ADVANCES in information and communication technologies

Transcription

1 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Use of Aalytical Performace Models for System Sizig ad Resource Allocatio i Iteractive Video-o-Demad Systems Employig Data Sharig Techiques Mary Y.Y. Leug, Joh C.S. Lui, Member, IEEE, ad Leaa Golubchik, Member, IEEE AbstractÐI desigig cost-effective video-o-demad (VOD) servers, efficiet resource maagemet ad proper system sizig are of great importace. I additio to large storage ad I/O badwidth requiremets, support of iteractive VCR fuctioality imposes additioal resource requiremets o the VOD system i terms of storage space, as well as disk ad etwork badwidth. Previous works have used ªdata sharigº techiques (such as batchig, bufferig, ad adaptive piggybackig) to reduce the I/O demad o the storage server. However, such data sharig techiques complicate the provisio of VCR fuctios ad dimiish the amout of beefit that ca be obtaied from data sharig techiques. The mai cotributio of this paper is a simple, yet powerful, aalytical modelig approach which allows for aalysis, system sizig, resource allocatio, ad parameter settig for a fairly geeral class of ªdata sharigº techiques which are used i cojuctio with the providig of VCR-type fuctioality. Usig this mathematical model, we ca determie the proper amout of resources to be allocated for ormal playback as well as for service of VCR fuctioality requests while satisfyig predefied system performace requiremets. To illustrate the usefuless of our model, we focus o a specific data sharig scheme which combies the use of batchig, bufferig, ad adaptive piggybackig, as well as allows for the use of VCR fuctios. We show how to utilize this mathematical model for system sizig ad resource allocatio purposesðthat is, how to distribute the available resources betwee the service of ormal playback ad VCR fuctioality requests uder various workloads ad resource price ratios, so as to obtai the lowest system cost. Idex TermsÐVideo-o-demad, data sharig techiques, resource allocatio, system sizig. æ INTRODUCTION ADVANCES i iformatio ad commuicatio techologies have made multimedia services feasible. Amog these services, iteractive TV ad video-o-demad (VOD) have bee idetified as two importat applicatios [8]. I a VOD system, video objects are stored o a storage server ad delivered to users upo request. Ulike textual data, video objects occupy large amouts of storage space ad require large amouts of I/O badwidth for real-time delivery. Thus, upo admissio of a user, sufficiet amouts of resources have to be reserved i order to provide a acceptable level of service to viewers. A sigificat amout of research effort has bee dedicated to desigig a fault-tolerat VOD server, as well as ivestigatig techiques for the efficiet use of resources i VOD servers [7], [], [4], [9], so that the amout of. M.Y.Y. Leug is with Acceture, 23/F Wig-O Ceter, Coaught Rd., Cetral, Hog Kog. mary.y.leug@acceture.com.. J.C.S. Lui is with the Departmet of Computer Sciece ad Egieerig, The Chiese Uiversity of Hog Kog, Shati, NT, Hog Kog. cslui@cse.cuhk.edu.hk.. L. Golubchik is with the Departmet of Computer Sciece ad UMIACS, Uiversity of Marylad, College Park, MD leaa@cs.umd.edu. Mauscript received 24 Mar. 998; revised 6 Dec. 999; accepted 3 Aug For iformatio o obtaiig reprits of this article, please sed to: tkde@computer.org, ad referece IEEECS Log Number resources eeded to satisfy predefied performace measures ca be reduced. For a survey of such approaches, please refer to [9]. Several techiques have bee proposed for icreasig the umber of cocurret viewers of popular objects through the sharig of data ad, cosequetly, the sharig of system resources, such as I/O badwidth ad buffer space. Oe such techique is batchig (e.g., as i [6]), i which oe I/O stream is dedicated to servicig several viewers, for istace, those that arrived withi a predefied ad relatively short time iterval. As log as viewers are willig to wait for a small amout of time, where the maximum waitig time is the legth of this time iterval, all requests for the same object arrivig withi the time iterval ca be serviced usig oe I/O stream, thereby reducig the overall amout of I/O badwidth ecessary. Aother techique for reducig the amout of eeded I/O badwidth is bufferig (e.g., as i [22]), where careful buffer replacemet algorithms isure that viewers ca fetch the eeded data blocks from the VOD system buffers istead of from the disk subsystem. Last, the adaptive piggybackig method [0] takes advatage of the fact that video display rates ca be modified (without the user oticig the chage i video quality) i order to dyamically merge ogoig displays of a popular object ito a sigle group which ca be served by oe I/O stream, thereby reducig the overall I/O badwidth requiremets. I geeral, the basic idea behid these /0/$0.00 ß 200 IEEE

2 2 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 approaches is to group users requestig the same object, so as to icrease the degree of data sharig ad, thereby, reduce the overall system I/O badwidth demad. Apart from ormal playback, it is also desirable to provide iteractive VCR fuctios, such as fast forward with viewig (FF), rewid with viewig (RW), ad pause/ resume (PAU). Part of the difficulty here is that the provisio of VCR fuctios ofte requires additioal system resources ad it is usually ot desirable to reserve such resources for each request i the system sice VCR resource reservatio o a per-request basis may be utilized ifrequetly ad, hece, these resources will ot be utilized efficietly. There are various schemes i the literature for supportig iteractive VCR fuctios i VOD systems, for istace, [3], [3], [5]. However, amog all these approaches, few