Inter-Class Resource Sharing using Statistical Service Envelopes

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1 In Proceedngs of IEEE INFOCOM 99 Iner-Class Resource Sharng usng Sascal Servce Envelopes Jng-yu Qu and Edward W. Knghly Deparmen of Elecrcal and Compuer Engneerng Rce Unversy Absrac Neworks ha suppor mulple servces hrough lnksharng mus address he fundamenal conflcng requremen beween solaon among servce classes o sasfy each class qualy of servce requremens, and sascal sharng of resources for effcen nework ulzaon. Whle a number of servce dscplnes have been devsed whch provde mechansms o boh solae flows and farly share excess capacy, admsson conrol algorhms are needed whch explo he effecs of ner-class resource sharng. In hs paper, we develop a framework of usng sascal servce envelopes o sudy ner-class sascal resource sharng. We show how hs servce envelope enables a class o over-book resources beyond s deermnscally guaraneed capacy by sascally characerzng he excess servce avalable due o flucuang demands of oher servce classes. We apply our echnques o several mul-class schedulers, ncludng Generalzed Processor Sharng, and desgn new admsson conrol algorhms for mul-class lnk-sharng envronmens. We quanfy he ulzaon gans of our approach wh a se of expermens usng long races of compressed vdeo. I. INTRODUCTION Fuure negraed servces neworks wll suppor heerogeneous Qualy of Servce (QoS) specfcaons and raffc demands. For example, a deermnsc servce [1] uses worscase resource allocaon o suppor applcaons requrng packe delvery whou losses or delay bound volaons; a sascal servce [] acheves a sascal mulplexng gan and provdes sascal QoS guaranees wh conrolled over-bookng of resources; a measuremen-based servce [3] suppors QoS by basng admsson conrol decsons on emprcal observaons of aggregae raffc behavor; bes-effor servces suppor applcaons wh less srngen QoS requremens such as bulk daa ransfer. Wh approprae admsson conrol and raffc schedulng, hese servces and ohers can co-exs n a sngle nework, as admsson conrol lms he number of admed raffc flows o ensure ha each class QoS requremens are me, and packe schedulers ensure ha packes are assgned he prory levels needed o mee her QoS objecves. In a lnk sharng envronmen as oulned n [4], raffc class k s allocaed capacy c k such ha whenever packes from class k are backlogged, he class receves servce a a rae of a leas c k. If class k s no backlogged, hen class k s unused capacy s dsrbued farly among backlogged sessons. Consequenly, classes can be assured o mee her respecve QoS requremens, regardless of he behavor of oher raffc classes, allowng any number of servces o co-exs n he nework. In he leraure, a number of servce dscplnes have been de- Ths work was suppored by NSF CAREER Award ANI , NSF Gran ANI-97314, and Noka Corporaon. The auhors may be reached va hp:// sgned o suppor such lnk sharng objecves [4], [5], [6]. For example, [5] develops a class of Herarchcal Packe Far Queueng algorhms focusng on an algorhm s farness, complexy, and ably o provde low end-o-end deermnsc delay bounds. Whle schedulng algorhms for effcenly and farly allocang excess capacy o backlogged classes are an mporan aspec of a lnk-sharng nework, an admsson conrol polcy ha enables one class of raffc o quanfy he mproved QoS wll receve due o capacy unused by oher classes has no been addressed. In addon o servce dscplnes, a number of admsson conrol algorhms have also been desgned boh for deermnsc servces whch do no explo sascal resource sharng [7], [8], as well as sascal [], [9], [1], [11], [1], [13] and measuremen-based servces [3] whch do. However, such admsson conrol algorhms consder raffc classes n solaon, and whle a sascal mulplexng gan s acheved whn a parcular raffc class, ner-class resource sharng s no addressed. In parcular, [1], [13] sudy sascal servce for Generalzed Processor Sharng (GPS) [8], and whle he solaon propery of GPS s exploed, ner-class sascal resource sharng s no addressed. Moreover, whle [1] allows