On the stability of flow-aware CSMA

On the stability of flo w-aw are CSMA T. Bonald a , M. F euillet b a T ele c om Par isT e ch, Pa ris, F r anc e b INRIA, R o c quenc ourt, F r anc e Abstract W e consider a w ir eless netw ork where eac h flow (instead of ea ch link) r uns its o wn C SMA (Carrier Sense Multiple Ac cess) algorithm. Sp ecifically , eac h flo w attempts to ac cess the radio c hannel after some random time and transmits a p ac ket if the c hann el is sensed idle. W e pr ov e that, unlike th e standard CSMA algorithm, th is simple distribu ted acc ess sc h eme is optimal in t he sense that the netw ork is sta ble for all traffic in tensities in the capacit y region of the net wo rk. Keywor ds: Wireless net wo rk, conflict graph, CSMA, flo w-lev el d y n amics, stabilit y , throughput p erformance. 1. In tro duction The CSMA (Carrier Sense Multiple Access) algorithm is one of the most common medium access sc h emes in to d a y’s net works, b oth wir ed (e.g. IEEE 802.3) and wireless (e.g. IEEE 802.11 ). H ow eve r, this a lgorithm is known to b e inh eren tly unfair, as illustr ated b y the t wo scenarios of Fig. 1. The first scenario relates to the do wnstream vs. upstream b andwidth sharing for a sin gle access p oint . In the presence of n activ e mob iles on the up stream, the access p oin t comp etes with n nod es for accessing the channel, resulting in a do wns tream to upstream band w idth ratio of 1 /n , indep end en tly on the n umb er of ac tive fl o ws on th e do wnstream. The s econd s cenario illustr ates the impact of in terference on bandwid th sharing. The cen ter access p oin t cannot transmit if one of the edge access p oin ts is activ e and th us gets m uch less transmission opp ortu nities. Moreo ver, th e r esulting b andwidth s h aring is in efficien t since the edge a ccess p oin ts can access the c hann el a lternately , prev en ting the cent er access p oint from send ing its traffic. Th us th e CS MA algo rith m is not able to fully utilize net wo rk capacit y , a statemen t that will b e made more pr ecise later in the pap er. (a) Do wnstream vs. up- stream (b) I nterfe rence Figure 1: Unfairness of standard CSMA. W e pr op ose a sligh t mo dification of the standard CSMA algorithm that consists in runn ing the alg orithm for ea c h flow instead o f eac h transmitter. In this pap er, w e refer to a flo w as Pr eprint submi tte d to Elsevier Novemb er 19, 2018 an y file tr an s fer from a source to a destination; it can typica lly b e identified through the u sual 5-uple: IP source and d estination addresses, source and destination p orts, proto col. F or a single access p oint, eac h flo w (either do wnstr eam or u pstream) ru ns the CSMA algorithm and thus gets the same band width share. The whole system can then b e viewed as a unique, ev enly shared wireless link. The focus o f the presen t pap er is rather on the second scenario where some links suffer from h igh inte rf er en ce. Sp ecifically , w e sh o w that the flo w-aw are CSMA algo rithm is optimal in the sense that it stabilizes the net w ork whenev er p ossib le. In the example of Fig. 1, the center access p oin t is likel y to access the channel when it has a high n umber of activ e flows; at the end of the corresp onding act ivity p erio d, the edge access p oint s can access the c hannel and will lik ely b e simultane ously activ e, wh ic h is a necessary condition for f ully utilizing net work capacit y . The main result of the pap er is to demonstrate that the fl o w-a wa re CSMA alg orithm is optimal f or any net w ork top olog y . W e consider a general mo d el consisting of an arbitrary n umb er of wireless links w hose m utual interference is represented by some co nfl ict graph. Flo w s of rand om size arrive at random at eac h link. In order to study the flow-l evel dynamics, w e calculate the throughp ut of eac h flow gran ted by the CSMA algorithm under the us u al time-scale separation assumption. W e then pro v e th at, pro vided there exists some sc hedu le of the links that stabilizes the net work, the flo w-a w are C SMA algorithm will do s o, in a p urely distributed and async h ronous w a y . The rest of the pap er is o rganized as follo ws. Related w ork is presen ted in the n ext section. W e th en present th e mo del and analyse it s sta bility under standard and fl o w-a wa re CSMA, resp ectiv ely . T he impact of net work load on th e mean throughpu t of eac h flo w u nder flo w-a ware CSMA is considered in Section 6. S ection 7 concludes the pap er. 2. Related w ork The p roblem of optimal b andwidth sharing in wireless net w orks has fi rst b een tac kled by T assiulas an d Ephremides, who s ho we d in [19] that