Efficient Medium Access Control Design A Game Theoretical Approach

Transcription

1 Effent Medum Aess Control Desgn A Game Theoretal Approah Ln Chen, Jean Leneutre Department of Computer Sene and Networkng Éole Natonale Supéreure des Téléommunatons {Ln.Chen, Jean.Leneutre}@enst.fr Abstrat In ths paper, we address the queston that how to desgn effent MAC protools n selfsh and nonooperatve networks, whh s rual n nowadays open envronments. We model the medum aess ontrol problem as a non-ooperatve game n whh the MAC protool an be regarded as dstrbuted strategy update sheme approahng the equlbrum pont. Under suh game theoretal framework, three MAC protools, the aggressve, onservatve and heat-proof MAC protool, are then proposed wth tunable parameters allowng them to onverge to desred soal optmal pont. The frst two MAC protools requre network partpants to follow the rules, whle the heat-proof MAC protool an survve the selfsh envronments where nodes are pure self-nterested. Based on our game theoretal analyss, we provde a general methodology for desgnng effent MAC protools n non-ooperatve and selfsh envronments. We beleve that the proposed methodology not only provdes a general way of desgnng stable and ontrollable MAC protools ahevng hgh performane even n selfsh envronments, but also provdes a general framework that an be extended to desgn effent protools n other non-ooperatve and selfsh envronments. I. Introduton Medum Aess Control (MAC) s rual for networks where the ommunaton medum s shared by network partpants ompetng for the hannel aess. Desgnng effent MAC protools s a hallengng task, espeally n wreless envronments where hannel sensng s muh less effetve than n wred medum. IEEE 82. DCF (Dstrbuted Coordnaton Funton), the most popular MAC protool for WLANs, uses the exponental bakoff (EB) mehansm where eah node doubles ts ontenton wndow (CW) upon a ollson untl CW max and sets t to the bas value CW mn upon a suessful transmsson. Ths MAC protool results n too many ollsons and thus leads to network sub-optmalty when the network sales. Moreover, t has short-term farness problem due to the EB mehansm appled after the ollson. Ths motvates us to address the fundamental queston that how to desgn effent MAC protools, what are the methodology and gudelnes to follow? An effent MAC protool should satsfy the followng propertes: ) Convergene and stablty: the protool should onverge to a stable equlbrum; 2) Soal optmalty and farness: the onverged equlbrum should be network-wde optmal or at least quas-optmal and eah partpant should get a far share of payoff at ths pont. Besdes the above requrements on performane, we pose another requrement: the survvablty of the MAC protool n selfsh envronments. Nowadays, networks beome more and more open. Hene network partpants may behave selfshly rather than ooperatvely,.e., they adapt the strategy that maxmzes ther own utlty, regardless of others. Thus we an not mpltly assume all partpants at ooperatvely by followng the desgned protools. Under suh rumstane, we requre an effent MAC protool to be survvable suh that t an gude the ndvdual nodes to operate on the desgned equlbrum pont even they are purely self-nterested and non-ooperatve. In other words, the MAC protool onssts of a strategy that eah selfsh ndvdual node has no nentve to devate. We ondut our work usng game theory, a powerful tool to study the nteraton among deson makers wth onfltng objetves. Our motvaton of usng game theoretal approah rather than global optmzaton approah s two-fold: ) Game theory s a powerful tool to model selfsh behavors and ther mpat on the system performane n dstrbuted envronments wth self-nterested players; 2) Game theory an model the features or onstrants suh as lak of oordnaton and network feedbak n dstrbuted envronments. In fat n suh envronments, selfsh behavor s often muh more robust and salable than any entralzed ooperatve ontrol, whh s very expensve or even mpossble to mplement. We begn by modellng the medum aess ontrol problem as a non-ooperatve game G MAC, where eah player hooses ts strategy, the hannel aess probablty, to maxmze ts utlty funton, defned as the dfferene between ts throughput and transmsson ost. In suh nonooperatves games, a Nash equlbrum (NE) s the strategy profle where no player has nentve to devate unlaterally. The MAC protool an be vewed as the dstrbuted strategy update mehansm approahng the NE. We then ondut an n-depth study on the NE of G MAC. We fnd that there always exsts a based NE and that under ertan ondton, there also exsts a unque nonbased NE, n whh we are speally nterested sne at the non-based NE, the farness s ensured among players. However, by studyng the network-wde utlty on the nonbased NE and the soal optmal pont, we fnd that the non-based NE s neffent. We then propose two prng shemes to mprove the effeny of the non-based NE. We show that by wsely hoosng the prng fators, the non-based NE an be approahed to the soal optmal pont. We then seek the MAC protool that an lead the network to onverge to the soal optmal pont. However, the onvergene s not guaranteed under the best response

