Statistical Admission Control Using Delay Distribution Measurements

Transcription

1 Statstcal Admsson Control Usng Delay Dstrbuton Measurements KARTIK GOPALAN State Unversty of New York at Bnghamton LAN HUANG IBM Almaden Research Center GANG PENG and TZI-CKER CHIUEH Stony Brook Unversty and YOW-JIAN LIN Telcorda Research Growth of performance senstve applcatons, such as voce and multmeda, has led to wdespread adopton of resource vrtualzaton by a varety of servce provders (xsps). For nstance, Internet Servce Provders (ISPs) ncreasngly dfferentate ther offerngs by means of customzed servces, such as vrtual prvate networks (VPN) wth Qualty of Servce (QoS) guarantees or QVPNs. Smlarly Storage Servce Provders (SSPs) use storage area networks (SAN)/network attached storage (NAS) technology to provson vrtual dsks wth QoS guarantees or QVDs. The key challenge faced by these xsps s to maxmze the number of vrtual resource unts they can support by explotng the statstcal multplexng nature of the customers nput request load. Whle a number of measurement-based admsson control algorthms utlze statstcal multplexng along the bandwdth dmenson, they do not satsfactorly explot statstcal multplexng along the delay dmenson to guarantee dstnct per-vrtualunt delay bounds. Ths artcle presents Delay Dstrbuton Measurement (DDM) based admsson control algorthm, the frst measurement-based approach that effectvely explots statstcal multplexng along the delay dmenson. In other words, DDM explots the well-known fact that the actual delay experenced by most servce requests (packets or dsk I/O requests) for a vrtual unt s usually far smaller than ts worst-case delay bound requrement because multple vrtual unts rarely send request bursts at the same tme. Addtonally, DDM supports vrtual unts wth dstnct probablstc delay guarantees vrtual unts that can tolerate more delay volatons can reserve fewer resources than those that tolerate less, even though they requre the same delay bound. Comprehensve trace-drven performance evaluaton of QVPNs (usng Voce over IP traces) and QVDs (usng vdeo stream, TPC-C, and Web search I/O traces) shows that, when compared to determnstc admsson control, DDM can potentally ncrease the number of admtted vrtual unts (and resource utlzaton) by up to a factor of 3. Categores and Subject Descrptors: C.2.1 [Computer-Communcaton Networks]: Network Archtecture and Desgn; D.4.2 [Operatng Systems]: Storage Management General Terms: Algorthms, Measurement, Performance Authors addresses: K. Gopalan, Bnghamton Unversty; emal: kartk@cs.bnghamton.edu; L. Huang, IBM Almaden Research Center; emal: lanhuang@us.bm.com; G. Peng, T.-C. Chueh, Stony Brook Unversty; emal: {gpeng,chueh}@cs.sunysb.edu, Y.-J. Ln, Telcorda Research; emal:yjln@research.telcorda.com. Permsson to make dgtal or hard copes of part or all of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or drect commercal advantage and that copes show ths notce on the frst page or ntal screen of a dsplay along wth the full ctaton. Copyrghts for components of ths work owned by others than ACM must be honored. Abstractng wth credt s permtted. To copy otherwse, to republsh, to post on servers, to redstrbute to lsts, or to use any component of ths work n other works requres pror specfc permsson and/or a fee. Permssons may be requested from Publcatons Dept., ACM, Inc., 2 Penn Plaza, Sute 701, New York, NY USA, fax +1 (212) , or permssons@acm.org. c 2006 ACM /06/ $5.00 ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006, Pages 1 24.

2 2 K. Gopalan et al. 1. INTRODUCTION Performance senstve applcatons such as Voce over IP (VoIP), vdeo conferencng, meda streamng, and onlne tradng, requre dedcated network, storage, and computatonal resources to meet ther strngent delay and throughput requrements. A powerful concept beng appled to meet ths emergng need s the vrtualzaton of physcal resources nto multple vrtual unts of resources. As an example, Internet Servce Provders (ISP) provson multple Vrtual Prvate Networks (VPN) wth dstnct QoS guarantees (or QVPNs) where each QVPN acts as a traffc trunk carryng performance senstve aggregated traffc. Technologes such as Multprotocol Label Swtched (MPLS) networks can map each QVPN to a long-term Label Swtched Path (LSP). For nstance, a QVPN could be a longterm Voce over IP (VoIP) trunk that carres aggregate traffc from several voce sessons rather than just one ndvdual voce sesson. QVPNs are set up and torn down over longer tmescales and carry aggregate traffc that s more stable than short-lved ndvdual connectons. Smlarly, Storage Servce Provders (SSP) ncreasngly use storage vrtualzaton technology to create a set of vrtual storage devces from a sngle physcal storage resource such as a Storage Area Network (SAN) or a Network Attached Storage (NAS). Each such vrtual dsk (VD) can have dstnct QoS guarantees (QVD) such as capacty, request throughput, and latency bound. QVDs serve as backend storage servers for separate enterprse functons such as Web servers, meda servers, or database servers. As n the case of QVPNs, QVDs can bundle multple vrtual unts for hgher aggregated I/O rates. The key challenge faced by xsps s to maxmze the utlzaton effcency of the physcal resource nfrastructure and stll support the strngent QoS requrements of each vrtual unt. Maxmzng utlzaton effcency calls for an effectve admsson control algorthm that admts as many vrtual unts as possble, whle allocatng the least amount of resources needed to satsfy ther QoS requrements. A smple approach of determnstc admsson control allocates all the resources needed to ensure that the QoS guarantees are never volated. Specfcally consderng delay guarantees, determnstc admsson control ensures that the delay n servcng each request (packet or I/O) never exceeds the worst-case delay bound guaranteed for each vrtual unt. On the flp sde, however, worst-case delays are rarely encountered n practce. Because determnstc admsson control errs on the sde of beng hghly conservatve, a large proporton of physcal resources reman underutlzed. Two specfc statstcal effects can help to mprove the resource usage effcency of these systems. (1) Tolerance to delay volatons. Most real-world delay-senstve applcatons can tolerate a small fracton of excess delays n request servce tmes [Wang and Zhu 1998]. For nstance, VoIP sessons can tolerate up to 10 3 fracton of ther packets experencng excess delays or losses wthout perceptually affectng audo qualty. If 99.9% of the packets are observed to experence at most 50% of ther expected worst-case delay, a network admsson control algorthm can potentally reserve only half of the resources that determnstc admsson control would reserve. (2) Statstcal multplexng along delay dmenson. Due to statstcal multplexng, typcally not all the vrtual unts can smultaneously experence ther peak request arrval rates. For nstance, packet bursts from all QVPNs (or I/O bursts from all QVDs) wll usually not arrve exactly at the same tme at ther servce queues and would generally be dspersed over tme. Consequently, request servce delays rarely approach the worst-case delays bounds that would occur only f all vrtual unts experence ther peak request burst smultaneously. To llustrate ths multplexng effect, we aggregated the ON OFF packet traces for dfferent numbers of recorded VoIP sessons (detals n Secton 4). Fgure 1 shows the complementary cumulatve dstrbuton functon of the fracton of VoIP sessons n an aggregate that are smultaneously n ther ON state. We observe that half the tme, less than 12% of the VoIP sessons are n ther ON state smultaneously, and ts almost never the case that more than 40% of the sessons are smultaneously actve. Smlar statstcal effects ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