vdeo on demand sysems o explo sascal gans from real me raffc flows, does no address general lnk sharng envronmens. In hs paper, we address he problem of ner-class sascal resource sharng. Our key echnque s o develop a framework of sascal servce envelopes o sudy he problem. Inspred by [7], [14], we defne a sascal servce envelope as a probablsc descrpon of he servce avalable o a raffc class as a funcon of nerval lengh. We use hs servce envelope o characerze he addonal capacy avalable o a raffc class beyond he mnmum deermnscally guaraneed capacy se asde by he lnk sharng rules. In hs way, we sascally capure he flucuang excess capacy lef unused by one raffc class so ha anoher class may explo an ner-class sascal mulplexng gan and poenally adm addonal raffc flows ha would no oherwse have been deemed admssble. Thus, we use he sascal servce envelope as a ool for overbookng ner-class resources n a conrolled manner, so ha a class can probablscally quanfy he addonal resources avalable n a lnk sharng envronmen. We apply hs framework of sascal servce envelopes o wo mul-class servce dscplnes, namely, Sac Prory (SP) and lnk-sharng GPS [4], [5]. We show ha whle he concep of a sascal servce envelope was mplcly used n prevous

2 sudes of SP [11], explcly compung he servce envelope of oher raffc classes provdes a smpler analyss and allows us o unformly rea deermnsc and sascal servce classes. For GPS, we concepually paron raffc classes no solaon classes and sharng classes dependng on wheher or no he raffc class wll explo he effecs of ner-class resource sharng n makng admsson conrol decsons. For example, a deermnsc servce s an solaon class as excess capacy from oher raffc classes s no guaraneed n he wors case and hence a sascal envelope of excess capacy canno mprove hs class admssble regon. We hen bound he servce receved by a raffc class wh an arbrary paron of he classes, and show ha he above paron no solaon and sharng classes can ghly approxmae he sascal servce envelope obaned by he sharng classes. In hs way, each sharng class can characerze he capacy avalable beyond s guaraneed rae, ncorporang he relave weghs and raffc demands of all oher raffc classes, and mprovng he class admssble regon. We llusrae he poenal ulzaon gans of our ner-class resource sharng scheme wh a se of race-drven smulaon expermens usng long races of MPEG-compressed vdeo. As an llusrave example wh a 45 Mbps lnk supporng equally weghed deermnsc and sascal servce classes wh he GPS servce dscplne, we fnd ha he average ulzaon of he lnk can be mproved from 47.7% o 84.6% by usng he sascal servce envelope o characerze he excess capacy of he deermnsc class. II. STATISTICAL SERVICE ENVELOPES: THEORY AND APPLICATIONS In hs secon, we defne sascal servce envelopes and develop her applcaons o ner-class resource sharng. In parcular, we frs sudy he delay dsrbuon for a sngle class usng sascal raffc envelopes and deermnsc servce envelopes. Nex, we exend hs analyss o nclude sascal raffc envelopes and sascal servce envelopes. Fnally, we llusrae he applcaon of sascal servce envelopes by dervng admsson conrol ess for SP schedulers usng hs heory. A. Sngle Class Queueng Model Throughou hs paper, we model a mulplexer by a dscreeme nfne buffer queue n whch flud flows no and ou of he buffer only a dscree nervals. For raffc class, le Xk denoe s aggregae arrvals n me slo k, and le X denoe P he oal arrvals beween slos j and k, such ha X = k =j X. Le represen he amoun of flud served for raffc class n me slo k, and denoe Y as he oal flud served beween me Yk P slos j and k, such ha Y = k =j Y. Denong Q k as he backlog of raffc class a he end of me slo k, Q k s obaned from he Lndley recurson as, Q k = max jk fx Y g (1) where he maxmum wll be reached a j f Q j 1 =. B. Deermnsc Servce Envelopes X m Deermnsc servce s suded n [7] usng deermnsc servce envelopes and deermnsc raffc envelopes. Here, we frs sudy sascal servce wh sascal raffc envelopes and deermnsc servce envelopes, and laer focus on sascal servce