2 2 and subgradent ontrol, the two bas dynam ontrol mehansms n game theory. We thus turn to more sophstated ontrol mehansms under whh the game s provably onvergent to the soal optmal pont. We propose three MAC protools: aggressve, onservatve and heat-proof MAC protool. We show that the proposed MAC protools have the followng desrable propertes: ) They onsst of nature aess probablty update shemes for ratonal players; 2) They provde tunable parameters wth whh one an ontrol the onvergent pont of the network; 3) They lead the network to a stable state where both the onvergene and the soal optmalty are ensured. The aggressve and the onservatve MAC protool requre network partpants to respet ther rules. In ontrast, the heat-proof MAC protool wth the orrespondent prng funton based on observable nformaton s desgned suh that the hannel aess probablty update sheme s the best response for the ratonal selfsh players when maxmzng ther utlty. Hene no player has nentve to devate from the heatproof MAC protool. As a result, the heat-proof MAC protool an survve the selfsh envronments where players are purely self-nterested and may break any protool rules f they an get more payoff than obeyng the rules. Based on our game theoretal analyss, we answer the posed queston by provdng the followng methodology n desgnng effent MAC protools n non-ooperatve and selfsh envronments: ) hoosng a natural hannel aess update sheme; 2) onfgurng the parameters n the hosen update sheme to ensure that the network onverges to the desgned optmal equlbrum under the update sheme; 3) dervng approprate prng funtons based on observable nformaton suh that no player has nentve to devate from the MAC protool. Our man ontrbutons an be summarzed as follows: We formulate the medum aess ontrol as a nonooperatve game and perform an n-depth analyss on the game, nludng the exstene, unqueness, onvergene and effeny of the NE; Three MAC protools, the aggressve, onservatve and heat-proof MAC protool, are proposed wth tunable parameters allowng them to onverge to desred soal optmal pont. The frst two MAC protools requre network partpants to follow the rules, whle the heat-proof MAC protool an survve the non-ooperatve and selfsh envronments. Based on our game theoretal analyss, we provde a general methodology for desgnng effent MAC protools n non-ooperatve and selfsh envronments. We beleve these ontrbutons are very relevant for the medum aess ontrol desgn ahevng hgh effeny and survvablty n selfsh and non-ooperatve envronments. II. Related Work Game theory has been employed wdely to study the non-ooperatve behavors at MAC layer. [6] studes the non-ooperatve equlbra of Aloha for heterogeneous users. [5] studes the stablty of mult-paket slotted Aloha wth selfsh users and perfet nformaton. [7] shows that the 82. MAC protool leads to neffent equlbra f users onfgure ther paket sze and data rate to maxmze ther own throughput. [8] shows that the exstene of small populaton of selfsh nodes leads to network ollapse. The authors thus propose a penalzng sheme to prevent the network from beng paralyzed. [2] reverse-engneers bnary exponental bakoff algorthm n game theory framework. In the feld of MAC protool desgn, muh reent work [] [] apples the network utlty maxmzaton (NUM) framework by vewng the network as an optmzaton solver and the MAC protools as dstrbuted algorthms solvng some global network utlty maxmzaton problem. Our work, however, s based on a game theoretal framework under whh the medum aess ontrol problem s modeled as a non-ooperatve game and the MAC protool s regarded as the dstrbuted strategy update sheme approahng the equlbrum. We argue that our work s more suted n selfsh non-ooperatve envronments suh as nowadays open aessed networks where partpants are purely self-nterested. [] also studes the MAC desgn from a game theoretal angle. It s foused on modellng a large lass of system-wde qualty of serves models va utlty funtons and dervng dstrbuted ontenton resoluton algorthm based on ontnuous feedbak sgnal rather than bnary ontenton sgnal to approah the NE (may not be soal optmal). In ontrast, our work fouses on provdng a methodology on how to desgn effent MAC protools n selfsh envronments that an gude the network to a stable equlbrum whh s network-wde optmal and at whh eah partpant gets a far share of payoff. III. System Model We onsder a LAN onsstng of a set N = {,2,,n} of nodes sharng the ommon medum. We base our study on a general and bas MAC layer model: Tme s dvded nto synhronzed slots. Eah node an send one paket n a slot. If a node has a new paket to send, t attempts transmsson durng the next slot wth probablty p alled hannel aess probablty. The hannel aess probablty p an be realzed va ontenton wndow n the ase where a bakoff mehansm s mplemented suh as CSMA. In the above smple model, we do not assume any ollson avodane or deteton mehansm although suh mehansms may faltate the MAC protool desgn. Bult on a bas MAC layer model wthout any added funtonaltes, our proposed MAC protools an be mplemented n almost all nowadays network systems, from slotted Aloha to CSMA. IV. Non-ooperatve Medum Aess Control Game In game theory, the utlty funton desrbes the satsfaton level of the player as the result of ther strateges. In our study, we onsder a utlty funton as follows U = p ( p j ) p

3 3 In the above defned utlty funton, node gets payoff for a suessful frame transmsson and no payoff f the transmsson fals due to a ollson. On the other hand, the transmsson of a frame also nurs the transmsson ost ( s normalzed), e.g., n terms of energy. In ths paper, for the reason of smplty, we assume that =. The utlty funton U thus represents the net beneft of a node when operatng on p. We now formulate the medum aess ontrol problem as a non-ooperatve game G MAC. Defnton : The non-ooperatve medum aess game G MAC s a trple (N,{A },{U }), where N s the player set, A s the strategy set of player, U s the utlty funton of player defned prevously. Eah player selets ts hannel aess probablty p A = [,] to maxmze ts utlty U. Formally, G MAC s expressed as: G MAC : max U (p, p ), p A A. Nash Equlbrum Analyss N For non-ooperatve games as G MAC, the most mportant onept s the Nash equlbrum (NE), where no player has nentve to devate from ts urrent strategy. The NE an be seen as optmal agreements between the opponents of the game. In the ase of the G MAC, we have the followng defnton