3 Statstcal Admsson Control Usng Delay Dstrbuton Measurements 3 1 Complementary CDF N=5 N=10 N=20 N=30 N=40 N= Fracton of VoIP streams smultaneously n ON state Fg. 1. Complementary CDF of the fracton of VoIP sessons n on state smultaneously as the number of VoIP sessons (N) n aggregate QVPN s vared. can be expected for other categores of real-tme network traffc such as vdeo conferencng and onlne fnancal transactons. Ths artcle proposes a practcal and effcent measurement-based technque, called Delay Dstrbuton Measurement (DDM)-based admsson control, that explots the prevous two statstcal effects to maxmze the number of vrtual unts admtted wth performance guarantees. The QoS parameters that the DDM algorthm supports nclude delay bound, delay volaton probablty bound, and the long-term average bandwdth. DDM s the frst measurement-based algorthm that smultaneously provdes all the followng features. Statstcal multplexng along delay dmenson. DDM s the frst measurement-based approach whch explots statstcal multplexng along the delay dmenson to ncrease resource utlzaton n comparson to purely determnstc admsson control. In contrast, the earler measurement-based approaches manly focused on statstcal multplexng along the bandwdth dmenson, that s, multplexng due to the fact that vrtual unts often request rates much below ther stated long-term bandwdth requrement. Dstnct per-vrtual-unt probablstc delay bounds. DDM supports vrtual unts for whch a certan percentage of delay bound volatons are tolerable. The key dfference from pror approaches s DDM s ablty to dfferentate among vrtual unts n terms of ther tolerance to delay bound volatons. Vrtual unts wth hgher tolerance to delay bound volatons are allocated fewer resources than those wth lower tolerance even though they may have the same delay bound requrement. Unfed support for probablstc and determnstc delay bounds. DDM provdes a sngle admsson control framework to support vrtual unts that may have probablstc or determnstc delay bounds. Determnstc delay bound requrements smply correspond to zero tolerance to delay volatons. ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

4 4 K. Gopalan et al. The prncpal challenge n provdng dstnct per-vrtual-unt probablstc delay guarantees s to determne the mappng between delay bound, delay volaton probablty bound, and resource requrements. DDM dynamcally measures the servce delay of each request, computes the rato between the actual servce delay and the worst-case delay that the request could experence, and derves a delay rato dstrbuton. Ths dynamcally measured delay rato dstrbuton s used to derve the bandwdth reservaton needed to support a gven probablstc delay bound. Once the DDM algorthm reserves an amount of bandwdth for a vrtual unt, a rate-based request scheduler (such as Vrtual Clock [Zhang 1991] or WFQ [Parekh and Gallager 1993]) guarantees the assgned bandwdth share. The DDM algorthm appled to network resource allocaton alone was frst ntroduced n our earler conference artcle [Gopalan et al. 2004]. In ths artcle, we addtonally descrbe how the concepts of the DDM algorthm are appled to perform effcent storage resource allocaton n a mult-dmensonal storage vrtualzaton system called Stonehenge [Huang et al. 2004]. We also present several addtonal performance results demonstratng the benefts of DDM for both network and storage resource allocaton. The rest of the artcle s organzed as follows. In Secton 2, we frst descrbe the DDM algorthm n the context of network resource allocaton for QVPNs. In Secton 3, we descrbe how the same prncples of the DDM algorthm are appled n the context of storage resource allocaton to support QVDs wth dstnct probablstc delay and bandwdth guarantees. Sectons 4 and 5 present performance evaluaton of the DDM algorthm for network and storage resource allocaton, respectvely. In Secton 6, we dscuss the pror work n statstcal admsson control n the areas of both network and storage resource allocaton. Secton 7 summarzes the man research contrbutons and outlnes future research drectons. 2. STATISTICAL NETWORK RESOURCE ALLOCATION USING DDM The prmary goal of network resource allocaton wth DDM s to maxmze the number of admtted QVPNs wth dstnct bandwdth, delay, and delay volaton probablty bounds. In other words, consder a QVPN F that carres aggregate real-tme traffc wth an average bandwdth of ρ avg and burst sze σ. Assume that F traverses a lnk l havng total capacty C l. It s guaranteed at admsson control tme that each of F s packets wll be servced by the packet scheduler at lnk l wthn a delay bound D,l and wth a delay volaton probablty no greater than P,l. For nstance, f D,l = 10ms and P,l = 10 3, t means that no more than a fracton 10 3 of packets belongng to the QVPN can experence a delay greater than 10ms. 2.1 Worst-Case Delay Bound We frst revew the classcal results for determnstc delay bounds usng rate-based schedulers. We assume that each QVPN s ncomng traffc s regulated by a token bucket wth bucket depth σ and token rate ρ avg. The amount of QVPN F traffc arrvng at the scheduler n any tme nterval of length τ s bounded by (σ + ρ avg τ). The job of a lnk scheduler s to prortze the transmsson of packets belongng to dfferent QVPNs over a common lnk. We assume that packets are servced by rate-based lnk schedulers, such as WFQ [Parekh and Gallager 1993] or Vrtual Clock [Zhang 1991]. It can be shown that the worstcase queung delay D,l wc experenced at a lnk l by any packet belongng to a QVPN F under the WFQ or Vrtual Clock servce dscplne s gven by the followng expresson. D wc,l = σ + L max + L max, (1) ρ,l ρ,l C l ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