envelopes. Frs, we formally defne boh deermnsc and sascal raffc envelopes and servce envelopes. We refer o an nerval [j; k] as class s backlogged nerval f Q m >, for m = j; ; k. Defnon 1 (Avalable Servce) For gven npu process of all raffc classes excep, we defne he avalable servce e Y as he oupu of he h class n he nerval [j; k] gven a mnmally backloggng npu process e X, whch s defned as he mnmal class npu such ha class s connuously backlogged hroughou nerval [j; k]. Noe ha avalable servce Y e s a funcon of he schedulng mechansm and npu process X m, m 6=, and s ndependen o he npu process n class ; whereas he acual oupu process Y s decded by all classes npus. By usng hs noaon of avalable servce, we decouple he mpac of class s npu on Y, and make Y e a pure descrpon of avalable nework resources, separae from he raffc ha s acually sen. Defnon (Deermnsc Servce Envelope) 1 A non-decreasng non-negave funcon s () s a deermnsc servce envelope of raffc class, f for any nerval [j + 1; j + ], he avalable servce sasfes ey j+1;j+ s (): To llusrae he concep of a deermnsc servce envelope, noe ha for a FCFS server wh capacy C, e Yj+1;j+ = s() = C. In a GPS server, a servce class wh guaraneed rae g, sasfes ey j+1;j+ s () = g : Defnon 3 (Deermnsc Traffc Envelope) [15] A nondecreasng non-negave funcon b () s a deermnsc raffc envelope of class, f for any nerval [j + 1; j + ], he npu raffc sasfes X j+1;j+ b (): Defnon 4 (Sascal Traffc Envelope) [] A sequence of random varables B () s a sascal raffc envelope of class, f for any nerval [j + 1; j + ], he npu raffc sasfes X j+1;j+ s B (): where X j+1;j+ s B () (sochasc nequaly) denoes P [Xj+1;j+ > z] P [B () > z] for all z: Denong Dk as he vrual delay experenced by a b arrvng a me slo k, he key QoS merc ha we consder s he probably of delay bound volaon, P [D > d ]. As long as EX1; s () lm < lm!1!1 1 Ths defnon s a slgh generalzaon of he one n [7]. Wh abuse of noaon, Y c for a consan c denoes P(Y c) = 1.

3 (he sably condon), and Xk s saonary and ergodc, P [Dk > d ] converges o a seady sae al probably P [D > d ]. Proof. P [D k > d ] converges o P [D > d ]. From Equaon (3), P [D k > d ] = P [max jk fx e Y +d g > ]: (5) Q D X 1,k d Y 1,k Fg. 1. Delay and Buffer Occupancy Fgure 1 shows he delay and buffer occupancy n erms of X 1;k and Y 1;k f he buffer s nally empy. The vrual delay D k s defned as [7] D k = mn f : and X 1;k Y 1;k+ g: () Lemma 1: For a delay bound d, he even of delay bound volaon n class a me slo k sasfes fdk > d g fmax jk fx Y e +d g > g: (3) Proof. By defnon fd k > d g fx 1;k Y 1;k+d > g = fmax jk fx Y +d g > g: Observe ha f max jk fx Y +d g >, hen max jk fx Y e +d g >. Ths s because f max jk fx Y +d g >, here mus exs an such ha s = maxfj : j < k and Q j = g max jk fx Y +d g = X s+1;k Y s+1;k+d ; [s + 1; k + d ] s a backlogged nerval of class, and ey s+1;k+d Y s+1;k+d ; snce e Y s+1;k+d s he mnmum backlogged servce. Thus fd k > d g fmax jk fx e Y +d g > g: Theorem 1: For a servce class, wh deermnsc servce envelope s () and sascal raffc envelope B (), he al probably of P [D > d ] s gven by P [D > d ] P [max fb () s ( + d )g > ]: (4) From Defnon 4 and Defnon, such ha max jk fx e Y +d g s max jk fb (k j + 1) s (k + d j + 1)g P [max jk fx e Y +d g > ] P [maxfb () s ( + d )g > ]: C. Sascal Servce Envelopes Theorem 1 enables us o explo he sascal mulplexng gan of flows whn a servce class. Whle he deermnsc servce envelope s () provdes solaon among servce classes and smplfes admsson conrol, precludes sascal nerclass resource sharng. In mul-class schedulers such as SP and GPS, he ulzaon gans avalable from explong ner-class resource sharng can be sgnfcan. Nex we nroduce a sascal servce envelope o sudy he ner-class resource sharng problem, and develop new heory o calculae he delay bound volaon probably usng sascal servce envelopes. In a mul-class server, he avalable servce for class, Y e, s a funcon of he npu raffc n oher classes, and of he parcular servce dscplne whch specfes how o schedule servces