Analysis of a Polling System Modeling QoS Differentiation in WLANs

Transcription

1 Analyss of a Pollng System Modelng QoS Dfferentaton n WLANs T.J.M. Coenen Unversty of Twente Department of Appled Mathematcs Enschede, The Netherlands t.j.m.coenen@utwente.nl J.L. van den Berg TNO Informaton and Communcaton Technology Delft, The Netherlands j.l.vandenberg@tno.nl Rchard J. Bouchere Unversty of Twente Department of Appled Mathematcs Enschede, The Netherlands r.j.bouchere@utwente.nl ABSTRACT Ths paper nvestgates a pollng system wth a random pollng scheme, a 1-lmted servce dscplne and determnstc servce requrement modelng WLANs wth QoS dfferentaton capablty. The system contans hgh and low prorty queues that are dstngushed va the probablty of beng served next. We propose a new teraton algorthm to approxmate the watng tme of customers n the hgh and low prorty queues. As shown by smulaton results, our approxmaton s accurate for lght to moderately loaded networks. Keywords: Pollng model, QoS dfferentaton, WLAN, IEEE e AMS subject classfcaton: 90B15, 90B18, 90B22, 68M20, 60K25 1. INTRODUCTION Wreless Local Area Networks (WLANs) have become wdely avalable for nternet access and there s currently a growng demand for the support of other applcatons, n partcular speech and vdeo. Specfc mechansms then need to be deployed n order to provde approprate QoS to the varous applcatons. A typcal approach to provde such QoS dfferentaton s for example by gvng a larger share of the avalable capacty to preferred users, or gvng prorty to preferred classes. Introducton of such mechansms requres nsght nto ther performance. Ths paper nvestgates the nfluence of prortzaton of the packet delay handlng at the Medum Access Control (MAC) layer n WLANs. In IEEE WLAN prortzaton appears n the support of dfferent QoS classes. These QoS classes are mplemented va dfferent settngs of MAC layer parameters, lke ther access tme, the maxmum and mnmum value for ther Correspondng author: Tom Coenen, Unversty of Twente, Postbox 217, 7500 AE, Enschede, The Netherlands, t.j.m.coenen@utwente.nl Also afflated wth: Unversty of Twente, Department of Computer Scence, Enschede, The Netherlands Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. ValueTools 2008, October 21 23, 2008, Athens, GREECE. Copyrght c 2008 ICST ISBN # back-off counter or the number of consecutve packets that may be transmtted, see [10] for an overvew of IEEE e that ncorporates these mechansms. QoS provsonng for IEEE systems has been nvestgated manly va dscrete event smulatons. Analytcal models yeldng robust nsght nto system behavour are scarce. To a large extent, such models are based on the poneerng work of Banch [1], n whch a basc system wth persstent sources,.e. sources that always have packets ready to be transmtted, s modeled and analysed usng a Markov chan approach and valdated va smulaton showng excellent agreement wth actual system behavour. Extensons to nclude physcal layer detals are gven n e.g. [9],[16]. The extenson to non-persstent sources s provded n [3],[14], where a flow level model s ntroduced that s analysed usng a Processor Sharng queueng model. Comparson wth dscrete event smulaton shows that ndeed the MAC layer can be adequately modeled va the Processor Sharng mechansm. Extensons to multple traffc classes wth dfferent QoS requrements, as e.g. n e, are among others presented n [17],[18],[19]. Although the flow level modelng of [3],[14],[17],[18],[19] captures the resource sharng behavour of the MAC layer of protocols, the essental behavour at the packet level s not captured. At that level a flow conssts of a seres of packets that are transmtted one by one, where transmssons of dfferent flows are ntertwned. Especally for real tme applcatons, such as speech/telephony, the packet level s of hgh mportance. In [4], a packet level analyss for non-persstent sources s presented, extendng the Markov model of Banch to nclude the probablty of the node gong nto an empty backoff state. We take a further step to analyze the packet level by modelng the MAC layer as a pollng model where the server works off packets at dfferent queues. The essental characterstcs of the QoS aware MAC protocol are ncorporated va the frequency at whch the server vsts the dfferent nodes. In partcular, we gve the server a hgh probablty of vstng a node wth hgh prorty packets. In our pollng model, we consder two types of queues, vz. hgh and low prorty queues, each type wth a dfferent probablty of the server movng to t. Upon departure from a queue, the server randomly selects a queue accordng to these probabltes, whch mmcs the behavour of the MAC layer n systems. Note that we do not clam to accurately model the behavour of the IEEE e protocol, but analyse a mathematcally nterestng model that provdes nsght nto the effect of prortzaton such as used n