of NE. Defnton 2: A hannel aess probablty vetor p = (p,,p n) s a NE of G MAC f no player an mprove ts utlty by unlaterally devatng from p : U (p, p ) U (p, p ), p, N We use the onept of Pareto-optmalty and soal welfare optmalty to haraterze the effeny of dfferent strategy profles. Defnton 3: The strategy profle s s Pareto-optmal f there does not exst another strategy profle s suh that for eah player, t holds that U (s ) > U (s). Defnton 4: The strategy profle s s soal welfare optmal f t maxmzes the aggregated payoff N U. Theorem studes the NE of G MAC. Theorem : G NP C admts at least one NE. Proof: It an be verfed that the strategy set of eah player A = [, ] s a nonempty ompat onvex subset of Euldan spae. The utlty funton U s ontnuous and onave w.r.t. p on A. Hene, by Theorem n [3], there exsts at least one NE. Sne the utlty funton U s onave, p s ether on the border of the strategy spae or satsfes U =. We p all a NE p a non-based equlbrum f, for all nodes, p satsfes U =, and based equlbrum otherwse. p Theorem 2 provdes a more n-depth nsght on the NE of G MAC. The proof s straghtforward and s omtted here. Theorem 2: If, then G MAC has only one based NE {p = } and no non-based NE; If <, then G MAC has n based NE NE = {p =,p j = (j )}( =,,n) and one non-based NE {p = n }. Remark : s the trval ase where the transmsson ost s so expensve that all players hoose to keep slent. If <, the based NE orresponds to the stuaton that one player aptures the hannel and others always defer ther transmsson. The non-based NE s the ase where eah player gets a far share of the hannel. We are manly nterested n the non-based NE n our study and any effent MAC protool should not lead the network to the based NE. In the rest of the paper, we fous on the non-trval ase where <. Remark 2: We an analyze the non-based NE from another angle: onsder eah player has two pure strateges: transmt or wat. The non-based NE s thus the mxed strategy NE of G MAC and suh NE s guaranteed to exst. Remark 3: From an eonom pont of vew, an be regarded as the pre for player operatng on p. The NE s thus the pont where the margnal gan p ( p j) equals to the pre. From the players s pont of vew, operatng at hgher p nreases the gan p at the expense of payng more n terms of pre. Hene, to searh the NE s atually to seek a ompromsed pont between the gan and the ost. V. Ineffeny of the Non-based NE of G MAC The non-based NE dsussed n last seton provdes a soluton where no player an nrease ts utlty any further through ndvdual effort. A natural queston we pose s that whether the non-based NE s effent,.e., Paretooptmal and soal welfare optmal. In ths seton, we answer ths queston by omparng the utlty at above nonbased NE and the soal welfare optmal pont. Let P = { p } denote the soal welfare optmal pont of G MAC maxmzng the global network utlty N U, we nvestgate the far soal welfare optmal pont where p = p for all N n the followng lemma. Lemma : Under the ondton that n 2 and < <, there s a unque far soal welfare optmal pont where p = p for all N. Moreover, t holds that. p s the root of n( p) n (n )( p) n 2 =. 2. < p < /n 3. p < p Proof: It s easy to verfy the ase where n = 2. We onsder the ase where n 3. Let Q(p) = N U ({p = p}), by mposng Q(p) =, we get p Q(p) = n( p) n (n )( p) n 2 =. It follows Q (p) = n(n )( p) n 2 + (n )(n 2)( p) n 3 Hene Q(p) s monotonously dereasng w.r.t. p n (, 2 n ) and monotonously nreasng n ( 2 n, ). Notng that Q() = <, Q( ) = < and Q() = >, we n obtan that Q(p) = admts a unque soluton p (, n ). Moreover, Q(p) < when p ( p, ) and Q(p) > when p (, p). Hene, p s the unque maxmzer of N U.

4 4 On the other hand, followng < <, we have Q(p ) = (n )( n ) <. Ths leads to p > p. Consder the utlty funton at the non-based NE p, we have U (p ) = for all player. Sne all players are self-nterested and ratonal and would never aept a negatve payoff, operatng at p atually mnmzes both the ndvdual and network-wde utlty. On the other hand, at P, we have N U = n ( p( p) n p ). By expressng by p, after some mathemat operatons, we get N U = n(n )( p) n 2 p 2 >. The non-based NE s not Pareto optmal ether. If all players swth from the non-based NE to the soal optmal pont, both the ndvdual and the network utlty nrease. Ths s due to the fat of lak of ooperaton and the nentve to operate at soal optmal pont. The followng theorem summarzes our result of ths seton. Theorem 3: The non-based NE of G MAC s neffent,.e., nether Pareto-optmal nor soal welfare optmal. Fgure and Fgure 2 show the non-based NE and the soal optmal pont p as a funton of. poly. The non-ooperatve medum aess game wth ths prng sheme G MACP s thus formally expressed as G MACP : max p U (p, p ) = U (p, p ) b p, N We rewrte the utlty funton as U = p ( p j ) p, where = + b. It an be shown that f <, G MACP admts n based NEs and a unque non-based NE {p = p = ( ) n }. By mposng p = p, the nonbased NE ondes to the soal optmal pont. Theorem 4: By settng the prng fator b = ( p) n, G MACP admts a unque effent non-based NE whh s also the soal optmal pont. In the analyss of G MAC, we an nterpret as the pre for player operatng on p. Here n G MACP, the above pre beomes + b = ( p) n. As the pre nreases, eah player tends to derease ts p at the non-based NE. Fgure 3 and Fgure 4 show b as a funton of (n = ) and n (=.). non-based Nash equlbrum Fg n= n= p as a funton of soal welfare pont Fg n= n= bp as a funton of prng fator Fg b as a funton of prng fator Fg b as a funton of n n VI. Non-ooperatve Medum Control Aess Game Wth Prng Prng s a powerful tehnque n game theory to motvate selfsh players to adopt desrable behavors. In our ontext, we turn to prng to let the network onverge to the soal optmal pont. From Lemma, at the NE, players tend to operate at hgher p than