5 Statstcal Admsson Control Usng Delay Dstrbuton Measurements 5 Cumulatve Dstrbuton Functon Rato of actual to worst-case delay Fg. 2. Example of cumulatve dstrbuton functon (CDF) of the rato of actual delay to worst-case delay experenced by packets. X-axs s n log scale to hghlght the rato dstrbuton n the low-rato range. 39 VoIP QVPNs traverse a 10Mbps lnk. ρ avg = 256Kbps. Delay bound=10ms. Delay volaton probablty = where σ s F s burst sze at lnk l, L max s the maxmum packet sze, ρ,l s the reservaton for F at lnk l, and C l s the total capacty of lnk l. The frst component of the delay s flud far queung delay, the second component s the packetzaton delay, and the thrd component s scheduler s nonpreempton delay. We are nterested n rate-based schedulers snce, n ther case, the relatonshp between delay bound and the amount of bandwdth reserved for a QVPN can be explctly specfed. Furthermore, as we wll see n Secton 2.2, rate-based schedulers enable us to dfferentate among QVPNs n terms of ther delay volaton probablty requrements. In contrast, for nonrate-based schedulers, such as Earlest Deadlne Frst (EDF), the resource-delay relatonshp s dffcult to determne, whch n turn makes the admsson control process more complcated. Hence, even though nonrate-based schedulers can potentally provde hgher lnk utlzaton, t s dffcult to guarantee delay volaton probablty bound on a per-qvpn bass. 2.2 Delay to Resource Mappng Probablstc delay guarantees assst n reducng the bandwdth reservaton for each QVPN by explotng ther tolerance to certan level of delay volatons. Due to statstcal multplexng, packet bursts from dfferent QVPNs F tend to be temporally spread out and rarely occur at the same tme. As a result, worst-case delay s rarely experenced by packets traversng a lnk. Assume that the request for a QVPN F specfes ts average rate ρ avg, burst sze σ, requred delay bound D,l, and delay volaton probablty P,l at lnk l. Each QVPN F traversng the lnk s assgned a bandwdth reservaton ρ,l ρ avg, whch satsfes both the delay requrement (D,l, P,l ) as well as the average rate requrement ρ avg. Note that ρ avg s the long-term average rate of F, whereas the bandwdth reservaton ρ,l s used by the scheduler to determne the runtme preference for F s traffc over other QVPNs. In ths secton, we derve the correlaton functon that maps F s specfcaton (ρ avg, σ, D,l, P,l ) to ts bandwdth reservaton ρ,l. ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

6 6 K. Gopalan et al CDF Constructon. Assume that for each packet k, the system tracks the runtme measurement hstory of the rato r k, whch s the actual packet delay experenced D,l k to the worst-case delay D,l wc, that s, r k = D,l k /Dwc,l, where r k ranges between 0 and 1. We can use these measured samples of rato r k to construct a cumulatve dstrbuton functon (CDF) Prob(r). The dstrbuton Prob(r) gves the probablty that the rato between the actual delay encountered by a packet and ts worst-case delay s smaller than or equal to r. Conversely, Prob 1 (p) gves the maxmum rato of actual delay to worst-case delay that can be guaranteed wth a probablty p. Fgure 2 shows an example of a CDF constructed n ths manner for a specfc smulaton scenaro of 39 VoIP QVPNs. (Smulaton detals follow n Secton 4.) To construct the CDF n practce, we partton the rato range from 0 to 1 nto a number of subranges, and then, for each subrange, keep updatng the count of packets transmtted whose rato r k falls wthn the subrange. The CDF can be constructed by computng the accumulated count of packets from the lowest subrange to each subrange. The CDF would typcally be mantaned over a sldng measurement wndow. The duraton of the measurement wndow partly determnes how aggressve the admsson control algorthm can be n admttng new QVPNs. The mpact of dfferent wndow szes on the admsson process s evaluated n Secton Resource Mappng. The CDF curve Prob(r) concsely quantfes the level of statstcal multplexng along the delay dmenson. For nstance, Fgure 2 ndcates that most of the packets experence less than 1/4th of ther expected worst-case delay. Thus, reservng resources to cover for the worstcase delay s wasteful snce t s rarely encountered n practce. In ths secton, we descrbe how we can explot the statstcal multplexng nformaton quantfed by Prob(r), n addton to each QVPN s tolerance to delay volatons, to reduce the amount of per-qvpn bandwdth reservaton. Gven the measured estmate of functons Prob(r) and Prob 1 (p), the followng expresson determnes the delay-derved bandwdth reservaton ρ delay,l requred to satsfy QVPN F s probablstc delay requrement (D,l, P,l ). ( ) σ + L max D,l = + L max Prob 1 (1 P ρ delay,l ). (2) C l,l Equaton (2) states, that n order to obtan a delay bound of D,l wth a delay volaton probablty bound of P,l, we need to reserve a mnmum bandwdth of ρ delay,l whch can guarantee a worst-case delay of D,l wc = D,l /Prob 1 (1 P,l ). Conversely, the delay-derved bandwdth requrement ρ delay,l of a QVPN F at lnk l s ρ delay σ + L max,l = D,l Prob 1 (1 P,l ). (3) L max C l The actual reservaton requred to satsfy QVPN F s QoS requrement (ρ avg, D,l, P,l ) s ρ,l = max{ρ avg, ρ delay,l }. In other words, the actual bandwdth reservaton for a QVPN s dctated by the tghter of two QoS requrements one mposed by ts average bandwdth requrement ρ avg, and the other mposed by ts probablstc delay requrement (D,l, P,l ). It s worth pontng out once more that ths resource mappng functon explots statstcal multplexng along the delay dmenson rather than along the bandwdth dmenson as n earler approaches. Ths s a drect consequence of the fact that DDM measures the dstrbuton of actual to worst-case delay rato. Specfcally, f ρ delay,l happens to be larger than ρ avg for all QVPNs, then the resource allocaton wll be guded by statstcal delay requrements rather than determnstc bandwdth requrements. ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

