Optimizing the IEEE b Performance using Slow Congestion Window Decrease

Transcription

1 Optimizing the IEEE 802.b Performane using Slow Congestion Window Derease Jérôme Galtier Frane Teleom R&D 905 rue Albert Einstein F-0692 Sophia Antipolis edex, Frane Phone: , Fax: ABSTRACT We investigate a new proess for the derease of the IEEE 802.b Contention Window CW) alled CW derease, introdued by Ni et al. We give some mathematial analysis of the model that explains the experimental results previously obtained. The analysis allows us to refine the MAC optimization and reah the optimal asymptotial saturation throughput when the number of stations inreases. We also investigate other parameters suh as the average waiting time, that lead us to introdue another variant of the CW derease that we all additive - as opposed to the multipliative original sheme. We provide some experimental results based upon simulation to validate our analysis. Surprisingly our global goodput rate in the basi mode is kept from 7.4 Mbit/s with 5 stations to more than 7.3 Mbit/s with 00 stations, overperforming by 40% the original basi sheme and by 8% the sheme with the RTS/CTS enhanement. Keywords. Multiple aess, IEEE 802.b, MAC layer, DCF, CSMA/CA, Goodput. Introdution The 802.b norm has reeived in the past years a growing interest for its ability to offer relatively wide-band radio networking. Appliations over a large area of domains inluding omputer network wireless infrastrutures, and high speed Internet aess for rural areas. The underlying mehanism of the norm[] is a 2- layer protool whose first part relies on a derivative of the Binary Exponential Bakoff protool BEB). The system works as follows. Eah station stores a ontention window CW) size variable, whih is an integer W varying from CW min to CW max. Before transmitting, a station observes the hannel during a silent seurity period alled DIFS for Distributed Inter-Frame Spaing, 50µs). Then the station hooses a random number k in {0,..., W }. The station waits during k slots eah of 20µs). If the hannel is oupied during one of these slots, the station stops derementing k, waits until a new DIFS silent period is observed and then ontinues derementing k if new slots are empty. One the k empty slots have been waited, the station sends its paket. The only way to know whether the paket was reeived is the reeption of an ACK, that is sent after a SIFS for Short Inter-Frame Spaing, 0µs) period by the reeiving station. If a transmission is unsuessful i.e. no ACK is reeived and a DIFS period is observed immediately after emission), the CW size W of the sender is multiplied by a onstant fator, denoted as P F the persistene fator, denoted as η throughout this paper), up to CW max. In the 802.b variant of the norm, we have P F = 2, but other fators are onsidered in the 802.e release. In all the variants, in ase of suess, the norm proposes to reset W to CW min, whih is highly aggressive to the hannel, espeially if it is lose to ongestion. Therefore an alternative approah [] is then to set: W := maxcw min, RF W ), where RF is an appropriate redution fator. This is the basi mehanism that we study in this paper. A typial value is RF = /P F but other alternative features will be onsidered. An alternative mehanism we will onsider is to add a onstant value ω to W in ase of failure, and withdraw it in ase of suess, while staying within [CW min, CW max ]. At this point we need to say that an additional mehanism, alled RTS/CTS for Request- To-Send/Clear-To-Send), is inluded in the IEEE 802. families of norms. With this feature, the

2 sender first emits a small RTS paket to warn that it will send a paket. If another station attempts to emit at the same time, noone an hear any of the messages, and the hannel is free after a DIFS period. Otherwise, the reeiving station sends a CTS after a SIFS period, and all the stations let the sender emit its paket till ompletion. Therefore, while onsuming a share of the bandwidth, this mehanism allows to pay a lower prie for the ollisions. We need to stress that the lower the paket size is, the less effiient it is, whih makes it inappropriate for real-time voie appliations for instane. Already many researh work has been done to model the 802. window derease proess. Strong simplifying assumptions are at the basis of some models [6], while others fous on an individual station while onsidering that the effet of the others on the hannel an be represented by an oupany probability p see [3, 3, 4]), following an earlier popular approah on CSMA [0, 2]. In fat, a losed loop effet naturally takes