the optmal pont p, we enourage the players to derease ther p va prng to approah the soal optmal pont. In ths new ontext, we develop a non-ooperatve game wth prng denoted by G MACP = ( N, {A }, {U } ), where the utlty funton U ( ) s defned as U = U + τ (p ), where τ : A R s the general form of the prng funton. In ths paper, we nvestgate the followng two prng shemes. A. Prng Sheme Motvated by the fat that, at the non-based NE of G MAC, players operate at hgher p than the soal optmal ase, we mpose a lnear prng funton τ = b p whh s monotonously dereasng w.r.t. p by settng b > to enourage the players to derease ther p. b p an be regarded as extra pre to players mposed by the prng Under our model, when =, we get bp = /n, the aggregated utlty beomes the network throughput S. From Lemma we have S = P N U = ( /n) n < /e and lm n S = /e. The result s oherent to the tradtonal performane bound of slotted-aloha. B. Prng Sheme 2 In ths prng sheme, we mpose a gan dsount on the utlty funton. Under ths rumstane, U 2 = d p ( p j) p, where d < s the dsountng fator appled to dsourage players to nrease ther p. The non-ooperatve medum aess game wth ths prng sheme G 2 MACP s formally expressed as G 2 MACP : max U 2 (p, p ) = d p ( p j ) p, N p For G 2 MACP, n based NEs and a unque non-based NE {p = p 2 = ( ) n } exst f <. By mposng p 2 = d d p, the non-based NE ondes to the soal optmal pont. Theorem 5: By settng the prng fator d = ( p) n, G2 MACP admts a unque effent non-based NE whh s also the soal optmal pont. Fgure 5 and Fgure 6 show d as a funton of (n = ) and n (=.2). Dfferent from the frst prng sheme ahevng the goal by nreasng the ost of the transmsson from to, the seond prng sheme attans the same goal by dereasng the gan of the suessful transmsson from p ( p j) to d p ( p j).

5 5 prng fator Fg d as a funton of prng fator Fg d as a funton of n VII. Approahng the Non-based NE Untl now, we have studed the NE of G MAC and two prng shemes to mprove the effeny of the non-based NE. The MAC protool an be vewed as the dstrbuted strategy update mehansm to onverge to the NE. To study suh game dynams, we onsder the repeated play of G MAC, and look for update mehansm n whh players repeatedly adjust strateges n response to observatons of other player atons so as to approah the non-based NE. In ths seton, we study the onvergene of two bas strategy update mehansms wdely used n game theory: the best response update and the subgradent update. A. Best Response Update In game theory, the smplest strategy update mehansm s best response update: at eah teraton, every node hooses the best response to the atons of all the other nodes n prevous teraton. Mathematally, at teraton t +, player updates ts hannel aess probablty as = r(p t ) := argmax U (p, p t ) p Clearly, f the above dynam reahes a stable state, ths state s a NE. The onvergene to the NE under best response update s also guaranteed. For our medum aess game wthout or wth prng, the onvergene under the best response update s not guaranteed. Take G MAC as an example, the best response s = r(p t ( p j) > ) = ( p j) < p [,] ( p j) = Starng by p =, N, we have pt = f t s even, pt = f t s odd. Hene, the best response update of the medum aess games may not onverge to the NE. B. Subgradent Update An alternatve strategy s the subgradent update. Compared to the best response update, subgradent update an be vewed as the better response update n whh every player adjusts ts hannel aess probablty n the gradent dreton suggested by observatons of other player atons. Mathematally, player updates ts strategy aordng to = p t + λ t U p=p p t where λ t > s the stepsze of player at teraton t. The subgradent update sheme an be nterpreted from an eonom pont of vew. If the margnal gan s greater than n ontenton pre,.e., U p=p >, nreases the aess probablty p, otherwse, dereases p. Sne at eah p t teraton, players update hannel aess probabltes by a small amount, they usually experene smooth trajetores. We take G MAC as an example to study the onvergene to the NE under the above subgradent update sheme. For G MAC, the subgradent update an be derved as = p t + λ t ( p t j) where the fxed pont of the subgradent update s p = n whh s also the non-based NE. Consder a smple ase where n = 2, startng from p = (p, p 2 ) = ( + ɛ, ɛ) where ɛ s a small postve value, we have p t (,) as t + ; startng from p = ( ɛ, + ɛ), p t (, ) as t +. Hene, the subgradent update may not onverge to the non-based NE. VIII. Medum Aess Control Desgn Gven the fat that the two bas update shemes studed above do not guarantee the onvergene to the non-based NE n the medum aess game wth or wthout prng, we nvestgate more sophstated medum aess ontrol mehansms and propose the followng three medum aess ontrol shemes wth provable onvergene to the desred equlbrum. A. Medum Aess Control Sheme : Aggressve Control The frst medum aess ontrol sheme s defned as: = ( p t j) + βp t ( p t j) () where < <, β <. One nterpretaton of the sheme s that at eah teraton player sets ts hannel aess probablty p to the maxmum value wth probablty ( pt j ) dependng on the hannel aess probablty of other players n the last teraton, whle redues p by a fator β otherwse. If one teraton orresponds to one slot, s the expeted hannel aess probablty of the followng update sheme based on hannel ondton: f the hannel s not ouped by other players durng the last slot, then player sets p to for the omng slot; otherwse t redues p by β. We refer ths sheme as the aggressve medum aess ontrol as players set ther hannel aess probablty to the maxmum value one the hannel s not ouped by others. The followng theorem studes the dynams under the aggressve medum aess ontrol { sheme. Theorem 6: If max (n ), β + ( ( ) (n ) n )pmax } β <, the update sheme defned n () admts a unque fxed pont p f = {p f } and t holds that:) < p f < ; 2) Startng from any