7 Statstcal Admsson Control Usng Delay Dstrbuton Measurements Cumulatve dstrbuton functon CDF_old CDF_new CDF_est CDF_unform Rato of actual to worst-case delay Fg. 3. Example of dfferent CDF curves for one smulaton scenaro. X-axs s n lnear scale to hghlght the dfference between measured and estmated CDF curves. The Y-axs range shown s from 0.99 to 1.0 whch corresponds to the typcal tolerance range for delay volatons (below 10 2 ). 2.3 Admsson Control Usng DDM In ths secton, we descrbe the DDM admsson control algorthm for admttng a new QVPN F N that arrves at a lnk l on whch N 1 QVPNs have already been admtted. The prncpal challenge of admsson control les n estmatng the mpact of F N s traffc on the guarantees provded to already admtted QVPNs. If F N s admtted, t wll cause an ncrease n traffc load carred by the lnk and consequently larger actual delays experenced by packets from all QVPNs. Specfcally, the CDF of actual to worst-case delay rato wll tend to become more conservatve by shftng to the rght after F N becomes actve. Hence t s mportant that, even before F N can be admtted, DDM must estmate and account for the mpact of the new QVPN on the delay dstrbuton of exstng QVPNs. The DDM algorthm conssts of two phases. The frst phase estmates the expected delay dstrbuton assumng QVPN F N s admtted. The second phase performs the actual admsson control usng the estmated CDF from the frst phase and computes future resource requrements of all QVPNs (ncludng the new one). F N s admtted only f each QVPN s resource requrement can be satsfed wthn the avalable lnk capacty Sgnfcance of CDF Evoluton. If the new QVPN F N s admtted, the lnk wth a fnte capacty C l has to shoulder the addtonal traffc load from F N. As a result, packets, for all QVPNs traversng the lnk wll experence larger delays on average. More specfcally, the addtonal load from F N could mpact the CDF curve shown n Fgure 2 by shftng t to the rght. In other words, for the same delay volaton probablty p, fr 1 = Prob 1 old (1 p) before admttng F N and r 2 = Probnew 1 (1 p) after admttng F N, then r 2 r 1. Because a larger value of Prob 1 new (1 p) translates nto-hgher bandwdth requrement n Equaton (3), CDF new s sad to be more conservatve than CDF old snce CDF new can admt fewer QVPNs than CDF old. Fgure 3 provdes an example of CDF old and rght-shfted CDF new ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

8 8 K. Gopalan et al. for one smulaton scenaro n the Y-axs range from 0.99 to 1.0 (snce ths range happens to be of most nterest). If we smply use CDF old to derve the bandwdth reservaton for F N, and the actual CDF new turns out to be sgnfcantly more conservatve than CDF old, F N may be assgned a much smaller bandwdth than what t actually needs to meet ts probablstc delay requrement. The key research challenge of the DDM algorthm thus les n how to predct the mpact of the new QVPN F N on the delay dstrbuton of (N 1) exstng QVPNs wthout assumng any apror traffc model. The mpact of new QVPN F N on CDF old depends on several factors. In general, tght QoS requrements, such as a small delay requrement D N,l, a low tolerance to delay volaton P N,l, a large average rate ρ avg N, or a bg burst sze σ N, all lead to larger rato of actual to worst-case delay and a more conservatve CDF. Furthermore, the ncrement from Prob 1 old (1 p) toprob 1 new (1 p) could be dfferent for dfferent values of volaton probablty p. Fnally, the magntude of a new QVPN s relatve load contrbuton to a lnk s traffc affects the amount of dfference between the CDFs before and after the new QVPN s admtted. n Predctng CDF Evoluton. Gven the multtude of factors that nfluence the evoluton of CDF, t s dffcult (f not mpossble) to exactly predct CDF new usng CDF old and QVPN F N s QoS requrements. The DDM algorthm uses a heurstc approach to approxmate CDF new. Let τ be the length of a movng tme wndow over whch the delay dstrbuton CDF old of exstng N 1 QVPNs s measured. Let m be the number of packets generated by N 1 QVPNs that traverse the lnk n duraton τ. In a tme nterval τ, F N can potentally transmt a maxmum of n = σ N /L mn + ρ avg N τ/l mn number of packets, where L mn s the mnmum packet sze. Assume that these n addtonal packets experence a unform dstrbuton of actual to worst-case delay rato. A unform dstrbuton s a very conservatve estmate of delay dstrbuton (though not the most conservatve one) whch assumes that packet delays for the new QVPN F N are expected to be unformly dstrbuted over the range of ratos from 0 to 1 and that all packets are of sze L mn. In realty, a large majorty of packets experence small packet delays (as shown n Fgure 2) and are of sze greater than L mn. To characterze CDF new, we frst combne the unform delay rato dstrbuton for F N obtaned prevously wth a weght of n+m and the delay rato dstrbuton CDF old wth a weght of to obtan a dstrbuton called CDF unform, whch represents an estmate of the cumulatve dstrbuton that would result f F N were fully loaded and the delay rato of the packets from F N were dstrbuted unformly between 0 and 1. CDF unform can be constructed usng the technque descrbed n Secton 2.2, but wth the dfference that, before computng the accumulated sum for each rato subrange, we add n/r to the count of rato samples n each subrange, where R s the number of subranges between 0 and 1. In other words, n delay ratos are assumed to be unformly dstrbuted over all rato subranges. Emprcally, CDF unform s a very conservatve estmate of the dstrbuton CDF new because both the unform delay rato dstrbuton assumpton and the full load assumpton are too pessmstc. As a result, CDF new les somewhere between CDF old and CDF unform as prevously constructed. We further approxmate CDF new by constructng CDF est, whch n turn s a weghted combnaton of CDF old and CDF unform. Specfcally, m n+m Probest 1 (1 p) = α Prob 1 unform (1 p) + (1 α)prob 1 old (1 p). (4) The factor α s the mpact factor that determnes how far the dstrbuton curve CDF est s from CDF unform and CDF old. For a new QVPN that mposes a relatvely large load on the lnk wth respect to an exstng load, CDF est should be close to CDF unform snce the latter s more conservatve n admttng QVPNs. On the other hand, for a new QVPN that mposes a relatvely small load wth respect, to an exstng load, CDF est should be closer to CDF old snce, n ths case, the new QVPN has a relatvely smaller mpact on ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