the so-called maximal weight sc hedulin g p olicy , wh ic h activ ates a set of links that maximizes the total bac k log of activ e links, stabilizes an y net work whenev er p ossible. A num b er of distributed imp lemen tations of this p olicy ha v e then b een prop osed, all relying on some message p assing proto col b et w een n o des, see e.g. [11, 16]. S imple heuristics based on greedy algorithms that require limited or no message passing ha ve also b een studied, most selecting schedules of maximal size (in terms of n u m b er of links) in stead of maximal we ight and, as such, b eing sub optimal [4 , 6 , 8, 9, 13, 21]. A new appr oac h to optimal sc h eduling has r ecen tly b een prop osed b y J iang and W alrand, who in tro du ced in [7] a d istributed CS MA algorithm where at eac h link, the attempt rate is adapted to the arriv al rate and service r ate so as to meet the demand. The r esult is based o n a time-scale separation assumption whereby the activit y states of th e links, whic h dep end on the C SMA algorithm, evolv e m uch faster than the atte mpt rates o f the links. In practice, the algorithm u sed for adapting the attempt rates must b e carefully designed in order to guarant ee conv ergence and optimalit y [7, 14]. Similar pr oblems arise for those adaptiv e CS MA algorithms w here the attempt rates are functions of the q u eue lengths instead of s ome slowly v aryin g estimates of the arriv al rates and serv ice rates [12, 15]: the algorithm con v erges only for some sp ecific c h oices of these fun ctions. In all these pap ers, optimalit y is d efined either in terms of stabilit y , assuming exoge nou s random pac kets arriv als at eac h link, or in terms of utilit y maximization, cf. [7, 14]. The 2 flo w-lev el dynamics are not considered, w hereas they are ke y to u nderstandin g n et work p er- formance [18]. In particular, it can b e argued that the v ery n otion of c ongestion sh ou ld b e defined at the flow lev el [1]. In a recen t pap er, v an d e V en, Borst and Shneer ha ve shown that the maximal w eigh t sc hedu ling p olicy , whic h is kno wn to stabilize the net work at the pac ket lev el, may b e unable to stabilize the net work at the flo w lev el, whic h highligh ts the difference b et wee n th e t w o n otions of stabilit y [20 ]. The main con tribution of the present pap er is to pro vide an algorithm that sta bilizes the net wo rk at flow level whenever p ossible. Wi th this ob jectiv e in mind, it is v ery natur al to think of flo w-a ware CSMA. The fac t that it suffices for eac h flo w to run its own CS MA algorithm is far from obvious, h o w eve r. It is for instance w ell-kno wn that maximizing the total thr oughput of the net work at an y time may mak e the net wo rk u nstable at flo w lev el [3]. It turns out that th e fairness imp osed by th e pr op osed flo w-a ware CSMA is ind eed suffi cient to ac hiev e stabilit y . Sp ecifically , the fl o w-a w are CSMA a lgorithm selects eac h feasible sc hedu le in prop ortion to its weight , wh ere the w eight of a sc hedu le is the pr o duct of the n umb er of fl o ws on the corresp ondin g links. F or a large n umb er of flo w s, the selected sc hedu les are close to the corresp ondin g maximal weig ht sc hedule (with pro du ct w eigh ts in stead of add itiv e weigh ts), a p olicy that turns ou t to b e optimal. W e note that a similar prop erty is used by Ni, Bo and Srik an t in [12] for proving the stabilit y of queue-length b ased CSMA at pac ke t lev el. The constrain ts im p osed b y the pac k et lev el, like the ab o ve men tioned p r oblem of time- scale separation that restrict s the set of eligi b le w eigh t functions, mak e their a lgorithm v ery differen t from ours, ho wev er. Our mo d el is purely asynchronous and stateless, the num b er of activ e flo ws at eac h link b eing determined b y the pac k et headers in the corresp ondin g buffer; moreo ve r, the time-scal e separation assumption is ve ry n atural in our case since th e attempt rates are adapted at the flo w time-scale, whic h is t ypically m uch s low er than the pac k et time-scale. 