Quality of Service Evaluations of Multicast Streaming Protocols *

Transcription

1 Quaity of Service Evauations of Muticast Streaming Protocos Haonan Tan Derek L. Eager Mary. Vernon Hongfei Guo omputer Sciences Department University of Wisconsin-Madison, USA {haonan, vernon, Department of omputer Science University of Saskatchewan, anada ABSTRAT Recenty proposed scaabe on-demand streaming protocos have previousy been evauated using a system cost measure termed the required server bandwidth. For the scaabe protocos that provide immediate service to each cient when the server is not overoaded, this paper deveops simpe anaytic modes to evauate two cient-oriented quaity of service metrics, namey () the mean cient waiting time in systems where cients are wiing to wait if a (we-provisioned) server is temporariy overoaded, and (2) the fraction of cients who bak (i.e., eave without receiving their requested media content) in systems where the cients wi toerate no or ony very ow service deays during a temporary overoad. The modes incude nove approximate MVA techniques that appear to extend the range of appicabiity of customized AMVA to incude questions focussed on state probabiities rather than on mean vaues, and to systems in which the operating points of interest do not incude substantia cient queues. For exampe, the new AMVA modes accuratey estimate the server bandwidth needed to achieve a baking rate as ow as one in ten thousand. The anaytic modes can easiy be appied to determine the server bandwidth needed for a given number of media fies, anticipated tota cient request rate and fie access frequencies, and target baking rate or mean wait. Resuts show that (a) scaabe media servers that are configured with the required server bandwidth defined in previous work have ow mean wait but may have unacceptaby high cient baking rates (i.e., greater than one in twenty), (b) for high to moderate cient oad, ony a % increase in the previousy defined required server bandwidth is needed to achieve a very ow baking rate (e.g., one in ten thousand), and (c) media server performance (either mean wait or baking rate) degrades rapidy if the actua cient oad is more than % greater than the anticipated oad.. INTRODUTION A probem of considerabe current interest is that of providing scaabe Internet services to potentiay vast numbers of cients. Particuary chaenging are scaabe deivery protocos for appications that require on-demand streaming of mutimedia, such as news services, distance education, or entertainment-ondemand. Recenty proposed protocos [3, -7, 9-7, 2, 22, 23] use broadcast or muticast to reduce the required server and To appear in Proc. AM SIGMETRIS 22 Int. onf. On Measurement and Modeing of omputer Systems, Marina de Rey, A, June 22. network bandwidth from inear in the request rate, to subinear. They achieve this bandwidth reduction by dynamicay aggregating cients that make requests cosey spaced in time, so that eventuay these cients share the same stream(s). Three of the recent muticast streaming protocos, namey patching [, 6, 4, 6, 22], dynamic skyscraper [9, ], and hierarchica stream merging (HSM) [3, 7, -2] provide scaabe on-demand streaming without requiring cients to wait for some period of time as in a periodic broadcast system [3,, 7, 2, 23] or a batching system. These three protocos and a variant of HSM caed bandwidth skimming [] have been compared using a server cost or provisioning measure termed the required server bandwidth [, 2]. This measure is the average concurrent server bandwidth used if the server has infinite bandwidth and each cient request is satisfied immediatey. The previous resuts show that the required server bandwidth for HSM systems, for fu fie requests and very genera assumptions about the cient arriva process, grows ony with the ogarithm of the cient request rate, whereas the required server bandwidth for patching systems with Poisson arrivas and fu fie requests grows with the square root of the cient request rate [, 4]. In practica terms, if cients request the most popuar media content at a rate greater than requests per time that it takes to pay the content, the HSM system has ower required server bandwidth than patching. The dynamic skyscraper system has required server bandwidth that grows with the ogarithm of the cient request rate, but with a arger constant factor than for HSM. Since HSM aso supports interactive cient requests more efficienty than the dynamic skyscraper protoco, we consider ony the HSM and patching protocos in this paper. A key open question addressed in this paper is how we the patching and HSM systems perform when the server has fixed (we provisioned) bandwidth and two cient-oriented, quaity of service metrics are considered, namey () the mean cient wait if cients wait when the server is temporariy overoaded, and (2) the fraction of cients who eave without receiving their requested media content if cients wi toerate no or ony ow service deays during the temporary overoad. The atter quantity is caed the baking rate in the remainder of this paper. Specificay, for each protoco, we investigate the foowing questions:. What is the mean waiting time, or the cient baking rate, when the server is configured with the required server bandwidth defined above for a given set of popuar media objects and a given cient oad? This work was partiay supported by the U. S. Nationa Science Foundation under grants R-99744, ANI-78, and EIA-2787, and by the Natura Sciences and Engineering Research ounci of anada under Grant OGP-264.