have addressed the problem of icorporatig VCR fuctioality i a system which uses data sharig techiques for the service of ormal playback requests. The basic problem with combiig VCR type fuctioality with data sharig techiques is as follows: All data sharig techiques (such as batchig, bufferig, ad adaptive piggybackig) improve the cost-effectiveess of a VOD system by servig groups of cliets requestig the same video with a sigle set of resources. However, they all suffer from the problem of where to obtai resources whe oe of the viewers i a group requests a VCR-type fuctioality ad, thus, is o loger part of the data sharig group sice, i this case, resources are ot reserved o a per-cliet basis but oly o a per-group basis. Eve if some resources are reserved for performig VCR-type fuctios, aother problem (ad perhaps a more difficult oe) is where does the system obtai resources for a viewer that is ready to go back to the ormal playback but is o loger part of a group. We refer to such viewer as a falle out viewer. The basic approaches to dealig with this problem iclude: ) combiig data sharig techiques (such as batchig ad bufferig) i order to icrease the probability that a cliet that left a group to perform VCR-type fuctioality will rejoi aother group oce they are ready to retur to ormal playback (e.g., as i [8], [7]) ad 2) settig aside a relatively small pool of resources to provide for cliets who eed to perform VCR-type fuctioality (e.g., as i [3], [2]) ad/or for those that are ot able to rejoi aother group. Oce these basic approaches are employed, the mai difficulty is i the settig of parameters, i.e., determiig: a) the size of the pool of resources (for VCR-type fuctioality), b) the batchig iterval (or the umber of I/O streams) to use per video object, c) the amout of buffer space to use per video object (or per group), etc., such that the probability of allowig VCR-type fuctioality ad the cost-effectiveess of the whole system (give some QoS requiremets) are optimized. Related work o allowig users i a data sharig group to perform VCR fuctioality is as follows: I [25], the authors propose a method for allowig users i a data sharig group to pause ad evetually resume back to ormal playback. This method uses additioal buffer space resources to gai improvemets i the amout of I/O badwidth eeded to support pause/resume of batched requests. I [2], the authors propose a scheme for providig iteractive fuctioality to differet multicast groups (each multicast group ca be viewed as a batch group). The mai idea there is to utilize the cliet-side resources, i.e., provide eough buffer space i the viewer's set-top box i cojuctio with reservig some emergecy chaels i the etwork durig high loads such that the system is able to service a iteractive request by a viewer with a reasoably high probability. I [8], the authors propose a protocol which splits a iteractive user from a batch ad serves the user usig a dedicated I/O stream. This iteractive viewer is the ªattachedº back to a batch of requests by usig additioal buffer space eeded to bridge the gap betwee a batched group ad this iteractive stream. However, oe of the approaches metioed above [2], [8], [25] solve the problem of system sizig, i.e., how oe should set desig parameters ad, hece, allocate differet resources to differet types of display modes (either ormal or iteractive) so as to provide a certai level of QoS ad, at the same time, reduce the overall system cost. The cotributios of this paper are as follows: The mai cotributio is our aalytical model which is used for system sizig ad for determiig the amout of resources eeded for supportig ormal playback ad VCR fuctioality, while satisfyig predefied system performace requiremets. This model allows us to maximize the beefits of data sharig techiques i the presece of iteractive VCR fuctios. Cosequetly, our paper uses the aalytical model for system sizig purposes i iteractive VOD systems usig data sharig techiques. Specifically, we divide the overall resources ito those required for ormal playback ad those required for the service of a VCR request. The goal is to reduce the probability of viewers ªfallig outº of a group (after resumig to ormal playback), thereby icreasig the beefits of data sharig, but without sigificatly icreasig the system costs. We use our model to calculate the amout of resources required for ormal playback such that, upo resumptio to ormal playback, additioal resources (which were dedicated to the VCR fuctio) ca be released with a prespecified probability, which reflects the system's QoS requiremets. To further icrease this probability, we use the adaptive piggybackig techique to facilitate the joiig of a existig group by the falle out viewers. Hece, although a falle out viewer may have to hold o to additioal resources to resume ormal playback, we attempt to reduce this resource holdig time. All this implies that the total amout of resources reserved for service of VCR requests ca be reduced. Thus, aother cotributio of our work is a ovel data sharig techique that combies batchig, bufferig, ad adaptive piggybackig i order to improve the cost-effectiveess of a iteractive VOD system. The orgaizatio of the paper is as follows: I Sectio 2, we preset the backgroud iformatio ad a brief survey of batchig, bufferig, ad adaptive piggybackig techiques. I Sectio 3, we describe our system ad itroduce the model for VCR display. I Sectios 4 ad 5, we cotiue with the expositio of our aalytical model ad describe details of resource allocatio for ormal playback, as well as VCR modes. I Sectio 6, we describe how to apply our model to solvig system sizig problems i iteractive VOD servers which employ data sharig techiques. Last, our coclusios are give i Sectio 7.