among compeng classes. The nerference among classes s refleced n Y e, and n some cases, s possble ha he avalable servce s far greaer han he mnmally guaraneed servce,.e., Y e s (k j + 1). Thus we defne a sascal servce envelope o descrbe he avalable servce beyond he deermnscally guaraneed s (). Defnon 5 (Sascal Servce Envelope) A sequence of random varables S () s a sascal servce envelope of raffc n class, f for any nerval [j + 1; j + ], he avalable servce sasfes ey j+1;j+ s S (): Noce ha whle a deermnsc servce envelope s () descrbes he servce of a class n solaon, he sascal servce envelope S () descrbes ner-class resource sharng. We employ S () n he delay dsrbuon calculaon wh he followng heorem. Theorem : For a servce class, wh sascal servce envelope S () and sascal raffc envelope B (), he al probably of P [D > d ] s gven by P [D > d ] P [max fb () S ( + d )g > ]: (6)

4 Proof. From Equaon (5), P [D k > d ] = P [max jk fx e Y +d g > ]: (7) From Defnon 4 and Defnon 5, so ha max jk fx e Y +d g s max jk fb (k j + 1) S (k + d j + 1)g P [max jk fx e Y +d g > ] P [maxfb () S ( + d )g > ]: Below we employ Theorem o devse admsson conrol algorhms for mul-class servers ha explo ner-class sascal resource sharng. D. Sac Prory Admsson conrol for sac prory schedulers was suded n [11], [16], here we approach he problem usng servce envelopes. Consder an SP scheduler wh N prory queues, lnk speed C, and he aggregae raffc n class bounded by B () and b (), wh = 1; : : : ; N denong he prory level from hgher prory o lower prory. The sascal servce envelope for class s S () = (C X 1 j=1 The deermnsc servce envelope for class s s () = (C X 1 j=1 B j ()) + (8) b j ()) + (9) where b () = P jc b j (); B () = P jc B j (); and b j () and B j () are he sascal and deermnsc envelopes of he jh flow n class. Lemma : Consder an SP scheduler wh N prory queues and lnk speed C. For each servce class, raffc s bounded by B () and b (), wh QoS parameers (d ; P ), where d s he delay bound, and P s he delay bound volaon probably. The QoS for all servce classes n hs mul-servce SP scheduler s sasfed f for all deermnsc servce classes wh P =, maxfb () + X 1 k=1 b k ( + d ) C( + d )g and for all sascal servce classes wh P >, P [max fb () + X 1 k=1 B k ( + d ) C( + d )g > ] P : Proof. For sascal servce classes, Equaon (8) gves B () S ( + d ) s B () C( + d ) + X 1 k=1 B k ( + d ); and applyng Theorem requres P [max fb () S (+d )g > ] < P. Thus, f P [max fb () + P 1 k=1 Bk ( + d ) C( + d )g > ] P ; hen he sascal servce n he h servce class s sasfed. For deermnsc servce classes, he proof s smlar. Noe ha ner-class nerference n an SP scheduler s n a sngle drecon, only from hgher prory classes o lower prory ones. For GPS, we wll see ha every class affecs every oher class such ha he sascal servce envelope for one class becomes a funcon of he raffc envelopes and relave weghs of all oher classes. III. INTER-SERVICES RESOURCE SHARING IN LINK-SHARING GPS In Secon II, we developed ools for managng mul-class servces usng sascal servce envelopes, consderng SP as a specfc example. Here we sudy a lnk-sharng GPS server, agan usng he framework of sascal servce envelopes, wh a goal of ncreasng he oal ulzaon of he mul-class GPS server by explong ner-class resource sharng. A. Generalzed Processor Sharng Flows Classes Lnk g 1 g g N De. Sa. MBAC Sa. MBAC Fg.. Sysem Model for Admsson Conrol Bes-Effor Fgure shows he sysem model for admsson conrol n a mul-class GPS scheduler (see [4] for example). There are N servce classes n he sysem, each allocaed a wegh. Each servce class provdes eher deermnsc, sascal, measuremen-based, or bes-effor servces. 3 The admsson conrol algorhm should adm a new flow only f he QoS of all classes can be sasfed. Ths mul-class servce model can also suppor flow-based servces, n whch some servce classes serve only one flow. Whou consderng ner-class resource sharng, 3 Here, we sudy mulple deermnsc and sascal servce classes and leave sudy of measuremen-based servce o fuure work.

