Feasibility and Optimization of Delay Guarantees for Non-homogeneous Flows in IEEE WLANs

Transcription

1 Feasblty and Optmzaton of Delay Guarantees for Non-homogeneous Flows n IEEE 8. WLANs Yan Gao,CheeWeTan, Yng Huang, Zheng Zeng,P.R.Kumar Department of Computer Scence & CSL, Unversty of Illnos at Urbana-Champagn Department of Computer Scence, Cty Unversty of Hong Kong Department of Computer Scence, Unversty of Illnos at Urbana-Champagn Abstract Due to the rapd growth of real-tme applcatons and the ubquty of IEEE 8. MAC as a layer- protocol for wreless local area networks (WLANs, t s of ncreasng nterest to support qualty of servce (QoS n such WLANs. In ths paper, we develop a smple but accurate enough analytcal model for predctng queueng delay n non-homogeneous random access based WLANs. Ths leads to tractable solutons for meetng queueng delay specfcatons of a number of flows. Usng ths model, we address the feasblty problem of whether the mean delays requred by a set of nelastc flows can be guaranteed n WLANs. Based on the model and feasblty analyss, we further develop an optmzaton technque to mnmze the delays for nelastc flows. We present extensve smulaton results to demonstrate the accuracy of our model and the performance of the algorthms. I. INTRODUCTION The recent rapd growth of real tme applcatons has led to a strong need to provde qualty of servce (QoS for moble computers and portable devces n wreless local area networks (WLANs. Ths has to be supported over the IEEE 8. snce t has ganed wdespread popularty and become the de facto WLAN standard. However, the mechansms employed n the IEEE 8. MAC, namely random access and the dstrbuted coordnaton functon (DCF, render t substantally more dffcult to ensure delay guarantees because of the channel contenton and the random back-off mechansm. Therefore, as the frst task confrontng researchers n ths feld, t s necessary to characterze the delays n such networks. Second, t s mportant to devse solutons that provde the requred delay performance. We address both ssues n ths paper. Exstng studes on the performance analyss of the IEEE 8. MAC have focused on ts throughput capacty n networks wth saturated traffc; see Banch [], Cal, Cont, and Gregor []. In [], a dscrete-tme M/Geo/ queueng analyss under network saturaton s studed. Models for unsaturated homogeneous networks have also been reported. For example, Medepall and Tobag [4] present a unfed model for multhop networks that approxmates each queue by an ndependent M/M/ queue. However, ths approxmaton s not accurate for Ths materal s based upon work partally supported by NSF under Contracts CNS-578, CCF-997, CNS-9597, CNS-54, USARO under Contract Nos. W9NF-8--8 and W9NF-7-87, AFOSR under Contract FA955-9-, and CtyU Project detaled delay analyss n WLAN. Tkoo and Skdar [5] present a G/G/ queueng model for delay analyss n homogeneous networks. Ther focus s on performance analyss of the standard IEEE 8. DCF. Varous studes have also been conducted on provdng QoS support n WLANs, and most use centralzed pollng technques based on the pont coordnaton functon (PCF. For example, Coutras, Gupta and Shroff [6] analyze the performance of PCF n support of voce servces. However, they do not address the fact that both best-effort traffc and real-tme traffc coexst n WLANs, and IEEE 8. DCF s the de facto settng used n most WLANs. Provdng QoS requres networks to support servce dfferentaton under non-homogeneous traffc dynamcs. Networks should also reallocate lmted resources from the over-provsoned flows to the under-provsoned flows. IEEE 8.e has been proposed to enhance the orgnal standard to support QoS. However, IEEE 8.e classfes flows only by ther applcatons (e.g., voce, vdeo, etc. and provdes the same servce to flows that fall n the same class. Moreover, t only dfferentates prorty among flows, but does not actually provde delay guarantees. A non-homogeneous and adaptve WLAN s hghly preferred over one that operates n a fxed homogeneous manner. However, an accurate model of nonhomogeneous flows n random access WLANs, especally ther delay characterzaton, s stll elusve. We develop a smple but suffcently accurate analytcal model based on an M/G/ queue for non-homogeneous unsaturated IEEE 8. networks. We characterze the channel access delay wth respect to the contenton wndow and the probablty that the queue s nonempty. The latter n turn depends on the channel access delay. Both ths probablty of beng non-empty and the access delay can be jontly obtaned by solvng a coupled system of nonlnear equatons through a fxed pont teraton wth a carefully chosen ntal pont so that t converges to a fxed pont. Moreover, we show that n random access networks, the second moment of the access delay s determned only by ts frst moment f the packet sze s suffcently large. Ths approxmaton smplfes the formula of the queueng delay. Thereby, we analytcally determne whether the network can provde the delay guarantees requred by the QoS flows. The contrbutons of the paper are summarzed as follows:

2 We derve a smple but accurate model for queueng delay n non-homogeneous IEEE 8. MAC based networks. We use t to determne the feasblty of usng a random-access based WLAN to serve a set of real-tme flows wth mean delay requrements. We provde characterzaton of the average delay and access rate, and propose fxed pont algorthms to compute them. A lnear system approxmaton s derved to complement the analyss. We provde an algorthm to mnmze the delays for a set of nelastc flows whle meetng mean delay requrements. 4 We valdate our algorthm to provde performance guarantees through extensve NS- smulatons. We motvate the non-homogeneous IEEE 8. flows problem n Secton II. In Secton III, we characterze the channel access delay and queueng delay. In Secton IV, we study fxed pont teratons related to the queueng model. We show how to optmze the delay performance n Secton V. Numercal results are gven n Secton VI. II. PROBLEM STATEMENT A. Non-homogeneous IEEE 8. network In IEEE 8. DCF networks, each node wth a packet to transmt randomly selects a back-off tmer BC from [,CW ], where CW denotes the contenton wndow. If the channel s sensed dle, these nodes decrement ther tmers untl one of them expres. Then that node attempts to access the channel and the remanng nodes pause ther tmers. The count-down resumes when the channel s dle agan. If more than one node attempts n the same slot, a collson occurs. A collded transmsson s retred up to a retransmsson lmt before t s dscarded. In the standard IEEE 8. network, the contenton wndow of each node s set to be the same. Ths homogeneous or unform settng works well for best-effort traffc where farness s to be taken nto account. However, the ncreasng need to support heterogeneous QoS requrements of dfferent flows requres networks to have the ablty to provde servce dfferentaton to real-tme flows. We consder a WLAN wth nodes that are capable of changng the backoff parameters by tunng only the contenton wndow CW. Thus, our scheme s IEEE 8. standardscompatble. We analytcally show that CW alone can effectvely be used for resource allocaton and performance dfferentaton. B. Soft Deadlne We are nterested n the soft deadlne that s the mean delay of a flow. Soft-deadlne guarantees are mportant for several real-tme applcatons such as voce over IP, onlne games and IP-TV, snce they often requre a fxed bt rate but are senstve to mean delays. We now formulate the problem. Consder a WLAN where N nodes are actve and each has a QoS flow to the access pont (AP. These flows dffer n rate and delay requrements. Assume that for each node the packet arrvals form a Posson process and the nter-arrval tme s exponentally dstrbuted wth mean /λ. Note that the Posson arrval model has been used n the lterature, e.g. []. Let λ =[λ,λ,..., λ N ] T be the arrval rate vector. Addtonally, each flow has a soft deadlne D to meet. The average queueng delay of the packet for flow s requred to be less than D. Let D =[D,D,..., D N ] T denote the target delay vector. Gven both λ and D, the queston we address s the followng: Does there exst an assgnment of contenton wndows CW = [CW,CW,..., CW n ] T such that all deadlnes are met? Ths queston s especally mportant for admsson control, whch needs to decde whether t s feasble to accommodate a new flow n the network wthout hurtng the performance of hgh prorty flows (.e., the already exstng flows. Furthermore, f t s ndeed feasble, to acheve all these mean delays, then how should one assgn CW for each node? We provde analyss to answer these two key questons n the followng. III. ANALYTICAL MODEL OF NON-HOMOGENEOUS IEEE 8. NETWORK A. Meda Access Delay We now analytcally address the meda access delay of a non-homogenous WLAN. We do not employ the exponental back-off algorthm mplemented n the standard protocol because our standalone scheme conssts of choosng a fx contenton wndow for each flow so as to meet ts mean delay. Imposng the redundant CW adjustment mechansm, e.g., exponental back-off algorthm, s an unnecessary layer of adaptaton that s not needed n our scheme. We note that smlar approaches have been adopted n the lterature. The schemes proposed n [7], [8] dsable the exponental backoff, and drectly adjust the contenton wndow. However, ther goal s to maxmze the throughput, whle ours s to provde the delay guarantees for heterogeneous flows. We consder an access rate for node that s equal to /CW. Ths corresponds to IEEE 8. DCF wth BC chosen randomly from [,CW ] [], [], [7], [9]. Snce our flows are not saturated, queues may be empty,.e., they do not transmt any packet. Let NE denote the event queue not empty and E denote the event queue empty. Then, the uncondtonal channel access (CA probablty of node s P[CA] =P[CA E]P[E]+P[CA NE]P[NE]. It s obvous that P[CA E] s equal to zero because the node has no packet to transmt when empty. We next approxmate P[CA NE] by CW, whch s only an approxmaton when the random backoff s chosen unformly wthn [,CW ], buts not an approxmaton f t s attempted after an exponentally dstrbuted nterval. Denotng p = /CW, and ρ as the probablty that the queue s not empty, we have P[CA] = ρ = ρ p. ( CW Let p be the vector [p,p,..., p N ] T, notng that p. Lkewse, ρ := [ρ,ρ,...,ρ N ] T. Now, we compute the probablty PI that the channel s dle when node has a packet to send, the probablty PS

