Extended Abstract. University of Virginia. Charlottesville, VA fas4r j Abstract

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1 S GPS: Slow-Stat Genealzed Pocesso Shan Extended Abstact Anastasos Stamouls Jo Lebehe Depatment of Compute Scence Unvesty of Vna Chalottesvlle, VA 903 Emal: fas4 j jo@cs.vna.edu Abstact Pacet scheduln methods that appoxmate Genealzed Pocesso Shan GPS) ae cuently the focus of much eseach n the aea of Qualty-of-Sevce QoS) netwos. he ablty of GPS schedules to povde ate uaantees as well as delay uaantees meets the demand of many netwo applcatons. hs pape addesses a shotcomn of GPS whch has been ven lttle attenton, howeve, whch can have sncant mpact on the sevce povded by GPS. Snce, wth GPS, the sevce ate eceved by a sesson s popotonal to the numbe of bacloed sessons n the system, the sevce ate of a sesson may chane abuptly f some othe sesson becomes actve. hs may esult n abupt nceases of delay of consecutve pacets. In ths pape we popose a new schedule, called Slow-Stat GPS S GPS), whch allevates the poblem of abupt delay nceases when new sessons stat tansmttn. S GPS s a modcaton of GPS whee sessons do not eceve the uaanteed sevce ates mmedately afte they become actve. Instead, the sevce ates of such sessons ae adually nceased. hs We wll show that ths pevents an abupt delay ncease of the othe sessons. We deve delay bounds fo sessons constaned by leay bucets and we expess quanttatvely the advantaes of the S GPS dscplne. 1 Intoducton he Genealzed Pocesso Shan GPS) scheduln method s nown fo povdn suppot fo solaton and shan n a Qualty-of-Sevce QoS) netwo [4, 7, 8]. Even thouh GPS can povde ate uaantees to the sesson t sevces, a sesson that has been actve fo a lon peod of tme can expeence a sudden educton n ts sevce ate when some othe, pevously dle, sesson becomes actve. he decease of the sevce ates can be qute lae, esultn n a possbly sncant ncease of the delay of consecutve pacets of an actve sesson. o allevate ths poblem, we popose a modcaton to GPS, called Slow-Stat Genealzed Pocesso Shan S GPS) that pevents abupt ate and delay nceases by acefully deadn the sevce ate of actve sessons. hs s accomplshed by the follown modcaton to the onal GPS scheduln method. Wheneve a sesson becomes actve and stats sendn pacets, ths sesson s not assned the full bandwdth at once, but adually. he name \slow-stat" should ndcate that the sevce ate of a newly actve sesson s nceased slowly when the sesson stats tansmttn. As a esult, all sessons that have been actve see the sevce ates deceased smoothly.

2 he emande of ths extended abstact s stuctued as follows. In Secton we evew GPS and demonstate the poblem of dastc delay nceases n GPS. In Secton 3 we pesent the new Slow-Stat GPS schedule that allevates ths poblem. In Secton 3 we pesent an analytcal esult on the wost-case delays n S GPS. In Secton 4 we dene the pacetzed veson of S GPS and show how t can be mplemented usn the concept of vtual tme. We pesent conclusons n Secton 5. he Poblem: Abupt Delay Inceases n GPS A GPS schedule [7] s a wo-consevn schedule that seves all ncomn avals at a xed ate. Each sesson s chaactezed by aweht. Let S ; t) be the pocessn tme ven to sesson n the nteval ; t]. hen, a GPS schedule s dened as one fo whch: S ; t) S j ; t) j fo any pa of sessons and j that ae actve thouhout the nteval ; t]. If Bt) s the set of actve sessons at tme t, then evey sesson Bt) s seved at the nstantaneous sevce ate of: t) = P jbt) j 1) We efe to t) asthefa shae of sesson at tme t. A sesson s uaanteed a mnmum sevce ate of = P Nj=1 ) j whee N s the maxmum numbe of sessons that ae ben sevced by the GPS schedule. GPS s an dealzed schedule n that t wos unde the assumpton that all woload s nntely dvsble and that all bacloed sessons can be seved smultaneously. Snce a pacet schedule can tansmt only one pacet at a tme, appoxmaton technques ae needed that emulate the dealzed GPS system. Seveal appoxmatons of GPS have been poposed ecently, e.., [1, 3, 4, 5, 7, 1, 9, 10, 6, 13, 14]. An advantae of scheduln method deved fom GPS s that wthn cetan estctons) end-to-end delay bounds can be calculated wth elatve ease [8]. A poblem wth GPS: he follown example wll demonstate how the sevce ate of a sesson can decease unde GPS due to the actvty of othe sessons, and how, as a esult, the delay can ncease abuptly. Suppose that we have a vdeo tansmsson system ove an AM Pemanent Vtual Connecton PVC) wth a bandwdth of 1Mbps. On ths PVC, thee MPEG move tansmssons ae multplexed usn PGPS scheduln, a pacet-by-pacet veson of the GPS scheduln method [7]. he move tansmssons esult fom two softwae-compessed MPEG taces [?] Sessons 1 and send dentcal steams). Sessons 1 and have awost-case delay bound of 400 ms and sesson 3 has a wost-case delay of 00 ms. Sessons 1 and stat tansmttn at tme t = 0 sec. Fue 1a) depcts the delay of Sessons 1 and wthout tansmsson fom Sesson. In Fue 1b) we assume that Sesson stats tansmttn at tme t = 3 sec. Note n Fue 1a) that even wthout Sesson the bustness of the MPEG taces esults n abupt delay nceases; Fo example, the delay ncease at tme t =:8 sec s only due to bustness of the MPEG taces.