5 one could vew each servce class as a FCFS server wh capacy g, whch s he guaraneed servce rae g = P C; as m m defned by he GPS servce dscplne. However, whle explong hs solaon propery of GPS smplfes admsson conrol, does no corporae poenal ulzaon gans due o ner-class sascal sharng. Sesson 1 Sesson Sesson N 1 X X N X 1 Q Q N Q Fg. 3. GPS Sysem Y 1 Y N Y Fgure 3 llusraes he GPS sysem n he vew of npus, oupus and buffers. The aggregae raffc n each class s vewed as a sesson, and he noaon for npus, oupus and queues are as defned n Secon II. For 1 N, le Y be he amoun of class raffc served durng [j; k]. By defnon of GPS, Y Y m ; m = 1; ; : : : ; N (1) m for any class backlogged durng [j; k]. Snce each class has a guaraneed rae g whenever s backlogged, he deermnsc servce envelope of class s s () = g : B. Sascal Servce Envelopes n GPS Our goal s o calculae he sascal servce envelope for class whch s a lower bound for class s avalable servce. Frs, we lower bound class s servce n a backlogged nerval as follows. We defne (A X 1) = P [C (k j + 1) ma 1 m ]; (11) C na Y n wh he N classes separaed no wo arbrary subses, A 1 and A, such ha A 1 [ A = f1; ; Ng, A 1 \ A s an empy se, and A 1. From Equaon (1), we have Y (A 1) (1) f class s backlogged hroughou [j; k]. Ths propery enables us o esmae he backlog servce for class, usng (A 1) wh an arbrary paron of A 1. For each nerval [j; k] wh a leas one backlogged class, one could n prncple dynamcally paron he N classes no wo subses: subse B conanng all classes ha are connuously backlogged hroughou [j; k], and subse U conanng classes ha are no connuously backlogged hroughou [j; k] (alhough hey may be backlogged for a sub-nerval n [j; k]). For any B, by defnon, Y = (B): (13) We also clam ha f A 1 \U s empy se, hen Y = (A 1); oherwse Y (A 1). Snce he exac dsrbuon of class s backlogged servce can be very dffcul o compue due o he dynamcs of he ses B and U, we nex lower bound he avalable servce Y e for any paron. Lemma 3: The avalable servce for class n nerval [j; k],, always sasfes e ey Y (A 1) (14) for an arbrary paron A 1. Proof. If class s backlogged hroughou [j; k], hen from Equaon (1), Y e = Y (A 1). If class s no connuously backlogged hroughou [j; k] wh npu raffc X, consder suffcen class raffc X e, such ha class s backlogged hroughou [j; k], whle all Q m j 1 and Xm, for m 6=, reman he same. The oupus of he N classes wll be rearranged accordng o he GPS servce dscplne and he new npus, and wll change from Y m ;m o Y, for all m = 1; ; N. For he new oupus, we can agan consruc wo subses, a backlogged subse B and an unbacklogged subse U, such ha ey = Y ; = ; (B ) = PmB m [C (k j + 1) X nu Y ;n ]: For any oher paron of A 1, Y e ; (A 1). Snce Y ;m Y m, for m 6=, ; (A 1) (A 1), hus, we have shown ha Y e. Equaon (14) enables us o sascally lower bound Y e usng he dsrbuon of (A 1) wh an arbrary paron of A 1 3, S () = P X ma 1 m [C ()]; (15) na B n ou where Bou n () s he sascal raffc envelope for he oupu raffc Y n. By delberaely seng A 1 and A, we can oban a gh sascal lower bound for Y e. The echnque of choosng A 1 s explored n deal below. C. Mul-Class Admsson Conrol n GPS Equaon (15) esablshes a way o calculae sascal servce envelopes wh an arbrary paron of classes, ye a gh lower bound s requred o fully explo ner-class resource sharng. We devse a echnque for hs purpose as follows.