plae in the 802. features. Coarsely speaking, when more ollisions our, the CW size in eah station inreases, so that the ollisions are less frequent. It turns out, as derived for instane in [3], that this norm, however, is not effiient when the number of stations inreases. As more and more stations aess the system, the time spent to ollisions and auto-regulation of the ontention window inreases drastially, to the point that the total amount of goodput tends to zero as the number of stations grows, resulting in a very redued bandwidth offered to eah individual. The main reason for that is that no memory is kept on the number of stations present in the system, and as soon as a station emits one paket suessfully, it forgets any information on the past and aesses again the hannel with a probability that do not onsider the number of stations present in the system. As a result, a number of ollisions is neessary to regulate the aess probability, whih wastes the bandwidth for this simple purpose. A key point in that diffiulty is to know or not at any point of time the number of stations that are ative, i.e., urrently transmitting. Of ourse, some simple paradigm would onsist to handle the problem in a entralized way suh as done in [5], but then the approah looses preisely the flexibility brought by the DCF mehanism in the IEEE 802. family of norms as opposed to the PCF mehanism that direts the transmissions also in a entralized basis). A series of works have addressed the problem of estimating the number of ative stations. The seminal work an be found in [4] and [6]. In [4] the assumption is first made that all the stations are transmitting with the same CW size W, and an estimate is then done based on the number of observed oupations B) during an observation period of B slot times. The number of stations n is then estimated by E[B)]W + ) n +, 2B and in return, the size of the window W is fixed aording to n. However, some instability problems oblige the authors to introdue some triky funtions that make the protool more omplex to implement and to analyze. The question beomes even more diffiult when the number of ative stations varies. The alternative approah of [6] onsists in observing the total amount of idle slots in a transmission time and setting: n T otal Idle p p = W B B), using the above notation, and setting B equal to one transmission time. Here again, some smoothing fators are neessary to make the protool stable. The authors argue that the apaity is globally preserved but the resulting protool implementation is again quite omplex. Not only those approahes are somehow more ompliated, but performane parameters beome terribly diffiult to analyze. For instane, it turns out that most of these protools - if not all - seem to solve the bandwidth sharing problem by offering almost all of the apaity to one station for some possibly long) periods of time. Some authors have defended this behavior as positive and proposed to implement it in the protool [3]. Others view it as a negative side-effet that should be avoided [8]. In any ases, it is desirable to have a protool that an be analyzed and tuned orretly to that respet. In partiular, as will be shown in this paper, it turns out that for some versions of the protool, a station an stay waiting for an infinite amount of time - whih should be avoided. Another more onfusing aspet addresses the fairness aspets of the protool. Indeed, as a station gets further from the others, its transmission rate dereases from Mbit/s to 2 Mbit/s while the remaining stations maintain their respetive rates from one to eah other. As observed by [9], the atual IEEE 802. norm solves the problem on a Max-Min basis, that is, gives the same throughput to all the stations, regardless of the fat they an transmit to a lot more than 2 Mbit/s. We notie, by the way, that all the above approahes will have the same behavior sine they do not onsider the effetive rate of transmission or the transmission time of a paket) of the stations. Without addressing primarily this problem in the present paper we will only provide individual average waiting times of the stations) we stress the importane of having a protool that an be analyzed properly.

