Opportunistic Weighted Fair Queueing

Transcription

1 Opportunstc Weghted Far Queueng Knda Khawam, Danel Kofman GET/ENST Telecom Pars Emal: {khawam, Abstract The ongong evoluton towards the generalzaton of wreless access to mult-servce networks stresses on the need for optmzng the control of rado resources and n partcular, for desgnng effcent schedulng approaches. The schedulng mechansms proposed for wrelne lnks do not carry over to wreless nterfaces due to the varablty of rado lnks capacty. Effcent schedulers for the rado channel have to take nto account nformaton relatve to the channel state when allocatng resources to the varous connected equpments. For the case of delay tolerant traffc, the scheduler may take advantage of termnals dversty. We propose n ths paper a modfed verson of the Weghted Far Queueng (WFQ) algorthm, termed OWFQ (Opportunstc WFQ), that notably ncreases the average system performance through opportunstc schedulng whle fulfllng users QoS needs n terms of mnmum realzed throughput. I. INTRODUTION The number of devces accessng the network through wreless nterfaces s growng fast and wll overtake, n the comng years, the number of wrelne connected devces. Ths evoluton exacerbates the dffcultes rased by the scarcty of rado resources and ncreases the need for optmal utlzaton of the latter, especally n the case of nowadays mult-servce networks. Schedulng s a key tool for optmzng resource allocaton under the QoS constrants mposed by multservce networks. The algorthms that have been proposed for adaptng far queueng to the wreless doman (e.g. [5]) make the assumpton that the channel capacty s constant and try to make short bursts of channel errors transparent to flows by a dynamc reassgnment of channel allocaton over small tme scales. In ths paper we follow a dfferent approach: rather than falsely assumng that the shared capacty s constant, we propose to enhance the rado lnk utlzaton by allowng the scheduler to make ts decsons based on the knowledge of the varous termnals channel state. More precsely, we propose a modfed verson of the Weghted Far Queueng (WFQ) algorthm [1], termed OWFQ (Opportunstc WFQ), that notably ncreases the average system performance through opportunstc schedulng whle fulfllng users QoS needs n terms of mnmum realzed throughput. Our mechansm can enhance the performance of exstng and future systems, as those based on the ([6], [7]) technology. The rest of the papes organzed as follows. In Secton II-, we present the proposed OWFQ mechansm and n Secton III, we prove analytcally that OWFQ guarantees farness among competng flows. Secton IV s devoted to performance comparson between WFQ and OWFQ n a wreless envronment mpacted by fast fadng. We conclude n Secton V. II. OPPORTUNISTI WEIGHTED FAIR QUEUEING Most schedulng algorthms desgned for rado access dvde the flows n a smplstc and bnary way nto two categores: good channel-state flows that can be scheduled and bad channel-state flows that cannot be scheduled and consequently relnqush ther bandwdth to the former category. The scheduler keeps track of the excess bandwdth obtaned by goodchannel state flows n order to restore t to bad channel-state flows. Exstng algorthms dffer manly by the compensaton process they propose. The man lmtaton of these works s that channel condton s modelled as ether good or bad whch s too smple to characterze realstc wreless channels. ontrary to exstng schemes, our approach profts from the channel fluctuatons by way of opportunstc schedulng, seekng to augment overall Throughput under users QoS requests. In ths secton, we present the rado resource model and then ntroduce our schedulng mechansm. A. The rado model We consder a wreless system where the schedulng s performed by a Base Staton (BS) servng a multtude of flows n a downlnk channel of mean capacty. Let x be the..d. random varables (of unt mean) representng the effect of fast fadng experenced by flow, then the channel capacty of ths flow at tme t s: (t) = x (t) (1) As we