zheng. K.-J. Park s wth the Department of Computer Scence, Unversty of Illnos, 201 N. Goodwn Avenue, Urbana, IL USA (e-mal: kjp@uuc.edu). J. C. Hou was wth the Department of Computer Scence, Unversty of Illnos at Urbana-Champagn at the tme of ths work. She s now deceased. T. Başar s wth the Department of Electrcal and Computer Engneerng and the Coordnated Scence Laboratory, Unversty of Illnos, 1308 West Man Street, Urbana, IL USA (e-mal: tbasar@control.csl.uuc.edu). H. Km (correspondng author) s wth the School of Electrcal Engneerng, Korea Unversty, Seoul Korea (e-mal: hnkm@korea.ac.kr). Dgtal Object Identfer /TWC /09$25.00 c 2009 IEEE can be allowed n the network. In prncple, spatal reuse cannot be arbtrarly exploted due to a tradeoff between spatal reuse and nterference mtgaton. As the level of spatal reuse s ncreased by allowng more concurrent transmssons, the fracton of successful transmssons decreases due to the ncreased nterference among concurrent transmssons. Snce the overall network capacty depends on the total number of successful concurrent transmssons, there exsts a certan level of spatal reuse that maxmzes the network capacty. The level of spatal reuse s prmarly characterzed by physcal carrer sense as follows: Before each transmsson, a sender lstens to the channel and determnes whether or not the channel s busy by comparng the receved sgnal strength wth ts carrer sense threshold. If the receved sgnal strength s below the carrer sense threshold, the sender consders the channel to be dle and starts ts transmsson. Otherwse, the sender consders the channel to be busy and defers ts transmsson. Snce the receved sgnal strength s proportonal to the transmt power of the correspondng transmtter, both the carrer sense threshold and the transmt power are major control knobs for explotng the level of spatal reuse. The ssue of transmt power control for explotng spatal reuse has been extensvely studed n the context of topology control by graph-theoretc approaches, where the man objectve s to reduce transmt power to mtgate MAC nterference whle preservng graph-theoretc network connectvty (a comprehensve survey on topology control can be found n [2]). Transmt power control has also been broadly studed from the vewpont of maxmzng network capacty (see [1] and the references theren). The common basc dea n these studes s to adopt an approprate transmt power n order to explot spatal reuse and ncrease the overall network capacty. In contrast, n spte of ts mportance, the ssue of tunng the carrer sense threshold has not been pad much attenton untl recently, when a number of studes have been carred out [3] [7]. Nevertheless, there have been few analytcal studes on how to control the carrer sense threshold n a decentralzed manner. The am of ths paper s thus to nvestgate the problem of mprovng network capacty by carefully tunng the carrer sense threshold n CSMA wreless networks. In partcular, we buld an analytcal framework and a dstrbuted algorthm theren for tunng the carrer sense threshold. We frst derve the collson probablty of each node by takng nto account the essental feature of physcal carrer sense. Then, analogously as the power control problems n wreless networks [8], [9], a noncooperatve game-theoretc framework naturally be-

2 PARK et al.: NONCOOPERATIVE CARRIER SENSE GAME IN WIRELESS NETWORKS 5281 comes a very effcent tool for dstrbuted control of the carrer sense threshold. In order to develop reasonable utlty and prcng functons, we dentfy the man consequences when a node ncreases ts carrer sense threshold as follows: () The chance for channel access of the node wll ncrease because t wll care for fewer nodes. () Both the nterference from the node to others and that from others to t wll become more severe. In order to characterze the frst property, we ntroduce a nondecreasng concave functon of each node s carrer sense threshold as a utlty functon. Then, to balance out these conflctng factors, we further ntroduce a prcng functon by ncorporatng the collson probablty of each node. The overall cost functon of each node s then defned as the dfference between the prcng and the utlty functons. A detaled explanaton on the problem formulaton wll be gven n Secton III. We prove that the noncooperatve carrer sense game admts a unque Nash equlbrum (NE) for unformly strctly concave utlty functons under some techncal assumptons. We also derve suffcent condtons that ensure the convergence of synchronous and asynchronous versons of a gradent-based algorthm for updatng the carrer sense threshold. Based on the analyss, we propose a fully dstrbuted algorthm, called noncooperatve carrer sense update algorthm (NCUA), for tunng the carrer sense threshold. NCUA has the followng desrable propertes: () NCUA s fully dstrbuted and does not requre nformaton exchange among nodes; () NCUA s adaptve to network envronment n ts nature; () By ncorporatng the collson probablty nto the cost functon, NCUA sgnfcantly mproves the overall network capacty compared to the standard CSMA. The smulaton study ndcates that NCUA outperforms