2 the IEEE MAC layer. In our model, we wll take the probablty of movng to a hgh prorty (HP) queue to be α tmes as hgh as movng to a low prorty (LP) queue. The servce tme of a packet s consdered to be determnstc as the packet szes n the system are equal for all queues and the channel speed s assumed to be constant at all tmes. As a queue s only allowed to transmt one packet when obtanng the channel, the servce dscplne s 1-lmted. Ths paper analyzes the steady state watng tme for ths 1-lmted pollng system wth random pollng. For the 1-lmted pollng model, general results are avalable n lterature. In [6] Fuhrmann and Cooper derve the well known decomposton result for queues wth server vacatons, whch s very useful for analyzng pollng models. For symmetrc queues, so wth dentcal arrval and servce rates at the queue, and a cyclc pollng order, [7] extends ths result to gve analytcal results on the average watng tme of packets n the queues. In [2], Boxma gves a pseudoconservaton law for the mean watng tme n a pollng system wth Markovan pollng, that ncludes random pollng. Ths law provdes an exact expresson for a weghted sum of the mean watng tmes at all queues, whch need not be symmetrcal. However, results for ndvdual queues cannot be derved from ths law when the network s not symmetrc. The man contrbuton of ths paper s an analyss of the steady state margnal dstrbuton of the watng tme of packets for dfferent types of queues n a 1-lmted asymmetrc pollng model. We consder the dfferent queues n the system ndvdually and model a partcular queue as a queue wth server vacatons, where these vacatons depend on the state of the other queues. To obtan the steady state watng tme dstrbuton, we propose an teraton algorthm. The algorthm computes the margnal steady state dstrbuton of the number of packets at a tagged queue, assumng a steady state at all other queues. Iteratng ths approach over the queues, for varous settngs, we obtan the steady state watng tme dstrbuton for packets at the dfferent queues. The remander of ths paper s organsed as follows. Secton 2 descrbes the queueng networks under consderaton and the analytcal approach for determnng the dstrbuton of the watng tme of customers per queue. Numercal results of the proposed algorthm are compared wth smulaton n Secton 3, and Secton 4 concludes the paper. 2. MODEL DESCRIPTION AND ANALYSIS Consder a pollng model consstng of queues Q 1,..., Q n wth fnte buffer B and a sngle server S vstng the queues. Customers arrve at a queue Q accordngtoapossonprocess wth rate λ. The servce process at the queues s determnstc wth servce tme τ and there s no swtchover tme between the queues. The routng polcy for the server s random, meanng there s a probablty p that the server moves to queue Q upon departure from queue Q j,j=1,..., n. For a hgh prorty queue, ths probablty s α tmesashghas for a low prorty queue, that s p HP = αp LP. The servce polcy s assumed to be 1-lmted, meanng at most one customer s served at each vst of the server, and customers are served FCFS at each queue. When the server reaches an empty queue, t wll mmedately proceed to the next. When all queues are empty, the server wats at the last queue to nstantly move to the frst queue that receves a customer. To ensure stablty of the system we assume that ρ = n λτ <1. In the followng, we derve expressons for the average watng tme of a packet for both types of queue. We start by consderng one hgh prorty queue surrounded by n low prorty queues. At both queues, packets arrve accordng to a Posson process. The server wll move to the HP queue α n+α wth probablty and to a certan LP queue wth probablty. We present an algorthm to approxmate the 1 n+α watng tme of a packet for both types of queue. Ths algorthm consders queues separately as served by a server wth vacatons. The length of the vacatons depends on the number of customers at the other queues. Startng wth an arbtrary dstrbuton of the number of customers at the other queues, the steady state of the number of customers n the consdered queue s determned, usng the vacaton tme dstrbuton. Ths process s terated over the dfferent types of queues repeatedly, untl convergence occurs. For specfc cases, beng that ether the HP or LP queues are saturated, meanng they always have packets ready to be transmtted, exact results are presented. Exact results are also gven for the case where all queues have equal prorty. 