consder Raylegh fadng, the random varables x are exponentally dstrbuted. However, we choose to bound the mnmum value taken by x to x mn to obtan a stable system. Indeed, snce E[ 1 x ] =, the mean servce tme s also nfnte and the system s not stable. Thus, we replace (1) by the followng: (t) = max ( x (t), x mn ) (2) B. Remnder on Weghted Far Queung WFQ s a packet schedulng technque allowng guaranteed bandwdth servces. It s an approxmaton of the Generalzed Processor Sharng (GPS) schedulng whch allows dfferent flows to have dfferent servce shares. In a lnk of capacty, WFQ guarantees to each flow of weght, a mnmum rate R gven by: R = j r j

2 The WFQ scheduler assgns a start tag and a fnsh tag to each arrvng packet and serves packets n the ncreasng order of ther fnsh tags. We denote by p k the k th packet of flow, by A(p k ) ts arrval tme and by S(pk ) and F (pk ), respectvely, the start and fnsh tags assgned to p k. The WFQ behavous defned by the followng equatons: S(p k ) = max ( V (A(p k )), F (p k 1 ) ) (3) F (p k ) = S(p k ) + Lk (4) where L k s the sze of packet pk and V (t) s the vrtual tme at tme t defned by: dv (t) dt = B(t) wth B(t) beng the set of actve flows at tme t.. Opportunstc Weghted Far Queung In order to account for the varablty of the channel capacty, we propose a new scheduler, named Opportunstc WFQ (OWFQ), defned as follows (as n WFQ, packets are scheduled n ncreasng order of ther fnsh tags): S(p k ) = max ( V (A(p k )), S(p k 1 ) + Lk 1 ) Whenever the BS fnshes servng a packet at tme T S, n order to schedule the next packet, t computes the fnsh tag for HOL (Head Of Lne) packets of actve flows accordng to the followng equaton: F (p k ) = S(p k ) + L k x (T S) The ratonale behnd OWFQ s as follows: the better the channel qualty experenced by a flow, the greater the value taken by x and the lower the value taken by ts fnsh tag. As a result, ts chance to be scheduled wll ncrease. Thus, potental canddates for accessng the channel may be compelled to pass ther turn n favour of flows wth better channel state whch wll obvously ncrease global Throughput. Besdes, flows that mssed ther turn wll not wat too long even f they experence persstng bad channel state because we do not change the defnton of the start tag and therefore the state of the channel only mpacts one of the terms defnng the fnsh tag. Indeed, the start tag s stll computed as n the orgnal model and reflects how frequently the present flow s served n respect to ts weght. For that reason, the penalzed flow wll eventually have the smallest fnsh tme among contendng flows despte ts large vrtual servce tme. As we consder fast fadng, the approach reaches ts optmal behavour when the packet duraton tme L s n the same or lower order of magntude of fast fadng varatons. Ths s the case when large capactes are provded to flows. Otherwse, (5) (6) (7) fragmentaton of packets wll enhance performances. III. FAIRNESS GUARANTEE Dfferent metrcs of farness can be defned. In ths secton, we use the weghted farness concept as defned n [4] and prove that, when usng OWFQ, s W(t1,t2) Wj(t1,t2) r j bounded for any nterval [t 1, t 2 ] n whch both flows and j are backlogged, where W (t 1, t 2 ) s the aggregate servce (n bts) receved by flow n the nterval [t 1, t 2 ]. The proof s obtaned by establshng an upper and lower bound on W (t 1, t 2 ) n Lemmas 1 and 2 respectvely (detaled proofs are found n VI). Lemma 1: In an OWFQ server, f flow s backlogged n the nterval [t 1, t 2 ], then: W (t 1, t 2 ) (v 2 v 1 ) L max L max x mn Where v 1 = V (t 1 ) and v 2 = V (t 2 ). We conclude that OWFQ guarantees a mnmum throughput for each flow. Lemma 2: In an OWFQ server, durng any nterval [t 1, t 2 ], we have that: W (t 1, t 2 ) (v 2 v 1 ) + L max Theorem 1: For any nterval [t 1, t 2 ] n whch two flows and j are backlogged durng the entre nterval, the dfference n the servce receved s bounded by: W (t 1, t 2 ) W j(t 1, t 2 ) L max( )+ r j r j L max x mn max( 1, 1 r j ) The result s easly obtaned from Lemmas 