3 that the channel s successfully carryng a packet of node, and the probablty PO that node sees the channel as busy though node does not transmt a packet successfully. Note that PI + P S + P O =for all. All these quanttes are to be computed as a functon of the vector p. Note that node competes for channel access only when t has a packet to transmt. Under ths condton, node fnds the channel dle n a tme slot f t does not attempt and no other node attempts at the begnnng of ths slot. Hence, N PI =( p ( ρ j p j. ( Node successfully transmts a packet f t attempts, and no other node attempts n the same slot. Ths probablty s N PS = p ( ρ j p j. ( Otherwse, node sees the channel occuped by other actvtes, consstng of successful transmssons of other nodes or collded transmssons. Note that the collded transmssons consst of both the transmssons nvolvng node as well as those not nvolvng node. Thus, we have N PO = PI PS = ( ρ j p j. (4 Defne the servce tme x of a packet (also referred to as channel access delay here as the tme from the nstant the packet reaches the head of the queue n the node tll the nstant t successfully departs from the queue. Ths servce tme ncludes two parts, namely the channel contenton delay and the packet transmsson tme. For smplcty, we assume that all packets are of the same sze and all nodes adopt the same bt rate to transmt. Thus, they have the same packet transmsson artme denoted by T. More precsely, n the IEEE 8. network, a packet transmsson artme s gven by T :=DIFS+PACKET+SIFS+ACK, (5 where DIFS denotes the duraton of the dstrbuted nterframe space, PACKET denotes the transmsson tme of a data packet, SIFS denotes the duraton of the short nterframe space, and ACK denotes the transmsson tme of an acknowledgement. There are two access modes used n IEEE 8. DCF, namely the basc access mode and the RTS/CTS access mode. In ths paper, we model the system only for the basc access mode. The RTS/CTS access mode s usually dsabled n practce due to ts large overhead. In the basc access mode, a collson s detected when a node does not receve an ACK wthn an ACK-tmeout perod. The ACK-tmeout s defned to be the tme to transmt an ACK frame plus SIFS. Thus, we assume that the artme spent on a collded transmsson s the same n duraton as that of a successful transmsson. We denote a slot-tme duraton by τ. Let t k denote the tme nstant when the k-th dle slot begns,.e., the nstant that the channel s dle at the begnnng of the correspondng slot. There are two possble events followng ths nstant: a the channel contnues to be dle for a duraton of τ untl the next dle slot begns; b at least one of the nodes attempts to transmt n ths slot, whch results n a T tme unt channelbusy perod. We assume that the ntervals S (k =t k+ t k are ndependent and dentcally dstrbuted random varables and refer to these ntervals as vrtual slots. Assume that the tme nterval from the tme the packet reaches the head of the queue at node to the tme t starts to depart from the queue conssts of K vrtual slots, where K s a random varable ndependent of S. Its dstrbuton s gven by P[K = n] =( PS n PS for n =,,,... (6 It follows that E[K ]=PS n( PS n = P S P. (7 n= S For node, ts servce tme s therefore K x = S (k+t, (8 k= where the S (k are Bernoull random varables that are ether equal to τ f the channel s dle or equal to T f a transmsson of a node other than occurs: P τ wth probablty I S (k = T Then, we have wth probablty PS P O. PS E[S ]= P I τ + P O T PS. ( One can see that both E[S ] < and E[K ] <. Fromthe ndependence of S and K, we can apply Wald s equaton [] to obtan X := E[x ] = E[K ]E[S ]+T. ( Substtutng (7 and ( nto ( gves X = P I τ + P O T PS + T. ( Note that ( captures an nterestng relatonshp between the expected servce tme and the access rate n CSMA-based random access WLAN. Snce the network s unsaturated, we need to determne the probablty ρ that the queue s nonempty. However, snce each node s an M/G/ queue, we have ρ = λ X. ( Substtutng ( nto (, we have N equatons wth N unknowns [x,x,,x N ]. Solvng ths N dmensonal vector fxed pont problem gves us the servce tmes for the nonhomogeneous random access flows. We summarze ths relatonshp that allows us to compute the mean servce tmes for non-homogeneous random access WLANs: Gven the contenton wndows CW, the mean servce tmes are gven by (, where PI, P S and P O are gven by (,,4, wth p defned by (. The quanttes ρ s n ( satsfy (. (9

4 B. Queueng Delay In the prevous secton, we have derved an analytcal model that can be used to compute the servce tme f the access rates of all nodes are gven va ther contenton wndows. Snce many real-tme applcatons such as onlne games, VoIP and IPTV requre strct lmts on jtter and mean delay, n ths secton, we study how the non-homogeneous contenton wndow settngs and the non-homogeneous throughput requrements jontly affect the average queueng delay. Defne the queueng delay of a packet to be the tme from the nstant that the packet arrves at the queue to the nstant that the packet successfully departs from the queue. The average queue sze of the M/G/ queue s gven by []: E[Q ]=λ X + λ E[x ] ( λ X, (4 where Q denotes the queue sze and E[x ] s the second moment of the servce tme. Usng Lttle s law, the average queueng delay Y s Y = E[Q ] = X + λ E[x ] λ ( λ X. (5 To determne the average queueng delay (5, we need to determne the second moment of the