3 To that respet, the slow CW derease approah is really attrative sine the atual size of the windows keeps an impliit memory of the stations present in the system, avoiding any ompliated estimation proess. One an verify - either on simulations and on a mathematial analysis - that the throughput is effiiently distributed and tune the protool to address the points mentioned above. On top of that, no more information on the system is required than the suess or the failure of eah individual attempt of the station to aess the hannel, whih fits exatly to the IEEE 802. norm assumptions. The paper is organized as follows. The two next setions are devoted to the mathematial analysis of the protools, with a partiular emphasis on the goodput and waiting time when the number of stations inreases. The last one applies the theory into some onrete simulation senarios - applying restriting parameters speified by the norm. 2 The multipliative slow CW derease model Following [], we represent the ontention window derease sheme by a Markov hain model where eah state is indexed by two integers i, j). The first integer i represents the stage of the proess a lass of fixed ontention window size W i ) and j the number of slots to wait before transmission. An additional parameter, g = logrf )/ logp F ) reall η = P F ), allows to regulate the derease of the ontention window size. We set W 0 = CW min and W m = CW max. We also add the orretion of [4] on the fat that a station derements its bakoff ounter only when no paket is heard. We use for this phenomenon the variable. Note that results following the model of [] are obtained for =, while the real ase is addressed by = p. The reader an also onsider that oming from one system of formulas to the other is equivalent to inrease the header of a paket by one slot of 20µs.) The transition probabilities are then given as follows: P [i,k i,k+]= P [i,k i,k]= P [0,k i,0]= p)/w 0 P [i g,k i,0]= p)/w i g P [i,k i,0]=p/w i P [m,k m,0]=p/w m k {0,...,W i 2}, i {0,...,m}, k {,...,W i }, i {0,...,m}, k {0,...,W 0 }, i {0,...,g }, k {0,...,W i g }, i {g,...,m}, k {0,...,W i }, i {,...,m}, k {0,...,W m }. We plot this proess in Fig. in the ase g = - whih will be the most important one studied in this paper. Let {st), bt)} be a bi-dimensional stohasti proess that follows these laws. Markovian properties show that, in this ase, the proess onverges almost surely to a stationary distribution, given by π i,k = lim t P [st) = i, bt) = k], i {0,...,m} k {0,...,W i }. The analysis allows to derive the following formulas: π = ) g 0,0 π i,0 = j=0 j,0, pπ i,0 + )π i+g,0 for 0 < i m g, π i,0 = pπ i,0 for m g < i < m, pπ m,0 = )π m,0. 2. Closed expressions We derive some lose expressions for small values of g. If we set g =, then we have: π m,0 = p p π m,0 π i,0 = pπ i,0 + )π i+,0, 0 < i < m. ) So we write π m j,0 = ar j + brj 2, for a, b IR, where r and r 2 are the roots of the polynomial We then derive: px 2 X + ). Result For g =, the distribution π, follows the equation : ) j π m j,0 = π m,0, 0 j m. 2) p It is then not diffiult to see that, for all i {0,..., m}, k=w i k=0 π i,k = + W ) i π i,0. 2 ) and from that value we an derive the value of π m,0, given that the ongestion window size is given by W i = η i W 0 for some number η >. Result 2 For g =, we have: 2 p m ) π m,0 =. 2 2p p)m+ p m+ p) ) + W m+ ηp) m+ 0 η+)p 3)

4 A losed expression for the probability τ that the terminal attempts to send a paket an be derived: Result 3 For g =, the probability that a station transmits in a randomly hosen slot time is given by: τ = 2 + W 0 2 ) p) m+ ηp) m+. 4) p) m+ p m+ 2p η+)p Proof. We simply derive the formula from τ = m j=0 π j,0. A similar analysis an be done for all the values of g. The losed expression an be expeted till g = 4, after what several mathematial diffiulties an our, as stated by the Galois theorem. Anyway, the analysis is still valid after that value via a numerial extration of the roots of a polynomial. For instane, for g = 2, the harateristi polynomial is given by: whih leads to: px 3 X 2 + ), Result 4 For g = 2, the distribution π, is given by : π m j,0 [ /π m,0 = p 3p+ ) p+ +3p) p) 2p ) + p p +3p) p) 3p+ 2p 2.2 Asymptotial behavior ) j ) j ]. 5) In the remaining of this paper, we will study more preisely the ase g =. As our main interest in the slow derease proess is the memory aspets of it, we an onsider the ase m. The asymptotial funtion an be rewritten as: 2 ) τ = 2 + W 0 2p η+)p for p < η +, 6) and 0 otherwise. We plot the τ funtion in Fig. 2. We note that the value for p = 0 does not depend on η, but the funtion abruptly reahes 0 when p = /η + ). Following the analysis of [3], let n the number of ative stations in the hannel, we seek for the intersetion point of one of 4) or 6) with the funtion giving the probability that the hannel is free, given that all the remaining n stations have the same probability of aess τ, and all the aesses are independent proesses whih is obviously an