standard CSMA wth respect to the pernode throughput by 10 50%, when the latter operates over a wde range of the carrer sense threshold. The rest of the paper s organzed as follows. In Secton II, we ntroduce the network propagaton and nterference models as well as the notons used n our analyss throughout the paper. Based on the model, we derve the condtonal collson probablty of each node as a functon of carrer sense thresholds. Then, n Secton III, we formulate the problem of updatng the carrer sense threshold as a noncooperatve carrer sense game. Followng that, we derve n Secton IV asuffcent condton for the exstence of a unque NE, and show the convergence propertes of synchronous and asynchronous algorthms. In Secton V, based on the analyss, we propose NCUA and evaluate ts performance va a smulaton study. Fnally, the paper ends wth the concludng remarks n Secton VI, and an Appendx. II. NETWORK MODEL The CSMA wreless network model used n our analyss conssts of a set of N nodes, denoted by N = {1,,N}. For each node N,letr() N denote the correspondng recever. Let P and P r() denote the transmt power of node and the receved power at r(), respectvely. Then, P r() s expressed as P r() = G r(), F r(), P,whereG r(), and F r(), respectvely represent the path loss and the Raylegh fadng from sender to recever r(), whch s a wdely used model for wreless channel envronments [10]. In addton, let τ denote the channel access probablty of node, whch denotes the probablty that node attempts to transmt a packet when t detects the channel dle. For further use, we also ntroduce G r(),j and F r(),j as the correspondng quanttes effectng recever r() where the transmtter s sender j (whch contrbutes to nterference at r().) As the magntude of Raylegh fadng, F r(), s ndependent across nodes, and s exponentally dstrbuted wth unt mean. Here, we assume that the nterference from other senders s much larger than the background nose n the recevers, and thus do not consder the nose n our analyss. As a necessary condton for recever r() to correctly decode the symbols, P r() should be larger than or equal to the receve senstvty of r(), denoted by β r(),.e., P r() = G r(), F r(), P β r(). (1) In addton, for successful recepton at the recever r(), the receved power P r() should be large enough so that the nterference from other nodes does not prevent the recever from correctly decodng the symbols. Ths condton s usually characterzed by the sgnal to nterference rato (SIR) as follows: SIR r() = P r() γ r(). I r() where I r() = j = G r(),jf r(),j P j and γ r() s called the SIR threshold of the recever r(). The collson set of recever r(), denoted by C r(), s defned as a set of nodes whose smultaneous transmsson wth node, taken one at a tme, wll prevent r() from correctly decodng the symbols of node as follows: C r() = { j P r() <γ r() G r(),j F r(),j P j }. Now, let x denote the carrer sense threshold of node. If the sgnal strength perceved at node s larger/smaller than x, the channel s consdered busy/dle by node. Foragven node, the carrer sense set S (x ) s defned as S (x )={j G,j F,j P j x }. Thus, node wll be slenced f any node n S (x ) transmts. In a smlar manner, let L denote the slenced set of node, whch s defned as L = {j G j, F j, P x j }. Hence, every node j L wll be slenced whle node transmts. Note that the collson set C r() and the slenced set L are ndependent of the carrer sense threshold x by ther defntons, whle the carrer sense set S (x ) s a functon of x, whch satsfes S (y) S (x) f x<yfrom ts defnton. In fact, what each node does by carrer sense s not to prevent other nodes from transmttng, but to detect the status of the channel before each transmsson attempt. Hence, even when C r() S (x ), node s transmsson may stll fal f nterrupted by a sngle smultaneous transmsson n C r() L or multple concurrent transmssons n (N C r() ) L.The overall notaton s summarzed n Table I. Now, we look nto the collson probablty of a node when t attempts to transmt. There are two sources of transmsson collsons, namely, collson ncurred by smultaneous

3 5282 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER 2009 Notaton N r() P P r() G r(), G r(),j F r(), F r(),j τ β r() γ r() x C r() S L Defnton TABLE I NOTATIONUSEDINANALYSIS. Set of nodes n the network Correspondng recever of sender N Transmt power of sender Receved power at r() of sender Path loss from sender to ts recever r() Path loss from sender j to recever r(), whch contrbutes to nterference at r() Raylegh fadng from sender to ts recever r() Raylegh fadng from sender to recever r(), whch contrbutes to nterference at r() Channel access probablty of node Receve senstvty of recever r() SIR threshold of recever r() Carrer sense threshold of sender Set of nodes whose smultaneous transmsson wth node wll result n collson Set of nodes whose transmsson wll make node slenced Set of nodes whch wll be slenced whle node transmts transmssons n C r() and collson ncurred by multple concurrent transmssons outsde C r() that accumulatvely volates the SIR constrant at the recever r(). Let q (x) denote