3 Howeve, the delay ncease caused by the busty natue of MPEG s much smalle than the delay caused by the aval of Sesson 3 at tme. Fom Fue 1b) we can see that when sesson 3 stats tansmttn at t =:8 sec. Hee, the delay of Sessons 1 and s nceased to ove 50 ms, coespondn to a 66% ncease of delay. hs example shows that even wth vey busty types of tac le MPEG), the abupt delay ncease when new sessons stat tansmttn can be damatc. hs poblem s esolved by the S GPS schedule pesented n the next secton Delay secs) 0.15 Delay secs) Aval me secs) a) Only Sesson 1 and tansmt. Fue 1: Abupt Incease of Delay Aval me secs) b) All sessons 1,, and 3) tansmt. 3 Slow-Stat GPS S GP S) In ths secton we popose a modcaton to the GPS scheduln method that pevents abupt delay nceases such as the one shown n the pevous secton. he delay nceases ae a dect esult of Equaton 1) fo calculatn the sevce ate t) fo a sesson. Wheneve the set of actve sessons, Bt), chanes, the sevce ate t) of a sesson s chaned also. heefoe, when a new sesson becomes actve,.e., the set Bt) ows, the sevce ate of all actve sessons s mmedately educed. As a soluton to mtate ths eect, we pesent the Slow-Stat GPS S GPS) scheduln method. In S GPS, wheneve a sesson becomes actve, ts sevce ate s nceased lnealy ove an nteval of lenth. hus, when a `new' sesson becomes actve, the sevce ate of the `old' sessons decease lnealy. Fo each sesson, > 0 speces the amount of tme that has to pass befoe sesson s assned ts fa shae of the bandwdth. If a sesson becomes actve at, the S GPS schedule nceases ts sevce ate lnealy n the nteval [ ; + ]. At tme +, the sesson s assned the same shae of bandwdth that s allocated by the GPS schedule and ven n Equaton 1)). If we use ^ t) to denote the sevce ate fo sesson that s allocated by the S GPS schedule at tme t, wehave: ^ t) = mnf t); t, t) : t 3) whee t) s the sevce ate allocated to sesson unde GPS. he ate ncease of sesson n the tme nteval [ ; + ] can be vewed as a slow-stat phase of ths sesson. At tme t = +, the slow-stat phase fo sesson s ove, and sesson wll be assned ts full fa shae of the bandwdth. In Fue we llustate the deence between the ate allocaton of GPS and S GPS. he ue depcts the sevce ate of a sesson as a functon of tme. We assume that sesson 3

4 Sesson becomes actve Sesson j becomes nactve end of busy peod Rate Allocated to Sesson t) GPS ^ t) S GPS t 1 t t 3 Fue : Sevce ate of a sesson as a functon of tme n GPS and S GPS. becomes actve at tme t 1 and that some pevously actve sesson j 6= becomes nactve at tme t. Unde GPS, the sevce ate t) schaned mmedately wheneve the set of actve sessons s chaned. Unde S GPS, the sevce ate adapts slowly wth a lnea slope. Note that the slope of the ate ncease afte tme 1 s completely detemned by and t); Howeve, the slope of the ate decease afte tme t addtonally depends on Bt), the set of actve sessons. 4 Analyss of S GPS he slow-stat phase ntoduced by S GPS can ncease the wost-case delays of a sessons, f compaed to a GPS schedule. o an nshts nto the delay pefomance of the S GPS we have deved analytcal wost-case delay bounds. Smla to [7], we assume that the tac of a sesson s constaned by a leay bucet. hat s, fo each sesson we have a pa of paametes ; ) and the maxmum tac that the sesson can submt to the schedule s constaned by A t) = + t. he devaton of the wost-case delay bounds too complex to be pesented hee. A complete devaton s avalable n [11]. Hee, we only pesent delay bounds n the follown theoem. heoem 1 InaS GPS schedule the wost-case delay foaleay bucet constaned sesson wth paametes ; ) and wth uaanteed ate s ven by = 8 >< = >: q : 1 + 3, )), 1 3, ) : +, ), 1 +, 1 q and,, 0 and 0 1 3, : 0 1 3, ) ) : 1, )+ 1 + ) : 1 + ) +, :, : othewse and 1 ) < and 0 ) 0, ) +, ), ) 4) 4