approximation): or, with respet to τ p = τ) n 7) τ = ) n. 8) Result 5 If is a non-dereasing funtion of p, the funtions 8) and 4) have a unique intersetion point. We note, in partiular, that both the ases = p and = are handled by the theorem above. Proof. Obviously the funtion 8) is ontinuous, stritly inreasing on [0, ] from 0 to. The funtion 4) an be rewritten as follows: τ = + 2 ) + + W p) pη p) p )i + + p) pη )m ) i + + p) ). p ) m In order to show that the intersetion point exists and is unique, it suffiient to show that τ is noninreasing. Sine is non-dereasing with p and W 0, it is suffiient to show that the funtion x + +ηx)i + +ηx) m + +x i + +x is greater than and nondereasing as long as m η. Then use the following notations: f k x) = j=m j=k+ i=k x j a k g k x) = x i a i i=0 and notiing that g k x) + af k x) = g k+ x) + f k+ x), we have + + ηx) i + + ηx) m m + + x i + + x m = k=0 g k x) + af k x) g k x) + f k x). It is then suffiient to show that eah member of the right-hand produt is greater than and inreasing. We an easily see it is greater than. We note also that the derivate has the sign of η )f k x)g kx) g k x)f kx)). Reall that η and f k x)g kx) g k x)f kx)) j=m i=k = jx j a k+ + 0, j=k+ j=m i= j=k+ j i)a k+ x i+j and therefore the funtion is non-dereasing. We leave to the reader the proof of the following result:

5 Result 6 The funtions 8) and 6) have a unique intersetion point. From that on, we haraterize the intersetion point p, τ ) when the number n of stations beomes large. From equation 6) we see that the intersetion point p will verify p < / + η). Substituting in 8) we have ) η τ n < 0, η + when n beomes large. As a result, Result 7 Under our model s assumptions i.e. supposing that the stations proesses to aess the hannel are independent), and with an infinite maximum window size, the onvergene point verifies p / + η) when n beomes large, and τ = ) ) η + n ln + o. 9) η n This result is extremely important from the theoretial point of view. It shows that naturally the proess tends to share the bandwidth between the proessors. We show in the next setion that this indeed governs the saturation throughput when the number of stations beomes large. 2.3 Saturation throughput Following the analysis of many authors, suh as [0, 3], we an study a renewal proess drawn in Fig. 3, that an be desribed as follows: the W state is when the system has an idle slot wait), the Q state is when the system has had at least one query, but it is not known if a onflit has ourred query), the P state is when the system suessfully transmits a paket payload). Theoretially speaking, it is a semi-markov proess, and the disrete) transition probabilities are given by the arrows. Noting π w, π q and π p the stationary probabilities of the orresponding disrete Markov hain π w + π q + π p = ), we have the following transition equations: ] π w = π p + τ) n π w + [ nτ τ)n τ) π n p, π q = [ [ τ) n ]] π w, π p = nτ τ) n τ) π n q. 0) Therefore, if we note µ w, µ q and µ p the expeted time spent in a state before a transition during the proess, the proportion of time spent in the stage P noted T hr as it is the effetive saturation throughput of the hannel) will be given by for more details see for instane [2, pp ]): T hr = µ p π p µ w π w + µ q π q + µ p π p ) Solving 0) and substituting in ) we obtain: T hr = 2) + µw+µq µq τ)n µ pnτ τ) n The optimal τ + for the throughput is the one that minimizes the denominator and after some manipulation, we an see that if it exists it verifies: µ w + µ q µ q τ + ) n nτ + µ w + µ q ) = 0 3) From that we an derive that τ + = ) µ q τ + ) n n µ w + µ q n, and τ + = K n + o n ) for some K 0 and the K onstant verifies: K)e K = µ q µ w + µ q. The solution an be found quite easily by introduing the Lambert s w funtion or the Omega funtion) satisfying wz)e wz) = z and that an be developed as wx) = p) p x p. p! p Then the optimal τ + an be approximated by ) + w µ q τ + e µ w+µ q = + o n n ) 4) where e = exp)) and the orresponding saturation throughput tends to T hr + µ q /µ p = w e µ q µ w+µ q ) ). 5) Making the onnetion between the equations 9) and 4), we obtain:

6 )/W 0 0, 0 0, 0, 2 0, W 0 2 0, W 0 p/w )/W 0, 0,, 2, W 2, W p/w 2 )/W 2, 0 2, 2, 2 2, W 2 2 2, W 2 p/w 3 )/W 2 p/w i )/W i i, 0 i, i, 2 i, W i 2 i, W i p/w i+ )/W i p/w m )/W m m, 0 m, m, 2 m, W m 2 m, W m p/w m Figure : The Markov hain in the ase g =. tau eta=.2 eta=.4 eta=.6 eta=.8 eta=2.0 eta=2.5 eta=3.0 eta=4.0 eta=5.0 W τ) n nτ τ)n τ) n Q nτ τ) n τ) n P p Figure 2: Different values of τ W 0 = 32, = p). τ) n Figure 3: General figure of the slotted CSMA/CA proess.