1 and 2. We conclude that OWFQ provdes farness guarantees. IV. NUMERIAL EXPERIMENTS The values chosen n the followng numercal analyss are nspred by the propertes of systems. We consder a wreless channel of mean capacty equal to = 1M bps subject to Raylegh fadng. Packet sze follows the Bounded Pareto (BP) dstrbuton whch s commonly used n analyss because t can exhbt the hgh varance property as observed n the nternet traffc [8]. We denote the BP dstrbuton by BP (p, q, α) where p and q are respectvely the mnmum packet sze (5 bytes) and maxmum packet sze (15 bytes) and α s the exponent of the power law. The probablty densty functon of the BP dstrbuton s: f BP (x) = α pα ) α x α 1, p x q, α 2 1 ( p q

3 We take α = 1.16 and therefore the mean packet sze equals 155 bytes. We consder three ndependent permanent Posson processes of ntensty λ/3 wth the followng weghts r =.5, r 1 =.25 and r 2 =.1. The global arrval process s therefore a Posson process of ntensty λ. In order to vary the load n the cell, we vary λ. We take x mn =.1 and we normalze the mean capacty to = 1.. The buffer capacty of each flow s lmted to 1 6 packets. New packets generated when the buffes full are lost. We consder two ndependent models, n the frst, flows are served accordng to the WFQ algorthm and n the second, flows are served accordng to our Opportunstc WFQ. The total load ρ s defned as the rato of the total arrval rate, λ, over the average capacty. We frst analyze and compare, for both models, the mpact of total load on the throughput, the packet loss rato and the percentage of packets served at the mnmum rate for each ndvdual flow. Then we analyse the mpact of total load on delay. Fg. 1. Mean Throughput per flow as a functon of ρ As we can see n Fgure 1, for ρ.2, the OWFQ does not realze any gan n terms of throughput compared wth WFQ whch s natural snce the probablty of havng more than one bottleneck flow s neglgble and therefore no advantage s taken from takng nto account the rado channel state. Any schedulng polcy based on opportunsm wll brng no mprovement n comparson wth a non-opportunstc scheme f there are not at least two actve users present n the system when the schedulng decson s made. For ρ.3, OWFQ does not realze any gan n terms of throughput for flows and 1 snce only flow 2 has bottlenecked packets wth non neglgble probablty; accordng to smulaton results, the mean number of packets enqueued for flows and 1 s strctly nferor to 2 for ths range of the total load. Nevertheless, whle the number of packets served at the mnmum rate n WFQ remans constant when ρ vares n the cted range, (approxmately equal to 9.6% of the total number of served packets), the latter decreases n OWFQ as load ncreases passng from 8% for ρ =.2 to 6% for ρ =.3. As for flow 2, the mean number of packets enqueued s around 2 whch means that the lattes often actve and served smultaneously ether wth flow or flow 1. As a result, flow 2 realzes a gan n mean throughput of approxmately 2%. For.4 ρ 1., the gan obtaned from mult-user dversty s tangble: for flow, the gan n mean throughput as compared to WFQ vares from 6% for ρ =, 4 to 128% for ρ =.4. For flow 1, the gan vares from 6% to 127% and for flow 2, the gan vares from 25% to 128%. Moreover, at ρ = 1., flow and flow 1 lose respectvely 3% and 7% of ther packets n WFQ due to buffer overflow whle no packets losses have been observed durng the smulaton tme n the OWFQ case. Besdes, 9.6% of the packets are served at mn n WFQ aganst respectvely 4% and 2% n OWFQ. As for flow 2, the number of lost packets, at ρ = 1., s 85% n WFQ aganst 55% n OWFQ whle the number of packets served at mn s lmted to.4% n OWFQ (ther number remans the same n WFQ for all flows and equals 9.6% whatever the load s). Besdes, the dfferentaton n the servce receved by flows and 1 appears n OWFQ at ρ >.8 whle t appears earlen WFQ at ρ =.7 whch hghlghts the effcent allocaton of resources n OWFQ. For 1. < ρ 1.5, the gan n mean throughput for flow s around 137%, whle at ρ = 1.5 