the condtonal collson probablty of node gven that a transmsson attempt s made. Then, we have q (x) =P [Smultaneous tx n C r() ]+ P [No smultaneous tx n C r() ] P [SIR r() <γ r() No smultaneous tx n C r() ] =s +(1 s )h (x), (2) where s := P [Smultaenous tx n C r() ], h (x) := P [ ] SIR r() <γ r() No smultaneous tx n C r(), and tx s an abbrevaton for transmsson. Here, h (x) can be further expressed as follows: [ h (x) =P P r() k Γ (x) G r(),kp k F k <γ r() ], (3) where Γ (x) denotes the set of nodes whch concurrently transmt wth node after no smultaneous transmsson n C r() has occurred. III. PROBLEM FORMULATION: NONCOOPERATIVE CARRIER SENSE GAME We now present a noncooperatve game-theoretc framework for decentralzed control of physcal carrer sense. Frst, let U denote a utlty functon of node, whch the node seeks to maxmze. Then, one possble approach s to formulate the problem of maxmzng the network utlty n a centralzed fashon as max x X N U (x), =1 (P1) where x denotes the carrer sense threshold of node wth x X = [x mn,x max ], X = N =1 X, and x = (x 1,...,x N ). Note that (P1) s formulated from the system perspectve, and hence the computaton of ts soluton wll naturally requre a centralzed algorthm. However, due to the lack of a pre-nstalled centralzed nfrastructure n CSMA wreless networks, t s more practcal to formulate the problem from the perspectve of ndvdual nodes. We frst consder the case where each node smply attempts to maxmze ts own utlty functon wthout carng for other nodes. Ths stuaton corresponds to the followng optmzaton problem for each node : max U (x, x ), N, (P2) x X where x = (x 1,...,x 1,x +1,...,x N ). Snce the utlty functon U s generally nondecreasng wth respect to x for gven x, (P2) admts the unque soluton x = [x max,,x max ], whch s trvally also the unque NE. In other words, f each node does not care for others, the best strategy for each node s to ncrease ts carrer sense threshold as much as possble n order to ncrease ts own utlty. Ths selfsh behavor results n an equlbrum of x =[x max,,x max ], whch wll gve poor network performance because of sgnfcantly ncreased nterference among the nodes. 1 Ths undesrable selfsh behavor can be resolved by mposng a prcng functon n (P2). In order to develop reasonable utlty and prcng functons, we need to characterze the man effects of the carrer sense threshold on network performance. We dentfy the followng man consequences as node ncreases ts carrer sense threshold x : () The chance for channel access of node wll be ncreased because node wll care for fewer nodes and thus wll sense an dle channel more frequently. () In the meantme, the nterference from node to others wll be more severe because of more aggressve channel access of node. Furthermore, the nterference from other concurrent transmssons to node wll be ncreased snce the average number of sensed nodes s decreased. By takng nto account the frst property, we ntroduce as a utlty functon a nondecreasng concave functon of each node s carrer sense threshold. Furthermore, to balance out these conflctng consequences of ncreasng the carrer sense threshold, we adopt a prcng functon P (x) := x x mn {q (ξ,x ) q } dξ by ntroducng a threshold q on the collson probablty for each node. Snceq s ncreasng n x for gven x, t can be easly verfed that P s convex n x and the mnmum s attaned when q = q for gven x. Thus, by comparng the collson probablty q wth q, the prcng functon attempts to mantan a reasonable carrer sense threshold n order to keep q around q. 2 Now, we provde a game-theoretc framework wth the ntroduced utlty and prcng functons. Frst, node s overall cost functon J s defned as J (x, x )=P (x, x ) U (x, x ). (4) 1 A smlar selfsh behavor has been addressed n the power control problem of wreless networks n [11]. 2 Note that heterogeneous values for q s can further mprove the overall network performance. An ntutve approach s that each node teratvely reduces q n a large tme scale as long as q remans around q.however,a more rgorous analyss s requred for ths knd of algorthms. It wll be an ssue of future research to develop an effcent algorthm for tunng q n an analytcal framework.

4 PARK et al.: NONCOOPERATIVE CARRIER SENSE GAME IN WIRELESS NETWORKS 5283 Wth (4), we have the followng NE computaton. mn J (ξ,x ) J (x, x ), N. (P3) ξ X One may devse a cooperatve soluton, whch makes (P3) equvalent to (P1). In fact, (P3) wll correspond to (P1) f P (x), N, were chosen such that P (x) = j N {} U j (x), (5) whch can be obtaned by equatng J / to N j=1 U j/. However, mplementng a cooperatve soluton based on (5) requres a feedback mechansm for conveyng all the nformaton on U j /, j N {}. Ths nformaton exchange wll be an expensve overhead n wreless networks. Hence, we consder a noncooperatve carrer sense game n order to develop a fully dstrbuted control algorthm. In our framework, U wll depend only on x, and thus be denoted as U (x ). We further ntroduce the followng two techncal assumptons for U (x ). A1. U (x ) s chosen to be twce contnuously dfferentable, nondecreasng and unformly strctly concave n