7 Result 8 Under our model s assumptions i.e. supposing that the stations proesses to aess the hannel are independent), the optimal saturation throughput of the hannel will be reahed automatially in aordane with the number of stations by setting along with m =. η = µq µ w+µ q ) w µ q e µ w+µ q 6) At first glane, one an be surprised that W 0 plays no role to reah the asymptotial saturation throughput. This fat is not so strange beause as the number of stations inreases, the proess rebalanes the weights in higher states than W 0 and reahes this equilibrium. In Fig. 5, we plot the funtion x / x/w x/e) ) that governs the value of η in funtion of µ q /µ w + µ q ). We see that the ondition η > turns to translate to µ q /µ w + µ q ) > , a ondition that is easily full-filed, sine the query time implies that an aknowledgment is sent, and therefore µ q > 2µ w. Anyway, we show in the next paragraph that this mehanism has nevertheless some signifiant drawbaks that make its use signifiantly sensitive. 2.4 Asymptotial waiting time An important parameter to study is the asymptotial waiting time when the number of station inreases. Naturally, it is lear that the more stations are present in the system, the less probability there is that a station will aess the system. We show, however, that the asymptotial behavior of the present mehanism is muh worse. Result 9 Suppose a station follows a slow CW derease mehanism with m =, and observes a ollision probability of p / + η 2 ). Then the average number of waited timeslots before emission is infinite. If p < / + η 2 ), the average number of waited timeslots is given by: E[W ] = 6 ) η 2 +)p 2p 2 2p + W p 0 η+)p. 7) Else if m <, then the number of waited timeslots is given by: p) m+ η 2 p) m+ E[W ]= p η 2 p)m+ p m+ ) +)p 2p 6 2p 2p p)m+ p m+ p) )+W m+ ηp) m+. 0 η+)p 8) Equation 7) shows that the waiting time tends to infinity when p / + η 2 ). As a result, when the number of stations inreases, p grows from 0 to / + η), leaving E[W ] infinite. This result learly shows a major drawbak of the slow CW derease mehanism, sine there is then no guarantee that a paket will be emitted within a short amount of time. The surprising point with this result is that sine /η 2 + ) > /η + ), a finite number of stations is suffiient to reah this behavior. Proof. The expeted waited time before leaving a state i, k), k 0, is given by i + )p i ) = / ), i=0 and therefore the general expeted waited time is i=m E[W ] = k= i=0 k=w i k=0 kπ i,k / ). 9) Given that for k, π i,k = π i,0 W i k)/w i / ) and k=i k 2 = 2i3 3i 2 + i, 6 and assoiating to equation 2) we obtain: E[W ] = i=m i=0 π 0,0 p ) 2 ) i [ W 2 i 6 ]. 20) We set W i = η i W 0. We note that for p /η 2 + ); /η + )), the series does not onverge when m. Also, reintroduing 3), we obtain 8) after some manipulation. For p < /η 2 +), sine the series is bounded in absolute value, letting m gives 7). Hene the result. This last result advoates for a mehanism that will respet waiting times more fairly. In the next setion we present an alternative mehanism that keeps the good properties of the slow derease paradigm while avoiding the presented drawbaks. 3 An additive inrease/derease model In this setion, we simply set W i = W 0 +ω i instead of W i = η i W 0 ) and show the behavior obtained by this assumption. We take here the opposite ourse to some fashion that was introdued by the MIMD mehanisms that work well indeed under some speifi onditions [7]. We argue that, however, in our ase the onditions are somehow different. Indeed