3. Mo del Wir eless network. W e consider the general m o del describ ed in [7]. There a re K links in the net wo rk, where e ac h link is a n ordered transmitter-rec eive r pair. The net work is asso ciated with a conflict graph G = ( V , E ), where V is the set of ve rtices (eac h r epresen ting a link) and E is the set of edges (eac h repr esen ting a conflict). T w o links k, l can b e sim ultaneously activ e if and only if t hey do n ot conflict, that is if ( k, l ) 6∈ E . W e refer to a fe asible sche dule as any set of links S ⊂ V (p ossibly empt y) that do not conflict with eac h other. W e denote b y N the num b er of distinct feasible sc h edules and b y S i the set of activ e links in sc hedule i , for a ll i = 1 , . . . , N . By co nv ention, sc hedule 1 corresp onds to the sc h ed ule wh ere all no des are idle, that is S 1 = ∅ . Consider the netw ork of K = 3 links depicted by Fig. 2 for instance. Tw o links conflict if and only if the d istance b et ween the transm itter or r eceiv er of one li n k and the trans- mitter or receiv er of the other link is less than some fi xed thresh old. The conflict graph is linear and there are N = 5 feasible sc hedules, corresp ondin g to the s ets of activ e links ∅ , { 1 } , { 2 } , { 3 } , { 1 , 3 } . Cap acity r e gion. L et ϕ k b e the ph ysical rate of link k when sc hedu led, in bit/s. T he through- put of link k when eac h sc hedu le i is selected with probability p i , with P N i =1 p i = 1, is giv en b y: ∀ k = 1 , . . . , K, φ k = ϕ k X i : k ∈ S i p i . (1) 3 2 3 1 2 3 1 Figure 2: A 3-link net work and its conflict graph. Let φ b e the corresp onding ve ctor. W e refer to the c ap acity r e gion as the set of vecto rs φ generated b y all probabilit y measures p 1 , . . . , p N . Flow-level dynamics. Assum e that flo ws arriv e according to a Po isson pr o cess of inte ns ity λ k > 0 at link k a n d hav e exp onential flo w sizes of mean σ k > 0, in bits. W e denote by ρ k = λ k σ k the traffic intensit y at link k (in bit/s) and by ρ the corresp onding v ector. Let x k b e the num b er of activ e flo ws at link k . W e refer to the vecto r x as the net wo rk state. W e sh all consider random access algorithms that select eac h schedule i with some prob a- bilit y p i ( x ) that dep ends on the net w ork s tate x , with P N i =1 p i ( x ) = 1. Un der the time-scale separation assumption, the s c hedules c hange at a v ery high frequency compared to the flo w - lev el time-scale, so that the through p ut of link k in state x is giv en by: φ k ( x ) = ϕ k X i : k ∈ S i p i ( x ) . (2) The ev olution of the n etw ork s tate then defines a Marko v p ro cess X ( t ) with transition rates λ k from state x to state x + e k and µ k ( x ) = φ k ( x ) /σ k from state x to state x − e k (pro vided x k > 0), where e k denotes the K -dimensional unit v ector on component k . Stability c ondition. W e are in terested in the stabilit y of the net work in the sense of the p ositiv e recur rence of the Marko v pro cess X ( t ). A necessary condition is that the vec tor traffic in tensities ρ lies in the capacit y region. W e lo ok f or distributed access sc hemes th at stabilize the n et w ork whenever possib le, that is for all vecto rs of traffic in tensities ρ in th e in terior of the capacit y region. Suc h acce ss sc hemes are referred to as optimal . F or the sak e of complete n ess, w e first give an example showing the sub op timalit y of standard CSMA, that realizes some form of ma ximal size sc hed u ling. W e then pro ve the optimalit y of flo w-a ware CSMA. 4. Standard CSMA Algor ithm. W e first consid er a stand ard C S MA algorithm where eac h link w aits for a p erio d of random duration referred to as t h e b ackoff time b efore eac h transmission at tempt. If the radio c hannel is sensed idle (in the sense that no conflicting link is activ e), a pack et is transmitted; otherwise, the link w aits for a new back off time b efore the next attempt. Pa c kets ha v e r andom sizes o f mean θ k bits at link k and are transm itted at the physical r ate ϕ k ; the bac k off times are rand om with mean τ k at link k . W e denote b y α k = θ k / ( ϕ k τ k ) the ratio of mean pac ket transmission time to mean bac k off time at link k . 4 Equivalent sche duling. W e look for th e steady-state probab ility p i ( x ) that the set of activ e links corresp onds to sc hedule i in state x . W e assu me that, in state x , eac h link k such that x k > 0 tak es all opp ortunities offered by the C S MA algo rithm to transmit pac ket s; an y other link remains idle. If the pac k et sizes and the bac ko ff times had exponential distributions and there w ere n o conflict, the ev olution of the set of activ e links S would form a reversible Mark o v p ro cess. A stationary m easure of this Mark o v pro cess is giv en by 1 if S = ∅ and: Y k ∈ S α k 1 ( x k > 0) otherwise. By rev ersibilit y , the actual stationary measure induced by the conflict graph is the truncation of this measure to the s et of feasible sc hed u les. Sp ecifically , the w eigh t w i ( x ) of feasible sc h edule i in the stationary measure is giv en b y: w 1 ( x ) = 1 , w i ( x ) = Y k ∈ S i α k 1 ( x k > 0) for all i = 2 , . . . , N . W e deduce that sc hedu le i is selected in state x with probabilit y: p i ( x ) = w i ( x ) P