2 2. How much additiona or ess server bandwidth is needed to achieve a given target cient baking rate, such as one in ten thousand, or a given target mean waiting time, such as.t, where T is the requested media paying time? 3. How sensitive is the mean wait, or baking rate, to cient arriva rate higher than the anticipated rate that the server is configured for? For patching systems these questions are particuary reevant for modest cient oads, in which case the previousy defined required server bandwidths for patching and HSM are simiar. A key goa is to evauate the QoS measures over a wide range of system configurations and cient workoads. Simuation is timeconsuming, and error-prone (e.g., due to ack of vaidation of the streaming protoco impementation, difficuty in choosing simuation run engths for estimating ow baking rates, etc.). Thus, another question addressed in this paper is whether simpe, efficient anaytic methods can be deveoped to accuratey estimate the QoS metrics of interest. Previous work [8] has proposed anaytic methods for estimating mean wait and baking rate for reativey simpe batching systems (in which fu media fie transmissions start at fixed times) and for FFS unicast streaming systems. Deveoping accurate anaytic methods for the patching and HSM systems considered here is substantiay more compex for at east three reasons. First, the muticast stream duration (or service time) is correated with the number of active streams for the object, and the precise form of the correation is not easy to characterize. Second, the cient baking rate cannot be computed using a known waiting time distribution or known stream start times. Third, when cients wait for service, the cients waiting for the same media fie batch together for a stream with a state-dependent start time. Previous techniques that accuratey mode state-dependent behavior such as genera state-dependent service times, cient baking, or cients seectivey batching whie waiting for service, are primariy based on state-space anaysis (e.g., Markov hain anaysis), which has high computationa cost for arge and compex systems. Aternativey, Approximate Mean Vaue Anaysis (AMVA) techniques have the advantages of very ow computationa cost and intuitive equations that readiy permit heuristic extensions for modeing compex system features. However, it is not cear that AMVA can be appied to systems where cients batch whie waiting for service or to the statedependent stream durations that occur in scaabe streaming systems. Furthermore, AMVA seems to be inherenty focused on predicting the impact of queueing on mean performance metrics such as mean waiting times or mean queue engths. It is not cear a priori, that AMVA techniques can be appied to questions such as how to provision a system so that the probabiity that a cient has to wait is cose to zero. Such questions may become more important (for exampe) as Internet services mature and become more utiity-ike. By deveoping customized AMVA techniques for estimating mean wait or cient baking rate in on-demand streaming systems, we hope to obtain (a) insight into scaabe media server provisioning questions, and (b) specific techniques and approaches that may be appicabe in other contexts. The main contributions of the paper are: Efficient, highy accurate customized AMVA modes for estimating mean cient waiting time or cient baking rate in patching and HSM systems. A server configured with the previousy defined required server bandwidth for a given protoco and anticipated moderate to high cient arriva rate wi have mean cient wait under.t for HSM or under.2t for patching, where T is the time to pay the media fie. A patching or HSM system configured with the previousy defined required server bandwidth wi have unacceptaby high cient baking rate (i.e., greater than one in twenty) if cients are unwiing to wait for service. However the resuts aso show that very ow baking rate (e.g., /,) can be achieved with a reativey modest increase in server bandwidth (i.e., on the order of a % % increase for high to moderate cient oad). Scaabe media server performance (either mean wait or baking rate) degrades rapidy if the actua cient oad is more than % greater than the anticipated oad. The modes for cient baking rate incude customized AMVA techniques that are capabe of accuratey estimating the server bandwidth needed to achieve a wide range of target baking rates (from - to beow -4, for exampe), suggesting that customized AMVA techniques may be appicabe to a significanty broader range of metrics and system design questions than has previousy been recognized. The modes for mean cient waiting time, aso based on AMVA, are aso interesting in that simpe modes are found to be quite accurate, even though scaabe streaming systems have a number of compex features (such as stream ength being strongy correated with the number of active streams for a fie, and batching of requests for the same fie whie waiting in a queue) that might suggest a need for more compex modes. The remainder of the paper is organized as foows. Section 2 provides background on patching, hierarchica stream merging, the bandwidth skimming variant of HSM, and the previous anaysis of required server bandwidth for these protocos. Sections 3 and 4 deveop modes for estimating cient baking rate and mean cient waiting time, respectivey. Section vaidates the modes and appies them to evauate patching and HSM system configurations with respect to the cient QoS metrics. oncusions are presented in Section BAGROUND The notation used in this section and in Section 3 is summarized in Tabe. In this section we review the patching protoco (Section 2.), the HSM and bandwidth skimming protocos (Section 2.2), and the system assumptions that are made in deveoping the new modes in this paper (Section 2.3). 2. Patching The basic operation of patching [, 6, 4, 6, 22] is as foows. In response to a given cient request, the server deivers the requested media in a singe muticast stream. A cient that submits a new request for the same fie soon after this stream has started istens to the muticast, buffering the data received. Each such cient is aso provided a new unicast stream (i.e., a patch stream) that deivers the data that was deivered in the muticast stream prior to Our protoco descriptions and anayses assume that cients have sufficient oca storage to buffer a data received in advance of its payback time. The protocos can easiy accommodate cient buffer constraints, as described in the cited earier work.