8 the maximum desirable CW size is a ritial parameter sine a too high value will indue huge delays for some of the terminals, and a restrited one will erase the memory effets of the mehanism, whih is what we try to stimulate in the present study. In that ontext, lowering the aess to high values of CW seems to be reasonable. 3. The model studied We slightly hange the previous model as follows. We replae the previous parameter p by two parameters p and q, and study the following variant of the proess. When the proess is in a state i, 0), it goes with probability p to the upper stage, with the probability q on the lower stage, and stays in the same stage with probability q). The evolution equations an be rewritten as follows with g = ): P [i, k i, k] = P [i, k i, k + ] = P [0, k 0, 0] = )/W 0 P [i, k i, 0] = q/w i P [i, k i, 0] = p/w i P [i, k i, 0] = q)/w i k {,...,W i }, i {0,...,m}, k {0,...,W i 2}, i {0,...,m}, k {0,...,W 0 }, k {0,...,W i }, k {0,...,W i }, i {,...,m}, k {0,...,W i }, i {,...,m }, P [m, k m, 0] = q)/w m k {0,...,W m }. 2) We plot this proess in Fig. 4. Let {st), bt)} be a bi-dimensional stohasti proess that follows these laws. Again, markovian properties show that the proess onverges almost surely to a stationary distribution, given by π i,k = lim t P [st) = i, bt) = k], i {0,...,m} And similar formulas an be derived: pπ 0,0 = qπ,0, p + q)π i,0 = pπ i,0 + qπ i+,0 for 0 < i < m, qπ m,0 = pπ m,0. We then derive: k {0,...,W i }. 22) Result 0 The distribution π, follows the equation : ) j p π j,0 = π 0,0, 0 j m. 23) q Then the basi state is obtained as: Result We have: π m,0 = [+ W 0 2 p))q p)+ 2 p)]q pw m+ p m q p) 2 Proof. i=m = [+ W 0+mω 2 p) )q p)+ qω 2 p)]p m+ p m q p) 2. 24) Similarly as before, we have i=0 + W 0 + iω 2 ) ) ) m i q π m,0. p And the result omes from the simple fat that the reader an hek by indution) that: i=m ix i = x m + )xm+ + mx m+2 x) 2. i= And onsequently we have the similar result: Result 2 If we denote as before τ the probability that a terminal will aess the hannel, given that it observes a probability p of failure, we have: τ = + W0 2 ) + ω p 2 ) q p. m+) p q ) m +m p q ) m+ p q ) m+ 3.2 Optimal saturation throughput 25) If then we tend to an asymptotial result, i. e. m =, we have for p < q: τ = + W0 2 p + ) ω p 2 ) q p. 26) In Fig. 6 we draw various urves for τ depending on ω. As expeted, τ tends to zero as long as p = q. The exat value of ω is a ritial parameter for the dynamiity of the proess. A too small value will lead to a slow onvergene of the rates and therefore many additional ollisions before regulating the value). A too high one will generate for at least some of the proesses exessively high levels of CW, and therefore high levels of delay. The idea for the use of q is to turn it into an ation in ase of suess. To that purpose, we an write q = p) δ), where δ [0; ] is a parameter that drives between the ation of dereasing the window or leaving it at the same size. The ondition p = q turns to beome p = δ)/2 δ). Given that, we look for an asymptotial intersetion point

9 )/W 0 0, 0 0, 0, 2 0, W 0 2 0, W 0 p/w q/w 0 q)/w, 0,, 2, W 2, W p/w 2 p/w i q)/w i i, 0 i, i, 2 i, W i 2 i, W i p/w i+ q/w i p/w m q/w m m, 0 m, m, 2 m, W m 2 m, W m q)/w m p Figure 4: The Markov hain for the additive model. /-x/w-x/e)-) w=4 w=8 w=6 w=32 w=64 w=28 w= Figure 5: Optimal urve for η in funtion of µ q /µ w + µ q ). x Figure 6: Different aess probabilities for W 0 = 32, = p, and q =.