5 5 S PGPS: A Pacet-by-pacet Veson of S GPS Smla to GPS, S GPS assumes a ud model whee tac s nntely dvsble. In eal lfe, howeve,aschedule can only seve one pacet at a tme. Analoous to the wo n [7], we dene a pacet appoxmaton of S GPS, called Slow-Stat Pacetzed GPS o S PGPS. We can show that S GPS can closely appoxmate the ud model. In fact we have shown that a S PGPS system cannot fall behnd the coespondn S GPS system by moe than the tansmsson tme of a pacet wth maxmum sze. We have devsed an alothm that mplements S GPS usn the concept of `vtual tme', a concept that was poposed n [4, 7] fo mplementn GPS systems. In [4, 7], the vtual tme V t) = R t Pd t 1 s used as a measue of poess n the system n the tme nteval [ 1 ; ). Bt) When a pacet aves, the schedule assns t a `vtual nshn tme' and then seves pacets n nceasn ode of the vtual nshn tmes. Fo the p-th pacet of the -th sesson, the vtual stat tme S p) and vtual nsh tme F p) ae dened as [7]: S p) = maxff p,1) ;Vt p) ) 5) whee t p) F p) = S p) + Lp) 6) s the aval tme of pacet p and L p) s the lenth of pacet p. he dculty of ndn a pacet-by-pacet veson of S GPS, as compaed to the onal GPS and PGPS systems, s that the weht of a sesson s not constant f sesson s n the slowstat phase. We have solved ths poblem by assnn to sessons that ae n a slow-stat phase an \eectve" weht p) ef f and set the vtual nshn tme equal to Sp) Equaton 6) to calculate the vtual nshn tmes. he eectve weht p) ef f + Lp). hen we can use p) ef f can be ntepeted as the aveae weht of the pth pacet, aveaed ove the tansmsson tme of the pth pacet. If we set p) ef f to: p) ef f = NP p) L =1 ;6= w p, w p,1 ), L p) whee w p s an auxlay vaable ven teatvely by wth w,1 = 0): v u w p = tl p) P N=1 7) +w p,1 ) 8) we can pove the follown theoem [11] whch s the analoue to the esult n [7] fo the elaton on GPS and PGPS. heoem Let d p) ;S and dp) PGPS ;S denote the depatue tmes of the pth pacet of sesson GP S unde S GPS and S PGPS. hen we have d p) ;S PGPS, dp) ;S GP S L max 5 8p; 9)

6 In othe wods, a S PGPS system cannot fall behnd a S GPS system by moe than one maxmum pacet sze. We have pefomed smulaton expements [11] that demonstate how well S GPS educes the ate at whch delays can chane n a GPS system. Due to space lmtatons, the esults of the smulatons ae not ncluded n ths abstact. If accepted, we wll show the smulaton esults at the woshop. Refeences [1] J.C.R. Bennett and H. Zhan. Heachcal pacet fa queuen alothms. In ACM Scomm '96, Auust [] J.C.R. Bennett and H. Zhan. WFQ: Wost-case Fa Wehted Fa Queuen. In Poc. IEEE Infocom '96, Mach [3] J. Davn and A. Heybey. A smulaton study of fa queuen and polcy enfocement. Compute Communcaton Revew, 05):3{9, Octobe [4] A. Demes, S. Keshav, and S. Shene. Analyss and Smulaton of a Fa Queuen Alothm. In Poc. Scomm '89, paes 1{1, [5] S. J. Golestan. A self-cloced fa queuen scheme fo boandband applcatons. In Poc. IEEE Infocom '94, paes 636{646, [6] P. Goyal, H. M. Vn, and H. Chen. Stat-tme Fa Queuen: A Scheduln Alothm fo Inteated Sevces Pacet Swtchn Netwos. In ACM Scomm'96, paes 157{168, [7] A. K. Paeh and R. G. Gallae. A Genealzed Pocesso Shan Appoach to Flow Contol n Inteated Sevces Netwos: he Snle-Node Case. IEEE/ACM ansactons on Netwon, 13):344{ 357, June [8] A. K. Paeh and R. G. Gallae. A Genealzed Pocesso Shan Appoach to Flow Contol n Inteated Sevces Netwos: he Multple Node Case. IEEE/ACM ansanctons on Netwon, :137{150, Apl [9] D. Saha, S. Muhejee, and S. H. pah. Cay-Ove Round Robn: A Smle Cell Scheduln Mechansm fo AM Netwos. In IEEE Infocom'96, paes 630{637, [10] S. Sheedha and G. Vahese. Ecent fa queuen usn dect ound obn. IEEE ansanctons on Netwon, 43):375{385, June [11] A. Stamouls and J. Lebehe. S GPS: Slow-Stat Genealzed Pocesso Shan. echncal Repot R96-xx, Unvesty of Vna, In pepaaton. [1] D. Stlads and A. Vama. Desn and Analyss of Fame-based Fa Queuen: A New ac Scheduln Alothm fo Pacet-Swtched Netwos. In ACM Smetcs '96, paes 104{115, May [13] O. Yaon and M. Sd. Genealzed Pocesso Shan Netwos wth Exponentally Bounded Bustness Avals. Jounal of Hh Speed Netwos, 3:375{387, [14] Z.L. Zhan, D. owsley, and J. Kuose. Statstcal Analyss of Genealzed Pocesso Shan Scheduln Dscplne. In ACM Scomm'94, paes 68{77, Auust