x,.e., U (x ) 0, 2 U (x ) x 2 η<0, x, N for some η>0. A2. In order to ensure an nner NE, U (x ) s further chosen to satsfy J < 0 at x = x mn, J > 0 at x = x max, x, N. In the followng secton, we derve condtons on the exstence of a unque NE and the convergence of an teratve update algorthm for (P3). IV. NASH EQUILIBRIUM AND CONVERGENCE ANALYSIS A. Exstence and Unqueness of Nash Equlbrum The frst step for solvng the noncooperatve game of (P3) s to check the exstence and the unqueness of an equlbrum. Here, a Nash equlbrum (NE) s defned as a set of carrer sense thresholds, denoted by x =(x 1,...,x N ), whch satsfes the followng property: no node can beneft by devatng from ts carrer sense threshold whle those of other nodes are kept fxed. In the mathematcal sense, x s a NE f x, N, satsfes x =argmn x X J (x, x ).We have the followng result on the NE of (P3): Theorem 1 Under A1 and A2, (P3) admts a unque nner Nash equlbrum f Lτ max η>, e(1 τ max )x mn where L =ln(1+γ max P max /β mn ), γ max := max γ r(), β mn := mn β r(), τ max := max τ, and P max s the maxmum transmt power. Proof: See the appendx. B. Convergence of an Iteratve Update Algorthm for Carrer Sense Thresholds Next, we nvestgate the convergence propertes of synchronous and asynchronous update algorthms for the carrer sense threshold. Consder a dscrete-tme update scheme where each node updates ts carrer sense threshold for solvng (P3) as follows: J (x(t)) x (t +1)=x (t) α, N, (6) where t =1, 2, denotes the update tme nstants, and α s the step sze. Under the synchronous update scheme, all the nodes update ther carrer sense thresholds smultaneously. Theorem 2 below provdes a suffcent condton for the convergence of (6) n the synchronous case. Theorem 2 The synchronous update algorthm J (x(t)) x (t +1)=x (t) α, N converges to the unque NE, x := [x 1,,x N ],onx f, N, e(1 τ max )x mn α < and e(1 τ max )x mn u,max + L(N 1)τ max η> L(N 1)τ max, ex mn where u,max := 2 U (x mn )/ x 2, and L, τ max are defned n the same way as n Theorem 1. Proof: The proof follows lnes smlar to those n [12, Proposton 1.11 p. 194]. Frst, we derve a suffcent condton on the step sze α.letf (x) := J (x)/. Then, for the convergence of (6), the step sze α should satsfy 0 <α < 1/K where K s a postve constant such that f / K, x X. From (2) and (3), we have f (x) = 2 P (x) x 2 2 U (x ) x 2 u,max + h (x), (7) where u,max := 2 U (x mn )/ x 2. From the appendx, we have h (x) L k N {} P[k Γ (x)] L(N 1)τ max e(1 τ max )x mn. From (7) and (8), the step sze α should satsfy e(1 τ max )x mn 0 <α <. e(1 τ max )x mn u,max + L(N 1)τ max In a smlar manner, from (17) and (18), we have j N {} h (x) x j L j N {} P[j Γ (x)] x j L(N 1)τ max, (9) ex mn and from (9), the condton f / > j = f / x j n [12, Proposton 1.11 p. 194] s satsfed f η > L(N 1)τ max /(ex mn ). (8)

5 5284 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER 2009 A natural generalzaton of the synchronous algorthm s the asynchronous algorthm where only a random subset of nodes update ther carrer sense thresholds at a gven tme nstant. Ths s more realstc snce t s dffcult for nodes to synchronze ther update nstants n a practcal settng. In fact, the convergence analyss n Theorem 2 drectly apples to the asynchronous case, and we have the followng result for the convergence n the asynchronous case. Theorem 3 Let U(k) denote a random subset of nodes updatng ther carrer sense threshold at tme k. Then, the asynchronous update algorthm { J x (t) α x (t +1)=, f U(t +1); (10) x (t), otherwse, converges to the unque NE on X f, N, e(1 τ max )x mn α < and e(1 τ max )x mn u,max + L(N 1)τ max η> L(N 1)τ max, ex mn where u,max, L, and τ max are defned n the same way as n Theorem 2. Proof: Frst, we defne a sequence of nonempty, convex, and compact sets X(k) :=[x 1 δ(k),x 1 + δ(k)] [x 2 δ(k),x 2 + δ(k)] [x N δ(k),x N + δ(k)], where δ(k) := x(k) x. From Theorem 2, δ(k+1) <δ(k). Hence, X(k +1) X(k) X. Now recall two well-known suffcent condtons for the asynchronous convergence of a nonlnear teratve mappng x(t +1)=T (x(t)) [12, p. 431]. (Synchronous Convergence Condton) We have T (x) X(k +1), k and x X(k). Furthermore, f {y k } s a sequence such that y k X(k) for every k, then every lmt pont of {y k } s a fxed pont of T. (Box Condton) Gven a closed and bounded set X n R, for every k, thereexstsetsx (k) X such that X(k) =X 1 (k) X 2 (k) X N (k). Here, X s defned as [x mn,x max ],andx (k) := [x δ(k),x + δ(k)]. Thus, the box condton s satsfed by the defnton of X(k). Also,δ(k) s monotoncally decreasng n k by Theorem 2. Consequently, the convergence of the asynchronous update algorthm of (10) mmedately follows from [12, p. 431]. V. SIMULATION STUDY A. Noncooperatve Carrer Sense Update Algorthm (NCUA) Based on the analyss n Secton III and Secton IV, we propose a fully dstrbuted algorthm, called noncooperatve carrer sense update algorthm (NCUA), and evaluate ts performance va J-Sm smulaton [13]. The pseudo code of NCUA s gven n Algorthm 1. In the ntalzaton procedure of Algorthm 1, the ntal value for x s gven. Then, n the update procedure, the carrer