servce tmes. In (8, we have characterzed the servce tme x by a sequence of vrtual slots S plus a transmsson artme T. Takng squares on both sdes of (8, we have K x =( S [k]+t k= K K k K = S [k]+ S [k]s [l]+t S [k]+t. k= k= l= k= Applyng Wald s equaton agan, we get E[x ]=E[K ]E[S ]+E[K K ]E [S ] +TE[K ]E[S ]+T. Usng the dstrbuton of S n (9, we compute (6 (7 E[S ]= τ PI + T PO PS. (8 To determne E[K K ], we frst obtan the moment generatng functon of K from (6 as follows: M K (B = B n ( PS n P PS S = ( P. (9 n= S B It s easy to verfy that d M K (B d B B= = n(n B n ( PS n PS B= = n= n(n ( PS n PS = E[K K ]. n= Hence, from (9 and (, we get ( E[K K ]= ( P S (P S. ( Thus, substtutng (7, (, ( and (8 nto (7, we get the second moment of the servce tme for node as follows: E[x ]= τ P I + T P O P S + (τp I + TP O (P S ( +T τp I + TP O PS + T. Substtutng ( and ( nto (5, we therefore obtan the average queueng delay wth respect to vector p. Queueng delays as a functon of contenton wndows Y (p: Consder a non-homogeneous random access WLAN wth contenton wndows CW and packet transmsson tme T. Then the average queueng delay s gven by (5, where E[x ] s gven by (, E[x ] s gven by (, P I, P S and PO are gven by (,,4, and ρ =[ρ,ρ,,ρ N ] T s a fxed pont of (. C. Queueng Delay and Channel Access Delay Substtutng ( nto (, we have E[x ] = (X T +T (X T +T + P I τ + PO T PI τ + P O T (X T. ( Snce lm τ E[x ]=(X T +T (X T +T + T (X T, f we assume that the packet transmsson artme T s suffcently large compared to slot-tme τ, then we get a smplfed formula for E[x ] as follows: E[x ]=(X T X. (4 Note that (4 mples that the second moment of x s determned by ts frst moment only. Ths demonstrates an nterestng property nherent n the random access mechansm. Therefore, the average delay s Y = ( λ T X ( λ X. (5 Now, (5 s equvalent to Y X =. (6 λ T +λ Y Note that (6 captures an mportant property: The mean queueng delay n a random access network s determned only by the mean channel access delay. IV. ANALYSIS OF FIXED-POINT PROBLEMS A. Nonlnear Characterzaton of Delay and Access Rate We have shown that when the transmsson artme T s suffcently large compared to the slot-tme τ, the queueng delay Y s determned by X. Thus, we need to characterze X,.e., to analyze (. Based on (, we derve a set of fxed pont equatons: T p X +( p (T τ = ( λ. (7 jx j p j There are two problems of nterest, namely analyss and desgn. We now express the fxed pont problems of both problems n the form of (7. The analyss or performance

5 evaluaton (PE problem conssts of determnng the delay, gven the access rates. The desgn or access rate assgnment (ARA problem s the reverse; t conssts of determnng the access rates for the flows so as to meet all the delay constrants. Performance Evaluaton (PE: We fx the access rate p, and evaluate the channel access delay X. For node, ts delay can be wrtten as X = I PE (X T := p ( λ jx j p j ( p (T τ (8. p We denote by X a fxed pont of (8, assumng one exsts. We consder the followng fxed pont teraton to solve (8: X(k +=I PE (X(k. (9 From a protocol desgner s vewpont, t s nstead more nterestng to compute the access rate assgnment such that all the flows meet ther requred delays: Access Rate Assgnment (ARA: We want to adjust the access rate p such that all the delays X are fulflled. For node, the access rate s gven by p = I ARA (p := T (X T + τ ( λ jx j p j T τ X T + τ. ( We denote by p a fxed pont of (, assumng one exsts. We consder the followng fxed pont teraton to solve (: p(k +=I ARA (p(k. ( B. Lnear System Approxmaton Note that λ X p < for all f the system s stable. Now, usng the fact that /( z +z for nonnegatve z<, we can lower bound the RHS of (7 by an affne expresson. In partcular, we have p X +( p (T τ T ( + λ j X j p j. ( Note that we can approxmate the nequalty n ( by an equalty f we assume small p X for all, and apply Taylor s expanson theorem to the RHS of (7. Ths leads us to consder the followng fxed pont equaton: p X λ j Tp j X j = p T +( p τ. ( Now, we can consder two dfferent lnear fxed pont equatons n the form of (: One n terms of X for performance evaluaton assumng fxed p, and the other n terms of p for access rate assgnment assumng fxed X. The followng results show that each of these two lnear fxed pont teratons has a unque soluton. Theorem 4.: If [p,p,..., p N ] T are gven, ( has a unque soluton for [ X, X,..., X N ] T. Proof: Let y denote the vector [p X,p X,,p N XN ] T and b denote the vector [p T +( p τ,p T +( p τ,,p N T +( p N τ] T. Then we represent ( by y = F y + b, (4 where F s an rreducble nonnegatve matrx wth entres: {, f l = j F lj = (5 λ T, f l = j. We now apply nonnegatve matrx theory to characterze the soluton to (4. Let Λ A denotes the spectral radus of a nonnegatve matrx A. By the Collatz-Welandt theorem (see, e.g., [], Λ F max λ j T< λ T<, (6 where the last nequalty follows from the necessary condton that the M/G/ system s stable only f the workload s strctly less than,.e., λ T <. Next, we state the followng result from []. Lemma 4.: A necessary and suffcent condton for a soluton z, z = to exst to the equatons (I Az = c, for any c, c = s that Λ A <. In ths case there s only one soluton z, whch s strctly postve and gven by z =(I A c. Applyng Lemma 4. to (4, ths mples (I F b has a unque postve soluton. Ths proves the theorem. Lemma 4.: Assume X s gven. If p s the fxed pont of (, and p s the fxed pont of (7, then we have, component wse, p < p. Proof: Suppose the followng holds: p X +( p (T τ =T ( + λ j X j p j, (7 p X +( p T (T τ = ( λ jx j p. (8 j For each, we subtract (7 from (8 to obtan (X T + τ(p p T = ( λ jx j p j T ( + λ j X j p j >T( + λ j X j p j T ( + λ j X j p j (9 = λ j TX j (p j p j. Let u denote a vector wth the th entry as X (p p. Now, (9 for all can be wrtten n matrx form as (I Cu = v >, (4 where v denotes some postve vector (wth the postve slack of nequalty (9 as ts th entry, and C s a postve matrx wth entres { (T τ/xl, f l = j C lj = (4 λ T, f l = j. Snce C s a postve matrx, usng the Perron-Frobenus theorem, Λ C s strctly postve. Now, Λ C satsfes Λ C (a max ( λ jt + (b < max ( T X j + T τ X T τ X (c < T (d X <, (4

6 where nequalty (a s due to the Collatz-Welandt theorem, nequalty (b s due to the servce rate /X beng strctly larger than the arrval rate λ (as (6 enforces ths constrant, nequalty (c s obvous, and nequalty (d s due to the necessary stablty condton for a M/G/ queue. Applyng Lemma 4. to (4, shows that u s strctly postve. Ths proves the lemma. C. Convergence Theorem 4.4: If p exsts, then startng from p, theara algorthm produces a monotone ncreasng sequence of vectors p(k that converges to a fxed pont. Proof: By Lemma 4., we know p < p. Note that I ARA (p s a monotone non-decreasng functon. Thus, startng from p, wehavep( = I ARA ( p < I ARA (p and p( = I ARA ( p p. Suppose p( p( p(n p. Then monotoncty mples p = I ARA (p I ARA (p(n = p(n + I ARA (4 (p(n = p(n. That s, p p(n + p(n. Hence, the sequence p(n s nondecreasng and bounded above by p.itmplesthatp(n converges to a fxed pont. One can use a smlar approach to prove the convergence of the PE algorthm (9, and the proof s omtted. V. APPLICATIONS A. Feasblty Problem To demonstrate the utlty of the proposed model, we use the above algorthm to address the followng mportant ARA queston: In an IEEE 8. network, f the arrval rates λ and the requred delays D =[D,D,..., D N ] T are gven, does there exst a set of access rates [p,p,,p N ] T such that the resultng delay for each node s guaranteed to be smaller than D? We refer to ths problem as the average delay feasblty problem. More formally, we say that {(λ,d, (λ,d,, (λ N,D N } s feasble f there exst [p,p,,p N ] T such that Y (p D. (44 We argue that f there exsts a p such that the equalty holds (.e., Y D, for =,,..., N, then {(λ,d, (λ,d,..., (λ N,D N } s feasble. We mplctly assume n the followng that f a vector of delays s feasble, then any set of component-wse larger set of delays s also feasble. Equvalently, we have the expected channel access delay as D X =, (45 λ T +λ D where we substtute Y = D. Note that both D and λ are nputs, and hence X s completely determned by them. Consequently, ρ = λ X s also determned. Substtutng ρ nto ( yelds a fxed pont problem to determne contenton wndows p. One can use the ARA algorthm proposed n the prevous secton to solve ths fxed pont problem. After obtanng the fxed pont p,f <p < for all, then we can assert and conclude that the flows are feasble, and a feasble contenton wndow CW s then the maxmum nteger that s smaller than /p. Otherwse, we conclude that the flows are not feasble because f the fxed pont had exsted, the ARA algorthm s guaranteed to converge. We wll provde examples n smulatons. B. Mnmzaton of Delay We now consder a scheme for the delay mnmzaton problem that s solved by a central controller, e.g., access pont n a WLAN, whch collects the QoS requrements {(λ,d, (λ,d,..., (λ N,D N } from all nodes. Based on ths nformaton, the WLAN frst solves the feasblty problem n Secton V-A, and then optmzes the delay performance. Assume that the th node has a cost functon f (Y that s dfferentable, non-decreasng and strctly convex. Now, from (5, Y s convex n X. We substtute (5 nto f (Y to yeld a convex functon n X, whch we denote as f (X. Ths leads us to consder the followng optmzaton problem: N mn f (X (46 = s.t. X ˆX D := λ T +λ D, (47 X = I PE (X(p, (48 <p, (49 Varables: X,p. (5 Above, constrant (47 guarantees that the average delay s less than the requred delay. However, the constrant (48 that relates p to X s nonconvex. Hence, the optmzaton problem (46 s nonconvex. In ths paper, we use the barrer method [] to compute a local optmal soluton. The barrer method s an nteror-pont method whch, when started from a feasble pont, yelds a soluton n the nteror of the feasble regon. Ths property s useful for fndng a feasble soluton as t s crtcal to meet the delay requrements,.e., the delay constrants (47 as they are satsfed at all tmes. Based on (47 and (49, we consder the barrer functon: B (p := ˆX X (p + +. (5 p p Note that the barrer functon ncreases to + as any of the constrants approaches ts boundary. Let ε be a postve weght assocated wth B (p for all. Then we consder the optmzaton problem: N N max J(p := f (X (p + ε B (p. (5 We present the followng algorthm based on the gradent method to solve (5 []. Gradent Algorthm Obtan an ntal pont p by solvng the feasblty problem as dscussed n Secton V-A. We assume that each node has only one QoS flow for the AP.