6 Lnk sascal servce envelope for any class S, 4 s 1 () s() S () S () = P ms m [C X ni B n ()]: (16) Flows Classes De. Sa. MBAC Isolaon Sa. MBAC Sharng Bes-Effor Fg. 4. An Isolaon/Sharng Model for Admsson Conrol Frs, we llusrae an solaon/sharng model for admsson conrol n Fgure 4. In hs model, some servce classes wll use her deermnsc servce envelope s () n admsson conrol. These servce classes may suppor deermnsc servces, n whch deermnsc raffc envelopes b () are used. Or hey may suppor less aggressve sascal servces whch do no wsh o explo spare capacy from oher classes. In vew of servce envelopes, we refer o hese servce classes as solaon classes. Apar from hese solaon classes, oher servce classes wll explo ner-class resource sharng usng her sascal servce envelope S () o adm an ncreased number of flows no he raffc class. We refer o hese servce classes as sharng classes. Sharng classes canno suppor deermnsc servces, bu can suppor sascal, measuremen-based, and bes-effor servces. Reurnng o he problem of consrucng a gh sascal servce envelope, from Equaon (13), we know ha when class s backlogged hroughou [j; k], f A 1 \ B s an empy se, hen (A 1) = Y e. If we move any unbacklogged class no A 1, hen (A 1) < Y e. In hs sense, we should ensure ha all unbacklogged classes are n A. When class s no backlogged hroughou [j; k], from Equaon (14), we know ha f oo many classes are n A, (A 1) wll be small agan. For example, a greedy bes-effor class should never be pu no A. In order for o closely approxmae Y e, we propose paronng all sharng classes no A 1, and all solaon classes no A. We refer o hs paron of A 1 and A as ses S (sharng) and I (solaon). Anoher ssue wh Equaon (15) s ha he sascs of he oupu raffc, Bou n () for n A, are dffcul o compue, and he bound n [], Bou n () s Bn n ( + dn ), can be que loose n pracce. Consequenly, for n I, we approxmae Bou n () by Bn n (), because for hese solaon classes, admsson conrol s based on he wors case servce s n (), whle he acual servce receved s ypcally hgher han he wors case scenaro. Consequenly, hese classes are no backlogged mos of he me, and he dsorons of he oupus o npus are relavely small and can be negleced, such ha we can approxmae Bou n () wh he npu raffc envelopes B n (). Thus, we propose he followng We conclude by descrbng he complee admsson conrol algorhm for a mul-class GPS server. Each class provdes raffc parameers b () and B (), and QoS parameers d and P. Each class has a wegh and guaraneed rae g, wh guaraneed servce envelope s () = g. For deermnsc servce classes, f max fb () s ( + d )g ; hen he deermnsc QoS for flows nsde class s guaraneed. For solaon sascal servce classes, f P [max fb () s ( + d )g ] P ; hen he sascal QoS of class s sasfed. For sharng sascal servce classes, he sascal QoS s sasfed f P [max fb () s ( + d )g ] P ; or f P [max fb () S ( + d )g ] P exss for a sascal servce envelope S () obaned va any paron A 1, A usng Equaon (15). For smplcy, we use Equaon (16) nsead of esng all parons of A 1 and A. IV. COMPUTATIONAL AND EXPERIMENTAL INVESTIGATION In Secons II and III, we suded he delay bound volaon probably usng sascal raffc and servce envelopes for SP and GPS schedulers. In hs secon, we address he compuaonal aspecs of hese admsson conrol algorhms and perform race-drven smulaons o quanfy he ably of our approach o explo ner-class resource sharng. The workload consss of a se of 3-mnue races of MPEG compressed vdeo from [17]. A. Compung he Delay Tal Probably To approxmae each flow s raffc descrpor B j (), we use he rae varance envelopes n [11], where RV () = var Xs;s+ 1 T m = EX=T, and T s he lengh of he me slo, such ha EfB j ()g = m j and varfb j ()g = RV j (). When flows are mulplexed, he aggregae raffc envelope for he h class approaches a Gaussan envelope wh B () havng mean P jc m j, and varance P jc RV j () [11]. In pracce, raffc flows can specfy polcng parameers, and use [18] o compue such sascal raffc envelopes from he deermnsc parameers. To calculae P [max fb() S( + d )g > ] n Equaon (6) we ulze he maxmum varance approxmaon of [9]. Le = varfb() S( + d )g; = EfB() S( + d )g ; := nf : 4 For he specal case n whch all classes are sharng classes, a gher bound wh alernave paron of A1, A may exs. We leave sudy of hs ssue o fuure work. ;