2.1 General case To determne the average watng tme of a packet n the queue, we consder the queues separately, as f they are n solaton. From the pont of vew of a queue, the server s ether present and servng a packet, or away whle servng an other queue. We thus can consder a queue as an M/D/1/B queue wth vacatons (c.f. [5],[11],[12]), where the absence of the server whle servng other queues are the vacatons. The length of these vacatons, whch depends on the number of customers at the other queues, nfluences the watng tme of the packets n the queue. For llustratory reasons, we frst gve the analyss for the scenaro where there are two queues, one hgh prorty and one low prorty queue, whch as we show later can be extended to any number of queues Two queues In the two queue scenaro, each queue can be consdered separately as a queue wth a server that goes on vacaton. The duraton of a vacaton now depends on the state of the other queue. We approxmate the dstrbuton of the length of the vacaton V x, gven the number of customers N y at the other queue (HP or LP) usng the followng recurson: q y B j= 1 P (V x = kτ N y = ) = (1) P (V x =(k 1)τ N y = j)p (A y = j +1), k 1 { P (V x =0 N y = ) = 1, =0 (1 q y), =1,..., B where V x s the length of the vacaton seen by the queue x, N y,q y and A y are the number of customers at queue y, the probablty of the server pollng queue y and the number of arrvng customers at queue y durng the servce tme at queue x, respectvely. Note that length of a servce perod s known to be τ due to the 1-lmted servce dscplne, hence we wll denote ths as a servce tme. The varable x can be the HP or LP queue and y s the other type of queue. The

3 vacaton length dstrbuton s then determned usng P (V x = kτ) = B P (V x = kτ N y = )P (N y = ), k 0 =0 (2) As the steady state dstrbuton P (N y = ), s not known, we start wth an arbtrary dstrbuton, for example an always empty queue. Usng ths dstrbuton, the vacaton dstrbuton for the other queue s obtaned. We derve the steady state dstrbuton of the number of customers n the queue usng the vacaton tme dstrbuton, so that by usng Lttle s law we acqure the expected watng tme of a packet. The queue under consderaton can be seen as an M/D/1/B queue wth vacatons (c.f. [5],[11],[12]). To analyze the steady state of ths queue, we frst focus on the state of the system at embedded ponts, whch are after the departure of a customer or the end of a vacaton. The probablty p n that an embedded pont s the completon of a servce and the departng customer leaves n customers behnd, and the probablty q n that an embedded pont s a vacaton termnaton wth n customers n the system are related n the followng manner n+1 p n = g n k+1 q k, n =0, 1,.., B 2 B 1 n=0 p n + p B 1 = q n = q B = B q n =1 n=0 k=1 B k=1 g C B kq k n h n k p k + h nq 0, n =0, 1,.., B 1 k=0 B 1 k=0 h C B kp k + h C Bq 0 where g j and h j denote the probablty of j customers arrvng durng a servce and vacaton tme, respectvely, gj C and h C j denote the probablty of j or more customers arrvng. As these probabltes are known, ths set of equatons can be solved, gvng the steady state dstrbuton at the end of an nterval (ether a servce or vacaton). To determne the contnuous tme steady state dstrbuton, we note that the number of tmes a departng customer leaves a certan number of customers behnd equals the number of tmes an arrvng customer fnds ths number of customers n the system. We have to take nto account, however, that an arrvng customer can fnd B customers n the system n whch case the customer s dscarded and leaves. Let P B denote the probablty that an arrvng customer fnds the system full. To evaluate ths expresson, observe that P B = ρ ρ ρ where ρ = λτ, λ = λ s the offered load and ρ s the carred load, ρ (1 b)τ = bev +(1 b)τ where EV denotes the expected vacaton tme and b denotes the probablty that an embedded pont s a vacaton termnaton pont, b = B q n. n=0 Let σ denote the multplcatve nverse of the average nterval between consecutve embedded ponts, that s σ 1 = bev +(1 b)τ then (1 b)σ P B =1. λ The queue length dstrbuton at arrval epochs, π n, n = 0,..., B s π n = P (Arrval sees n packets Arrval s accepted)(1 P B) + P (Arrval sees n packets Arrval not accepted)p B = p n(1 P B)+P B1(n = B) where { 1fn = B 1(n = B) = 