N j =1 w j ( x ) . (3) By the ins ensitivit y prop ert y of the und erlying loss net work, th is is also the probabilit y that sc hedule i is selected in s tate x for arbitrary phase-t yp e distributions of p ac ket size s and bac k off times with the same means; su c h d istributions are kno wn to f orm a den se subset within the set of all distributions with real, n on-negativ e supp ort [2]. Sub optimality. W e p ro vide s im p le examples showing the su b optimalit y of the standard CSMA algorithm. W e consider unit ph ysical rat es, that is ϕ k = 1 for all links k . F or a single link, the optimal stabilit y condition is ρ 1 < 1. In view of (2) and (3), the throughput is given by: φ 1 ( x ) = α 1 1 + α 1 . W e deduce the actual stabilit y c ond ition: ρ 1 < α 1 1 + α 1 . This loss of efficiency is du e to the bac koff times, that m ust b e c hosen sufficien tly small to limit the o verhead of the CSMA algorithm. No w consider the example of Fig. 2 with K = 3 li nk s . The optimal stabilit y condition is giv en b y: ρ 1 + ρ 2 < 1 and ρ 2 + ρ 3 < 1 . Assume for simplicit y all links ha v e the same mean pac ke t sizes a nd mean bac koff times, so that α 1 = α 2 = α 3 = α for some α > 0. I n view of (2) and (3), the throughp ut of th e links in state x are giv en by: φ 1 ( x ) =      α 1+ α if x 2 = 0 , α 1+2 α if x 2 > 0 , x 3 = 0 , α + α 2 1+3 α + α 2 if x 2 > 0 , x 3 > 0 , 5 and φ 2 ( x ) =    α 1+ α if x 1 = 0 , x 3 = 0 , α 1+2 α if x 1 > 0 , x 3 = 0 , or x 1 = 0 , x 3 > 0 , α 1+3 α + α 2 if x 1 > 0 , x 3 > 0 . The throughput of link 3 follo ws by symm etry . As for a single link, the bac koff times must b e c hosen sufficientl y small to limit the o verhead of the algorithm. In the limit α → ∞ , we get: ( φ 1 ( x ) , φ 2 ( x ) , φ 3 ( x )) =        (1 , 0 , 1) if x 1 > 0 , x 3 > 0 , (1 / 2 , 1 / 2 , 0) if x 1 > 0 , x 2 > 0 , x 3 = 0 , (1 , 0 , 0) if x 1 > 0 , x 2 = 0 , x 3 = 0 , (0 , 1 , 0) if x 1 = 0 , x 2 > 0 , x 3 = 0 , (4) the other cases follo wing by sym m etry . Note that link 2 is n ot serv ed when b oth links 1 and 3 are activ e. Th is is due to the fact that link 2 is in conflict w ith b oth links 1 and 3 and th us cannot acce ss the c hann el for a n infi nitely small b ac k off time. Th is r esults in a sub optimal stabilit y regio n: Prop osition 1. The st ability r e gion is given by: ρ 1 < 1 + ρ 3 2 , ρ 3 < 1 + ρ 1 2 , ρ 2 < π 0 + π 1 , 3 2 , or ρ 1 < 1 + ρ 3 2 , 1 + ρ 1 2 ≤ ρ 3 < 1 + ρ 1 2 + 1 − ρ 1 2 π 2 , 1 , ρ 2 < 1 − ρ 1 2 , or ρ 3 < 1 + ρ 1 2 , 1 + ρ 3 2 ≤ ρ 1 < 1 + ρ 3 2 + 1 − ρ 3 2 π 2 , 3 , ρ 2 < 1 − ρ 3 2 , wher e π 0 , π 1 , 3 , π 2 , 1 and π 2 , 3 ar e t he r esp e ctive pr ob abilities that: • b oth link s 1 and 3 ar e id le when link 2 is alw ays active; • one of the links 1 or 3 is id le when link 2 is always active; • link 2 is id le given that link 1 is id le, w hen link 3 is always active; • link 2 is id le given that link 3 is id le, w hen link 1 is always active. Mor e pr e c isely, the Markov pr o c ess X ( t ) is p ositive r e curr ent if the v e ctor of tr affic intensities ρ lies in this r e gion and tr ansient if it l ies outside its closur e. The pr o of is giv en in the App end ix. Note that, when one of the lin k s is alw ays activ e, the t wo other links form a coupled system of t w o queues as considered b y F ay olle and Iasno- goro d ski [5]. In particular, the sta b ility region can b e calculat ed exactly . In the symmetric case ρ 1 = ρ 3 , the stabilit y condition red u ces to ρ 1 < 1 , ρ 2 < π 0 + π 1 , 3 / 2 . Fig. 3 sho ws that the corresp onding stabilit y reg ion for equal mean fl o w sizes. 6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Traffic intensity at link 2 Traffic intensity at link 1 Optimal Actual Figure 3: Stability cond ition for the netw ork of Fig. 1 u nder standard CSMA ( ρ 1 = ρ 3 ). 5. Flo w-aw a re C SMA Algor ithm. W e n o w consider the flo w-aw are CSMA algorithm wh ere eac h flow (instead of eac h link) wait s for a random bac koff time b efore eac h transmission attempt. If the r ad io c hannel is sensed id le (in the sense that n o conflicting link is activ e, nor any other flo w on the same link), a pac ket of this flo w is transmitted; otherw ise, the fl o w remains idle for a new random bac ko ff time b efore the n ext attempt. The bac k off times ha ve rand om d urations o f mean τ k for eac h act ive fl o w at link k . W e stil l denote by α k = θ k / ( ϕ k τ k ) the rat io of mean pac k et transm iss ion time to mean bac k off time at link k . Equivalent sche duling. Again, we look for the steady-state pr obabilit y p i ( x ) that the set of activ e lin ks corresp onds to sc hedu le i in state x . W e assume that all activ e flows take ea ch opp