half of packets are lost n WFQ and only 1% n OWFQ. The number of packets served at mn for flow s around 5% n OWFQ. For flow 1, the gan s around 136% and whle the number of lost packets s only lowered from 9% to 8% at ρ = 1.5, the number of packets served at mn s around 1% n OWFQ. As for flow 2, the gan n throughput s around 129% and although the blockng probabltes are alke n both scenaros, the number of packets served at mn s only.1%, at ρ = 1.5, n OWFQ. We conclude that, n addton to notably ncreasng performances n terms of realzed throughput, our approach protects flows at much hgher rates than n plan WFQ. The last remark allows predctng that Opportunstc WFQ wll lead to a reducton n average sojourn tmes. We analyse now the mpact of the total load on the relatve devaton dt of the Mean Sojourn Tme n the WFQ case, termed T W F Q, from the Mean Sojourn Tme n the OWFQ case, termed T OW F Q, whch we defne as dt = T W F Q T OW F Q /T W F Q 1. We can see from Fgure 2 that the Mean Sojourn Tme n OWFQ s dramatcally reduced as compared to WFQ. Therefore, the gap n the realzed mean throughput between OWFQ and WFQ wll wden even more n a realstc model based on the TP protocol where the delay experenced by packets has severe negatve repercussons on the overall performances. We analyse now the Total Throughputs. We denote by W F Q and OW F Q the Total throughput for the WFQ and the OWFQ cases respectvely. From Fgure 3, we can see

4 dt dt dt 1 dt 2 OWFQ T h T h j OW F Q TABLE I Load Fg. 2. dt as a functon of ρ WFQ T h T h j W F Q TABLE II Total Throughput Fg. 3. Load owfq wfq Total Throughput as a functon of ρ to the usage of an opportunstc approach. To show that the gan obtaned ncreases wth the number of users (mult-user dversty), we run three sets of smulatons wth respectvely n = 6, 1 and 14 flows at overload. For every value of n, flows are dvded nto two categores wth equal number of flows ( n 2 ). All flows of a gven category have the same weght and the weght of flows of the frst category s twce the weght of flows of the second category. We compute the Total Throughput W F Q and OW F Q for both schedulers, gven by (T h 1 + T h 2 ) n 2, where T h 1 and T h 2 denote, respectvely, the average rate of flows of categores 1 and 2. the remarkable gan realzed n terms of Total throughput n OWFQ n comparson wth WFQ. We notce also that W F Q converges to.36 whch s very nferor to the mean capacty, gven by the followng: E[(t)] = max(x,.1) e x dx whch n our experments s approxmately equal to 1. (we chose = 1.). To nterpret ths result, we observe that our model s equvalent to a model of constant capacty fed by the same arrval process as n the orgnal model but wth packets of length L x. We thus have the followng stablty condton: λ E[ L ] (8) x From (8), we have that: λ E[L ] E[ 1 x ] λ E[L ] E[ 1 x ] Hence, the actual mean capacty of the system s: E[ 1 x ] = 1 max(x,.1).36 (9) e x dx whch explans the result obtaned n WFQ. In OWFQ, the behavour of the system s much more complex so we cannot provde a smlar analytcal evaluaton of the capacty. The gan shown n Fgure 3 s of course due Frst, we notce that OW F Q ncreases sgnfcantly wth the number of flows. We also observe that W F Q s approxmately equal to.36 for all values of n, whch was expected from formula (9). To estmate the gan obtaned n terms of realzed Total Throughput, we compute the relatve devaton of OW F Q from W F Q by applyng the followng formula G n = W F Q OW F Q / W F Q. We get the subsequent results G , G and G Hence, we can see that the acheved gan s sgnfcant and that t ncreases wth the number of served flows, whch means that our scheduler takes advantage of flows dversty. Moreover, we see that for both schedulers, we have T h 2 T h j and thus, the dfferentaton n the realzed throughputs s acheved. V. ONLUSION Far queueng has long been a popular paradgm for guaranteeng mnmum throughput for users or flows sharng a wrelne lnk. In ths paper, we propose a new scheduler that couples opportunstc schedulng wth weghted far queueng to enhance overall performances n the case of wreless lnks by takng advantage from the rado channel varatons. We proved analytcally that the Opportunstc WFQ guarantees farness among users and we showed through smulatons that our soluton provdes sgnfcantly better performances than the WFQ approach. In future work, we wll evaluate the nteractons of the proposed scheduler wth the dynamcs of TP and we wll ntegrate to ts mechansm prevous proposed deas allowng to deal wth the case where certan flows are