3 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE BACKGROUND ON DATA SHARING AND VCR SUPPORT I this sectio, we briefly describe several basic approaches to reducig I/O demad o a VOD server through data sharig. Note that these techiques are orthogoal, i.e., they ca be combied i order to better utilize the I/O resources i a VOD system. Last, we also briefly survey several approaches to providig VCR fuctioality i cojuctio with data sharig schemes.. Data sharig via batchig (e.g., as i [6]). Oe way to satisfy idepedet playback requests is to dedicate a I/O stream to each request. However, this (potetially) requires a large umber of I/O streams. Give that there will be a relatively large umber of requests for popular videos, we ca batch together requests separated by relatively small time itervals ad serve the etire batched group usig a sigle I/O stream. As log as the viewers are willig to tolerate a certai latecy, batchig ca help to achieve a substatial reductio i the required server I/O badwidth capacity. However, this reductio i I/O streams will icrease the waitig time of the viewers, thus, the trade-off is betwee the I/O badwidth utilizatio ad the waitig time of viewers. There are may batchig policies that oe ca cosider. Oe simple approach is to start the video periodically at predetermied time itervals. I this case, we ca easily evaluate the savig i I/O badwidth. For istace, if is the average arrival rate of requests for a particular video ad b is the batchig time iterval for that video, the the average fractio of I/O savigs is b b, provided b >. 2. Data sharig via bufferig (e.g., as i [5], [2], [20], [23]). Bufferig is a techique which attempts to bridge the temporal gap betwee ªsufficietly closeº successive requests for the same video usig buffer space. This is achieved by retaiig the video frames fetched by the previous stream i the buffer space ad allowig streams correspodig to subsequet requests for the same video to be served from the buffer space rather tha by usig a I/O stream. The basic idea is to use buffer space to promote data sharig ad, hece, reduce the demad for I/O badwidth. Sice eve retaiig a few miutes of the video i the buffer is costly, it is importat to determie proper buffer replacemet algorithms. There are several existig research results for the use of bufferig techiques. I [5], [2], [20], [23], the authors explore the beefits of cachig cotiuous media data to reduce the utilizatio of I/O badwidth. By makig efficiet use of the data already fetched ito the buffer space, the umber of cocurret viewers ca be icreased. I additio, i [4], the cost-performace trade-offs of various bufferig ad cachig strategies is studied by varyig the buffer size, disk utilizatio, ad disk characteristics. 3. Data sharig via adaptive piggybackig (e.g., as i [], [0]). Adaptive piggybackig exploits the fact that video display rates ca be altered without a user-perceived loss i the quality of that video, to ªmergeº requests for the same video ad, thus, allow the correspodig viewers to share a sigle I/O stream, thereby dyamically reducig the I/O demad o the VOD system. Although the reductio i I/O demad is ot as high as that of ªtraditioalº batchig, the advatage is that viewers do ot experiece the additioal latecy associated with batchig or is there a eed for additioal buffer space required by bufferig techiques. This approach assumes that the storage server is capable of alterig the display rate of a video. For istace, sice the display proceeds at a fixed rate, the slow dow i the effective display rate ca be doe by isertig additioal frames; similarly, the display rate ca be effectively icreased by removig frames. It has bee poited out that differeces as large as 5 percet i the display rates will ot be perceivable by the viewers ad, thus, will ot result i a degradatio i video quality [0]. Thus, the basic idea behid this approach is to dyamically merge two (or more) I/O streams ito oe. Justificatios of the feasibility ad evidece of the beefits of this approach are give i [0]. As already metioed, these basic data sharig techiques are orthogoal ad ca be combied ito a variety of data sharig policies. As a example of oe combiatio of batchig ad bufferig, we ow briefly describe the static partitioig scheme proposed i [22]. Here, we choose this particular scheme as this is our poit of departure for the remaider of the paper. 