3 the new cient s request. The patch stream terminates when it reaches the point where the cient joined the fu-fie muticast. Both the muticast stream and the patch stream deiver data at the fie pay rate so that each cient can pay the fie in rea time. Thus, the achievabe aggregate transmission rate to a cient must be at east twice the fie pay rate. To keep the patch streams short, when the fraction of the fie deivered in the most recent muticast exceeds a given threshod, the next cient request triggers a new muticast rather than a patch stream. In optimized patching [6, 4], the threshod for each fie is chosen to minimize the required server bandwidth (as defined in Section ), assuming known fie request rate, Poisson arrivas, and that each cient requests the fu fie. We anayze optimized patching in this performance study, but note as in previous work that optima choice of the threshod parameter is difficut to achieve in practice. Letting N i denote the average number of requests for fie i that arrive during the fie pay time T i, the optima threshod is ( 2N i + ) / Ni, and the required server bandwidth for deivery of fie i, in units of the media pay rate, can be derived for Poisson request arrivas as [4]: B i, patching ( Ni ) = 2Ni +. () 2.2 HSM and Bandwidth Skimming Hierarchica stream merging protocos [3, 7, -2] initiate a new muticast transmission stream for each new cient request. In the simpest case, each cient aso istens to the cosest earier stream for the fie that is sti active [2], so that its own stream can terminate after transmitting the data that was missed in the earier stream, as iustrated in Figure. In the figure, cients A through D request the media fie at times T, T2, T3, and T4, respectivey. At time T4, cient D istens to the stream that starts at T4 as we as the stream that was initiated for cient at time T3. At time T, the stream for cient D can be terminated, and cients and D are merged. When cients merge, they isten to the cosest earier stream that is sti active, and so on. A number of more compex HSM poicies have very simiar performance [3, 7, 2]. Bandwidth skimming is a variant of hierarchica stream merging that uses ony a sma skim of the achievabe transmission rate to a cient to support the merging. This is usefu in cases where the quaity of the media content is such that a singe stream at the pay rate consumes more than haf of the achievabe transmission bandwidth to the cient. Bandwidth skimming poicies [] use the hierarchica stream merging strategy together with a mechanism for effecting each merge that does not require cients to isten to two pay rate streams concurrenty. For exampe, in the Partition poicy, streams are divided into substreams, and a cient A merges with another cient B by istening to successivey increasing numbers of B s substreams, whie istening to successivey decreasing numbers of its own substreams. Letting b denote the required aggregate transmission rate to a cient, in units of the media pay rate, the required server bandwidth for a given fie under HSM (b = 2) or bandwidth skimming (b < 2) and Poisson request arrivas can be approximated by the foowing formua []: N i B i, HSM / bandwidth skimming ( Ni ) b n +, (2) b Position in Media Stream Stream and progress for cient A T T2 Merged stream and progress for cients A and B Stream for cient B T3 T4 T Time where η b is the positive rea constant that satisfies the foowing equation: b b b =. b + Progress for cients and D Merged stream for cients and D Progress for cient D Stream for ient D Figure : HSM Exampe For exampe, for HSM with b=2, the required server bandwidth can be reasonaby approximated by []: B i, HSM ( Ni ).62 n ( Ni /.62 + ). (3) 2.3 QoS Motivation and System Assumptions Figure 2 pots the required server bandwidth for a media fie as a function of cient request rate for the fie, N i, as computed from equations () (3) above. Previous work [2] has provided the rationae as we as resuts from simuations that show that an HSM server containing a coection of media fies with given cient request rates has average cient waiting time cose to zero if configured to support a maximum number of concurrent streams equa to the sum of the required bandwidth for each fie given in equation (3). This paper deveops, vaidates, and appies modes to determine more precisey, for patching and bandwidth skimming servers as we as HSM servers, the mean cient baking rate or mean cient waiting time for various system configurations. In particuar, the Required Server Bandwidth ient Request Rate (N i ) Patching BWSkim, b=.2 HSM Figure 2: Required Server Bandwidth for Fie i ( B i )

4 customers / ide channes Figure 3: ient Baking Mode modes can be used to determine how much server bandwidth is needed to achieve a given target cient baking rate or target mean wait for a given set of media fies and cient access rates. The next section deveops the baking rate anaysis; Section 4 deveops the mean cient waiting time anaysis. In both cases, to enabe insight across a broad spectrum of cient oads and server capacities, we assume that a media fies on the server are of equa (but arbitrary) pay rate and duration, the request arriva process is Poisson, and each cient requests the entire media fie. The resuts derived under these assumptions can be refined for systems that have partia fie accesses, media fies with unequa durations or pay rates, and/or other arriva processes, as noted in Section LIENT BALING RATE ANALYSIS We consider a media server that is configured to deiver a maximum number of concurrent streams, denoted by, each at the media pay rate. If a new cient request arrives when there are aready streams serving other cients, the new cient baks. That is, the cient eaves the system without receiving the requested content. In the foowing, the unit of server capacity sufficient for deivering a singe stream is caed a channe. Beow we deveop a mode and four aternative anaytic methods for estimating the fraction of cients that bak (P bak ), as a function of, the tota cient arriva rate, and the reative popuarity of each of the media fies. The soutions are vaidated against system simuations in Section.. 