sense threshold x s updated Algorthm 1 Noncooperatve carrer sense update algorthm (NCUA) 1: // ntalzaton 2: x x nt 3: // update procedure for carrer sense threshold 4: Reset tmer t Δ 5: N t 0 and N c 0 6: whle (True) do 7: f node transmts a packet then 8: N t N t +1 9: f transmsson fals then 10: N c N c +1 11: end f 12: end f 13: end whle 14: q N c/n t 15: x x α (q q u /x ) TABLE II PARAMETERS USED IN THE SIMULATION STUDY. Parameter Value Defnton N 100 Number of nodes P length 1024 bytes Payload sze R (18, 54) Mb/s Data rate CW 63 Contenton wndow sze d mn 100 m Mnmal transmsson range d max 120 m Maxmal transmsson range θ 4 Path loss exponent x mn 73 dbm mnmum carrer sense threshold x max 84 dbm maxmum carrer sense threshold u Utlty parameter n NCUA α Update step sze of NCUA Δ 5 seconds Update nterval of NCUA x nt 75 dbm Intal value of x n NCUA q 0.2 collson probablty threshold every Δ seconds. Frst, the collson probablty s estmated by countng the total number of unsuccessful transmssons durng the nterval of Δ. Then, x s updated based on (6) wth U (x )=u ln x. In Algorthm 1, x and T are n pw and b/s, respectvely. There are a few ponts worth mentonng. Frst, NCUA s fully dstrbuted and does not requre nformaton exchange among nodes. Second, n order to mplement NCUA, t was requred to choose a specfc utlty functon U (x ) n (6). Here, we adopt a logarthmc utlty functon parameterzed by u for node as above. The choce of other structures s a subject of future work. B. Smulaton Setup In the smulaton study, we use IEEE a as the MAC protocol. Table II gves the default values for the parameters used n the smulaton. In partcular, the receve senstvty γ s set to the default value n IEEE a specfcaton. Also, to precsely quantfy the mpact of adjustng carrer sense thresholds on the throughput performance, we dsable the bnary exponental back-off (BEB) mechansm n IEEE

6 PARK et al.: NONCOOPERATIVE CARRIER SENSE GAME IN WIRELESS NETWORKS 5285 Fg. 1. Y coordnate (meter) X coordnate (meter) The network topology used n the smulaton study DCF n order to decouple ts effect. Instead, we use a fxed contenton wndow sze of 63 as gven n Table II. Fnally, the same value of u, denoted by u, s used for all the nodes. To calculate the suffcent condton n Theorem 2, from Table II, we have τ max =2/(CW +1) = 1/32, x mn = 84 dbm, and x max = 73 dbm. In addton, from the logarthmc utlty functon of U (x )=u log x,wehave η =mn u /x 2 max. Wth these values, the convergence condtonntheorem2gvesα < and u > Snce these values are qute suffcent rather than necessary, we set α =10 12 and u =10 11 as the default values n our smulaton study. Note that we also perform smulaton wth dfferent values of u to verfy ts effect on network performance. Fgure 1 shows the network topology used for the smulaton study. A total of N (= 100) nodes are dstrbuted n an area of m 2. Frst, half of the nodes are randomly located as senders. Then, for each sender, a recever s randomly located nsde a crcle wth center at the sender and radus d mn. As already explaned, the transmt power used by each node s randomly chosen to gve a transmsson range from d mn up to d max. All sources generate CBR traffc at ther full data rates. For NCUA, every node sets the ntal value of ts carrer sense threshold to 75 dbm. C. Smulaton Results Fgure 2 gves the average per-node throughput (averaged over all the source nodes) versus the carrer sense threshold for R =18Mb/s and 54 Mb/s, respectvely. Note that the x-axs n Fg. 2 only apples to the standard CSMA because NCUA dynamcally updates the carrer sense threshold. As shown n Fg. 2, NCUA acheves better throughput performance than standard CSMA over the entre range for both the data rates of 18 and 54 Mb/s. Specfcally, NCUA acheves 11.76% and 16.13% hgher throughput than the maxmal throughput acheved by the standard CSMA at R = 18 Mb/s and 54 Mb/s, respectvely. It should be noted that, n practce, t s extremely dffcult to set an optmal carrer sense threshold under the standard CSMA. Furthermore, as shown n Fg. 2, f an napproprate carrer sense threshold s chosen, standard Fg Carrer sense threshhold (dbm) Standard NCUA (a) Data rate R =18Mb/s Carrer sense threshhold (dbm) Standard NCUA (b) Data rate R =54Mb/s Average node throughput versus the carrer sense threshold. CSMA gves nferor throughput performance. Consequently, the performance of NCSU wll be much better than that of the standard CSMA n practce. For example, about 40% and 50% for R =18Mb/s and R =54Mb/s when the carrer sense threshold s set to 72 dbm for standard CSMA. Fgure 3 gves the temporal behavor of how the carrer sense threshold of each node adapts under NCUA when R = 54 Mb/s. In partcular, t shows the tme traces of the carrer sense thresholds of four randomly selected nodes. Note that the traces of the carrer sense thresholds for other nodes exhbt smlar trends and hence are not shown. As gven n Fg. 3, the carrer sense threshold of each node ntally starts from the default ntal value of 75 dbm and becomes stablzed. Fgure 4 depcts the relaton between the throughput performance and the utlty parameter u. Although the suffcent condton derved from our analyss provdes a gudelne for settng the utlty parameter u n terms of the convergence, t s worth to look nto the actual performance wth respect to u. The utlty parameter u can be nterpreted as the ncentve for ncreasng the carrer sense threshold. Hence, f u s too small, the carrer sense threshold s set to an napproprately small value, whch may result n degradaton of the throughput performance. However, we can verfy from Fg. 4 that the performance of NCUA s qute robust to changes n u over a wde range. Note that t s very dffcult, though not mpossble, to derve an explct relaton between the network performance and the utlty parameter u because of the complex problem