4. Data sharig via static partitioig (e.g., as i [22], [7]). Static partitioig is oe of the partitioed buffer maagemet strategies proposed i [22], which combies the use of batchig ad bufferig ad is developed based o a buffer refreshig process. This method is similar to batchig schemes i which a I/O stream for a particular video is restarted periodically at predefied time itervals. I additio, some amout of buffer space, termed a partitio, is associated with each of these I/O streams. This allocated buffer space allows for the retaiig of the video frames i memory for a certai period of time, which is termed the viewer erollmet widow. Thus, this scheme allows for the sharig of data betwee requests which arrive durig the viewer erollmet widow; these requests are served usig a sigle set of resources, i.e., a sigle I/O stream [6] plus the buffer space associated with it. This facilitates a reductio i the additioal latecy associated with ªpureº batchig at the cost of additioal buffer space. Fig. illustrates a sceario for the static partitioig scheme, where each batch of viewers is served usig a I/O stream ad a partitio of buffer space. Viewers that arrive before the closig of the viewer erollmet widow read the frames from the buffer partitio (we call these type 2 viewers). Viewers that arrive after the widow is closed (we call these type viewers) are

4 4 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig.. Differet types of viewers i static partitioig. queued up to wait for the ext restart of the video. The logest time a viewer has to wait occurs whe he/she arrives just after the viewer erollmet widow is closed. I this case, the maximum waitig time equals the batchig iterval b (the iterval betwee iitiatios of two cosecutive I/O streams) mius the legth of the viewer erollmet widow y. A iterestig approach which provides a optimal value of the viewer erollmet widow whe VCR fuctioality is ot used is give i [23]. Let us briefly survey schemes for combiig data sharig techiques with VCR fuctioalities. 5. Data-sharig with VCR support (e.g., as i [25], [2], [8]). I [25], the authors propose a method for allowig users i a data sharig group to perform pause/resume type fuctioality oly. This method uses additioal buffer space resources to gai improvemets i the amout of I/O badwidth eeded to support pause/resume fuctioality for batched requests. I [2], the authors propose a scheme for providig iteractive fuctioalities to differet multicast groups, where the basic techique is to utilize cliet-side resources, i.e., to provide buffer space i the viewer's set-top box i cojuctio with reservig some emergecy chaels i the etwork durig high loads such that the system is able to service a iteractive request by a viewer with a reasoably high probability. I [8], the authors propose a protocol which ªsplitsº a iteractive user from a batch ad serves him/her by usig a dedicated I/O stream. This iteractive viewer is the ªattachedº back to a batch of requests by usig additioal buffer space eeded to bridge the temporal gap betwee a batched group ad this iteractive stream. We should ote that part of the motivatio for us cosiderig the additio of the adaptive piggybackig scheme to the set of data sharig techiques used (refer to Sectio 5) is to be able to deallocate such additioal resources (i.e., such as those used to aid a viewer i joiig a existig batch) before the user exits the system ad, thereby, improves the cost-effectiveess of the system. Most of these approaches metioed are evaluated through extesive simulatios to cover a wide rage of possible desig parameters. Thus, partly what is desirable here is a better methodology for makig desig decisios ad evaluatig resultig schemes. 3 SYSTEM MODEL For ease of expositio as well as ease of illustratio of the mathematical model, we focus o the partitioed buffer maagemet strategy termed static partitioig [22] as a specific combiatio of data sharig techiques. The static partitioig scheme, proposed i [22], addresses ormal playback fuctioality oly. However, VCR fuctios such as fast-forward (FF), rewid (RW), ad pause (PAU) are importat ad ecessary features, which eed to be provided to the viewers. A fudametal problem i providig such iteractive features i cojuctio with data sharig techiques is the eed for additioal resources ot oly for servicig VCR requests, but also for allowig viewers to resume from a VCR mode to ormal playback. The difficulty here is that, at this resume poit, viewers are o loger part of a group of requests sharig the I/O ad buffer space resources, i.e., the resources are reserved o a per-group rather tha per-viewer basis. I this paper, we preset a model which facilitates the calculatio of distributio of resources betwee ormal playback ad VCR fuctioality, so as to reduce the additioal load resultig from the users resumig from a VCR operatio to ormal playback. I additio, this model aids i makig system sizig decisios, such that the overall cost-effectiveess of the system ca be improved. 