6 6 ntal pont p = {p } where < p <, the teraton defned by () onverges to p f. Proof: We frst show that wth any ntal pont < p <, t holds that < p t <, N, t >. We prove t by showng that f < p t <, t, then < <, N. Notng that ( < βp t < pt <, we get < < ( pt j ) + ) ( pt j ) =. We then show that () admts a unque fxed pont by usng the followng lemma onernng the fxed pont of a ontraton [4]: Lemma 2: If the update sheme defned n () s a ontraton, then t admts a unque fxed pont; Moreover, startng from any ntal pont, the teraton under t onverges to the unque fxed pont. The ontraton s defned n [4] as follows: let (X,d) be a metr spae, f: X X s a ontraton f there exsts a onstant k wth k < suh that d(f(x), f(y)) kd(x, y) x, y X where d(x,y) = x y = max x y. The key pont to establsh the unqueness of the fxed pont s thus to show the update sheme defned n () s a ontraton. We have d(f(x),f(y)) = f(x) f(y) f x y = f x x d(x,y) If the Jaoban f k <, f s a ontraton. In our x ontext, we show that the update sheme of () s a ontraton by provng J k, where J = {J j } s the Jaoban of the update sheme of () defned by J j = pt+ At teraton t (t ), we have { (βp t J j = ) l N,l,l j ( pt l ) j β( l N,l ( pt l )) = j Notng that < βp t < pt <, we have J = max J j = β( ( p t l)) N j N l N,l (βp t ) ( p t l) p t l) l N,l,l j < β( ( p t l)) + l N,l = β + l N,l ( p t l) l N,l,l j l N,l p t j ( p t l) p t β l Let p t = {p t }, Q(pt ) = β + l N,l l N,l. ( p t β, l we bound Q(p t ) by Q max. To ths end, we rewrte Q as Q(p t j) = β + ( p t l) l N,l,l j l N,l,l j p t β ( p t j) + l It follows that Q attans Q max when p t j = or pt j =. Performng the same analyss for all j N, we show that Q attans Q max at the border of the strategy spae. Let N( N n ) be the number of players wth the aess probablty at Q max, t follows that Q(N) = β+( ) N ( ( Imposng Q =, we obtan N N 2 + ( n β N + n N ) N ( ) = ) ) β Q We an further verfy that N = at N = ( ) ( ) 2 n β + β + 4pmax 2 ( ), If N n, Q N < n [, N Q ), N > n (N, n ]. Hene Q s mnmzed at N = N and maxmzed at N = or N = n. If N < or N > n, Q has no loal maxmzer n (, n ) and attans ts maxmum { at border. } In both ases, we have Q max = max Q(), Q(n ) = { ( ) (n max (n ),β + ( ) n )pmax } β { Let k = max (n ), β + ( ( ) (n ) n )pmax } β, f the ondton n the theorem holds,.e., k <, we have J k <. The update defned n () s a ontraton. It admts a unque fxed pont and the update onverges to the fxed pont,.e., lm p t = p f t. Sne we have shown that < p t <, we have < p f <. Ths onludes our proof. Reall that our goal of the medum aess ontrol desgn s to enourage the players to operate stably at the soal optmal pont, to ths end, we mpose p = p f. The followng theorem s mmedate. Theorem 7: Under the ondton of Theorem 6, by tunng β and suh that p = p f, or p = ( p) n + β p ( ( p) n ), the proposed aggressve medum ontrol sheme s onvergent to the soal optmal pont, whh s also the non-based NE of the medum aess game wth prng. Theorem 6 and Theorem 7 provde gudelnes for hoosng parameters for aggressve MAC sheme. From Theorem 6, we an see that small and large β help the

7 7 network operate at a stable pont. Theorem 7 further quantfes and β to approah the stable onvergent pont to the soal optmal pont. As an example, f n s large (( n )n e ),, then = n + ɛ where ɛ s a postve small number, e.g. ɛ = O(n 2 ), β = nɛ (e ) s a possble settng. B. Medum Aess Control Sheme 2:Conservatve Control In ths seton, we propose a more onservatve medum aess ontrol sheme defned as: = p t f(p t ) ( p t j) + p mn ( p t j) (2) where f(p t ) = + p t δ, < p mn < <, < p mn δ p mn. We pose the above onstrant of δ to en- sure that wth any p mn p t, t holds that p mn p t f(pt ). One nterpretaton of the above update sheme s that at eah teraton player sets p to the mnmum value p mn wth probablty ( pt j ), whle nreases p by a fator f(p t ) otherwse. f(pt ) s speally desgned suh that f(p t ) = + δ at p mn, f(p t ) = at, < f(p t ) < + δ and s lnearly dereasng w.r.t. p t n (p mn, ). The nreasng fator s thus adaptable based on the urrent hannel aess probablty p t. If one teraton onssts of one slot, beomes the expeted hannel aess probablty of the followng update sheme: f the hannel s ouped by other players durng the last slot, then sets p to p mn for the omng slot; otherwse t nrease p by + δ. We refer ths sheme as the onservatve medum aess ontrol as players set the hannel aess probablty to the mnmum value one the hannel s sensed ouped by others. We next nvestgate the dynams under the above onservatve medum aess ontrol sheme. Theorem 8: If < δ p mn and (n )( ( ) p mn )( p mn ) n 2 δ + ( p mn ) n < p mn, the update sheme defned n (2) admts a unque fxed pont p f2 = {p f2 } and t holds that:) p mn < p f2 < ; 2) Startng from any ntal pont p where p mn < p <, the teraton defned by (2) onverges to p f2. Proof: We follow the same way as the proof of Theorem 6 by showng that the Jaoban for (2) J k <. To ths end, we ompute J j as ( ( p mn p t + δ p t )) p mn ( p t l) j J j = ( l N,l,l j + δ 2p t p mn ) l N,l ( p t l) = j Notng that < δ p mn, we an show that J = max N j N J j s maxmzed at p t =, p t j = p mn for j, we thus have J (n )( p mn )( p mn ) n 2 + ( ) δ ( p mn ) n p mn If the ondton n the theorem( holds,.e., k = ) (n )( p mn )( p mn ) n 2 δ + ( p mn p mn ) n <, J k <. (2) s a ontraton. Both the unqueness of the fxed pont and the onvergene are guaranteed. It s further easy to show that the fxed pont p mn < p f2 <. We then study the onvergene of the onservatve medum aess ontrol to the soal optmal pont. Theorem 9: Under the ondton of Theorem 8, by tunng δ, and ( p mn suh that p = p f2, or p = pf( p)( p) n + p ) mn ( p) n, the