ortun ity offered by the CSMA algorithm to transmit pac k ets. If the pac ket sizes and th e bac k off times had exponential distributions and there w ere n o c onfl ict, the ev olution of the set of activ e links S wo uld aga in form a r ev ersible Ma rko v pro cess. S ince there a re x k flo ws attempting to access the c hannel at link k , a stationary measure of this Mark o v pro cess is giv en b y 1 if S = ∅ and: Y k ∈ S α k x k otherwise. By r ev ersibilit y , the actual stationary measur e ind uced by the conflict graph is the truncation of this mea sur e to the set of feasible sc h edules. The w eigh t w i ( x ) o f feasible sc hedule i in the s tationary measure is giv en b y: w 1 ( x ) = 1 , w i ( x ) = Y k ∈ S i α k x k for all i = 2 , . . . , N . Sc hedu le i is then selected with prob ab ility p i ( x ) giv en b y (3) in state x . By the insensitivit y prop erty of the un derlying loss net work, this probability remains the same for arb itrary phase- t yp e distrib utions of pac ket sizes and bac koff times w ith the same means, cf. [2]. Optimality. W e now giv e the main resu lt o f the paper, that demonstrates the optimalit y of the ab o v e flow-a w are C SMA alg orithm. 7 Theorem 1. The network is stable for al l ve ctors of tr affic intensities ρ in the i nterior of the c ap acity r e gion. Pr o of. W e apply F oster’s criterion. Sp ecifically , we lo ok for some Ly apun o v function F ( x ) suc h that the corresp onding drift, given by: ∆ F ( x ) = K X k =1 λ k ( F ( x + e k ) − F ( x )) + X k : x k > 0 µ k ( x )( F ( x − e k ) − F ( x )) , satisfies: ∆ F ( x ) ≤ − δ for some δ > 0, in all states x but some finite num b er. If the vecto r of t raffic i ntensities ρ lies in the i nterior of the capac ity region, there exists some ǫ > 0 and some pr obabilit y measure q 1 , . . . , q N on the set of feasible sc h edules suc h that q i > 0 for all i = 1 , . . . , N and: ∀ k = 1 , . . . , K, ρ k = (1 − 2 ǫ ) ϕ k X i : k ∈ S i q i . (5) Define: F ( x ) = X k : x k > 0 σ k ϕ k x k (log( α k x k ) − 1) . W e g et: ∆ F ( x ) = G ( x ) + X k : x k > 0 ρ k ϕ k ( x k + 1)(log (1 + 1 x k ) − 1) + X k : x k > 0 φ k ( x ) ϕ k ( x k − 1)(log (1 − 1 x k ) + 1) , (6) with: G ( x ) = X k : x k > 0 ρ k − φ k ( x ) ϕ k log( α k x k ) . Noting that, for any p robabilit y measure p 1 , . . . , p N on the s et of feasible schedules: X k : x k > 0 X i : k ∈ S i p i log( α k x k ) = N X i =1 p i log( w i ( x )) , w e get using (5): G ( x ) = − ǫ N X i =1 q i log( w i ( x )) + N X i =1 ( q i (1 − ǫ ) − p i ( x )) log ( w i ( x )) . W e then need the follo wing lemma. Lemma 1. L et: w ( x ) = max i =1 ,...,N w i ( x ) . Then, for al l sta tes x but some finite numb er, N X i =1 p i ( x ) log ( w i ( x )) ≥ (1 − ǫ ) log ( w ( x )) . 8 Pr o of. T he pro of is similar to that of [12, Prop osition 2]. Let: I ( x ) = n i = 1 , . . . , N : log ( w i ( x )) ≥ (1 − ǫ 2 ) log ( w ( x )) o . W e ha ve: N X i =1 p i ( x ) log ( w i ( x )) ≥ (1 − ǫ 2 ) log ( w ( x )) X i ∈ I ( x ) p i ( x ) . Moreo ver, X i 6∈ I ( x ) p i ( x ) = P i 6∈ I ( x ) w i ( x ) P N i =1 w i ( x ) , ≤ ( N − | I ( x ) | ) w ( x ) 1 − ǫ 2 w ( x ) , = N − | I ( x ) | w ( x ) ǫ 2 . Since w ( x ) tends to + ∞ when | x | = P K k =1 x k tends to + ∞ , this qu an tit y is less than ǫ/ 2 for all state s x but some finite n umber. W e deduce that in all states x but some fin ite n um b er: N X i =1 p i ( x ) log ( w i ( x )) ≥ (1 − ǫ 2 ) 2 log( w ( x )) ≥ (1 − ǫ ) log ( w ( x )) . ✷ In view of Lemma 1, w e ha v e for all states x b ut some finite num b er: G ( x ) ≤ − ǫ N X i =1 q i log( w i ( x )) + (1 − ǫ ) N X i =1 ( q i log( w i ( x )) − log( w ( x ))) . Since w i ( x ) ≤ w ( x ) for all states x , w e deduce that for all states x but some fi nite num b er: G ( x ) ≤ − ǫ N X i =1 q i log( w i ( x )) . Since q i > 0 for all i = 1 , . . . , N , this expression t end s to −∞ when | x | = P K k =1 x k tends to + ∞ . Th e other terms of ∆ F ( x ) in (6) b eing b ounded, we dedu ce that there exists δ > 0 suc h that ∆ F ( x ) ≤ − δ for all states x but some finite num b er. ✷ 6. Throughput p erformance This section is dev oted to the throughp ut p erformance of flow-a ware CSMA, und er the stabilit y condition. W e are interested in the me an thr oughput , defi n ed as the ratio of the mean flo w size to the mean flow duration. By Little’s la w, the mean throughput at link k is giv en b y: γ k = ρ k E[ x k ] . (7) W e c onsid er unit p h ysical rates, that is ϕ k = 1 for all links k . 