3. VCR Fuctioality The service of VCR requests ca be divided ito the followig two phases. Phase. Displayig the VCR-versio of the video. Whe a viewer issues a VCR requests, other tha a pause, additioal resources have to be allocated so that the viewer ca watch the VCR-versio of the video. Let V CR deote the mea time spet i phase. Phase 2. Resumig to ormal playback. Whe resumig from a VCR operatio to ormal playback, the possibility of releasig the resources allocated durig Phase depeds o the time poit i the video at which the viewer resumes. If the frames the viewer eeds are already i the VOD system buffer whe the viewer resumes, i.e., i oe of the existig partitios, the the viewer ca read the data directly from the partitio i which he/she resumes. I this case, the viewer is said to joi that partitio, ad the resources allocated i Phase ca be released. We deote the probability of a viewer resumig from a VCR operatio at a existig partitios by P. O the other had, if the viewer caot resume at ay of the existig partitios, the to cotiue ormal playback without ay delay, the viewer has to make use of the resources allocated i Phase util a viewer ca joi a existig partitio (if that is possible). Thus, a viewer is said to have a hit if he/she ca joi a existig partitio whe resumig to ormal playback after a VCR request, such that additioal resources allocated to that viewer i Phase ca be released i Phase 2. Otherwise, the viewer is said to have a miss if he/she

5 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 5 Fig. 2. Hits ad misses for viewers resumig from VCR requests. resumes at a gap betwee partitios. The hit ad miss evets ad their correspodig probabilities are illustrated i Fig. 2. Let hold deote the mea time before the resources allocated i Phase ca be released. I this paper, the allocatio of resources for ormal playback esures that there is a miimum probability of P for the viewers to resume at a existig partitio after a VCR request. Therefore, with the probability of P, the viewer must hold o to the resources iteded for VCR requests whe he/she resumes. Thus, the expected time spet i Phase 2 is give by P hold. 3.2 Normal Playback As already stated, the allocatio of resources i Phase of performig a VCR operatio is usually ievitable. O the other had, whether or ot these allocated resources ca be released depeds o the resumig positio of the viewer. Therefore, our goal is to maximize the probability of releasig the allocated resources at a resume poit so as to reduce the cosumptio of reserved resources. Cosequetly, more resources will be available for servig future requests, such as forthcomig VCR requests or eve requests for opopular videos. Clearly, the size ad the umber of partitios used for servicig ormal playback, together with the duratio of VCR requests will greatly affect the probability of resumig at a existig partitio. The ormal playback scheme, derived based o the static partitioig techique, is used to calculate the amout of resources allocated to each video so as to esure the followig: Whe a viewer resumes from a VCR request, there is a probability of at least P that the additioal resources allocated i Phase ca be deallocated i Phase 2. Sice our goal is to maximize the probability of a hit, it is atural to cosider the behavior of VCR requests i determiig the cofiguratio of the system, i.e, i determiig how much buffer resources ad I/O streams should be reserved for ormal playback. We study the behavior of each VCR operatio as a fuctio of its respective probability desity fuctio (pdf), f x, where x is the amout of video time spet i a VCR request. The pdf of VCR requests ca be obtaied by collectig statistics ad is defied i the iterval 0; lš, where l is the legth of the video (i uits of time). Note that a pause of x time uits where x > l is equivalet to a pause of x mod l time uits, e.g., the situatio where l ˆ 20 miutes ad x ˆ 30 miutes is equivalet to pausig for 0 miutes sice a video is restarted periodically. Based o the distributio of the duratio of VCR requests, we perform resource allocatio for ormal playbacks such that the probability of holdig o to the resources upo resume ca be miimized. This resource allocatio is calculated based o the mathematical model we propose, which is preseted i Sectio 4. I this model, we assume that the arrivals of requests for a popular video are distributed accordig to a Poisso process with rate. This is a reasoable model of the arrival process sice we expect the VOD system to have a large user populatio. Based o the above priciples, we determie the expected hit probability upo resume to ormal playback uder various system cofiguratios. Ituitively, icreasig the size of the partitios will icrease the hit probability because the fractio of a video residig i buffers ca be icreased. Cosider the extreme case where the etire video is buffered, thus icreasig the hit probability to. However, this may ot yield a system with the lowest operatig cost if relatively little time is spet i a VCR request, i which case, bufferig the etire video would be too costly. Therefore, we use our model to determie a ªcofiguratioº which yields the lowest cost for supportig a give umber of cocurret viewers for both ormal playback ad VCR fuctios. 3.3 Model for VCR Display I this paper, we model the additioal I/O resources eeded for VCR fuctioality by a M=M=m queue, so as to provide VCR service with a predefied level of QoS guaratee. For istace, we could esure that the average time that a viewer has to wait to switch to VCR mode is less tha three secods. The details of QoS guaratees are preseted at the ed of this sectio. The otatio related to ormal ad VCR cotrol is summarized i Table. For simplicity of presetatio, let us cocetrate o the derivatio of resource distributio based o a sigle video. We assume the arrival rate of VCR requests to be Poisso with a rate of V CR per miute for each viewer, i.e., if, o the average, a viewer issues z VCR requests per video of legth l miutes, we have: V CR ˆ z=l: TABLE Major Notatios Ivolved i VCR Model ad Normal Playback Model