onservatve medum ontrol sheme s onvergent to the soal optmal pont. C. Medum Aess Control Sheme 3: Cheat-proof Control The above MAC protools have followng desrable propertes: ) they onsst of natural aess probablty update shemes for ratonal players; 2) they provde tunable parameters wth whh one an ontrol the onvergent pont of the network; 3) they lead the network to a stable state where both farness and soal optmalty are ensured; The above MAC protools meet the requrements for effent MAC protools n terms of performane. However, they both requre network partpants to respet the rules. Hene, they annot survve selfsh and nonooperatve envronments beause n suh envronments, players only adopt strateges that brng the most benefts to them, regardless of the fat that the adopted strategy leads to soal optmalty or not. In suh envronments, we an not mpltly assume that all partpants at ooperatvely by followng the desgned MAC protools. Under suh rumstane, we requre an effent MAC protool to be survvable suh that t an gude the ndvdual nodes to operate on the desgned optmal equlbrum even they are purely self-nterested and non-ooperatve. In other words, the MAC protool onssts of a set of strateges that eah selfsh ndvdual node has no nentve to devate. To ths end, we apply the prng tehnque agan. On one hand, the prng sheme approahes the non-based NE to the soal optmal pont; on the other hand, the prng sheme enourages the ndvdual selfsh players to follow the MAC protool. In the followng part of ths seton, we propose the heat-proof MAC protool. The above desrable propertes are mantaned n the heat-proof MAC protool. Moreover, the mposed prng funton an enourage the players to follow the proposed protool. The hannel aess probablty update sheme n the heat-proof MAC protool s defned as = ( p t j) + p mn ( p t j) (3)

8 8 One nterpretaton of the above update sheme s that at eah teraton player sets ts aess probablty p to wth probablty ( pt j ) and p mn otherwse. If one teraton orresponds to one slot, p t beomes the expeted hannel aess probablty of the followng update sheme: f the hannel s ouped by other players durng the last slot, then player sets p to p mn for the omng slot; otherwse t sets p to. In ths sheme, s deoupled wth p t, whh s a neessary ondton for the followng demonstraton. (3) an not only be regarded as the update sheme for the hannel aess probablty, but also be vewed as the strategy update that mpltly maxmzes some utlty funton, as studed n the followng theorem. Theorem : Regard (3) as the best response funton for eah player N at eah teraton, the underlyng utlty funton that eah player tres to maxmze s U C = (p p mn ( p mn ) 2 ( p j )) + C where C > max {( p mn ( p mn )( ) 2, ( ) 2 } ) n ( p mn )( p mn ) n s a onstant large enough to avod the negatve utlty value. Proof: The proof s straghtforward notng (3) an be wrtten as = p mn + ( p mn ) ( p t j) Reall that n G MAC, the utlty funton U s U = p ( p j ) p We mpose the followng prng funton τ (p ) = U C U = (p p mn ( p mn ) + C p ( p j ) p ( p j )) 2 Next we defne the non-ooperatve game wth the above prng funton as G C MACP : max U C (p, p ), p N The above proposed MAC sheme wth prng s heatproof n that the aess probablty update sheme orresponds to the best response strategy of G C MAC, thus a ratonal player wll follow (3) to maxmze ts payoff. The followng theorem establshes the exstene, unqueness of the NE n G C MACP and the onvergene to the unque NE under the heat-proof MAC sheme. Theorem : Under the ondton that (n )( p mn )( p mn ) n <, G C MACP admts a unque NE. Startng from any ntal pont p, the heat-proof MAC sheme s onvergent to the unque NE. Proof: We use the followng theorem n game theory onernng the unqueness of NE [4]: Lemma 3: If the best response funton s a ontraton, then the game admts a unque NE; Moreover, startng from any ntal pont, the teraton under the best response onverges to the unque NE. The above lemma shows that atually the NE onssts of the fxed pont of the best response funton. We now prove that the update sheme (3) s a ontraton. Ths an be shown by notng that { Jj C (pmax p = mn ) ( pt j ) j = j J C = (n )( p mn ) ( p t j) (n )( p mn )( p mn ) n < Thus (3) s a ontraton. The theorem s proven Furthermore, f the ondton n Theorem s satsfed, let p C = {p C } be the unque NE, we an show that p mn < p C <. The followng theorem studes the effeny of the unque NE of G C MAC. Theorem 2: If p = p mn + ( p mn )( p) n, the unque NE s effent,.e., p C = p. Theorem and 2 provde suffent ondton on the onvergene to the NE under (3), whh an be regarded as the best response update. One draw bak s that the best response update often leads to large flutuatons that may ause temporary system nstablty. We address ths ssue by studyng the subgradent update n G C MACP. By settng the step sze suffently small, the subgradent update sheme experenes a smooth trajetory. Theorem 3 gves the suffent ondton on the onvergene of the subgradent update to the NE of G C MACP. The proof follows the smlar way as that of Theorem 6 and s omtted here. Theorem 3: Consder the subgradent update for heatproof MAC sheme defned as = p t + λ U C p = p t 2λ p t p mn ( p mn ) ( p t j) under the same ondton as Theorem 2, the subgradent update sheme onverges to the unque NE. The above subgradent update sheme s atually a mld verson of the heat-proof MAC sheme n (3). By ontrollng the step sze λ, players experene less varaton n ther strateges than the best response update (3). The system s thus more stable. As pre, the onvergene delay nreases. D. Implementaton Issues In the pratal mplementaton of the proposed MAC protools, players usually do not have aess to the aess probablty of others, so they an not dretly alulate ( pt j ) whh s needed to update pt.