7 For a fxed p k (output of the feasblty problem, run the PE algorthm tll convergence to some tolerance to obtan X k. For fxed p k and X k, obtan d J(pk from (5 and (48. d p k 4 Update p by p k+ = p k d J(p k β d p k. 5 Repeat from Step untl convergence to some small tolerance. We let β be a dmnshng stepsze []. Due to the nonconvexty, our gradent algorthm n general yelds a feasble soluton that s not the global optmal soluton of (46. However, by explotng the lnear system approxmaton n Secton IV-B, we obtan a relaxaton to (46 that yelds a lower bound to the global optmal value of (46: N mn f (X (5 = s.t. X ˆX, (54 X ((I F b(p /p, (55 <p,, (56 where (Ax denotes the th element of the vector Ax, and b(p =[(T τp + τ,(t τp + τ,, (T τp N + τ] T. Note that (5 s obtaned by relaxng the constrant (55 n (46 usng ( and some rearrangement. Now, (5 s stll nonconvex. However, by makng the change of varable p = log p for all, we obtan the followng convex problem that s equvalent to solvng (5: N mn f (X (57 = s.t. X ˆX, (58 X ((I F b(e p /e p, (59 p,, (6 where e a =[e a,e a,,e an ] T. Our smulatons show that (57 yelds a value that s slghtly smaller than the feasble soluton obtaned from the gradent algorthm, showng that our gradent algorthm can compute a near-optmal soluton. VI. SIMULATION RESULTS A. Smulaton Setup The smulaton envronment s created usng the NS- network smulator (verson ns. [4]. Table I summarzes the system parameters used n the smulaton. Throughout the smulaton, the only parameters that are changed are CW mn and CW max. Note that the model does not employ exponental back-off. Thus, after obtanng a CW from the analytcal model, we just set CW mn = CW max = CW to dsable the exponental back-off. These values of CW mn and CW max shown n Table I are referred to as the default settngs for comparson. Collocated topologes were created n whch all nodes can carrer-sense each other. Each sender node s attached to a Posson traffc generaton agent n whch packet nter-arrval tmes can be customzed. The nterface queues at each node used a Droptal polcy and the queue sze s set at 5 packets. Each smulaton was run for 4 seconds n smulaton tme. Two metrcs, namely the channel access Packet payload 4 bytes UDP header bytes MAC header 8 bytes PHY header 4 bytes ACK frame 8 bytes Channel bt rate Mbps PHY header bt rate Mbps Slot tme μs SIFS μs DIFS 5 μs CW mn CW max Retransmsson lmt 7 TABLE I SYSTEM PARAMETERS delays and the queueng delays, are measured for each flow. For the channel access delay, we measure the tme nterval from the nstant that the packet arrves at the head of the queue to the nstant that the packet successfully departs from the queue. For the queueng delay, the tme nterval from the nstant that a packet s sent by the applcaton layer (labeled by AGT n trace fles to the nstant that the packet s successfully receved s measured. B. Accuracy of the Analytcal Model The accuracy of the model s measured through three scenaros: channel access delays under saturated condtons, channel access delays under unsaturated condtons, and queueng delays. For each smulaton, both the smulaton results (denoted by smulaton and the theoretcal results obtaned from our model (denoted by theoretcal are plotted for comparson. Channel access delays under saturated condtons: Snce we clam our model s generc, t should apply to saturated condtons as well. In these smulatons, three lnks were examned. The sender of each lnk sends a saturated traffc to the recever. The theoretcal results are obtaned by applyng (, where ρ =due to saturated condtons. Two scenaros are studed. In the frst scenaro, CW of lnk s vared between 4 and 6 whle the contenton wndows of lnk and lnk are fxed wth CW = CW =. Fgure plots the smulaton results as well as the theoretcal results. One can observe that as CW s ncreased, lnk s access delays ncrease. Even though CW and CW are not changed, ther correspondng access delays decrease because CW s ncreased. In the second scenaro, CW was changed, whle holdng the fxed rato CW : CW : CW =::. The results are shown n Fgure. One can observe that except for the nonlnear part when CW s very small, the channel access delays agree wth the theoretcal values. The nonlnear ntal part of the curves s due to the fact that the collson probablty s extremely hgh when every node has a small CW for contenton resoluton.