3. ient Baking Mode The media server is simiar to an M/G// system that has Poisson arrivas, servers (i.e., the channes), and no additiona queue for cients to wait for service, except that service times are correated with the number of active servers (i.e., stream durations that are correated with the number of active channes). Thus, the system can be represented by the cosed two-center queueing network with customers depicted in Figure 3, in which customers at the queueing center represent ide channes that are waiting to serve a new cient request whie customers at the upper deay center represent active channes. The queueing center generates new active channes at rate equa to the cient request rate λ as ong as at east one channe is ide. The service times at the deay center (i.e., the active stream durations) depend on the streaming protoco and are correated with the number of active streams, with overa mean denoted by S. Once a stream ends, the S active channes Input λ α T Output P bak p i N i B i,p S X Q R U Definition server capacity or bandwidth, in units of the media pay rate average tota cient request rate number of fies avaiabe on the server parameter of Zipf fie access distribution fie pay duration fraction of cients that bak, baking rate fraction of requests for fie i, p i = (/i α )/Σ(/j α ) normaized cient request rate for fie i, N i = p i λt i required server bandwidth for fie i using protoco p overa mean stream duration system throughput, X = λ ( P bak ) mean number of ide server channes mean queueing center residence time average fraction of time a server channe is busy Tabe : Notation for ient Baking Mode channe joins the ide channe poo in the ower queue, waiting to be reactivated by a new cient request. Note that the cients that arrive (at rate λ) when the queue of ide channes is empty depart immediatey (i.e., bak) and have no impact on the operation of the channes modeed in Figure 3. Two features of the mode make the soution significanty more chaenging than a queueing system with genera independent service times at the deay center. First, the average stream duration, S, is a function of system throughput, X, which is an output of the anaysis. Second, the service time at the deay center is correated with the number of active streams (i.e., with the number of customers in the deay center). Since the dependence between stream duration and number of active streams is difficut to characterize, we first deveop soutions that ignore the correation, and then discover a soution that approximatey captures the impact of the correation. The notation for the baking mode in Figure 3 is defined in Tabe. Due to the one-to-one correspondence between channe activations and non-baking cient requests, the system throughput, denoted by X = λ( P bak ), can be interpreted as the channe activation rate. Thus, by Litte s aw [9] the foowing equation reates the 2-center system measures to the measure of interest, namey, the baking rate: = X = R + S ( Pbak ). (4) A second observation that enabes the anaysis is that S can be estimated from the (unknown) throughput of the queue, using Litte s aw, as B i, p( pi XT ) S i= X, () where B i,p is the required server bandwidth computed using equation (), (2) or (3) with N i repaced by p i XT. This is an approximation since equations () (3) were derived assuming Poisson stream initiations, which is not the case for the system with a finite number of channes.

5 Equations (4) and () provide three equations that reate four unknowns, namey: X, S, R, and P bak. Beow we propose four methods for obtaining further equations for soving the mode. The soutions in Sections 3.2 and 3.3 are motivated by simpicity, the soution in Section 3.4 is motivated by refinements that shoud improve the accuracy of the first two modes, and the mode in Section 3. incudes a further refinement to the mode in 3.4 that partiay captures the correation between expected stream duration and the number of active streams when a cient initiates a new stream. Section wi show that this fourth mode has the best overa accuracy for estimating cient baking rate in patching, HSM, and bandwidth skimming systems. 3.2 LIS AMVA Soution One of the simpest soutions to the mode in Figure 3 is to assume that stream durations are oad independent (i.e., not correated with the number of active streams) and to sove the resuting machine repair mode (with unknown S) using Schweitzer s approximation [, 2, 2] to estimate the mean time that a channe remains ide (i.e., the mean residence time at the queueing center): ( ) R ( + XR), (6) where, using Litte s aw, the mean queue ength at the queueing center, Q, has been repaced by XR. Equations (4), () and (6) define the LIS AMVA mode that can be iterativey soved by successive substitution or some other suitabe technique for soving a system of non-inear equations. Two sources of error in the above LIS AMVA soution motivate consideration of further soutions. The first is that baking rate may be inherenty difficut to estimate. In particuar, a sma percentage error in X, as might resut from the Schweitzer approximation in (6), coud yied a arge error in P bak as cacuated from (4). A second source of error is the assumption that average time at the deay center is independent of the number of customers at the deay center. 3.3 LIS I Soution This next approach assumes that service times at the deay center are oad independent, but does not invove estimating mean residence time at the queueing center. Instead, we directy estimate the probabiity that a channes are busy, which is the baking rate, P bak. A very simpe approximation that the states of the channes are independent, yieds P bak U. (7) Using U = S/(R+S) and the first equaity in (4) yieds which, by (), can be re-written as S U = = SX, (8) R + S U B i, p ( pi XT ). (9) i= Finay, the second equaity in (4) together with (7) gives X ( U ). () Equations (8) () can be iterativey soved to obtain the estimated baking rate, U. Athough it seems that the assumption of independent channe states coud be quite inaccurate, vaidation resuts discussed in Section. show that the LIS I soution provides more accurate resuts than the LIS AMVA soution for HSM systems. 3.4 LIS EL/MVA Soution A third soution is motivated either by eiminating the Schweitzer approximation used in the LIS AMVA mode or by a desire to improve the estimate of baking rate used in the LIS I soution. In either case, note that if the service times at the deay center are assumed to be oad-independent with known mean S, then the mode in Figure 3 is a separabe queueing network [9] that is equivaent to an ordinary M/G// queue with arriva rate λ and mean service time S. To improve the estimate of baking rate, note that by the weknown BMP resut [4], the separabe network has the same soution as the M/M// queue. That is, the soution for generay distributed deay center service times with mean S is the same as for exponentiay distributed service times with mean S. The baking rate is equa to the probabiity that the queue contains customers, which is given by Erang s oss (EL) formua for the M/M// queue [8]: ( S) /! Pbak. () ( S) i / i! i= Substituting the above into equation (4) yieds ( S) /! X. (2) i ( S) / i! i = Equations (2) and () can be soved iterativey to estimate S and X, and then equation () can be appied to obtain the baking rate for this LIS EL soution. Equivaenty, we can sove the mode by estimating the mean residence time at the queueing center using the arriva instant theorem for the number of customers in the queue at the arriva instant [9], rather than Schweitzer s approximation. Let X(c), R(c), and Q(c), respectivey, denote the throughput, mean residence time at the queueing center, and mean queue ength at the queueing center, when the network contains c customers, c =, 2,,. Using the arriva instant theorem and equation () with X=X(), the remaining set of equations for this LIS MVA ( customized MVA ) soution are: R ( c) ( + Q( c )) (3a) c X ( c) =, c =,, (3b) R( c) + S Q ( c) = X ( c) R( c), (3c) with the boundary condition Q() =. To sove the system numericay, S can be initiaized to some suitabe vaue, and then equations (3) foowed by equation () with the new estimate of X = X() can be repeatedy appied unti convergence.