0otherwse Combnng these results, we obtan (1 b)σ π n = p n, n =0, 1,..., B 1 (3) λ (1 b)σ π B =1. λ From PASTA we obtan that the contnuous tme steady state queue length dstrbuton s gven by π n, n =0,..., B. Note that (3) requres the average vacaton tme EV,and(2) the dstrbuton of the other queue to determne the vacaton tme dstrbuton. We may terate (2) and (3) to obtan an approxmaton of the steady state queue length dstrbuton. Algorthm 1. Iteraton 1. Intalze It := 1, x:= 1, y:= 2 P (N y = ) =γ, EN (0) = 0 for =0,..., B where EN (j) denotes the average queue length of queue n teraton j 2. Determne the vacaton tme dstrbuton at queue x from (2), and EV := EV x 3. Determne the queue length dstrbuton P (N x = n) = π n, n =0,..., B, from (3) and determne the average queue length EN x(it) 4. Set y := x, x := 3 y and repeat steps 2 and 3 for ths settng EN 5. If x(it) EN x(it 1) EN x(it) < 0.01 for both x =1, 2, then STOP Else y := x, x := 3 y, It := It +1GotoStep2 The algorthm approxmates n each teraton the number of customers found at the other queue to determne the vacaton tme for the tagged queue. When ths vacaton tme s underestmated, the server swtches back early

4 to the queue and starts servcng a packet at the consdered queue (when avalable), thus leavng the server busy. When however the vacaton tme s overestmated, the approach leaves the server at the other queue for too long a perod, where ths queue mght actually have become empty, thus leavng the server dle whle t could process jobs n the tagged (non-empty) queue. The presented approach hence underestmates the capacty of the server, but equally for both queues. The average queue length of all customers n the total system, whch for larger values of B approxmately can be seen as an M/D/1 queue as t s work conservng, s known and gven by ρ(2 ρ) EN total = where ρ =(λ LP + λ HP)τ, the load of the total system. The results obtaned by the teraton gve a hgher average queue length due to underestmaton of the server capacty. The queue length of each type of customer should hence be scaled down, so that the average queue length of all customers n the system s correct. Ths leads to an mproved estmaton of the average queue length of a customer per type of queue. Usng Lttle s law, we obtan the average watng tme for each type of queue. The algorthm can start wth an arbtrarly chosen steady state dstrbuton for the queue length of the HP queue. From ths, a new steady state s computed for the same queue. Startng from each ntal dstrbuton for the HP queue, Algorthm 1 converges to the steady state dstrbuton. Theorem 1 below states that ths convergence s monotone startng from ether an empty or full HP queue. We need stochastc orderng. Let X and Y be random varables wth dstrbuton F X(.) andf Y (.), respectvely. We say that X st Y ff F X(x) F Y (x) for all x 0 (c.f. [15], p.410). Theorem 1. For each ntal dstrbuton, Algorthm 1 converges monotoncally. Proof. Let X HP and X LP denote random varables for the queue length dstrbutons of the HP and LP queue after the th teraton and let Y LP and Y HP denote the random varables for the correspondng vacaton length dstrbutons. From (2) t follows that f X0 HP st X1 HP also Y0 LP st Y1 LP as a hgher queue length for the HP queue leads to a longer vacaton length for the LP queue. From (3) t follows that f Y0 LP st Y1 LP also X0 LP st X1 LP as a longer vacaton for the server of the LP queue leads to a hgher number of packets n the LP queue. Followng the same reasonng for the LP node, we have that X0 LP st X1 LP leads to Y0 HP st Y1 HP and Y0 HP st Y1 HP leads to X1 HP st X2 HP. It thus follows that X HP 0 st X HP Smlarly we have that X HP 0 st X HP as X0 HP st X1 HP. for any 1aslongasX0 HP st X1 HP. for any 1aslong From Theorem 1, an obvous approach s to start wth { P (X0 HP 1 for n =0 = n) = (4) 0 for n>0 snce t then holds that X0 HP st X for any X wth a nonnegatve dstrbuton. Let X denote the random varable followng an equlbrum dstrbuton, that s X = X+1. We then have that as X0 HP st X,alsoX HP st X,so the teraton process cannot jump past an equlbrum. In every teraton, the dstrbuton may change, movng closer towards the equlbrum dstrbuton. Smlarly, we can start wth the dstrbuton P (X HP 0 = n) = { 1 for n = B 0 for n<b where B s the maxmum number of customers n the queue. It then follows that X0 HP st X for any X, sothatafter every step we have that X HP st X as X0 HP st X. In ths case every teraton takes a step closer to the equlbrum from above. Usng Algorthm 1 startng