7 5286 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER e e e e e e e e e e-11 Utlty parameter (u) (a) Data rate R =18Mb/s e e e e e e e e e e-11 Utlty parameter (u) (b) Data rate R =54Mb/s Fg. 4. Average node throughput versus utlty parameter u. structure. Fnally, we evaluate the throughput senstvty wth respect to the number of nodes n the network. Fgure 5 shows the average node throughput of NCUA and standard CSMA wth respect to the number of nodes. Overall, we can verfy from Fg. 5 that NCUA performs better than standard CSMA n a wde range of the number of nodes. When the number of nodes s small, t s qute ntutve that the throughput performance s not very senstve to the carrer sense threshold. Ths relaton explans the small gap between NCUA and standard CSMA when the number of nodes s small n Fg. 5. It should be noted that the carrer sense threshold of standard CSMA n Fg. 5 s set to 75 dbm, whch s qute a favorable value as shown n Fg. 2. If a less favorable value s chosen (for example, 80 dbm), the throughput gap between NCUA and standard CSMA wll become larger. 3 VI. CONCLUSION AND FUTURE WORK We have presented a noncooperatve game-theoretc framework for dstrbuted control of the carrer sense threshold n CSMA wreless networks. We have shown that the noncooperatve carrer sense game admts a unque NE for unformly strctly concave utlty functons under some techncal condtons. We have also derved suffcent condtons that ensure the convergence of both the synchronous and asynchronous versons of a gradent update algorthm for the carrer sense threshold. Our analyss has led to a fully dstrbuted algorthm, called noncooperatve carrer sense update algorthm (NCUA). The smulaton study has shown that NCUA outperforms the standard CSMA wth respect to the per-node throughput by 3 The smulaton result for other values of the carrer sense threshold has been omtted due to the lmt on the number of fgures %, when the latter operates over a wde range of the carrer sense threshold. In our analyss, we have not consdered the effect of the dynamc contenton resoluton algorthm, such as the bnary exponental back-off (BEB) algorthm used n IEEE DCF. Snce the BEB algorthm has recently been shown to maxmze a certan utlty functon n a noncooperatve manner [14], t wll be an nterestng research avenue to study the nteracton between physcal carrer sense and contenton resoluton n a game-theoretc framework. In addton, snce the operatng pont wll be dfferent for each node, heterogeneous values for q wll further mprove the overall network performance. Hence, t wll be a subject of future work to devse a dstrbuted algorthm for adjustng q of each node. ACKNOWLEDGEMENT Ths work was supported n part by the Korea Research Foundaton Grant funded by the Korean Government (MOEHRD) (KRF D00102), n part by the Korea Scence and Engneerng Foundaton (KOSEF) Grant funded by the Korean Government (MOST) (No. R ), and n part by the IT R&D program of MKE/IITA [2008-F015-02, Research on Ubqutous Moblty Management Methods for Hgher Servce Avalablty] APPENDIX The proof follows a lne of arguments smlar to those n [9], [15]. Frst, we show the exstence of a NE. The set of feasble carrer sense thresholds s defned as X = N =1 X,where x X =[x mn,x max ], N. Then, X s closed and bounded, and thus compact. Also, X s convex and has a nonempty nteror. Furthermore, the cost functon J (x) n (4) s convex n x under the condton (12), whch s to be shown

8 PARK et al.: NONCOOPERATIVE CARRIER SENSE GAME IN WIRELESS NETWORKS Number of nodes Number of nodes Standard NCUA Standard NCUA (a) Data rate R =18Mb/s (b) Data rate R =54Mb/s Fg. 5. Average node throughput versus the number of nodes n the network when the carrer sense threshold of standard CSMA s set to 75 dbm. as part of the unqueness proof below. Thus, by a standard theorem n game theory (for example, [16, Theorem 4.4 p. 176]) together wth A2, there exsts an nner NE. Now, we consder the unqueness of the NE. Let A j := 2 J (x)/ x j and B := 2 J (x)/ x 2. Then, from the arguments used n [9], [15], the condton B > A j,, j = s suffcent to show the unqueness of NE. Hence, we derve a suffcent condton, under whch B > A j s satsfed. Snce the utlty functon U (x ) depends only on x, 2 U (x )/ x 2 η from A1, and the prcng functon P (x) s convex n x,wehave B = 2 P (x) x 2 2 U (x ) x 2 