6 6 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 3. Scearios for catchig up to viewers (a) i frot (b) behid. Sice we assumed that the arrival rate of the ormal playback follows the Poisso distributio of rate, the total arrival rate of VCR requests to the VOD system, v, is give by l V CR. Usig (), we have v ˆ z. The time spet i utilizig the servers i the M=M=m queue correspods to the time spet by the viewers utilizig additioal resources for servicig VCR requests ad for cotiuig ormal playback (i case of a miss) before these viewers ca joi a existig partitio. It is assumed to be expoetially distributed with mea v. As metioed i Sectio 3., a VCR fuctio is a two-phase process. Therefore, v is calculated by summig up the mea time spet i each phase: v ˆ V CR P hold : To simplify the calculatio of the average time for releasig the hold o resources, hold, we also assume that each positio of the video has the same probability to be the poit of resume. We obtai the probability of queuig (Prob queueigš), the expected queue legth (N q ) ad the expected waitig time for service (T q ) as follows: N q ˆ 0 Prob queueigš ˆ P m kˆ0 m m m! h i ; h P m kˆ0 m k k! m m m! m m m! m k k! m m m! T q ˆ N q = v ; i C A; where ˆ v v =m. I applyig the above results of the M=M=m queue i calculatig the additioal I/O resources eeded, we have to esure that the service performace of hadlig VCR requests reaches a certai satisfactory level. I other words, whe the viewer issues a FF/RW request or whe he/she resumes from ay VCR request, it is ot desirable if the viewer has to wait for a log time for the display of the VCR-versio of the video or for ormal playback to resume. Therefore, oe of the followig performace levels has to be satisfied. They are:. Whe the viewer issues a VCR request, the probability that a viewer has to wait for the VCR request to be serviced must ot be greater tha If the viewer has to wait, we esure that the expected waitig time is withi the tolerace level of the viewers, t. 3. The average umber of viewers waitig for their VCR requests to be serviced is o more tha l. Give (3), (4), ad (5), the above three performace requiremets ca be formulated usig the followig equatios: Prob queueigš ; T q t ; N q l : These three iequalities will esure that the system provides service at a predefied performace level. Note that the system utilizatio is must be less tha, therefore the smallest possible value of m is b v v c. To determie the umber of servers eeded to esure a performace level, the value of m is icremeted util oe of the above three iequalities is satisfied. 4 RESOURCE ALLOCATION FOR NORMAL PLAYBACK I this sectio, we preset the derivatio of our model for determiig the cofiguratio of the system for ormal playback. Specifically, we wat to determie the allocatio of resources which, uder various VCR operatios (e.g., FF, RW, PAU), will isure the probability of a viewer resumig from a VCR request at ay existig partitios to be greater tha or equal to P, where P is a give system desig parameter. To maitai the I/O reductio beefit of the data sharig techique, we use the adaptive piggybackig with bufferig techiques to joi a falle out viewer with a existig partitio. Let us first describe the criteria for oe stream to catch up with aother (for simplicity of illustratio, we will use streams rather tha partitios). A example situatio is illustrated i Fig. 3a. We deote R F F as the rate of fast forwardig ad R PB as the rate of ormal playback. Suppose viewers X ad Y are viewig the movie at a rate of R F F ad R P B, respectively, ad X lags behid Y by miutes (i movie time). Sice R FF is greater tha R P B, X will evetually catch up with Y at the catch-up poit, deoted by a cross i the figure. Similarly, Fig. 3b illustrates a sceario where viewer Y rewids to catch up with viewer X. Thus, the amout of movie time, t, through which viewer X must fast-forward (or viewer Y must rewid) before a catch-up is accomplished is as follows: 6 7 8