6 X i customers (one per cass) S/ (conditiona) /λ i Output W i W i Q i X i X Definition mean wait for fie i requests mean wait for fie i ead requests mean queue ength of fie i ead requests ( Q i <) throughput of fie i ead requests tota throughput of ead requests Tabe 2: Additiona Notation for Waiting Mode Figure 4: Waiting Mode for Lead ient Requests 3. MVA Soution The fourth soution method is derived from the observation that in the computationa structure shown in (3a c), the same mean stream duration, S, is used for every network popuation, whereas mean stream duration shoud increase when the network popuation (or average number of active streams) decreases. The MVA soution modifies equation () to estimate the expected stream duration for each network popuation, as foows: Bi, p( pi X ( c) T ) S( c) i= X ( c), c = 2,,, (4) and S() = T. In this approach, an iterative soution of equation (4) and equations (3a c) with S repaced by S(c) in (3b) is required at each popuation eve, c, in order to determine the vaue of S(c), which approximatey captures the correation between mean stream duration and the number of active streams. 4. LIENT WAITING TIME ANALYSIS We again consider a media server that is configured to deiver a maximum of streams concurrenty. For the modes deveoped in this section, the cients that arrive when there are aready active streams wait to be served. The first request to wait for any particuar media fie wi wait in a first-come-first-served (FFS) queue for server bandwidth to become avaiabe. If an arriving request finds at east one request for the same fie aready waiting in the queue, the new request batches with the request(s) aready waiting. Thus, the maximum number of request groups waiting for service is equa to the number of fies on the server. The modes deveoped in this section assume that whie a cient waits in the queue for a channe that wi deiver the beginning of the fie, the cient does not isten to and buffer any other on-going stream that may be serving the same fie. Section.3 provides simuation resuts that show the impact of istening whie waiting on mean cient wait. As in systems with cient baking, stream duration is correated with the number of active streams for the fie, and the overa mean stream duration, S, is a function of system throughput, which is an output of the anaysis. Furthermore, cients batching together whie waiting in the queue presents another chaenge to deveoping a simpe, accurate, and efficient anaytic method for computing the mean cient wait. To deveop the requisite new modes, we again proceed initiay by ignoring the correations between stream durations and the number of active streams. 4. Basic Mean Wait Anaysis We first estimate the mean waiting time for the ead requests that arrive for each stream. These ead requests are either served immediatey (if the server has unused capacity when the ead request arrives) or they are the first request to wait for a given new stream. The mean wait for other requests that batch with the ead requests wi be estimated from the mean wait of the ead requests, as discussed beow. The initiations of new streams by the ead requests are modeed with a two-center queueing network that has customer casses (one per fie) with each cass having a singe customer, as iustrated in Figure 4. The ower deay center in the network modes the mean time unti a new ead request for fie i arrives, which is equa to the inverse of the fie i request arriva rate. The upper FFS queue modes the time that the ead request waits in the server queue for an avaiabe server channe. As soon as a ead request obtains a server channe, the customer moves to the ower deay center to generating a new ead request. If the FFS queue is empty and at east one server channe is unused when a ead request arrives, then the service time in the FFS queue is zero. Whie the FFS queue is not empty, server channes become avaiabe to serve waiting ead requests on average every S/ units of time, where S is the mean stream duration. Using the notation defined in Tabes and 2, the mean residence time of a ead request for fie i at the FFS queue in Figure 4, or the mean wait for fie i ead requests, can be approximated as S W i ( U + Q j ). () j= j i Here U is used to approximate the probabiity that a ead request finds a channes busy, where U is the average channe utiization. Note that each ead request that is ahead of the fie i ead request in the queue contributes S/ to the ead request s mean waiting time. S can be approximated by B i, p ( X i T) S i= X, (6) where X i T is the average number of new streams initiated per time T and X is the tota throughput of the ead requests. U is given by SX U =, (7)

7 and Q i can be obtained from Litte s Law as Q i = X i Wi. (8) Note that, for any incoming request, Q i can be interpreted as the probabiity that there is aready a ead request for fie i waiting in the queue. Since a request for fie i batches with any other request waiting for that fie, X i and Q i must satisfy X i = ( Qi ) i. (9) Substituting equation (8) for Q i into (9) gives the foowing expression that can be derived directy from Figure 4: X i =. (2) W i + / i Either (2) or (9) together with equations () (8) yieds a system of non-inear equations that can soved iterativey to compute W i. The average number of fie i requests that batch with a ead request for fie i is equa to λ i W i. The mean waiting time of these requests that batch with eading requests depends on the distribution of the waiting times of the ead requests, which is not computed in the anaysis above. Due in part to the upper bound on the wait queue ength, one might expect the waiting time distribution to have ow variance. If the ead request waiting times are assumed to be (approximatey) deterministic, then the average waiting time of requests that batch with a ead request is W i /2, and the overa mean waiting time for fie i requests is W λ i iwi + Wi Wi = 2. (2) + λ iwi Athough the waiting times for ead requests are not deterministic, vaidations