9 Statstcal Admsson Control Usng Delay Dstrbuton Measurements 9 CDF old. Wth ths consderaton n mnd, we defne the mpact factor as the fracton of new QVPN F N s load on the total expected load. α = ρ N,l N =1 ρ. (5),l Here ρ,l s computed usng the dstrbuton CDF unform snce t s the only estmate of future delay dstrbuton we have at the tme of admttng F N. Snce we are practcally nterested n only the delay volaton probabltes P,l for exstng and new QVPNs, we only need to compute that porton of CDF est whch covers these delay volaton probabltes of nterest; typcally the volaton probabltes le n the range 10 2 to 10 6 whch corresponds to a small upper porton of the Y -axs n Fgure 2. An example of dfferent CDF curves s llustrated n Fgure 3 wthn the Y-axs range of 0.99 to 1 for one smulaton scenaro. We see that CDF est s the closest approxmaton to CDF new, although a bt more conservatve. CDF unform s the most conservatve of all. Note that constructng CDF est nvolves two levels of weghted combnatons, frst n constructng CDF unform from CDF old and a unform dstrbuton of new QVPN s packets, and second n constructng CDF est from CDF old and CDF unform. The dfference s that the CDF unform provdes a frst-cut conservatve estmate of CDF new, whereas ths estmate s further refned by constructng CDF est. In Secton 4, we valdate that ths technque for CDF estmaton ndeed relably captures the future delay dstrbuton of admtted QVPNs The Admsson Control Algorthm. Wth the delay-probablty-bandwdth correlaton functon n place, we now present the DDM admsson control algorthm n Fgure 1. The algorthm can be nvoked ether to admt a new QVPN F N or to perodcally recalculate the requrements of already admtted QVPNs. Wthout loss of generalty, the followng dscusson assumes the frst scenaro. Assume that N 1 QVPNs are currently beng served by the scheduler, and F N arrves for admsson. The algorthm frst calculates CDF unform usng the measured delay dstrbuton CDF old and QVPN F N s average rate requrement ρ avg. For each of the N QVPNs (ncludng the new one) the algorthm next N Algorthm 1 The DDM algorthm to determne whether a new QVPN F N can be admtted such that each QVPN F,1 N, can be guaranteed a delay bound D,l, delay volaton probablty P,l, and average rate ρ avg. 1: Input : (a) (D,l, P,l, ρ avg, σ ) for each QVPN F,1 N. 2: (b) The measured delay rato dstrbutons. 3: 4: Compute CDF old and CDF unform from delay rato dstrbutons. 5: 6: for = 1toN do 7: Compute ρ delay,l = B l (D,l, P,l, σ ) usng Equatons (3) and (4). 8: ρ,l = max{ρ avg, ρ delay,l } 9: end for 10: 11: /*Perform admsson checks*/ 12: f ( N =1 ρ,l > C l ) then 13: Reject QVPN F N and ext. 14: end f 15: 16: /*QVPN F N can be admtted*/ 17: for = 1toN do 18: Reserve bandwdth ρ,l for F. 19: end for ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

10 10 K. Gopalan et al. computes the delay-derved bandwdth requrement ρ delay,l usng Equatons (3) and (4). The actual bandwdth requrement ρ,l s the larger of the delay-derved requrement ρ delay,l and average requrement. The new QVPN F N s admtted only f followng condton s satsfed. ρ avg N ρ,l C l. (6) =1 Equaton (6) states that the sum of bandwdth requrements of all QVPNs under the estmated delay rato dstrbuton CDF est, should be smaller than C l. The QVPN F N s rejected f ths condton cannot be satsfed. If the new QVPN s accepted, the algorthm sets the bandwdth reservaton for each QVPN to ρ,l as computed prevously. The robustness of the DDM algorthm, n essence, depends upon the accuracy of estmatng CDF est before admttng a new QVPN F N. Ths s because the act of admttng F N results n alterng the reservaton ρ,l of already admtted flows F 1 to F N 1.ACDF est that s too conservatve can lead to underutlzaton of a lnk s resources, whereas one that s overly optmstc can lead to a potental volaton of QoS guarantees for all QVPNs at runtme. The prncpal challenge n the DDM algorthm les n accurately estmatng CDF est before admttng F N usng an approprate value of the mpact factor α n Equaton (4) a value that s nether too optmstc nor too conservatve. Experments n Secton 4 show that an mpact factor gven n Equaton (5) that equals the fractonal load mposed by the new flow provded a good estmate of CDF est. The admsson control algorthm descrbed provdes a unfed framework to support QVPNs wth both probablstc as well as determnstc delay requrements. Specfcally, QVPNs requrng determnstc delay bounds can smply be treated as requrng a volaton probablty of zero whch, n turn, can be easly factored nto the calculaton of ρ,l descrbed n Secton Tme and Space Complexty. The step for computng CDF old and CDF unform has O(R) tme complexty, where R s the number of subranges n the delay rato nterval from 0 to 1. The subsequent steps n the algorthm have O(N) tme complexty, where N s the number of QVPNs beng consdered. Thus the complexty of the DDM algorthm s O(N + R). In practce, the frst step of computng CDF old and CDF unform s the more domnant of the two components due to the larger number of subranges R. The algorthm tself s nvoked qute nfrequently, only when ether new QVPN requests arrve for admsson at the lnk or exstng QVPN reservatons need to be perodcally recomputed. The runtme computaton overhead of mantanng CDFs s also mnmal snce we only need a few arthmetc operatons to record the rato for each packet transmtted by the lnk. In terms of space cost, the only sgnfcant addtonal space requred s n the order of O(R) (about 400KB wth R = 100K) for mantanng CDF old, whch represents aggregate delay dstrbuton nformaton for all QVPNs. The values for CDF unform and CDF est can be derved as and when requred durng admsson control. In partcular, DDM requres no addtonal space for per-qvpn state mantenance when compared to any other algorthm that provdes per-qvpn QoS. In our context, QVPNs represent a lmted number of traffc aggregates (such as LSPs n MPLS), rather than ndvdual TCP/IP connectons, whch further reduces the space requrement to wthn practcal bounds. 3. STATISTICAL STORAGE RESOURCE ALLOCATION USING DDM We next descrbe how the DDM algorthm has been appled n the context of a multdmensonal storage vrtualzaton system called Stonehenge [Huang et al. 2004] that allows for the creaton of multple QoSguaranteed vrtual dsks (QVDs) over a common physcal storage nfrastructure. Stonehenge effectvely solates the logcal storage servers as f they are separate physcal storage devces, each havng the ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