9 Single link. W e fi rst analyse the impact of the mean bac k off time on the m ean throughput in the case of a single link. In the p resence of x 1 flo ws, the total throughput is giv en by: φ 1 ( x 1 ) = α 1 x 1 1 + α 1 x 1 . The num b er of flo ws then b eha ve s as the num b er of cus tomers in a p ro cessor-sharing qu eue with state- dep enden t service rate. Th e corresp onding s tationary d istribution is given by: π ( x 1 ) = π (0) x 1 Y n =1 ρ 1 φ 1 ( n ) , under the stabilit y condition ρ 1 < 1. T he m ean throughput then follo ws fr om (7 ). F or α 1 → ∞ , the throughput is constan t and equal to 1 and the mean throughput is giv en b y γ 1 = 1 − ρ 1 ; for α 1 = 1, the system corresp onds to a pro cessor-sharing queue with an additional p ermanen t customer repr esen ting the bac koff times and we ha v e γ 1 = (1 − ρ 1 ) / 2; in general, w e ha v e γ 1 → α 1 / (1 + α 1 ) w hen ρ 1 → 0. T hese results are illustrated by Fig. 4. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mean throughput Load Figure 4: Impact of the mean backo ff time on the mean throughput for a single link (ratio of mean pac ket transmission time to mean bac koff time α 1 = 0 . 1 , 1 , 10, from b ottom to top). Networks. In the follo wing, w e consider n et w ork scenarios and assu me that the mean bac ko ff time is the same for all flo ws and equal to the mean pac k et transmission time, so that α k = 1 for all links k . Flo ws ha v e unit mean flo w sizes. T he traffic in tensit y is the same on all links, equal to ρ 1 . W e refer to the net wo rk load as th e rati o of the p er-link traffic in tensity ρ 1 to its maxim um v alue, giv en by the stabilit y condition. Fig. 5 and 7 give the results obtained for the 3-link line of Fig. 2 and for the three 4-link n et w orks of Fig. 6, f or the s ame mean flo w sizes . The results are obtained b y th e simulati on of 10 7 jumps of the u nderlying Marko v pro cess, after a wa rm -up p eriod of 10 5 jumps. W e obs erv e that the throughput decreases from its maxim u m v alue 1 / 2 to 0 when the load gro ws from 0 t o 1; it is lo w er on links t hat are in conflict with many other links, just lik e in wired net works, the m ean th roughput is lo w er on long routes, where fl o ws go through man y links [3 ]. 10 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Mean throughput Load Edge links Center link Figure 5: Mean throughp ut in t h e 3-link line. 4 1 2 1 2 3 4 3 2 1 4 3 Figure 6: Conflict graphs of the square, the 4-link line and the 4-link star. 7. Conclusion The sta nd ard CS MA algorithm is inherently unfair and inefficien t. W e ha ve shown that the pr op osed flo w-a ware CSMA algorithm, where eac h flow (instead of eac h link) run s its o wn CSMA algorithm, is not only fair but efficien t, in the sense that the net work is stable whenev er p ossible. T o o ur knowledge , this is the first distribu ted algorithm th at i s pro v ably optimal in terms of flo w-lev el stabilit y . The consider ed pac k et-lev el m o del relies on a n u m b er of s im p lifying assum p tions that w e plan to relax in future w ork . These include the absence of co llisions and hidden no des. The in teraction with the usual bac k-off mec hanism of IEEE 80 2.11 sh ould a lso b e studied. One ma y also envisag e differen t implemen tations of the prop osed flow-a ware C SMA algorithm where the attempt rate of eac h link is equal to some increasing fun ction of th e num b er of flo ws and the transmission opp ortunities are shared in a f air wa y b etw een activ e flo ws, using a deficit roun d-robin sc h eduler for instance. F r om a more theoretical p ersp ectiv e, it w ould b e w orth relaxing the assump tion of exp o- nen tial flo w sizes and deriving b ounds or a pp ro ximations o n the throughput p erformance of the algorithm. 11 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Mean throughput Load Line (edge links) Line (center links) Square (a) Square and line 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Mean throughput Load Edge links Center link (b) S tar Figure 7: Mean throughput in 4- link netw orks. App endix Pro of of Prop osition 1 This example is similar to the one stud ied in [ 17, p274]. W e consider the fluid li mits of the Marko v pro cess X ( t ). Sp ecifically , we define X ( n ) ( t ) as the Marko v pro cess X ( t ) whose initial state is X ( n ) (0) = ( ⌊ β 1 n ⌋ , ⌊ β 2 n ⌋ , ⌊ β 3 n ⌋ ) for some non-n egativ e real n umb ers β 1 , β 2 , β 3 suc h that β 1 + β 2 + β 3 = 1. W e then define: ¯ X ( n ) ( t ) = 1 n X ( n ) ( nt ) . The flu id limits of the Mark ov pro cess X ( t ), if they exist, are the limiting p oin ts of this set of p ro cesses wh en n → + ∞ . It is easy to chec k that the Marko v p ro cess X ( t ) b elongs to the class ( C ) defined in [17, p241] and that the associated Prop osition 9.3 app lies. In particular, the set { ¯ X ( n ) ( t ) , n ∈ N } is tight an d the fluid limits are con