presented in Section. show that equations () (9) and (2) are sufficienty accurate for the HSM and bandwidth skimming systems, over a wide range of system configurations and cient oads. 4.2 Improved Anaysis for Patching For patching, we first note an issue with respect to choice of optima threshod vaues. Unike in [6, 4, 22], it is assumed impicity in expresssion (6) above, and in our simuations in Section, that the threshod for a fie i is computed based on the throughput of ead requests for the fie (i.e., on the rate of stream initiations), which may be ower than the arriva rate of requests for the fie (many of which may batch together in the queue). This poicy can substantiay improve performance, and we have observed no case in which it degrades performance. We aso note that queueing can substantiay impact patching, since by the time a new stream is initiated for a queued request, any previous stream serving the same fie may have passed the threshod. Simuation resuts discussed in Section. indicate that the basic mean wait mode above (equations () (9) and (2)) underestimates the mean waiting time in patching systems (e.g., see Figure ). The simuations further show that the queueing of cient requests decreases the burstiness of new stream initiations. A derivation of the required server bandwidth for patching in the case of deterministic request interrarriva times is given in the Appendix. Let Q denote the tota mean queue ength of ead requests as predicted by equations () (9), and et W i denote the mean wait of a fie i ead request assuming deterministic stream initiations. That is, Ẅ i is computed using equations () (9) with B i, patching in equation (6) repaced by B i, patching from equation (A.) in the Appendix. We have experimentay derived the foowing heuristic interpoation that more accuratey modes the mean wait for ead requests in a patching system: Q Q W = W i interpoation i + Wi, (22) which indicates that the arriva of ead requests to the server is ess bursty as the tota mean queue ength increases. The improved estimate of overa mean wait for fie i requests in a patching system is then computed using equation (2), with the mean wait for ead requests computed using equations () (9) and (22). The vaidation resuts in Section wi show that the above modes of mean cient wait are quite accurate. To obtain even greater mode accuracy, we coud consider using the arriva instant theorem for separabe networks to estimate the mean queue ength seen by an arriving ead request, or we coud deveop a more accurate approximation than U for the probabiity that a ead request finds a channes busy, as we did for estimating baking rate in Section 3. However, since mean cient wait is estimated sufficienty accuratey with the above simper modes, we do not consider more compex approaches further.. RESULTS Section. provides comparisons between resuts of the anaytic modes deveoped in Sections 3 and 4, and resuts computed by simuating the media server with the reevant streaming protoco, over a wide range of vaues of the tota cient request rate (N = λt), number of popuar media fies on the server (), and server bandwidth capacity (). The simuations incude Poisson request arrivas, the Zipf distribution of fie access frequencies, stream initiations when server bandwidth is avaiabe, stream terminations and cient merges dictated by the streaming protoco, and cient baking or queueing when there are active streams. The vaidation resuts wi show that for each protoco, the most accurate anaytic estimates of cient baking rate and mean cient waiting time can be used to evauate protoco performance for system configurations of practica interest. Foowing the mode accuracy assessments, Sections.2 through.4 provide new evauations of the protocos with respect to the cient QoS measures. Section.2 considers a potentia enhancement for the protocos. Section.3 evauates the protocos with respect to the server bandwidth needed to achieve a given target (ow) cient baking rate or target (ow) mean waiting time. Section.4 compares the poicies with respect to how we a system configured for a given anticipated cient oad performs under sma to moderate increases in the actua oad on the system. For simpicity, a of the experiments in this section assume the reative fie access frequencies are given by the Zipf distribution with parameter α equa to one (see Tabe ). Resuts coud as easiy be obtained for other vaues of α or for other distributions of reative fie access frequencies.

8 Baking Rate MVA LIS-AMVA Simuation LIS-EL/MVA LIS-I Server apacity () % Error in Server Bandwidth Requirement Target Baking Rate N = MVA (a) Fies Target Baking Rate N = LIS-EL/MVA Target Baking Rate N = (b) 2 Fies Figure : Exampe Baking Mode Resuts for Patching ( Fies, N = ) Figure 6: Baking Mode Accuracy for Patching Systems HSM b=.33 MVA Simuation LIS-I Baking Rate Server apacity () Server apacity () Server apacity () Server apacity () HSM b =.33 (a) N = (b) N = (c) N = Figure 7: Exampe Baking Mode Resuts for HSM and Bandwidth Skimming Systems ( Fies) % Error in Server Bandwidth Requirement - MVA, Fies Target Baking Rate MVA, 2 Fies (a) N = (b) N = (c) N = Figure 8: MVA Accuracy for HSM Systems % Error in Server Bandwidth Requirement MVA, 2 Fies MVA, Fies Target Baking Rate (a) N = (b) N = Figure 9: MVA Accuracy for Bandwidth Skimming. Anaytic Mode Vaidations Figure provides exampe resuts of cient baking rate as a function of the media server bandwidth,, estimated by system simuation and by the four anaytic methods deveoped in Sections , for a patching system with = popuar media fies and tota cient request rate, N, equa to an average of cient requests per period of time equa to the media payback duration T. Note that the MVA estimates agree extremey we with the simuation resuts. The LIS AMVA estimates are ess accurate than the LIS MVA estimates due to the high sensitivity of the baking rate measure to sma errors in the system throughput estimate. Figure 6 shows the percent error in each of the two most accurate anaytic estimates, compared with simuation estimates of the server capacity needed as a function of the target baking rate. Note that for 2 or fies, average tota cient request rates of or, and target baking rate varying from one in ten to one in ten thousand, the MVA estimates are within 2% of the simuation estimates for the patching systems. Simiar accuracy of the MVA estimates was aso observed for higher cient oads. Figure 7 provides exampe vaidation resuts, and Figures 8 9 summarize the percent error for the most accurate anaytic estimates of baking rate in HSM systems and in bandwidth skimming systems with b =.33. Note that for 2 or fies and