from both (4) and (5), we fnd our approxmaton Multple queues The approach for two queues can easly be extended to multple queues of any prorty class. The vacaton length of a consdered queue then depends on the state of all the other queues, and can be computed by analogy to (2). The vacaton length dstrbuton n ths case s gven by P (V x = kτ N y = y,y x) = (6) B P (V x =(k 1)τ N z = a z,z x) q y y x z x,y a z= z,z x,y a y= y 1 P (A z = a z z) P (A y = a y y +1) P (V x =0 N y = y,y x) = q x + y, y>0 qy Here q x denotes the probablty of the server jumpng to queue x. The vacaton length dstrbuton s found usng P (V x = kτ) = (7) B P (V x = kτ N y = y,y x)p (N y = y,y x) y=0,y x where agan the steady state queue length dstrbuton of the other queues s needed. Startng agan wth a random dstrbuton for all but one queue we fnd the vacaton tme for ths tagged queue and hence the correspondng steady state queue length dstrbuton of ths queue. Ths dstrbuton can now be used for all queues of the same class and the other class can be analyzed usng the steps of the algorthm. Note that the proof of convergence remans the same, as the analyss s done for each type of queue. In the case of multple queues wth balanced load, that s wth dentcal arrval rates at the queues, the random varables X HP, X LP, Y HP and Y LP can be used for all queues of the same type as they are ndentcal. When arrval rates at the queues are dfferent, the same reasonng can be used for all separate varables X HP j, X LP j, Y HP j and Y LP j,wherethe subscrpt j denotes a specfc queue of the type HP or LP. 2.2 Specal cases For a hgh prorty queue, t may be needed that a certan average watng tme can be guaranteed. To obtan the maxmal average watng tme n a network wth one HP queue and n LP queues, we gve results for the stuaton wth saturated LP queues. To analyze the mpact of prortzng the hgh prorty queue on the low prorty queues, we compare the average watng tme at the LP queues wthout an HP queue n the system, wth the case where the HP q x (5)

5 queue s saturated. For these specal cases, exact results are avalable, whch are gven n ths secton Saturated LP queues Consder one hgh prorty queue wth Posson(λ HP)packet arrvals and n saturated low prorty queues,.e. λ LP. Let the probablty q of vstng the hgh prorty queue be q = α n + α where α denotes the factor of mportance gven to the hgh prorty queue, meanng the probablty of vstng the HP queue compared to the LP queue s α tmes as hgh. For the HP queue, the vacaton length dstrbuton s then gven by the geometrc dstrbuton P (V = kτ) =(1 q) k q as any tme the server does not jump to the HP queue, t wll servce exactly one packet at an LP queue. As the average tme between arrvals of the server s τ and the server only q serves one customer at each vst, the HP queue s stable when q>λτ. Wth the exact dstrbuton of the vacaton length known, we can use the pgf of the number of customers n the queue as gven by 3 to determne the average number of customers n the HP queue. The average watng tme then easly follows from Lttle s law Empty and saturated HP queue We now consder the case where the low prorty queues are no longer saturated, but each have an arrval process of rate λ LP and a determnstc servce tme of value τ. Let queue n + 1 be the HP queue, the conservaton law (c.f. [8]) then states that n+1 n+1 ρ EW q = ρ λβ(2) where EW q denotes the average watng tme n the queue (not ncludng servce) and ρ = λ LP τ for = 1...n and ρ n+1 = λ HPτ, so that and ρ = nρ LP + ρ HP. Astheservce tme dstrbuton s determnstc for any queue, we have that β (2) = τ 2 and the total watng tme of a customer s EW = EW q + τ. Consder the case where there are only n LP queues, so the arrval rate at the HP queue s set equal to zero. The stablty condton s that ρ = nλ LP τ<1and t mmedately follows that n n ρ LP EW q LP = ρ λlp τ 2 EW q LP = ρτ (2 ρ)τ EW LP = Now consder the case where the HP queue s saturated. We have n dentcal LP queues, and from the perspectve of the LP queues the server ncurs a swtchover tme when t vsts the HP queue. The stablty condton for ths system nλσ s that < 1, where σ denotes the mean swtchover (1 ρ) tme, as ths s the number of arrvng customers durng the average cycle tme of a queue. Let p denote the probablty of jumpng to queue and s the average tme t takes to swtch to queue. We have a pseudo-conservaton law statng that (c.f. [2]) n ρ ρ [1 λ σ p 1 ρ ]EW = n λβ(2) + σ 1 ρ n ρ p n ρ s + ρ 2σ n p s (2), where for our model we have that λ = λ LP = λ, ρ = λτ, ρ = nλτ, β (2) = τ 2, p = 1 qτ, s = n 1 q s(2) and = q(q+1)τ (1 q) 2 σ = s as all swtchover tmes are equal. Here q denotes the probablty of the server pollng the HP queue. As the LP queues are statstcally dentcal, the expresson smplfes to EW LP = nλτ 2 + qτn + q+1 2qτ 2(1 nλτ) (1 q)(1 nλτ) 2(1 q) nλqτ [1 ] (1 q)(1 nλτ) and applyng Lttle s law the average total number of customers n the queue s obtaned. Note that ths approach can easly be extended to a case wth multple hgh prorty queues, as only the probablty of the server beng on vacaton changes, so only the values of s and s (2) need to be adjusted. 