7 Approxmang fb() S( + d )g as Gaussan, under condons (C1)-(C) n [9], and P [maxfb() S( + d )g > ] max P [B() S( + d ) > ] = () (17) P [maxfb() S( + d )g > ] e (18) R where () = p 1 1 e x dx. Proof of hese wo bounds s gven n [9], and we ulze boh n he expermens below. B. Admssble Regons n Mul-Class GPS The scenaro we consder s a lnk sharng GPS server wh a oal capacy of 45 Mbps. Dfferen weghs are gven o dfferen classes, whch requre eher deermnsc or sascal servces. In he expermens, some classes wll explo nerclass resource sharng, whle ohers wll no. In each expermen, we calculae he admssble regon for each class accordng o he flows raffc characerzaons and QoS requremens usng he admsson conrol algorhm descrbed n Secon III. We hen perform race-drven smulaons usng a GPS scheduler wh each flow havng a randomly shfed nal phase. Afer he smulaon, we measure he ulzaon and he expermenal delay bound volaon probably, and compare he smulaon resuls wh he requred QoS. In he frs expermen, we consder a GPS server wh wo servce classes. Class 1 requres deermnsc servce, wh d 1 = 1 msec, class requres sascal servce, wh d = msec and P = 1 4. In he admsson conrol ess, we use boh he lower bound of Equaon (17) and he upper bound of Equaon (18) o approxmae P [D > d]. Class LB,NoSharng UB,NoSharng LB,Sharng UB,Sharng Class Fg. 5. Admssble Regons for Deermnsc and Sascal Servces Fgure 5 shows he admssble regons for class 1 and under four dfferen condons: wh and whou ner-class sharng for class, and upper and lower bounds for P [D > d]. Noce he sgnfcan ncrease n he admssble regon due o explong ner-class resource sharng usng our framework of sascal servce envelopes. For example, usng he lower bound for P [D > d] and seng g 1 = g = C=, whou ner-class sharng, he admssble regon s (7; 31) flows and he oal ulzaon s 45:3%. In conras, wh ner-class sharng, he admssble regon s (7; 6) flows and he oal ulzaon s 8:%, an ncrease of 81%. We also observe ha he dfferences n he admssble regons usng he lower and upper bounds are merely 1 or flows. We nex perform race-drven smulaons and measure he expermenal delay bound volaon raes usng he admssble regon calculaed from he sharng ess. For he lower bound, he mean delay bound volaon rae for class s 5 1 4, whle for he upper bound, he mean volaon rae for class s approxmaely Snce he QoS parameer s P = 1 4, we observe ha he acual admssble regon mus be beween he LB and UB sharng curves, and ha he admssble regons calculaed usng boh bounds are very close o he rue ones. Class Class Sharng No Sharng Class Fg. 6. Admssble Regons for a Three-Class GPS Server In he nex expermen, we consder a hree-class GPS scheduler. Class 1 requres deermnsc servce wh d 1 = msec, class requres sascal servce wh d = msec and P = 1 4, and class 3 requres sascal servce wh d 3 = 3 msec and P 3 = 1 4. Class 1 and are solaon classes. We perform admsson ess wh and whou class 3 explong ner-class sharng, and use he lower bound of Equaon (17) o approxmae P [D > d]. The admssble regon s shown n Fgure 6, whch also llusraes he sgnfcan ulzaon gan of he approach. In he above wo expermens, he deermnsc servce class s exploed by he sascal servce class o allow ner-class sharng. In he nex expermen, we show ha our approach s also able o explo ner-class sharng among sascal servce classes. We consder a hree class GPS server wh each class provdng sascal servces wh he same QoS: d = msec and P = 1 4. Class 1 and are se o solaon classes. In Fgure 7, we show he dfference n he admssble regons by allowng class 3 o explo ner-class sharng. From Fgure 7, observe ha gnorng ner-class sharng leads o as many as 8 fewer flows admed n class 3, for a loss of approxmaely 1% of he resource ulzaon. In hs scenaro,