tinuous. The Marko v pro cess X ( t ) is then positiv e recurren t if t h er e exist s some finite time a fter whic h all fluid limits are n ull, cf. [17, Theorem 9.7 , p 259]; it is tr ansien t if there exists some in itial state β 1 , β 2 , β 3 suc h that, 12 after some fi nite time, some comp onents of the fluid limits grow a t least linearly to infin it y [10]. W e first calculate the flu id limit un til the fir st time where one comp onen t r eac h es 0, if any , for all p ossible initial states. The three comp on ents of the pro cess X ( n ) ( t ) b ehav e as th ree coupled M / M / 1 queues, with arriv al rates λ 1 , λ 2 , λ 3 and state- d ep enden t service rates. W e denote by µ k = 1 /σ k the maximum service rate of queue k , so that ρ k = λ k /µ k . T he Mark o v pro cess is p ositiv e recurrent if all queues empty in finite time in the limit and transien t if, starting fr om some initial state, at least one queue grows linearly to infinit y after some finite time. W e start w ith the case β 1 > 0, β 2 > 0, β 3 > 0. The th ree queues are then m utu ally indep end en t, with resp ectiv e s er v ice rates µ 1 , 0 , µ 3 . The sca ling pr op ert y of the M / M / 1 queue sho ws that the pro cess ¯ X ( n ) ( t ) w eakly con verges to the fu nction: ( β 1 + ( λ 1 − µ 1 ) t, β 2 + λ 2 t, β 3 + ( λ 3 − µ 3 ) t ) , unt il one of the comp onents reac hes 0, if an y . W e n o w consider the case β 1 = 0, β 2 > 0, β 3 > 0. In view of (4), queue 1 h as service rate µ 1 and is empt y with probabilit y 1 − ρ 1 . Q ueues 2 and 3 ha v e service rates 0, µ 3 with probabilit y ρ 1 and µ 2 / 2, µ 3 / 2 with probabilit y 1 − ρ 1 . Prop osition 9. 14 of [17] applies and the pro cess ¯ X ( n ) ( t ) w eakly con verges to the fu nction: (0 , β 2 + ( λ 2 − µ 2 1 − ρ 1 2 ) t, β 3 + ( λ 3 − µ 3 1 + ρ 1 2 ) t ) , unt il one of the comp onents reac hes 0, if an y . Next, we consider th e case β 1 = β 2 = 0, β 3 > 0. In view of (4), queue 1 has service rate µ 1 . Q ueue 2 has service rate µ 2 / 2 if queue 1 is emp t y and 0 otherw ise. Th is queue is stable if ρ 2 < (1 − ρ 1 ) / 2, which w e assu me. Queue 2 then remains empt y in the limit, and the serv ice rate o f queue 3 is equal to µ 3 with pr obabilit y ρ 1 + (1 − ρ 1 ) π 2 , 1 and to µ 3 / 2 otherw ise. W e deduce that the p ro cess ¯ X ( n ) ( t ) w eakly con verges to the fu nction: (0 , 0 , β 3 + ( λ 3 − µ 3 ( ρ 1 + 1 2 − 1 − ρ 1 2 π 2 , 1 )) t ) , whenev er comp onent 3 is p ositiv e. Finally , w e consider the case β 1 = β 3 = 0, β 2 > 0. In view of (4), the s er v ice rates of queues 1 and 3 are equal to µ 1 and µ 3 when b oth are non-empty and to µ 1 / 2 and µ 3 / 2 otherwise. This system is stable if ρ 1 < (1 + ρ 3 ) / 2 and ρ 3 < (1 + ρ 1 ) / 2, whic h we assume. Queues 1 an d 3 then remain emp t y in the limit. The service rate o f queue 3 is equal to µ 2 with probabilit y π 0 and to µ 2 / 2 with probability π 1 , 3 . T he pro cess ¯ X ( n ) ( t ) w eakly con verges to the function: (0 , β 2 + ( λ 2 − µ 2 ( π 0 − π 1 , 3 2 )) t, 0) , whenev er comp onent 2 is p ositiv e. T o conclude th e pro of, we consider the ev olution of the fluid limit in the follo win g five cases (the others follo w by symmetry): 1. Assum e ρ 1 < (1 + ρ 3 ) / 2 and ρ 3 < (1 + ρ 1 ) / 2. Note that this implies ρ 1 < 1 and ρ 3 < 1. Queue 1 and 3 empty in finite time, indep endentl y of queue 2. Q u eue 2 then empties in finite time if ρ 2 < π 0 + π 1 , 3 / 2; it gro ws linearly to infinity if ρ 2 > π 0 + π 1 , 3 / 2. 13 2. Assum e ρ 1 < (1 + ρ 3 ) / 2 an d ρ 3 > (1 + ρ 1 ) / 2. If ρ 1 ≥ 1 then ρ 3 > 1 and queue 3 gro ws linearly to infinit y . W e no w assume ρ 1 < 1. If ρ 2 > (1 − ρ 1 ) / 2 then queue 2 gro ws linearly to infi nit y . If ρ 2 = (1 − ρ 1 ) / 2 then starting fr om a state where β 1 = 0, β 2 > 0 and β 3 > 0, queue 1 sta ys empt y , queue 2 is constan t and queue 3 gro ws lin early to infinity . W e assume that ρ 1 < 1 and ρ 2 < (1 − ρ 1 ) / 2. S tarting from the initial state β 1 = β 2 = 0, β 3 > 0, queue 3 gro ws linearly to infin it y if ρ 3 > (1 + ρ 1 ) / 2 + π 2 , 1 (1 − ρ 1 ) / 2. W e assu me that ρ 3 < (1 + ρ 1 ) / 2+ π 2 , 1 (1 − ρ 1 ) / 2. Starting f rom the initial state β 1 = β 2 = 0, β 3 > 0, queue 3 then empties in fi nite time. It remains to pro ve th at, starting from an y initi al state, queues 1 and 2 empt y in finite time. W e firs t note that, since ρ 1 < 1 and ρ 3 < 1, queue 1 or qu eue 3 empties in fin ite time. Moreo v er, if b oth queues 1 and 3 are empt y but not queue 2, then queue 3 gro ws linearly . T hus w e can assume that q u eue 1 empties b efore queue 3. W e kn o w that queue 2 empties in finite time in this case. 3. Assum e ρ 1 < (1 + ρ 3 ) / 2 an d ρ 3 = (1 + ρ 1 ) / 2. Note that ρ 1 < 1 and ρ 3 < 1 in this case. Moreo ver, w e hav e π 0 = 0 a n d π 1 , 3 = 1 − ρ 1 , so th at the inequalit y ρ 2 < π 0 + π 1 , 3 / 2 is equiv alen t to ρ 2 < (1 − ρ 1 ) / 2. If the latter is satisfied, then if queue 1 is non- empt y then qu eue 2 empties in fin ite time ind ep endently of queu e 3. W e just ha ve to c onsid er the case where β 1 = β 2 = 0 and β 3 > 0. Because ρ 3 = (1 + ρ 1 ) / 2 < (1 + ρ 1 ) / 2 + π 2 , 1 (1 − ρ 1 ) / 2, queue 3 emp ties in finite time. If ρ 2 > (1 − ρ 1 ) / 2, w e c ho ose an initial state such that queue 1 empties b efore 3. When queue 1 is empt y , queue 3 is constan t and queue 2 gro ws linearly to infinit y . 