11 Statstcal Admsson Control Usng Delay Dstrbuton Measurements 11 STONEHENGE CLIENTS STORAGE SERVERS STORAGE MANAGER ADMISSION CONTROL QVD LAYOUT SCHEDULING AND FEEDBACK I/O REQUEST AND RESPONSE Fg. 4. Stonehenge clents communcate wth a centralzed management server and a set of storage servers that are connected through a ggabt network. standard attrbutes assocated wth any physcal dsk such as bandwdth, access latency, capacty, and avalablty. As a result, QVDs n Stonehenge are as tangble as physcal dsks but much more flexble and manageable. Each QVD V can be specfed n terms of (1) bandwdth ρ avg or the number of I/O requests per second IOPS avg, (2) worst-case delay bound requrement D per I/O request, (3) delay volaton probablty bound P, and (4) capacty C of the QVD. Gven a QVD specfcaton <ρ avg, D, P, C >, for rate-based QoS-aware dsk request schedulers, a correlaton functon F (.) maps the bandwdth reservaton ρ delay requred to acheve a worst-case delay bound D. Gven ρ delay = F (D ), one can then further reduce each QVD specfcaton to < max(ρ avg, ρ delay ),, P, C >. Fgure 4 shows the overall archtecture of Stonehenge. Stonehenge s a cluster-based SCSI storage system that conssts of a central management server and a set of storage server nodes connected va a ggabt ethernet network. The central manager server performs admsson control and allocates physcal dsk resources to satsfy each QVD s QoS requrements. At runtme, the management server uses a Vrtual Clock scheduler to determne the order n whch ncomng requests from dfferent QVDs are processed such that each QVD s QoS requrement s satsfed. At the ndvdual storage server nodes, another effcency and deadlne aware Vrtual Clock-based dsk scheduler s used to decde the actual order n whch I/O requests are servced by physcal dsks. In ths secton, we focus specfcally on how Stonehenge apples DDM to convert the latency bound requrement D and volaton probablty requrement P to a bandwdth requrement ρ delay. Whle the basc prncples behnd DDM admsson control reman the same for both dsk and network resource allocaton, mportant dfferences arse due to the physcal nature of the resources. In the rest of ths secton, we focus on how the DDM algorthm for admttng QVDs dffers n terms of the delay-toresource correlaton functon and the manner n whch t explots the runtme load nformaton. Other ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

12 12 K. Gopalan et al. major components of Stonehenge, such as a two-level dsk schedulng archtecture, the dsk servce tme predcton mechansm, and an effcency conscous real-tme dsk scheduler are descrbed n Huang et al. [2004] and Gopalan and Chueh [2001]. 3.1 Delay to Resource Mappng Stonehenge uses a varant of a Vrtual Clock scheduler to compute the fnsh tme for each I/O request. Equaton (1), whch provdes the delay bound D for a bandwdth reservaton ρ delay, s approprate for network resource allocaton and schedulng. To convert ths network latency bound expresson to one approprate for dsk latency bound expresson, we need to account for the dsk servce overhead assocated wth each request. The resultng delay bound expresson as appled to QVDs becomes: D (δ + L max + overhead C)/ρ delay + (L max + avg overhead C)/C (7) where δ = max pendng reqs avg req sze and overhead = avg overhead (max pendng reqs + 1), where, C s the total bandwdth of the underlyng system, ρ delay s mnmum bandwdth reservaton requred to guarantee a delay bound of D, max pendng reqs s the maxmum number of requests the queue can hold for a gven request sze, and avg overhead s the average dsk access overhead tme measured and computed at runtme. Compared wth the orgnal delay bound equaton, we expand the request sze by (overhead C) bytes to account for the access overhead for each request. By multplyng the measured average dsk access wth C, we translate t to the number of bytes that could be transferred durng the overhead tme. Seek delay and rotatonal latency play an ncreasngly sgnfcant part n dsk servce tme. Consequently, dsk request sze tself becomes relatvely unmportant, especally when most requests are small. Therefore, we can further smplfy the expresson for latency bound as follows: D (max pendng reqs + 1)/IOPS delay + 1/IOPS max, (8) where, IOPS delay s QVD V s request throughput (n number of I/O operatons per second) requred to guarantee a delay bound of D. Smlarly, IOPS max s the maxmum throughput the physcal storage system can support. In cases where the assumpton about dsk request sze s nvald, one can always use Equaton (7). 3.2 Explotng Load Informaton n Admsson Control As wth network resource allocaton, the DDM algorthm n Stonehenge explots statstcal multplexng along the delay dmenson to ncrease the total number of QVDs that can be admtted nto a physcal storage system. Equaton (8) converts a delay bound to ts equvalent throughput requrement based on the worst-case delay bound assocated wth the Vrtual Clock scheduler. In practce, ths proves to be too conservatve because not every dsk request experences the worst-case delay. Therefore, Stonehenge also measures the CDF Prob(r), that s, the cumulatve probablty dstrbuton of the rato between the actual delay experenced by a request and the worst-case delay of the QVD wth whch the request s assocated. Prob(r) depends on the number of QVDs n the system because the delay a request actually experences depends on the actual load n the system whch s correlated wth the number of QVDs. Wth Prob(r), the delay bound expresson used to decde whether to admt the Nth QVD becomes: D N ((max pendng reqs + 1)/IOPS delay N + 1/IOPS max ) (Prob 1 (1 P N ) + s), (9) where Prob 1 (.) s the nverse functon of Prob(r), P N s the probablty bound that the Nth QVD s delay bound could be volated, and s s an adjustment factor that accounts for the mpact of the new QVD on ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