10 in p, τ) with equation 8), where n is the number of terminals. Similarly as before, when n beomes large, we obtain p δ)/2 δ) and τ = ) n ln2 δ) + o. 27) n At that point we see that the value of q direts the aess probability to the hannel. Then the optimal mode an be reahed asymptotially as explained in the following. Result 3 Under an additive model to add/derease the ongestion window, and supposing the stations proesses to aess the hannel are independent, then the optimal saturation throughput an be obtained by flipping a oin that will transform an inrease ation into a stationary ation with probability: δ = 2 + w e µ q µ w+µ q ) 28) µ q µ w+µ q The orresponding saturation throughput is again given by the equation 5). 3.3 Asymptotial waiting time We here give some onsiderations on the waiting time that prove the orretness of the additive slow CW derease mehanism. In order to simplify the notations, we swith bak to the notations with q and give some analytial evidene on the properties in question. Result 4 A terminal following an additive slow CW derease Markov proess observing a probability p < q of ollision and with an infinite number m of states experienes an average waiting number of slots of 2 pq3p q) E[W ] = qq p)w ) + 2W 0 ωpq + ω q p ) ]. 3 ) [q 2 p) 2 + W0 + ωp As a result, when p 2, we have E[W ] ω 3 29) q p. 30) This result is really important sine it shows that asymptotially, the waiting times behaves like the inverse of q p, whih grows like the number of stations. Therefore, the bandwidth is not only effiiently shared, but eah of the stations will reeive on average a even part of it and will observe the same average waiting time. Proof. short) Summing up the waited slots by stages of the Markov proess, we obtain similarly as before: E[W ] = π i=m ) i 0,0 p Wi 2 ) 2 3) q 6 i=0 The essential ingredients are manipulations similar to those previously done. A useful formula that is neessary for the proof is simply: i=m i 2 x i = x+3x2 +m+) 2 x m+ x) 3 i= 2m2 +6m+3)x m+2 +mm+4)x m+3 x) 3. In the next setion we give some experimental evidene on the auray of these models. 4 A simulated slow CW derease mehanism ompared to the analysis We have implemented a small simulator of the 802.b mehanism, that takes into aount all the details of the norm. The hannel bit rate is set to Mbit/s, the paket payload to 500 bytes, short headers are used, and the propagation delay is negleted. Eah station is managed individually and aesses to the hannel in a slotted fashion with respet to its own ounter. We ompare in the test three algorithms: The basi 80.b sheme with P F = 2, RF = 0, CW min = 32, and CW max = 024. In ase of failure of transmission in state CW max, we hoose to stay in this stage. A slow CW derease multipliative mehanism with η = P F = /RF i.e. g = ), CW min = 32, and CW max = 024. Aording to the previous analysis, and based upon a rate of Mbit/s and an average paket size of 500 bytes, we set η = 5.5 on the basi mode, and η = 2 with RTS/CTS. A slow CW derease additive mehanism with ω = 32, CW min = 32, and CW max = 024. Aording to the previous analysis, and based upon a rate of Mbit/s and an average paket size of 500 bytes, we set δ = on the basi mode, and δ = with RTS/CTS. In ase of suess, with probability δ, we do not hange the CW size, and with probability δ, we derease it by ω. In ase of failure, CW is inreased by ω.

11 8e e+06 IEEE 802.b sustained IEEE 802.b global Slow CW derease multipliative sustained Slow CW derease multipliative global Slow CW derease additive sustained Slow CW derease additive global 7e e+06 IEEE 802.b sustained IEEE 802.b global Slow CW derease multipliative sustained Slow CW derease multipliative global Slow CW derease additive sustained Slow CW derease additive global 7e+06 Goodput bit/s) 6.5e+06 Goodput bit/s) 6.9e e+06 6e e e+06 5e Number of stations 6.75e Number of stations Figure 7: Simulation of the obtained rates with the basi mehanism. Figure 8: Simulation of the obtained rates with the RTS/CTS mehanism. Given the relatively small value of CW max, it is not realisti to present experiments with more than 00 stations. However, further experiments with inreased CW max showed that our approah is also satisfatory for several hundreds of stations. In the first experiment, we give to eah of the n stations a volume of 00 Mbytes/n to transmit. We let them the simultaneous aess to the hannel, and we ompute the total volume of goodput divided by the time neessary to omplete all the transmissions - we all that number the global rate. We also ompute the total number of pakets that were sent during a given period after stabilization of the proesses, that is between t = 50s and t = 00s - we all the obtained rate the sustained rate. The differene between these two parameters indiates the speed of onvergene of the proess. We plot the results in Figs. 7 and 8. For the basi mehanism Fig. 7), the gains obtained with the additive mehanism are of more than 40% for 00 stations ompared to the basi mehanism, and 3% with the RTS/CTS mehanism