9 Mean Waiting Time (fraction of T) Patching, improved Patching, basic Simuation Server apacity () (a) Patching HSM b= Server apacity () (b) HSM/Skimming Figure : Exampe Waiting Mode Resuts ( fies, N = ) % Error in Server Bandwidth Requirement - Patching, improved HSM b= Target Mean Wait (fraction of T) N = N = N = (a) fies Figure : Waiting Mode Accuracy (b) 2 fies Server apacty () b=.33 Patching HSM Baking Probabiity Figure 2: Impact of Non-Zero Maximum Wait (2 fies) W max= (anaytic) + W max=.t (simuation) Baking Probabiity (a) N = (b) N = tota cient oad varying from to, requests, on average, per time T, the MVA estimates of the HSM server bandwidth needed for a given target baking rate have % absoute error. For 2 fies, the percent error in the MVA estimates is sighty ower than for fies for arge N; thus, at extremey high cient request rate (e.g., N =,) for 2 fies, the MVA soution sighty underestimates (i.e., by 7%) the HSM server capacity needed to achieve the target baking rate. For ess extreme cient oad or a arger number of popuar fies, the MVA estimates are instead sighty pessimistic (i.e., overestimate required server bandwidth for a given desired baking rate) for the HSM system. Figure 9 shows that the MVA mode is sighty pessimistic for the bandwidth skimming (b =.33) systems as we. Figure provides exampe vaidations, and Figure summarizes the accuracy of the mean cient waiting time modes for each of the three protocos. The basic mode for HSM and bandwidth skimming, and the improved mode for patching, estimate the server bandwidth needed to achieve the target mean cient wait within % absoute error compared with the simuation estimates. Note aso that in the range of practica target cient waiting times (i.e.,.t to.t), and for a cient request rate for the coection of popuar fies significanty greater than, the anaytic estimates are within % of the simuation # Server hannes Patching HSM 2 7 Wait Ony (anaytic) + Wait & Listen (simuation) 2... Mean Waiting Time (fraction of T) Figure 3: Impact of Listen Whie Waiting ( fies, N = ) vaues for a three protocos. Furthermore, the anaytic estimates are ess optimistic for the HSM systems than for the patching systems. Hence, any anayticay estimated performance gains for HSM as compared with patching wi be conservative. Figure 2 considers the case in which cients are wiing to wait for a short time before baking. Specificay, if the wait exceeds W max, they bak. We obtain the resuts for W max =.T using simuation. As shown in the figure, the server bandwidth needed to achieve a given baking rate is simiar for W max =.T and W max =, except in patching systems with high N, where HSM significanty outperforms patching. Thus, the anaytica estimates of server capacity needed for a given target baking rate when W max = are just very sighty conservative for systems where cients bak after a short waiting time. For a arge fraction of the system configurations we simuated, we found the baking rate estimate from a simuation run ength that has baking events to be essentiay identica to the estimate from a run ength that has ony baking events. Thus, simuating more than baking events woud yied negigibe improvement in accuracy. The corresponding tota numbers of cient request arrivas aso yieded essentiay identica estimates of mean cient wait. These observations were used in determining simuation run ength. Athough a few curves have odd

10 Baking Rate ient Request Rate (N) Fies % Increase from Patching HSM b= ient Request Rate (N) ient Request Rate (N) ient Request Rate (N) Fies 2 Fies Fies (a) Target Baking Rate=. (b) Target Baking Rate= Figure 4: Baking Rate at = = B i, p ( Ni) wigges, we have verified that these are not artifacts of the run ength seection. It takes on the order of 2 minutes to simuate baking events for a singe system configuration with moderate to high cient request rate and ow baking rate. Thus, the highy efficient and accurate anaytic estimates offer a key advantage for rapidy exporing system design issues such as those expored beow..2 Possibe Protoco Enhancement For the mean waiting time resuts presented above and estimated by the anaytica modes, cients waiting for service do not isten to any on-going stream that may be serving the same fie. As shown by the simuation resuts in Figure 3, istening whie waiting has approximatey the same performance for either the patching or the HSM systems considered in this work. The ack of significant performance improvement for istening whie waiting may be expained by two factors. First, the required ength of a new stream under a isten whie waiting strategy is determined by when the ast request arrived in that batch. Second, isten whie waiting offers significant improvements ony if the time spent in the queue is reativey substantia, yet practica systems are designed for short mean cient wait..3 Server Bandwidth Needed for Target QoS In this section we address the questions (a) what is the mean waiting time, or the cient baking rate, when the server is configured with the required server bandwidth defined in Section 2 for a given set of popuar media objects and a given cient oad, and (b) how much server bandwidth is needed to achieve a given target cient baking rate, such as one in ten thousand, or a given target mean waiting time, such as.t, where T is the requested