13 Statstcal Admsson Control Usng Delay Dstrbuton Measurements 13 the delay behavor of exstng QVDs. When the system s lghtly loaded or N s small, Prob 1 (1 P), wth P equal to 0.05, for example, can be as low as 10%, whch means 95% of the requests experence a delay that s smaller than 10% of the worst-case delay. In contrast, a determnstc admsson control algorthm wll assume 100% of the requests experence the full worst-case delay. For a gven P, Prob 1 (1 P) grows closer to 1 wth ncreasng N. The value of s s largely workload-dependent and s 0.2 n Stonehenge. However, f the system s stable enough, the measurement-based feedback s able to detect a relatvely stable s value. In ths case, Equaton (9) can be used. Otherwse, Equaton (8) should be used f the workload s hghly unpredctable. The DDM admsson control algorthm for QVDs s smlar n operaton to the algorthm for QVPNs descrbed n Secton Assume that QVD V N wth requrement (IOPS avg N, C N, D N, P N ) arrves for admsson where (N 1) QVDs have already been admtted. DDM frst calculates IOPS = max(iops avg, IOPS delay ), 1 N, where IOPS delay s calculated usng Equaton (9). The QVD V N s admtted only f N =1 IOPS C. 4. PERFORMANCE OF DDM IN NETWORK RESOURCE ALLOCATION In ths secton, we study the performance of the DDM algorthm for admttng QVPNs n comparson to determnstc admsson control. We use the determnstc approach as a baselne nstead of one of the earler approaches for the followng reasons. Frst, earler measurement-based approaches manly address multplexng along the bandwdth dmenson, that s, multplexng due to the fact that QVPNs typcally transmt at rates much below ther stated long-term bandwdth requrement. In contrast, DDM explots multplexng along the orthogonal delay dmenson whch occurs even when ndvdual QVPNs transmt at ther stated bandwdth, that s, multplexng due to the fact that dfferent QVPNs transmt ther traffc bursts at dfferent tmes. Second, to the best of our knowledge, earler analytcal approaches that address probablstc delay guarantees ether assume a flud traffc model (as opposed to a packetzed model) or do not support dstnct per-qvpn probablstc delay bounds, but rather provde shared guarantees such as by multplexng QVPN traffc n a shared buffer. Thus the problem addressed by DDM s fundamentally dfferent from earler approaches and leaves determnstc admsson control as the baselne for comparson. The real traffc traces used n our smulatons are prncpally composed of VoIP sources. However, a note regardng applcablty of DDM to heterogeneous real-tme traffc s n order. Unlke voce, vdeo conferencng applcatons have relatvely hgher and more varable data rates (due to quantzaton va moton vectors and predcton algorthms), though wth smlar latency requrements. Onlne tradng applcatons, on the other hand, have much lower data rates wth tghter latency requrements. In the presence of dfferent categores of real-tme traffc, we stll expect sgnfcant potental gans n lnk utlzaton wth varyng degrees of statstcal multplexng. However, the DDM algorthm s equally applcable to mxes of all categores of real-tme traffc and nothng n the algorthm precludes any specfc traffc category. 4.1 Evaluaton Setup Usng the ns-2 smulator, we confgured a sngle lnk at 10Mbps, and packets arrvng at the lnk were served by a WFQ scheduler. Traffc for each QVPN was generated usng aggregated traffc traces of recorded VoIP conversatons used n Jang and Schulzrnne [1996] n whch spurt-gap dstrbutons were obtaned usng a G.729 voce actvty detector. In other words, packet szes and nterpacket arrval duratons wthn each QVPN followed the exact pattern as n real traffc traces. Each VoIP stream had an average data rate of around 13Kbps, peak data rate of 34Kbps, and packet sze of L max = 128 bytes. ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

14 14 K. Gopalan et al. Actual rate of delay volatons 1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1e-08 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 Desred delay volaton probablty bound Actual rate of delay volatons 1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1e-08 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 Desred delay volaton probablty bound Actual rate of delay volatons 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 Desred delay volaton probablty bound Fg. 5. The DDM algorthm satsfes dstnct per-qvpn delay volaton guarantees when (a) all other requrements are the same and (b) consttuent QVPNs have dssmlar delay bound, data rate, and burstness requrements. Plot (c) shows that determnstc admsson control, wth a pure oversubscrpton of lnk capacty by a factor of 2, cannot satsfy dstnct per-qvpn delay volaton guarantees. All plots nclude data ponts from 5 smulaton runs wth dfferent random seeds. We temporally nterleaved the 20 VoIP streams to generate aggregate traffc trace for each QVPN wth an aggregate data rate of ρ avg = 256Kbps. Each aggregated VoIP trace was 8073 seconds long. Every QVPN n our smulatons sent traffc for the entre lfetme of the smulaton wth the aggregate traffc trace repeated over ts lfetme. Traffc from each admtted QVPN passed a token bucket wth bucket depth of 1280 bytes (10 packets) and token rate of 256Kbps. Each new QVPN requred a guarantee on a delay bound and a delay volaton probablty. The admsson control algorthm decded whether to admt or reject the QVPN and how much bandwdth to reserve accordng to the algorthm n Fgure 1. Each QVPN was generated wth a perodc nterarrval tme of 10, 000 seconds. The reason we selected perodc nstead of exponental nterarrval tmes (as n other works) s that our QVPNs are long-lved and are expected to arrve farly nfrequently so that the measured CDF can stablze before beng used to admt another QVPN. Hence the request arrval pattern does not sgnfcantly mpact the admsson control decsons. The CDF was measured over a tme nterval of 10, 000 seconds between QVPN arrvals. Each smulaton run lasted for 1000, 000 seconds. For smulatons, we recorded the rato of actual to worst-case delay of every packet traversng the lnk wthn the current CDF wndow (although n a realstc scenaro, an ntellgent samplng mechansm would be more desrable). The observed ratos are accumulated nto a hstogram. The actual CDF s computed from the hstogram only when makng admsson decsons or recalculatng exstng reservatons. 4.2 Per-QVPN Probablty Bounds We start by valdatng that the DDM algorthm can ndeed provde dstnct guarantees on heterogeneous delay volaton probabltes for a mx of dfferent traffc types. In the frst experment, we consder a traffc mx n whch all QVPNs request the same delay bound of 20ms, the same average rate of 256Kbps, and the same burst sze of 10 packets, but requre dfferent guarantees on delay volaton probablty snce the requrement s unformly dstrbuted among the four values 10 2,10 3,10 4, and Fgure 5(a) plots the actual fracton of packets exceedng ther delay bound of 10ms aganst the desred volaton probablty for each QVPN that experences any excess delay. The fgure ncludes data ponts from 5 smulaton runs wth dfferent random seeds, and each data pont represents the rate of delay volaton experenced by one QVPN. Fgure 5(b) plots the same data when the consttuent QVPNs have heterogeneous delay bounds (10ms 30ms), data rates (256Kbps 2Mbps), and burst-szes (10 40 packets), n addton to heterogeneous volaton probablty requrements ( ). The lne through the graph marks the lmt above whch the actual rate of delay volatons would exceed the desred delay volaton probablty. The fact that all data ponts are below the lne ndcates that the actual delay volaton rate s smaller than the maxmum permssble for each QVPN. Furthermore the ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