4. Assum e ρ 1 ≥ (1 + ρ 3 ) / 2 an d ρ 3 > (1 + ρ 1 ) / 2. Then ρ 1 > 1 and ρ 3 > 1 so that queues 1 and 3 gro w linearly to infinity . 5. Assum e ρ 1 = ( 1 + ρ 3 ) / 2 and ρ 3 = ( 1 + ρ 1 ) / 2. Then ρ 1 = ρ 3 = 1 a n d π 0 = π 1 , 3 = 0 . If ρ 2 = 0, the vec tor ρ lies on th e b oundary of the stabilit y region. If ρ 2 > 0, queu e 2 gro ws linearly to infinit y . References References [1] Be n F r edj, S., Bonald, T., Pr ou ti` ere, A., R´ egni ´ e, G., Rob erts, J. W., 2001. Statisti - cal bandw idth s h aring: a stud y of congestion at flo w lev el. In: Pro ceedings of A CM SIGCOMM. pp. 111 –122. [2] Bo nald, T., 2007. Insensitiv e traffic mo dels for comm u nication net works. Discrete Ev ent Dynamic Systems 17 (3), 405–421 . [3] Bo nald, T., Massouli ´ e, L., 2001. Impact of fairness on Inte rn et p erformance. In: Pro- ceedings of ACM S I GMETRICS/P erform an ce. pp. 82–9 1. [4] Di makis, A., W alrand, J., 2006. Suffi cient c onditions for stabilit y of longest-queue-first sc heduling: s econd-order prop erties usin g flu id limits. Adv. in Appl. Probab. 38 (2), 505–5 21. [5] F a y olle, G., Iasnogoro dski, R., 1979. Tw o coupled pr o cessors: The redu ction to a riemann-hilb ert problem. Probabilit y Theory and Related Fields 47 (3), 325–351. [6] Gupta, A., Lin, X., Srik ant, R., 2009. L ow-co mp lexit y d istributed sc hedu lin g algorithms for wireless net works. IEEE/A C M T rans. Net w. 17 (6), 1846– 1859. 14 [7] Jiang, L., W alrand, J., 2008. A distrib uted CSMA algorithm for throughput and utilit y maximization in wireless n et w orks. In : the 46th Annual Allerton C onference on Com- m unication, Con trol, and Computing. [8] Joo, C., Lin, X., Shroff, N. B., 2009. Un derstanding the capacit y region of the greedy maximal sc h eduling algorithm in m u ltihop wireless net works. IEEE/A CM T rans. Net w. 17 (4) , 1 132–1145 . [9] Lec onte , M., Ni, J., Sr ik ant , R., 2009. Imp ro v ed boun ds on the th r oughput efficiency of greedy maximal sc h ed uling in wir eless net works. In: MobiHo c’09. ACM, p p. 165 –174. [10] Meyn, S ., 1995. T r ansience of multicla ss queueing netw orks via fluid limit mo dels. Annals of Ap plied Probabilit y 5, 946–9 57. [11] Mo d iano, E., Shah, D., Z ussman, G., 2006. Maximizing throughput in wireless net w orks via gossiping. S IGMETRICS P erform. Ev al. Rev. 34 (1), 27–38. [12] Ni, J., T an, B., Srik an t, R., 2010 . Q -CSMA: Queue-length based CSMA/CA algorithms for ac h ieving maxim um throughput and lo w d ela y in wir eless n et w orks. In : IEEE INF O- COM. [13] Pr outi ` ere, A., Yi, Y., Chiang, M., 2008. Throughput of random access w ithout mes- sage passing. In: the 46th An n ual Allerton C onference on Comm un ication, Control, and Computing. [14] Pr outi ` ere, A., Yi, Y., Lan, T., Chiang, M., 2010. Resource allocation o ver net wo rk dynamics without timescale s eparation. In: IEE E INF OC OM. [15] Ra jagopalan, S., Shah, D., 2008. Distributed algorithm and rev ersible netw ork. In : CISS . pp. 49 8–502. [16] Ra jagopalan, S., Sh ah, D., Shin, J., 20 09. Net w ork adiabatic theorem: an efficien t ran- domized protocol for con ten tion r esolution. In: SIGMETRICS ’09. ACM, pp . 133–144 . [17] Rob ert, P ., 2003. Sto c hastic Ne tw orks and Queues. Sto c hastic Modeling and Applied Probabilit y Series. Sprin ger-V erlag, New Y ork. [18] Rob erts, J. W., Massouli ´ e, L., 2000 . Bandwidth sharing and admission con trol for elastic traffic. T elecomm un ication Systems 15, 18 5–201. [19] T assiulas, L., Ephremides, A., 1992. Stabilit y prop erties of constrained queueing sys- tems and sc hedu ling p olicies for maxim um throughp ut in m ultihop radio netw orks. IEEE T ransactions on Automati c C on trol 37, 1936– 1948. [20] v an de V en, P ., Borst, S., Shn eer, S., 2009. Instabilit y of MaxWeig ht sc heduling algo- rithms. In: IEEE I NF OCOM. [21] W u, X., S rik ant, R ., Perkins, J. R., 2007. Scheduling efficiency of d istributed greedy sc heduling al gorithms in wireless net works. I EEE T r ansactions on Mobile C omputing 6 (6), 595–605. 15