media paying time. Figures 4 and provide answers to these questions for cient baking rate; Figures 6 and 7 provide answers for mean cient wait, over a wide range of cient oads. In Figures 4 and 6, at each tota cient request rate (N), the media server using a given protoco has bandwidth capacity equa to the sum of the required server bandwidth as given in Section 2 for each fie i (i.e., for protoco p = ( N,, α ) = B i, p ( Ni ) ). i= Figure : Server Bandwidth Needed for Low Target Baking Rate (% increase from = B i, p ( Ni ) ) Figure 4 shows that when the server is configured in this way, for patching, HSM, or bandwidth skimming, the cient baking rate ranges from over 2% to % as cient request rate ranges from to, requests per time T. The resuts are simiar if the server contains 2 media fies (not shown). Such baking rates are unacceptaby high for most systems, but Figures (a) and (b) show that ony a modest increase in server bandwidth is needed to achieve a baking rate as ow as one in one hundred or as ow as one in ten thousand, respectivey. For exampe, if the server contains popuar fies and the tota cient request rate for the popuar fies is requests per time T, then for each of the three protocos, server bandwidth equa to approximatey. achieves a baking rate of one in ten thousand. Figure 6 shows that when the server is configured with bandwidth equa to (N,,α) and the tota request rate N is greater than, the mean cient wait is a sma fraction of T. Figure 7 shows further that ony sma increases in server bandwidth are needed to achieve very ow mean cient wait. The sma changes in needed to achieve a ow baking rate or ow target mean wait iustrate the advantage of aggregating the cient oad for many popuar fies. During periods when one media fie needs more than its average server bandwidth, another fie may use ess than its average server bandwidth. Thus, the variation in the sum of the server bandwidths needed to give every cient immediate service is statisticay ower than the variation for each fie. A simiar resut is shown for mean waiting time over a much smaer range of system configurations in [2]. Athough the vaue of is different for each streaming protoco, Figures and 7 show that the percent increase in needed to achieve a given target baking rate is simiar for a three protocos. Note that as N increases, the required server bandwidth per cient decreases (due to the subinear growth in B i, p iustrated in Figure ) and the percent increase in needed to achieve a given ow baking rate or mean wait aso decreases (as shown in Figures and 7). This iustrates the significant advantages of serving arger cient popuations from a given scaabe media server.

11 Patching HSM b=.33 Mean Waiting Time (fraction of T)... ient Request Rate (N) % hange from - ient Request Rate (N) - - ient Request Rate (N) ient Request Rate (N) Fies Figure 6: Mean Wait at = = B i, p ( Ni) Fies 2 Fies Fies (a) Target Wait =.T (b) Target Wait =.T Figure 7: Server Bandwidth Needed for Low Target Mean ient Wait (% change from = B i, p ( Ni ) ) Baking Rate.. Patching b=.33 HSM 2 ient Request Rate (N) (a) Anticipated N = 2 Fies ( =. N=) ient Request Rate (N) (b) Anticipated N = Fies ( =.2 N=) Figure 8: Baking System Robustness (Target Baking Rate =. at anticipated N).4 System Robustness Figures 8(a) and (b) and 9(a) and (b) each consider the case in which a scaabe media server is configured with bandwidth equa to the bandwidth needed for a given anticipated cient oad (N) and a given ow target baking rate or mean wait, respectivey. For exampe, in Figure 8(a) the server is configured with the bandwidth needed for cient request rate N = and baking rate equa to one in ten thousand. As shown in Figure (b), =. for this target baking rate, 2 fies, and N =. For each such system, the cient request rate is varied up to.4. times the anticipated oad, to determine how much the system performance degrades when the actua cient oad is somewhat higher than anticipated. Figures 8(a) and (b) show that the HSM and bandwidth skimming systems can toerate a % increase in cient request rate without significant degradation in cient baking rate, but system performance degrades rapidy for increases in cient oad greater than %. Patching systems are somewhat more sensitive to cient oad increases, toerating coser to a % increase before service begins to rapidy degrade. Figures 9(a) and (b) show that mean waiting time is ony insensitive to increases up to % in cient oad if the server is configured for very ow target mean wait (e.g., target mean wait Mean Waiting Time (fraction of T).2. HSM b=.33 Patching 2 4 ient Request Rate (N) (a) Target W =.T ( = N=) ient Request Rate (N) (b) Target W =.T ( =. N=) Figure 9: Waiting System Robustness ( fies, Anticipated N = ) equa to.t). Mean wait degrades somewhat ess rapidy than baking rate, but performance sti degrades significanty for greater than % increases in cient oad. 6. ONLUSIONS This paper has deveoped simpe, highy accurate and efficient anaytic estimates of mean cient wait and the fraction of cients that bak when a scaabe on-demand media server is configured to deiver a given maximum number of streams concurrenty. The customized AMVA estimates are noteworthy in that they capture compex phenomenon that woud suggest that more compex state space anayses woud be needed. In particuar, the baking rate anaysis uses customized AMVA to estimate a probabiistic measure rather than the usua mean vaue measures, suggesting that customized AMVA techniques may be appicabe to a significanty broader range of performance metrics and system design questions than has been previousy demonstrated. The resuts in Section show that (a) scaabe media servers that are configured with the required server bandwidth defined in previous work have ow mean wait but may have unacceptaby high cient baking rates (i.e., greater than one in twenty), (b) for high to moderate cient oad, a % increase in the previousy defined required server bandwidth wi achieve a very ow baking