9 9 To solve ths problem, we apply the Idle Sense approah proposed n [9] allowng a player to estmate the hannel ondton by observng the average number of onseutve dle slots between two transmsson attempts. Let Pdle t = j N ( pt j ) be the probablty of an dle slot and n t dle be the number of average onseutve dle slots between two transmsson attempts durng teraton t, t holds that n t dle = P dle t Pdle t. It follows that ( p t j) = nt dle n t dle + p t. Thus eah player an update p t by observng n t dle. Another desrable feature of usng Idle Sense approah s that our MAC protools deouple aess ontrol from ollson perepton, thus are mmune to problems nurred by paket ollson perepton. Algorthm shows the derved heat-proof MAC protool. The aggressve and onservatve MAC protools an be derved smlarly. In the protool, a transmsson orresponds to an ouped hannel slot when only one player transmts (a suessful transmsson) or multple players transmt smultaneously (a ollson). Algorthm Cheat proof MAC Protool After eah transmsson do sum sum + n, ntrans ntrans + /* Idle sense: the player observes n dle slots before the transmsson */ f ntrans ntrans max then n t dle sum ntrans /* estmate nt dle */ n t dle p mn + ( p mn ) n t dle + p t /* update p */ sum, ntrans /* reset varables */ end f end In our work, we do not address how to realze the prng, whh s not trval at all. An approprate prng sheme n our ontext should be dstrbuted and heat-proof n ase where players may provde forged nformaton to get extra gan. In prevous part of ths seton, we have shown how to estmate ( pt j ) based on nt dle, whh s observable to all players. Next we provde a mehansm to estmate p t based on only observable nformaton. Ths s a rual ssue to mplement any prng sheme. The mehansm s extended from the Idle Sense approah. Let Pdle, t = ( pt j ) and nt dle, be the number of average onseutve dle slots between two transmssons durng teraton t, where -transmsson orresponds to an ouped hannel slot when only one player exept transmts or multple players transmt smultaneously. Pdle, t s n fat the probablty of the slot wth no -transmsson. It holds that n t dle, = P dle,. P dle, It follows that p t = P dle P dle, = nt dle, nt dle n t dle, ( + Hene, nt dle ). p t an be estmated based on nt dle whh s observable to all players, and n t dle, whh s observable to all players exept ( knows p t ). By employng the above mehansm, the prng sheme an be realzed n a dstrbuted and heat-proof way based on only observable nformaton. E. Methodology for Effent MAC Protool Desgn Based on the analyss on the medum aess game and the three proposed MAC protools, we ntrodue the followng methodology for desgnng effent MAC protools for non-ooperatve and selfsh envronments:. Choosng a natural hannel aess update sheme; 2. Confgurng the parameters n the hosen update sheme to ensure that the network onverges to the global optmal pont under the update sheme; 3. Dervng approprate prng funtons based on observable nformaton suh that no player has nentve to devate from the desgned MAC protool. We beleve that the proposed methodology not only provdes a general way of desgnng stable and ontrollable MAC protools ahevng hgh performane even n selfsh envronments, but also provdes a general framework that an be extended to desgn effent protools n other non-ooperatve and selfsh envronments. IX. Numeral Results In ths seton, we provde numeral results on the performane of the proposed MAC protools. Frst, we onsder a network of nodes. We set =.25, ntrans max =. We alulate the soal optmal pont to be p =.58. Based on Theorem 6 and Theorem 7, we set =.8, β =.33. The orrespondent aggressve MAC protool leads the network to the soal optmal pont. Smlarly, for the onservatve MAC protool, we set δ =.33, =.8, p mn =.5. For the heat-proof MAC protool, we set p mn =. and =.5. Under these parameter settngs, based on our analytal model, the network onverges to the soal optmal pont under the proposed three MAC protools. Ths s onfrmed by the numeral result shown n Fgure 7-9, whh plot the hannel aess probablty trajetores of eah player. Fgure plots the trajetores of the aess probablty under subgradent update, a mld verson of the heat-proof MAC protool. The trajetory onverges n a smoother way wth longer onvergene delay. hannel aess probablty Fg teraton Aggressve MAC hannel aess probablty Fg teraton Conservatve MAC We then fous on the heat-proof MAC protool. In the protool mplementaton, the number of network partpants n s requred to onfgure the protool parameters.