8 Theoretcal (CW.45 Theoretcal (CW.6 Theoretcal (CW =CW Channel Access Delay (s Theoretcal (CW Theoretcal (CW Smulaton (CW Smulaton (CW Smulaton (CW Channel Acess Delay (s Theoretcal (CW =CW Smulaton (CW Smulaton (CW =CW Smulaton (CW =CW.5. CW Fg.. Saturated condtons: Channel access delays v.s. CW, where CW = CW = CW Fg.. Saturated condtons: Channel access delays v.s. CW, fx the rato CW : CW : CW =::. Channel access delays and queueng delays under unsaturated condtons: For the unsaturated condtons, three scenaros are examned. The frst scenaro s ntended to study how traffc arrval rates affect channel access delays. The nterarrval tme of flow s vared, whle keepng the other two lnks arrval rates fxed. The fxed packet nter-arrval tmes are λ =. and λ =.5. For the contenton wndow, CW = CW = CW =s set. Fgure plots the results. The theoretcal results are obtaned by solvng ( usng the PE algorthm. In the second scenaro, t s examned how CW affects channel access delays. Traffc arrval rates, and CW and CW are fxed. Only CW s changed from to 44. The results are shown n Fgure 4. One can observe that as CW s ncreased, the delays of flow ncrease. As a sde effect, the delays of flow and flow drop. The thrd scenaro s used to demonstrate how channel access delays change n response to the number of nodes. Each lnk has the same traffc rate and the same CW. In partcular, λ =. and CW =for all. Only the number of lnks s changed. Fgure 5 plots the results. As expected, the access delays ncrease as the number of lnks grow. Queueng delays: We repeat the same three scenaros for queueng delays. One can observe smlar trends n these fgures to ther counterparts for the channel access delays. From the three scenaros, one can see that the theoretcal results do accurately match the smulaton results. The accuracy s not only reflected n the trend but also n the quanttatve values. C. Performance Evaluaton In the followng smulatons, two case studes are examned to demonstrate the applcablty of the model and evaluate the performance of the proposed algorthm. Each pont n the fgures s a tme-average of the queueng delay over every 5 smulaton seconds. Feasblty: In the frst case, when the capacty s nsuffcent, the default 8. settng cannot meet the delay guarantees of all the QoS flows. But, wth the proposed scheme, one can fnd an approprate settng at whch all delay requrements are met. The three requred delays are assumed to be. seconds. Note that ths delay requrement s realstc accordng to [5]. The data rates of the nelastc flows are fxed as follows: λ =.5, λ =.4 and λ =.. The ARA algorthm was run to obtan a set of feasble contenton wndows: CW =66, CW =and CW =8. Fgure 9 plots the smulaton results. One can see that the default IEEE 8. can guarantee the delays only for flows and, whereas the delay of flow s much larger than the allowed delay. However, the network can actually guarantee all the delays f contenton wndows are approprately adjusted. In fact, one does see that the delays of all flows are met when the network uses the contenton wndows that are computed as descrbed n ths paper. Mnmzng delays: In ths case study, the performance of the scheme that mnmzes the average delays for nelastc flows, whle preservng ther delay guarantees s evaluated. The partcular cost functon ˆf (Y = Y (6 λ s used. Inelastc flows have fxed arrval rates λ =.4, λ =.4 and λ =.. The delay requrements are stll. seconds. Compared to the nput of the frst case, one can observe that the network capacty s suffcent for ths nput. Thus, there should be room for the flows to mprove the performance (.e., queueng delays n ths case. Usng the gradent algorthm presented n Secton V-B, the optmal CWs are computed to be CW = 9, CW =, and CW =9. The comparsons are plotted n Fgure. One can observe that when confgured wth the CWs suggested n ths paper, the network does acheve the optmzed delays and provde a certan level of farness. In contrast, n the default IEEE 8., flow suffers from bad delay performance and experences serous unfarness. VII. CONCLUSIONS We have presented a smple but apparently accurate model for analyzng queueng delay n non-homogeneous IEEE 8. MAC based WLANs. The model allows us to analytcally study the feasblty problem of whether the network can provde the mean delay guarantees requred by several non-homogeneous QoS flows. Future n order to optmze the performance of QoS flows, we have proposed an optmzaton algorthm to mnmze the mean delays, as measured by