while losing % for small number of stations). We note that the obtained rate without RTS/CTS is on top of 8% of the norm using RTS/CTS. This shows that even if RTS/CTS uts the optimization, the improvement is suh that the basi mehanism without RTS/CTS beomes more effiient. In this ontext, only the hidden terminal problem justifies the use of RTS/CTS. In the seond experiment, we start with a benh of 00 stations having eah Mbyte to transmit. Then at time t = 50s a new series of 0 stations numbered from 00 to 09) arrive with a volume of 00 kbytes to transmit. We plot the time of ompletion of the proesses for the different stations in Figs. 9 and 0. We observe - generally speaking - that the IEEE 802. standard and the slow CW derease multipliative mehanism experiene a larger deviation than that of the additive mehanism. However, we notie that the slow CW derease additive mehanism observes a larger deviation for the first 00 stations with the RTS/CTS mehanism. This and Fig. 8 showing some larger differene between the global and the sustained rates give some insight that the proess does not onverge very quikly to the asymptotial behavior. This is probably due to the introdution of the q parameter that lowers the impat of suess and failure rates in the ongestion window evolution. It suggests that if this problem beomes ritial, more information on the hannel should be integrated in the proess. 5 Conlusion In this abstrat, we have derived exat formulas for small values of g for the slow ongestion window derease 802. WLAN mehanism, with multipliative and additive evolution mehanisms. Our analysis has shown that both an naturally reah the asymptotial optimal saturation throughput of the hannel regardless of the number of stations, but the multipliative mehanism an introdue infinite waiting times when CW max is infinite and the number of stations beomes large. Therefore we reommend the use of an additive mehanism that distributes more evenly the hannel, and has finite waiting times. Experiene has shown that even with limited CW max those two protools behave better than the IEEE 802.b one, to the goodput point of view and waiting time one. The additive mehanism keeps remarkably well the optimal saturation throughput of the hannel when the number of stations inreases. Many questions remain open. Spae is laking

12 60 IEEE 802.b Slow CW derease multipliative Slow CW derease additive 40 IEEE 802.b Slow CW derease multipliative Slow CW derease additive Arrival time s) Arrival time s) Station index Figure 9: Completion times for the stations with the basi mehanism Station index Figure 0: Completion times for the stations with the RTS/CTS mehanism. here to deal with the fairness question. Other mehanisms in the same family g >, W i = W 0 + i 2 ω) seem also promising to many respets. Hidden terminal questions, of ourse, remain an important issue. Referenes [] Higher-speed physial layer extension in the 2.4 ghz band. IEEE Std 802.b-999 Part : wireless LAN medium aess ontrol MAC) and physial layer PHY) speifiations. [2] S. Beuerman and E. Coyle. The delay harateristis of CSMA/CD networks. IEEE Transations on Communiations, 365): , May 988. [3] G. Bianhi. Performane analysis of the IEEE 802. distributed oordination funtion. IEEE Journal on Seleted Areas in Communiations, 88): , Marh [4] G. Bianhi, L. Fratta, and M. Oliveri. Performane evaluation and enhanement of the CSMA/CA MAC protool for 802. wireless LANs. In Pro. PIMRC, pages , Taipei, Taiwan, Otober 996. [5] A. Branhs, X. Perez-Costa, and D. Qiao. Providing throughput guarantees in IEEE 802.e wireless LANs. In Proeedings of ITC 8, [6] F. Cali, M. Conti, and E. Gregori. Dynami tuning of the IEEE 802. protool to ahieve a theoretial throughput limit. IEEE/ACM Transations on Networking, 86): , Deember [7] D.-M. Chiu and R. Jain. Analysis of the inrease and derease algorithms for ongestion avoidane in omputer networks. Computer Networks and ISDN Systems, 7): 4, June 989. [8] S. Garg, M. Kappes, and A.S. Krishnakumar. On the effet of ontention-window sizes in IEEE 802.b networks. Tehnial Report ALR , Avaya Labs, June [9] M. Heusse, F. Rousseau, G. Berger-Sabbatel, and A. Duda. Performane anomality of 802.b. In Proeedings of 22nd Conferene of the IEEE Communiations Soiety INFO- COM), [0] L. Kleinrok and F. Tobagi. Paket swithing in radio hannels: Part I - arrier sense multiple aess modes and their throughput-delay harateristis. IEEE Transations on Communiations, 232):400 46, Deember 975. [] Q. Ni, I. Aad, C. Barakat, and T. Turlletti. Modeling and analysis of slow CW derease for IEEE 802. WLAN. In PIMRC, Beijing, China, September [2] S. M. Ross. Stohasti proesses. J. Wiley & Sons, New York, 983. [3] H. Wu, Y. Peng, K. Long, S. Cheng, and J. Ma. Performane of reliable transport protool over IEEE 802. wireless LAN: analysis and enhanement. In Proeedings of INFOCOM 02, [4] Y. Xiao. Saturation performane metris of the ieee 802. ma. In Pro. of The IEEE Vehiular Tehnology Conferene IEEE VTC 2003 Fall), pages , Orlando, Florida, USA, Ot 2003.