7 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 7 ( t ˆ ; where ˆ RF F R F F R P B ; where ˆ RRW R P B R RW for fast forward for rewid: 9 I what follows, we derive the buffer ad I/O resource requiremets for ormal playback uder static partitioig. After formulatig the problem, we preset the derivatio of our mathematical model for predictig the probability of deallocatig a I/O resource upo a resume from a VCR operatio (which will be used later i the paper for system sizig). This probability is derived by first cosiderig the coditioal probability of a hit for a viewer at a particular poit i the movie, which is ucoditioed to obtai the geeral probability of a hit for ay type of a VCR request. The otatios used are summarized i Table. 4. Mathematical Model The static partitioig model adopts the idea of batchig requests for a popular movie such that I/O streams are iitiated periodically. If is the umber of I/O streams that are available, the for a movie with a legth of l miutes, oe way to guaratee that the viewers have the same maximum waitig time is to start the movie every l= miutes. Assume that the total amout of buffers allocated for ormal playback for a popular movie ca store B 0 miutes of the movie. The, the size of each partitio is B 0 =, ad, i the worst case, a viewer arrivig at the closig of the viewer erollmet widow has to wait for the ext restart of the movie. The legth of the viewer erollmet widow is equal to B 0 =, where is the reserved amout of buffer space, which isures that whe the first viewer i a partitio replaces the frames i the buffer, the system will ot overwrite the frames ot yet viewed by the last viewer i the same partitio [22]. Thus, the maximum waitig time, w, is equivalet to the gap betwee partitios, which is equal to l B0. Let B ˆ B 0, the we have: w ˆ l B ; where ˆ ; 2;... ; l w : 0 Note that ˆ l w would imply that B ˆ 0, which correspods to the pure I/O or pure batchig case, where each batch is served by a sigle I/O stream. However, i this case, the hit probability is always equal to zero (uless is ifiitely large). Rearragig the above equatio gives, the umber of I/O streams eeded is l w B w, where B l. The differece betwee usig bufferig i cojuctio with batchig ad pure batchig lies i the amout of I/O ad buffer resources required. Whe we dedicate B miutes worth of buffer space for ormal playback, the we ca save B w I/O streams which ca be used to service VCR requests or requests for other movies, where we cosider usig w miutes of buffer space as a tradeoff for oe I/O stream.. We will cosider the pure bufferig i the later sectio of the paper. Fig. 4. Relative positios of viewers at V c, V f, ad V l i a partitio. Let V c deote the positio of our tagged viewer curretly viewig the Vc th frame 2 of the movie. Furthermore, the positios of the first ad the last viewer that ca exist i a partitio are deoted by V f ad V l, respectively. These two viewers may be virtual, i the sese that there ot be ay actual viewers viewig the Vf th ad Vl th frame of the movie, yet they are the two extreme positios i the partitio i which a viewer ca possibly exist. The maximum differece (i time uits) betwee these two extreme viewers is B=, which is illustrated i Fig. 4. The idea behid this model formulatio is to determie the probability of a hit whe a viewer resumes from a VCR request. We defie P hitjv c ; V f be the coditioal probability of a hit give that a viewer is at positio V c ad that the first possible viewer i the same partitio is at positio V f. Amog the three types of VCR requests (FF, RW, ad PAU), let us first cosider the FF operatio. I order to determie P hitjv c ; V f give that the VCR fuctio is FF, we eed to kow the probability desity fuctio (pdf) of the duratio of a FF request. The mai difficulty i hadlig VCR requests lies i their iheretly odetermiistic ature [3], where the retrieval patter is ot kow. Rather tha assumig a particular distributio, we allow a geeral distributio i our model ad we let the pdf of the duratio of FF requests be deoted by f x, where x 2 0; lš. Note that, our goal is ot to obtai the exact distributio or model for VCR behavior, but rather we assume that the VCR behavior has a geeral distributio ad costruct a model which is able to hadle a geeral probability distributio ad, thus, ot be limited by ay particular distributio. Give this geeral probability distributio, f x, we derive the probability of a hit. We subdivide a hit ito the followig two mutually exclusive cases: ) a hit withi the same partitio i.e, withi the partitio i which the VCR operatio is iitiated ad 2) a hit i aother partitios. 4.. Hits Occurrig withi the ªSameº Partitio (hit w ) The evet hit w occurs if a viewer ca resume i the same partitio i which the VCR request is issued. Our aim is to calculate the probability of a hit w whe a viewer resumes from ay VCR request, deoted by P hit w. We begi by cosiderig the FF situatio. The logest fast forward duratio which ca still result i a hit w is the distace betwee the viewer, V c, ad the first possible viewer i his/ her partitio. This is illustrated i Fig. 5a i which the viewer must resume i the shaded regio of the partitio. The probability of a hit w, give that: ) the viewer is at 2. Note that we model the movie as a cotiuous segmet. We use the term frame for ease of expositio.