10 hannel aess probablty teraton Fg. 9. Cheat-proof MAC Fg.. Cheat-proof MAC: mld verson We now study the mpat of the estmaton error of n on the protool performane by allowng ertan estmaton error on n and studyng the performane under suh estmaton error. Fgure plots U at /U opt under estmaton error % 5%, where U at s the atual global utlty wth estmaton error, U opt s the optmal global utlty wthout error. We an see that our protool s qute robust n that the global utlty does not degrade dramatally w.r.t. the estmaton error of n, even when the estmaton error reahes 5%. Ths s a dsable feature when runnng the protool n dynamal envronments. U at /U opt est err=%.6 est err=3%.5 est err=5% hannel aess probablty n Fg. 2. Fg.. Cheat-proof MAC: performane wth estmaton error of n aggregated utlty teraton exponental bakoff heat-proof MAC X N U as funton of Fnally, we ompare the performane of our protool wth the EB based MAC protool wdely employed n nowadays WLANs. We set n = 5, =.. Sne the performane of the EB protool hghly depends on and p mn, we smulate the EB protool wth dfferent and p mn values and plot the maxmum aggregated utlty wth the aggregated utlty aheved by the heat-proof protool n Fgure 2. We also ompares the farness of the two protools by plottng the normalzed Jan farness ndex [2] n Fgure 3. We an see that our protool aheves hgher utlty wth better short-term farness. The result s due to the fat that the EB protool reles on an neffent ollson resoluton mehansm whh auses both network sub-optmalty on performane and the short-term farness problem. However, the heat-proof MAC protool deouples the aess ontrol from ollson perepton and players have muh less varaton n hannel aess probablty around the soal optmal pont. Hene t s not surprsng that the proposed MAC protool outperforms the EB protool n both performane and farness. X. Conluson and Future Work In ths paper, we address the queston that how to desgn effent MAC protools n selfsh and non-ooperatve networks, whh s rual n nowadays open envronments. To ths end, we ondut an n-depth study on the medum Jan ndex Fg heat-proof MAC.2 exponental bakoff normalzed wndow sze Farness omparson aess ontrol under game theoretal framework. Three MAC protools, the aggressve, the onservatve and the heat-proof MAC protools, are then proposed wth tunable parameters allowng them to onverge to desred optmal pont. The frst two MAC protools requre network partpants to follow the rules, whle the heat-proof MAC protool an survve the selfsh envronments where nodes are purely self-nterested. Based on our game theoretal analyss, we provde a general methodology for desgnng effent MAC protools for non-ooperatve and selfsh envronments. We beleve that the proposed methodology not only provdes a general way of desgnng stable and ontrollable MAC protools ahevng hgh performane even n selfsh envronments, but also provdes a general framework that an be extended to desgn effent protools n other non-ooperatve and selfsh envronments. As future work, we plan to develop a general pratal prng sheme to gude selfsh players at ooperatvely. Another dreton s to apply the methodology proposed n ths paper to the network and transport layers. Referenes [] L. Chen, S. H. Low and J. C. Doyle, Random aess game and medum aess ontrol desgn, Tehnal Report, hen/papers/rama.pdf [2] A. Tang, J.-W. Lee, J. Huang, M. Chang and A. R. Calderbank, Reverse Engneerng MAC, Workshop on Modelng and Optmzaton n Moble, Ad Ho and Wreless Networks (WOpt), Boston, Massahusetts, Aprl 26 [3] J.B. Rosen, Exstene and unqueness of equlbrum ponts for onave n-person games. Eonometra, vol. 33, July 965. [4] E. Anell and E.J.S. Carrera, Contraton of Best Response Funtons and Unqueness of Nash-Cournot Equlbrum (November 25). Avalable at SSRN: [5] A.B. MaKenze and S. B. Wher, Stablty of Multpaket Slotted Aloha wth Selfsh Users and Perfet Informaton. In Pro IEEE INFOCOM, 23. [6] Y. Jn, G. Kesds, Equlbra of a nonooperatve game for heterogeneous users of an ALOHA network. IEEE Comm. Letters, vol. 6, 22. [7] G. Tan, J. Guttag, The 82. MAC protool leads to neffent equlbra. In Pro IEEE INFOCOM, 25 [8] M. Cagalj, S. Ganerwal, I. Aad and J.-P. Hubaux, On Selfsh Behavor n CSMA/CA Networks. IEEE INFOCOM, 25. [9] M. Heusse, F. Rousseau, R. Guller and A. Dula, Idle sense: An optmal aess method for hgh throughput and farness n rate dverse wreless LANS, Pro. ACM Sgomm, 25. [] J.-W. Lee, M. Chang, and A. R. Calderbank, Utlty-optmal medum aess ontrol reverse and forward engneerng, IEEE Inforom 26, Apr. 26. [] J.-W. Lee, M. Chang and R. A. Calderbank, Utlty-optmal random aess ontrol, to appear n IEEE Trans. Wreless Comm., 27. [2] R. Jan,D. Chu,an d W. Hawe. A Quanttatve Measure of Farness and Dsrmnaton for Resoure Alloaton n Shared Computer Systems, DEC Researh Report TR-3, 984.