9 Channel Access Delay (s 7 x Theoretcal (flow, nterarv=. Theoretcal (flow, nterarv=.5 Theoretcal (flow Smulaton (flow, nterarv=. Smulaton (flow, nterarv=.5 Smulaton (flow Channel Access Delay (s 8 x Theoretcal (CW =,nterarv=. Theoretcal (CW =, nterarv=.5 Theoretcal (CW, nterarvl =.4 Smulaton (CW =,nterarv=. Smulaton (CW =, nterarv=.5 Smulaton (CW, nterarvl =.4 Average Channel Access Delay (s 4 x.5.5 Theoretcal Smulaton Mean Interarrval Tme of flow CW Number of Clents Fg.. Unsaturated condtons: Channel access delays v.s.,where =., =.5, λ λ λ and CW = CW = CW =. Fg. 4. Unsaturated condtons: Channel access delays v.s. CW,where =., =.5, λ λ =.4 and CW λ = CW = CW =. Fg. 5. Unsaturated condtons: Channel access delays v.s. the number of lnks, where all = λ. and all CW =. Queueng Delay (s Theoretcal (flow, nterarv=. Theoretcal (flow, nterarv=.5 Theoretcal (flow Smulaton (flow, nterarv=. Smulaton (flow, nterarv=.5 Smulaton (flow Queueng Delay (s Theoretcal (CW =,nterarv=. Theoretcal (CW =, nterarv=.5 Theoretcal (CW, nterarvl =.4 Smulaton (CW =,nterarv=. Smulaton (CW =, nterarv=.5 Smulaton (CW, nterarvl =.4 Average Queueng Delay (s 5 x Theoretcal Smulaton Mean Interarrval Tme of flow CW Number of Clents Fg. 6. Unsaturated condtons: Queueng delays v.s.,where =., =.5, and λ λ λ CW = CW = CW =. Fg. 7. Unsaturated condtons: Queueng delays v.s. CW,where =., =.5, = λ λ λ.4 and CW = CW = CW =. Fg. 8. Unsaturated condtons: Queueng delays v.s. the number of lnks, where all =. and λ all CW =. Average Queueng Delay (s flow (CW =66, nterarv=.5 flow (CW =, nterarv=.4 flow (CW =8, nterarv=. flow (default 8. flow (default 8. flow (default 8. Average Queueng Delay (s flow (Mn Delay flow (Mn Delay flow (Mn Delay flow (default 8. flow (default 8. flow (default Smulaton Tme (s Fg. 9. A feasble soluton: Illustraton of the queueng delay dynamcs Smulaton Tme (s Fg.. Mnmzng delays: =.. λ λ =.4, λ =.4 and a certan cost functon, for a set of nelastc flows whle preservng mean delay guarantees. Extensve NS- smulatons have been conducted to verfy the accuracy of the model and to evaluate the performance of the algorthms. REFERENCES [] G. Banch, Performance analyss of the IEEE 8. dstrbuted coordnaton functon, IEEE Journal on Selected Areas n Communcatons, vol. 8, no., pp , March. [] F. Cal, M. Cont, and E. Gregor, Dynamc tunng of the IEEE 8. protocol to acheve a theoretcal throughput lmt, IEEE/ACM Trans. on Networkng, December. [] A. Abdrabou and W. Zhuang, Servce tme approxmaton n IEEE 8. sngle-hop ad hoc networks, IEEE Trans. on Wreless Communcatons, vol. 7, no., pp. 5, 8. [4] K. Medepall and F. A. Tobag, Towards performance modelng of IEEE 8. based wreless networks: A unfed framework and ts applcatons, n Proc. IEEE INFOCOM, 6. [5] O. Tckoo and B. Skdar, Queueng analyss and delay mtgaton n IEEE 8. random access MAC based wreless networks, n Proc. IEEE INFOCOM. IEEE, 4, pp [6] C. Coutras, S. Gupta, and N. Shroff, Schedulng of real-tme traffc n IEEE 8. wreless LANs, Wreless Networks, vol. 6, no. 6, pp ,. [7] M. Heusse, F. Rousseau, R. Guller, and A. Duda, Idle sense: An optmal access method for hgh throughput and farness n rate dverse wreless LANs, Proc. ACM SIGCOMM, 5. [8] L. Jang and J. Walrand, A dstrbuted CSMA algorthm for throughput and utlty maxmzaton n wreless networks, n Allerton Conference on Communcaton, Control, and Computng, 8. [9] Y. Gao, D.-M. Chu, and J. C. Lu, Determnng the end-to-end throughput capacty n mult-hop networks: Methodology and applcatons, Proc. ACM SIGMETRICS Perform. Eval. Rev., 6. [] A. Wald, Sequental tests of statstcal hypotheses, Ann. Math. Stat, vol. 6, pp. 7 86, 945. [] L. Klenrock, Queueng Systems. John Wley & Sons, 975. [] E. Seneta, Non-negatve matrces and Markov chans, Sprnger, 6. [] D. P. Bertsekas, Nonlnear Programmng. Athena Scentfc, 999. [4] The network smulator-ns, [5] Csco, Understandng delay n packet voce networks, 6.