5 blocked due to very bad channel qualty. A. Proof of Lemma 1: VI. APPENDIX If (v 2 v 1 ) L max Lmax x mn, Lemma 1 holds trvally snce W (t 1, t 2 ). Hence, we consder the case where: v 2 > v 1 + L max + L max x mn (1) Let packet p k be the frst packet of flow to receve servce n (v 1, v 2 ). To observe that such a packet exsts, we consder the followng two cases: Packet p n such that S(p n ) < v 1 and S(p n ) + Ln > v 1 exsts: snce flow s backlogged n [t 1, t 2 ], we conclude that V (A(p n+1 )) v 1. From (6), we get ) = S(p n ) + Ln. Usng the fact that S(p n ) < v 1, we get that ) < v 1 + Lmax. Also, usng (1), we deduce: ) < v 2 (11) Snce ) = S(p n ) + Ln > v 1, usng (11), we conclude that ) (v 1, v 2 ). Packet p n such that S(p n ) = v 1 exsts: p n may fnsh servce at tme t < t 1 or t t 1. In ether case, snce flow s backlogged n [t 1, t 2 ], we conclude that V (A(p n )) v 1. Hence ) = S(p n ) + Ln. Usng the fact that ) < S(p n ) + Lmax and S(p n ) = v 1, we get from (1) that ) < v 1 + Lmax < v 2. Snce ) = v 1 + Ln > v 1, we conclude that ) (v 1, v 2 ). Snce ether of the two cases always holds, we conclude that packet p k such that S(pk ) (v 1, v 2 ) exsts. Furthermore, we have the addtonal followng result: S(p k ) < v 1 + L max (12) Let p k+m be the last packet to receve servce n the vrtual tme nterval (v 1, v 2 ). Thus, F (p k+m+1 ) v 2. From (6) and (7), we know that at tme T S: F (p k+m+1 ) = S(p k+m ) + Lk+m + Lk+m+1 x (T S) We deduce the followng result: (13) From (12) and (16), we get the followng result: m L k+j (v 2 v 1 ) L max L max x mn Snce W (t 1, t 2 ) m B. Proof of Lemma 2: Lk+j, Lemma 1 follows. The set of flow packets durng tme nterval [t 1, t 2 ] have start tags at least v 1 and at most v 2. Ths set can be parttoned n two subsets: Set D consstng of packets that have start tags at least v 1 and strctly nferor to v 2. Formally, D = {k v 1 S(p k ) < v 2 F (p k ) v 2}. For packets n D, usng (6) and (7), we get k D lk (v 2 v 1 ). Set E consstng of packets that have start tags equal to v 2 and fnsh tags strctly greater than v 2. It s obvous that at most one packet belongs to ths set, because S(p k ) = v 2 S(p k+1 Hence k E lk l max. We conclude that Lemma 2 holds. REFERENES ) = S(p k ) + Lk+1 > v 2 [1] A. Parekh. and G. Gallager, A generalzed processor sharng approach to flow control n ntegrated servces networks: the multple node case, IEEE/AM Trans. on Networkng, vol. 1-3, pp , June [2] P. Goyal, H. Vm, and H. hen, Start-tme Far Queung: A Schedulng Algorthm for Integrated Servces Packet Swtchng Networks, IEEE/AM Trans. on Networkng, vol. 5, pp ,Oct [3] H. Zhang, Servce dscplnes for guaranteed performance servce n packet-swtchng networks, Proceedngs of IEEE, October [4] S. Golestan, A self-clocked far queueng scheme for broadband applcatons, Proceedngs of the IEEE INFOOM 94, pp , [5] V. Bharghavan, S. Lu, and T. Nandagopal. Far queueng n wreless networks: Issues and approaches. IEEE Personal ommuncatons Magazne, Feb [6] A. Gosh, Broadband wreless access wth WMax/82.16: current performance benchmarks and future potental, IEEE ommuncatons Magazne. [7] [8] rovella M., Bestavros A., Self-Smlarty n World Wde Web traffc: Evdence and possble causes, IEEE/AM Transactons on Networkng, pp , December S(p k+m ) v 2 Lk+m r L max (14) x mn Usng the taggng scheme n Secton II-B, we can derve the followng: m 1 S(p k+m ) = S(p k ) + L k+j (15) Thus, from (14) and (15), we get that: m S(p k L k+j ) + v 2 L max (16) x mn