15 Statstcal Admsson Control Usng Delay Dstrbuton Measurements 15 Fg. 6. The predcted CDF est (the rghtmost curve) provdes a relable bound on future delay rato dstrbuton for each admtted QVPN (all other curves). The Y-axs range shown (from 0.99 to 1.0) corresponds to the tolerance range below fgure shows that QVPNs that have a hgher tolerance to delay volatons are more lkely to experence a hgher rate of volaton than QVPNs wth lower tolerance. The DDM algorthm s able to dstngush among QVPNs n terms of delay volaton rates because t assgns servce bandwdth ρ,l to QVPNs n the nverse proporton to ther tolerance to delay volatons. Ths translates to hgher dynamc preference for packets belongng to QVPNs wth low delay tolerance and vce versa. In the next experment, we show that pure oversubscrpton of lnk capacty cannot provde dstnct guarantees on heterogeneous delay volaton probabltes. We use the same parameters as n Fgure 5(a) except that, nstead of usng the DDM algorthm, we use determnstc admsson control and oversubscrbe the lnk capacty by a factor of 2.0 n order to admt the same number of QVPNs as the DDM algorthm (.e., 35 QVPNs) wth no oversubscrpton. Fgure 5(c) shows that regardless of the desred delay volaton bounds, all QVPNs experence smlar rates of actual delay volatons. In fact, QVPNs wth low tolerance (10 5 ) to delay volatons can experence an order of magntude hgher delay volatons than ther actual tolerance. Ths s because pure oversubscrpton does not correlate to delay volaton bound requrements for a QVPN wth ts bandwdth reservaton. We need more than just bandwdth oversubscrpton, specfcally, a delay-probablty-bandwdth correlaton functon. such as n Equaton (2), to guarantee dstnct per-qvpn probablstc guarantees. 4.3 Valdatng the CDF Estmaton Technque Next we valdate that the technque for predctng the future delay rato dstrbuton CDF est n Secton 2.3 ndeed relably bounds the delay rato dstrbuton of admtted QVPNs. Valdatng the CDF estmaton technque s mportant n establshng that the DDM algorthm does not underestmate the resource requrements for ndvdual QVPNs, resultng n excess delay volatons n the long-term. Fgure 6 shows a representatve smulaton scenaro n whch 19 consttuent QVPNs are admtted wth heterogeneous delay bound, data rate, and burstness requrements. The rghtmost curve marked CDF est shows the delay rato dstrbuton estmated by DDM before admttng the 19th QVPN, where the curves on the left represent the stable per-qvpn dstrbutons at the end of the smulaton lfetme. The fgure demonstrates the fact that the CDF est dstrbuton used at admsson control tme stll remans more conservatve than ndvdual QVPN dstrbutons n the long-term. Thus CDF estmaton ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.

16 16 K. Gopalan et al. Fg. 7. Number of admtted QVPN vs. delay bound. Delay volaton probablty = 10 5, burst sze = 10pkts, lnk capacty = 10Mbps. technque can effectvely reduce each QVPN s resource requrement to sut ther ndvdual tolerance to delay volatons wthout rskng, underestmaton of true requrements. 4.4 Delay Bound Varaton Next we compare the performance of the DDM algorthm aganst determnstc admsson control as the delay bound requrement vares. Wth the DDM the algorthm, the delay volaton probablty for each QVPN s 10 5, where determnstc admsson control consders a zero delay volaton probablty. Fgure 7 plots the number of QVPNs admtted as the delay bound requrement s vared from 3 to 50ms. The maxmum number of QVPNs that can be admtted on the 10Mbps lnk s lmted to 39 QVPNs due to the average rate requrement of 256Kbps for each QVPN. Fgure 7 shows that, for small delay bound requrements, the DDM algorthm admts around 3.0 tmes more QVPNs than determnstc admsson control when the delay volaton probablty as small as 10 5 s allowed. As the delay bound requrement becomes less strngent, the DDM algorthm stll admts more QVPNs and acheves better lnk utlzaton than the determnstc algorthm but wth smaller mprovements. Beyond a 45ms-delay requrement, both algorthms are lmted to admttng 39 QVPNs due to the average rate the requrement of 256Kbps for each QVPN. The gan for the DDM algorthm comes from the fact that the large majorty of packets experence just 1% to 3% of the worst-case delay dctated by ther reserved bandwdth. Ths statstc gets reflected n the CDF whch, n turn, helps to reduce the resource requrement for each QVPN. 4.5 Burst Sze Varaton Fgure 8 compares the DDM algorthm aganst determnstc admsson control as the burst sze σ for each QVPN s ncreased from 1 to 100 packets. Larger burst szes have the effect of ncreasng the average tme a packet spends watng n queue to be servced by the lnk scheduler. Up to burst szes of 40 packets, the DDM algorthm admts a sgnfcantly larger number of QVPNs than the determnstc algorthm. Ths s because the determnstc algorthm operates on the worst-case scenaro that bursts from all QVPNs arrve at the lnk smultaneously. On the other hand, DDM successfully explots the statstcal multplexng effect, that s, bursts from dfferent QVPNs are temporally dspersed and rarely ACM Transactons on Multmeda Computng, Communcatons and Applcatons, Vol. 2, No. 4, November 2006.