η. (11) From (11), a suffcent condton for B > A j s gven as follows: η> 2 P (x) x j. (12) Thus, what remans to be done s to obtan an explct expresson for 2 P (x)/ x j n terms of x. From the defnton of the prcng functon together wth (2), we have 2 P (x) x j = q (x) x j =(1 s ) h (x) x j. (13) Note that, f z 1,...,z n are ndependent exponentally dstrbuted random varables wth means E[z ]=1/λ,wehave the followng expresson from [17]: [ ] n n ( ) 1 P z 1 z =1. (14) 1+λ 1 /λ =2 =2 By ncorporatng (14) nto (3), h (x) n (13) becomes h (x) =E 1 1 k Γ (x) 1+ γ r()g r(),k P k (15) G r(), P E ( ln 1+ γ ) r()g r(),k P k, (16) G r(), P k Γ (x) where the approxmaton n (16) comes from x ln(1 x) for x 1. Let the ndcator functon I Γ(x)(k) be defned as { 1, f k Γ (x) ; I Γ(x)(k) = 0, otherwse. Then, h (x) =E ( ln 1+ γ ) r()g r(),k P k I Γ(x)(k) G r(), P k N {} = ( ln 1+ γ ) r()g r(),k P k P [k Γ (x)]. G r(), P k N {} (17) Now, we obtan an expresson for P [k Γ (x)] n (17). In order for any node to nterfere wth node s transmsson, ether ts transmsson should not have been sensed by node or t should not sense node s transmsson and attempts to transmt. Thus, P [k Γ (x)] = τ kp S C +(1 τ k )τ k P L C, (18) τ k P S C +(1 τ k ) where P S C := P [k N S ]=P[G r(),k P k F k x ]=1 e x/(g r(),kp k ) and P L C := P [k N L ]=P[G k, P F x k ]=1 e x k/(g k, P ). In the meantme, from (1), we have

9 5288 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER 2009 Carrer sense threshhold (dbm) Carrer sense threshhold (dbm) node1 node2 Tme (sec) node3 node4 (a) Data rate R =18Mb/s node5 node node1 node2 Tme (sec) node3 node4 (b) Data rate R =54Mb/s node5 node6 Fg. 3. Tme traces of carrer sense thresholds of sx randomly selected nodes under NCUA. G r(),k P k /(G r(), P ) P max /β r(),wherep max s the maxmum transmt power. Thus, by collectng (12), (13), (17), and (18) wth some algebrac manpulaton, the suffcent condton for the unqueness of NE s gven as ( 1+ γ maxp max β mn ) τ max η>ln, (19) e(1 τ max )x mn where γ max := max γ r(), β mn := mn β r(),andτ max := max τ. REFERENCES [1] J. C. Hou, K.-J. Park, T.-S. Km, and L.-C. Kung, Medum access control and routng protocols for wreless mesh network," n Wreless Mesh Networks: Archtectures Protocols, E. Hossan and K. K. Leung, Eds. Sprnger, Oct [2] P. Sant, Topology control n wreless ad hoc and sensor networks," ACM Computng Surveys, vol. 37, no. 2, pp , June [3] J. Zhu, X. Guo, L. Yang, W. S. Conner, S. Roy, and M. M. Hazra, Adaptng physcal carrer sensng to maxmze spatal reuse n mesh networks," Wley Wreless Communcatons and Moble Computng, vol. 4, pp , Dec [4] X. Yang and N. H. Vadya, On the physcal carrer sense n wreless ad hoc networks," n Proc. IEEE INFOCOM, Mar [5] H. Zha and Y. Fang, Physcal carrer sensng and spatal reuse n multrate and multhop wreless ad hoc networks," n Proc. IEEE INFOCOM, Apr [6] T.-Y. Ln and J. C. Hou, Interplay of spatal reuse and SINR-determned data rates n CSMA/CA-based, mult-hop, mult-rate wreless networks," n Proc. IEEE INFOCOM, May [7] Y. Yang, J. C. Hou, and L.-C. Kung, Modelng of physcal carrer sense n mult-hop wreless networks and ts use n jont power control and carrer sense adjustment," n Proc. IEEE INFOCOM Mnconferences, May [8] T. Alpcan, T. Başar, R. Srkant, and E. Altman, CDMA uplnk power control as a noncooperatve game," Sprnger Wreless Networks, vol. 8, no. 6, pp , [9] T. Alpcan, T. Başar, and S. Dey, A power control game based on outage probabltes for multcell wreless data networks," IEEE Trans. Wreless Commun., vol. 5, no. 4, pp , Apr [10] D. Tse and P. Vswanath, Fundamentals of Wreless Communcaton. Cambrdge Unversty Press, [11] J. Huang, R. Berry, and M. Hong, Dstrbuted nterference compensaton for wreless networks," IEEE J. Select. Areas Commun., vol. 24, no. 5, pp , May [12] D. P. Bertsekas and J. N. Tstskls, Parallel and Dstrbuted Computaton: Numercal Methods. Prentce Hall, [13] J-Sm." [Onlne]. Avalable: [14] J.-W. Lee, A. Tang, J. Huang, M. Chang, and A. R. Calderbank, Reverse engneerng MAC: a non-cooperatve game model," IEEE J. Select. Areas Commun., vol. 25, no. 6, pp , Aug [15] T. Alpcan and T. Başar, A hybrd systems model for power control n multcell wreless data networks," Performance Evaluaton, vol. 57, no. 4, pp , Aug [16] T. Başar and G. J. Olsder, Dynamc Noncooperatve Game Theory. SIAM Seres n Classcs n Appled Mathematcs, Jan [17] S. Kandukur and S. Boyd, Optmal power control n nterferencelmted fadng wreless channels wth outage-probablty specfcatons," IEEE Trans. Wreless Commun., vol. 1, pp , Jan Kyung-Joon Park receved hs B.S., M.S., and Ph.D. degrees all from the School of Electrcal Engneerng and Computer Scence, Seoul Natonal Unversty, Seoul, Korea n 1998,, and 2005, respectvely. He s currently a postdoctoral research assocate n the Department of Computer Scence, Unversty of Illnos at Urbana-Champagn, IL, USA. He worked for Samsung Electroncs, Suwon, Korea as a senor engneer n , and was a vstng graduate student, supported by the Bran Korea 21 Program, n the Department of Electrcal and Computer Engneerng, Unversty of Illnos at