3. VALIDATION In the followng we valdate our approxmaton approach by comparson wth smulaton results. For a wde varety of settngs, varyng the load of the system and the grade of prortzaton, the average watng tmes of packets at the ndvdual queues are determned. Note that the approach presented calculates the dstrbuton of the watng tme, but only the averages are used n the followng for comparson wth smulaton. Results for the scenaro wth one hgh prorty and one or two low prorty queues are consdered, together wth the specal cases. 3.1 General case Two queues Table 1 shows the average watng tme of packets n a queue computed by the algorthm compared wth smulaton results for dfferent loads of the system n the case of two queues, one HP and one LP queue. The table shows the mpact of varyng α, the relatve mportance of the HP queue compared to a LP queue. The load at the queues s balanced,.e. each queue has the same arrval rate of packets. The probablty of movng to the HP queue s q = α,whch n+α s α tmes as hgh as for the LP queue and the buffer sze s set to 15 for all cases. The mpact of the dfferentaton appears to be hgher when the load of the system ncreases. For a low load, the queues are often empty, thus makng t possble for the server to attend to packets drectly upon arrval. As the load ncreases, the queues wll be fuller and the watng tme depends more on the frequency at whch the server vsts the queues. We observe that the accuracy of the algorthm deterorates as the load of the system ncreases. For a hghly loaded system, the queues wll at tmes be fully loaded, causng arrvng packets to be lost. Ths effect s not taken nto account when usng the pseudoconservaton law to scale the obtaned results. Smulaton however shows that the mpact of ths approxmaton s lmted, as the average

6 Table 1: Average watng tme n a two node network wth balanced load Rates Smulaton Algorthm Error α λ LP λ HP LP HP LP HP LP HP number of packets n the system remans close to a system wth nfnte queues. In a smlar fashon Table 2 shows results for unbalanced arrval rates, wth the probablty q = 2 (α = 2) of vstng 3 the HP queue kept constant. For more unbalanced stuatons, the results deterorate, especally for hgher loads. For the node wth the lower arrval rate, the error made by the algorthm s bgger, as the average queue length s smaller. Comparng the mpact of ncreasng the load of the LP queue on the HP and vce versa shows that the ncrease n load of the HP queue has a bgger mpact on the average watng tme at the LP queue than ncreasng the load of the LP queue has on the HP queue. As an ncrease of the load wll cause the queue to be non-empty for a larger fracton of the tme, the mpact t has on the other queue by causng the server to go on a vacaton becomes larger. As a HP has a hgher probablty of beng vsted, ncreasng the load of ths queue has a bgger mpact than ncreasng the load at the LP queue Three queues In Table 3 we consder the scenaro wth three queues, one HP queue and two LP queues. The table shows the average watng tme of packets computed by the algorthm compared wth smulaton results for the stuaton wth balanced load. As for the stuaton wth two nodes, we observe that for hgher loads, the mpact of the prortzaton ncreases. Agan, the results deterorate as the load of the system ncreases. Comparson wth the results of Table 1 furhter shows that the mpact of prortzaton s hgher when more nodes are actve n the network. The decrease n the average watng tme of customers for the HP queue s stronger relatve to the decrease for the two node stuaton. Wth more queues present, the relatve ncrease n probablty of beng vsted s hgher when the value of α s ncreased. For example, ncreasng the value of α from 2 to 3 for both stuatons gves the followng relatve ncrease (r..): α =2 α = 3 r.. 2nodes q = 2 q = % 3 4 3nodes q = 1 q = 3 20% 2 5 For all settngs, no more than 15 teratons