8 1 Class Class Class Fg. 7. Increase n Admssble Regons for 3 Sascal Classes he nra-class sascs are fully exploed, and he gan comes solely from he ner-class sascs. In a hgh-speed GPS server, even f each class provdes sascal servce, when he number of servce classes s large, he ner-class resource sharng gan can be sgnfcan. V. CONCLUSIONS In hs paper, we developed mul-class admsson conrol algorhms ha explo ner-class sascal resource sharng. We developed a framework of sascal servce envelopes o sudy he problem and showed how such envelopes characerze he excess capacy avalable o a raffc class due o varyng resource demands of oher classes. We appled he approach o Sac Prory and Generalzed Processor Sharng schedulers and expermenally demonsraed ha our admsson conrol algorhms are able o exrac a sgnfcan ulzaon gan from ner-class resource sharng. 3 [7] R. Cruz, Qualy of servce guaranees n vrual crcu swched neworks, IEEE Journal on Seleced Areas n Communcaons, vol. 13, no. 6, pp , Aug [8] A. Parekh and R. Gallager, A generalzed processor sharng approach o flow conrol n negraed servces neworks: he sngle-node case, IEEE/ACM Transacons on Neworkng, vol. 1, no. 3, pp , June [9] J. Choe and N. Shroff, A cenral lm heorem based approach o analyze queue behavor n ATM neworks, IEEE/ACM Transacons on Neworkng, vol. 6, no. 5, pp , Oc [1] G. de Vecana and G. Kesds, Bandwdh allocaon for mulple quales of servce usng generalzed processor sharng, IEEE Transacons on Informaon Theory, vol. 4, no. 1, pp. 68 7, Jan [11] E. Knghly, Second momen resource allocaon n mul-servce neworks, n Proceedngs of ACM SIGMETRICS 97, Seale, WA, June 1997, pp [1] S.-K. Kweon and K. Shn, Vdeo-on-demand servce on packe-swched neworks usng a sascal raffc envelope, Preprn, [13] Z. Zhang, D. Towsley, and J. Kurose, Sascal analyss of generalzed processor sharng schedulng dscplne, IEEE Journal on Seleced Areas n Communcaons, vol. 13, no. 6, pp , Aug [14] R. Cruz, Qualy of servce managemen n negraed servces neworks, Techncal Repor, Proc. 1s Sem-Annual Research Revew, CWC, Unversy of Calforna a San Dego, June [15] R. Cruz, A calculus for nework delay, par I : Nework elemens n solaon, IEEE Transacons on Informaon Theory, vol. 37, no. 1, pp , Jan [16] J. Lebeherr, D. Wrege, and D. Ferrar, Exac admsson conrol for neworks wh bounded delay servces, IEEE/ACM Transacons on Neworkng, vol. 4, no. 6, pp , Dec [17] O. Rose, Sascal properes of MPEG vdeo raffc and her mpac on raffc modelng n ATM sysems, n Proceedngs of IEEE Conference on Local Compuer Neworks, Mnneapols, MN, Oc. 1995, pp [18] E. Knghly, Enforceable qualy of servce guaranees for bursy raffc sreams, n Proceedngs of IEEE INFOCOM 98, San Francsco, CA, Mar VI. ACKNOWLEDGMENTS The auhors are graeful o Tao Wu and Zh-L Zhang for her nsghful commens. REFERENCES [1] D. Wrege, E. Knghly, H. Zhang, and J. Lebeherr, Deermnsc delay bounds for VBR vdeo n packe-swchng neworks: Fundamenal lms and praccal radeoffs, IEEE/ACM Transacons on Neworkng, vol. 4, no. 3, pp , June [] J. Kurose, On compung per-sesson performance bounds n hgh-speed mul-hop compuer neworks, n Proceedngs of ACM SIGMETRICS 9, Newpor, RI, June 199, pp [3] S. Jamn, P. Danzg, S. Shenker, and L. Zhang, A measuremen-based admsson conrol algorhm for negraed servces packe neworks, IEEE/ACM Transacons on Neworkng, vol. 5, no. 1, pp. 56 7, Feb [4] S. Floyd and V. Jacobson, Lnk-sharng and resource managemen models for packe nework, IEEE/ACM Transacons on Neworkng, vol. 3, no. 4, pp , Aug [5] J. Benne and H. Zhang, Herarchcal packe far queueng algorhms, IEEE/ACM Transacons on Neworkng, vol. 5, no. 5, pp , Oc [6] I. Soca, H. Zhang, and T. Ng, A herarchcal far servce curve algorhm for lnk-sharng, n Proceedngs of ACM SIGCOMM 97, Cannes, France, Sep

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