Urbana-Champagn n Hs current research nterests nclude desgn of medcal-grade protocols for wreless healthcare systems, desgn and analyss of self-adjustng protocols for wreless envronments, and dstrbuted control of physcal carrer sense n wreless networks. Dr. Jennfer C. Hou passed away on December 2, 2007 n Houston, Texas at the age of 43. Hou receved her B.S.E. degree n Electrcal Engneerng from Natonal Tawan Unversty, Tawan, ROC n 1987, M.S.E degrees n Electrcal Engneerng and Computer Scence (EECS) and n Industral and Operatons Engneerng (I&OE) from the Unversty of Mchgan, Ann Arbor, Mchgan n 1989 and n 1991, and Ph.D. degree n EECS also from the Unversty of Mchgan, Ann Arbor, Mchgan n She was an assstant professor n Electrcal and Computer Engneerng at the Unversty of Wsconsn, Madson, Wsconsn from , and an assstant/assocate professor n Electrcal Engneerng at Oho State Unversty, Columbus, Oho from She joned the Unversty of Illnos Computer Scence faculty n She was a prncpal researcher n networked systems and served as the drector of the Illnos Network Desgn and Expermentaton (INDEX) research group. Her research nterests n networked systems ranged from ssues of Qualty of Servce n wreless networks to enablng software nfrastructure for asssted lvng. She pursued topcs n both the theoretcal protocol desgn and deployment aspects of wreless sensor networks. Hou was elected as an Insttute of Electrcal and Electroncs Engneers (IEEE) Fellow and an Assocaton for Computng Machnery (ACM) Dstngushed Scentst n She was a recpent of an ACM Recognton of Servce Award n 2004 and 2007, a Csco Unversty Research Award from Csco, Inc., 2002, a Lumley Research Award from Oho State Unversty n 2001, an NSF CAREER award from the Network and Communcatons Research Infrastructure, Natonal Scence Foundaton n 1996, and a Women n Scence Intatve Award from The Unversty of Wsconsn-Madson n

10 PARK et al.: NONCOOPERATIVE CARRIER SENSE GAME IN WIRELESS NETWORKS 5289 Tamer Başar (S 71-M 73-SM 79-F 83) receved the BSEE degree from Robert College, Istanbul, and the MS, MPhl, and PhD degrees n engneerng and appled scence from Yale Unversty. After holdng postons at Harvard Unversty and Marmara Research Insttute (Gebze, Turkey), he joned the Unversty of Illnos at Urbana-Champagn (UIUC) n 1981, where he currently holds the postons of Swanlund Endowed Char, Center for Advanced Study Professor of Electrcal and Computer Engneerng, Research Professor at the Coordnated Scence Laboratory, and Research Professor at the Informaton Trust Insttute. He s also the Interm Drector of the Beckman Insttute for Advanced Scence and Technology. He has publshed extensvely n systems, control, communcatons, and dynamc games, and has current research nterests n modelng and control of communcaton networks; control over heterogeneous networks; resource allocaton, management and prcng n networks; gametheoretc tools for networks; moble computng; securty ssues n computer networks; and robust dentfcaton, estmaton and control. Dr. Başar s the Edtor-n-Chef of Automatca and the IEEE Expert Now, Edtor of the Brkhäuser Seres on SYSTEMS & CONTROL, Managng Edtor of the ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES (ISDG), and member of edtoral and advsory boards of several nternatonal journals n control, wreless networks, and appled mathematcs. He has receved several awards and recogntons over the years, among whch are the Medal of Scence of Turkey (1993); Dstngushed Member Award (1993), Axelby Outstandng Paper Award (1995), and Bode Lecture Prze (2004) of the IEEE Control Systems Socety (CSS); Mllennum Medal of IEEE (); Tau Beta P Drucker Emnent Faculty Award of UIUC (2004); the Outstandng Servce Award (2005) and the Gorgo Quazza Medal (2005) of the Internatonal Federaton of Automatc Control (IFAC); and the Rchard E. Bellman Control Hertage Award (2006) of the Amercan Automatc Control Councl (AACC). He s a member of the Natonal Academy of Engneerng (of USA), a member of the European Academy of Scences, a Fellow of IEEE, a Fellow of IFAC, Presdent-Elect of AACC, a past presdent of CSS, and the foundng presdent of ISDG. Hwangnam Km receved hs Ph.D. degree n Computer Scence at the Unversty of Illnos at Urbana- Champagn n He receved hs M.S.E. degree from the Department of Computer Engneerng at Seoul Natonal Unversty, Seoul, Korea, n 1994, and receved hs B.S.E. degree from the Department of Computer Engneerng at Pusan Natonal Unversty, Busan, Korea, n He s currently an assocate professor n the School of Electrcal Engneerng at Korea Unversty, Korea. He was a senor engneer at Samsung Electroncs Co., Ltd., from 2005 to 2006, and had been a post-doctorate fellow n the Department of Computer Scence at the Unversty of Illnos at Urbana-Champagn, Illnos from 2004 to He had worked for Bytemoble Inc., Calforna, from to He had worked for LG Electroncs Ltd., Seoul, Korea, from 1994 to Hs research nterests are n the areas of network modelng and smulatons, wreless sensor networks, communty wreless networks, self-organzng networks, pervasve/ubqutous computng, and cyber physcal systems.