were needed by the algorthm wth the accuracy set n such a way that the last step gave an mprovement less that 1%. Longer runs wth hgher accuracy dd not mprove the results sgnfcantly. To run the teratons, the values of P (V x = kτ N y = ) for the two node case and P (V x = kτ N y = y,y x) for the three node case had to be computed once usng the teratons gven n (1) and (6), whch s tme consumng for large values of the buffer szes. For hghly flled buffers however, the geometrc dstrbuton can be used, as the probablty of the vacaton havng a duraton of kτ s then very close to the probablty of frst vstng k other queues before vstng the consdered queue, as the other queues wll not become empty durng the process. The tme needed for the teraton tself s very lmted, as (2) (or (7)) only encompasses the addton over all possble values of queue lenghts and (3) s a small enough system of equatons to be solved wthn seconds. 3.2 Specal cases Saturated low prorty queues For a user wth mportant traffc, the QoS dfferentaton s of hgh mportance. To get an dea of the mpact of the settngs for the dfferentaton, a worst case scenaro can be analysed to see the mnmal prortzaton that s needed to obtan a certan average watng tme for the hgh prorty packets. The worst case scenaro s when all other (low prorty) queues always have traffc to transmt. Fgure 1 shows the average watng tme of a packet n the HP queue, for dfferent values of n, the number of saturated low prorty queues n the system. The arrval rate at the HP queue s set to λ HP =0.01. The three lnes represent the results of the model for α = 2...4, the grade of prortzaton. It clearly follows from the fgure that where for a sparce network (low number of LP queues) the dfferentaton has a lmted effect and that for a dense network (hgh number of LP queues) gvng more prorty has a much bgger mpact Empty and saturated hgh prorty queue The dfferentaton between users s prmarly done to provde better performance for more mportant traffc. However, t also has to be taken nto account what the mpact s on performance of the less mportant traffc. If the prortzaton of the hgh prorty queue s too hgh, the low prorty queues mght be starved. To analyse the mpact on the low prorty queues, we compare the stuaton wthout the HP queue (or an empty HP queue) wth the stuaton that the HP queue always has packets to transmt. In the latter case, we vary the grade of prortzaton. Fgure 2 shows the average watng tme of a packet n an LP queue, for dfferent values of n, for dfferent settngs of the HP queue. The arrval rate λ LP s set to 0.01 for each of the n LP queues. In ths case the HP queue s ether absent (or empty) n whch

7 Table 2: Average watng tme n a two node network wth unbalanced load Rates Smulaton Algorthm Error λ LP λ HP LP HP LP HP LP HP Table 3: Average watng tme n a three node network wth balanced load Rates Smulaton Algorthm Error α λ LP λ HP LP HP LP HP LP HP alpha = 2 alpha = 3 alpha = 4 Empty HP alpha = 2 alpha = 3 alpha = avg watng tme avg watng tme number of LP nodes number of LP nodes Fgure 1: Average watng tme for the scenaro wth saturated LP queues Fgure 2: Average watng tmes for the scenaro wth an empty or saturated HP node case the complete network behaves as a standard M/D/1 queue where each separate queue has the same average behavour or the HP s saturated, wth dfferent values for α, the grade of prortzaton. For hgher values of α the server wll more often be processng HP packets, leavng less capacty for the LP queues. Ths shows from the fgure as the watng tme reaches hgh values already for lower values of n. When the network s sparce, we see there s already a substantal mpact of the dfferentaton on the watng tme of the low prorty packets. 4. CONCLUSION In ths paper we analyzed the mpact of QoS dfferentaton on the delay of packets for dfferent classes of queues usng a 1-lmted pollng model wth a random schedulng polcy and determnstc servce tmes, capturng the random nature of the MAC layer protocol. The model gves nsght n the effect of the parameter settngs on the QoS n a WLAN for the ndvdual classes of queues. We developed an approxmaton approach for the packet delay n a network wth hgh and low prorty queues. Comparson wth smulaton results shows that for low to moderately loaded systems, the approach works well. 5. REFERENCES [1] Banch, G., Performance analyss of the IEEE dstrbuted coordnaton functon, IEEE Journal on Selected Areas n Communcatons, vol. 18,

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