8 8 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 5. Two cases of hit w for viewer V c : (a) catchig up with all viwers ahead i the same partitio ad (b) catchig up with viewers betwee V c ad V t for V t < V c B. positio V c, 2) the first viewer is at positio V f, ad 3) the requested VCR fuctio is FF is as follows: P hit w jf F ; V c ; V f ˆ Z w 0 f x dx where w ˆ V f V c ad the upper limit is the time to eable a catch-up, as defied i (9). The remaider of this sectio is dedicated to determiig how to ucoditio o V f ad V c so as to obtai P hit w jff, which is the expected probability of a hit withi the same partitio after a viewer resumes from a FF request.. Ucoditioig o V f. Recall that V f is the positio of the first possible viewer that ca exist i the same partitio as the viewer at positio V c. Thus, we ca ucoditio o V f with respect to V c. Sice V c ca be ay positio withi the partitio, it ca rage from beig the positio of the first viewer i the partitio (i which case V f ˆ V c ) to beig the positio of the last viewer i the partitio (i which case V f ˆ V c B ). Note that there are boudary cases where the viewer at positio V c may ot be able to catch up with all the possible viewers ahead of him i the same partitio. As show i (9), the time t, elapsed before a catch-up is accomplished, is equal to the iitial distace,, times a costat greater tha. If the catch up poit is after the ed of the movie, or V c t > l, the the viewer at positio V c caot catch up with the target viewer before the movie eds. Thus, we divide the ucoditioig of V f ito two cases. Case A. Viewer V c ca accomplish a catch-up with all possible positios of the first viewer i the same partitio, i.e., the largest value of V f is equal to V c B. I this case, eve if V c correspods to the last possible viewer, he/she will be able to catch up with the first possible viewer, who is B uits of time ahead, before the movie eds. Thus, V f rages from V c to V c B, ad clearly, we have: P a hit w jf F ; V c ˆ Z Vc B V f ˆV c P hit w jf F ; V c ; V f P V f dv f ; 2 where P V f is the probability that the first viewer, who is i the same partitio as V c, is at the Vf th miute of the movie. Here, we approximated P V f to be equal to B=. Note that, this quatity is oly a approximatio sice those viewers who arrive before the ext restart of the movie all become ªpart ofº the first viewer of the partitio. Aother reaso why this is a approximatio is that after viewers have resumed from VCR requests, the positio of viewers may ot be uiformly distributed withi a partitio. Nevertheless, we will show i a later sectio that results obtaied usig our mathematical model match well with those obtaied through simulatio. Case B. Viewer V c caot catch up with the farthest possible positio of the first viewer i the same partitio, i.e., the largest value of V f that the viewer ca catch up to is equal to l Vc. I this case, we eed to hadle the boudary coditio which esures that V c t lðthat is, a catch-up to some viewers withi the same partitio is still possible before the ed of the movie. I other words, cosider the sceario where, give a viewer at positio V c, there exists a viewer at positio V t, (where V t V c ) such that whe the former viewer catches up to the latter, both viewers are at the lth miute of the movie. Usig (9), we ca describe a relatioship betwee V t ad V c by the equatio V c V t V c l, which gives the upper boud for V t : V t l V c : 3 Note that the viewer at positio V c will ot be able to catch up with a viewer at positio V f, where V f > V t for V f 2 V c ; V c B Š. The furthest positio V f with which the viewer at positio V c is able to catch up is V t, as give by (3). Fig. 5b illustrates the regio where a hit w is esured for V f V t. Therefore, we have a secod equatio for ucoditioig o V f. The first term below correspods to oe case where V t V f. The secod term correspods to the case where V t < V f V c B, ad the evet hit w is esured for the duratio 0 to V t V c. P b hit w jff; V c ˆ Z Vt V f ˆV c P hit w jf F ; V c ; V f P V f dv f Z Vc Z B Vt V c V f ˆV t 0 f x dxp V f dv f : 4

9 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 9 Fig. 6. A hit i aother partitios (hit o ). (a) A complete hit i o ad (b) a partial hiti o. Agai, we use the assumptio made i Case A, that is, P V f is equal to B=. 2. Ucoditioig o V c. As log as V c B is smaller tha V t, a viewer at positio V c would be able to catch up with a viewer at all possible positios of V f i the same partitio. Otherwise, the farthest possible positio of V f will be at V t (refer to Fig. 5b). Here, we also ucoditio o V c based o the two cases used for ucoditioig o V f. Case A. V c B V t (or V c < l B, usig (3)), we have: P a hit w jf F ˆ Z l B V cˆ0 P a hit w jff; V c P V c dv c : 5 Here, the oly ukow is P V c, which is the probability that a viewer is at the V c th miute of the movie. Sice a viewer ca be at ay positio of the movie, we assume that P V c ˆ l. I other words, we assume that all positios of the movie have a equal probability of beig viewed. As V c is always smaller tha or equal to V f, i the secod case, we have the farthest V f to which a viewer at positio V c ca catch up equal to V t. I this case, we ucoditio as follows: Case B (V c V f ad V f ˆ V t (or V c V t which gives V c l)). Therefore, ucoditioig o V c, where l B < V c l gives the followig equatio for P b hit w jf F, where P V c is equal to l, as i Case A. P b hit w jff ˆ Z l V cˆl B P b hit w jff; V c P V c dv c : 6 Hece, give that the VCR request is a FF, the probability of a hit withi the same partitio, P hit w jf F, is obtaied by summig up the two expressios i (5) ad (6) Jump to Other Partitios (hit o ) Beside the possibility of beig able to resume i his ow partitio, a viewer ca also resume i aother partitio such that the I/O resources allocated i Phase for servig his/ her VCR request ca be released. We refer to this evet as a hit o. As illustrated i Fig. 6, if a viewer is to resume at the ith partitio ahead of its curret positio, the he/she must fast forward log eough to at least catch up with the last possible viewer i the ith partitio ahead, amely, at positio V li. Let us deote by V fi the positio of the first possible viewer i the ith partitio, the the evet hit o ca be divided ito two cases, amely, the complete hit i o ad the partial hit i o. As illustrated i Fig. 6a, a complete hiti o is a evet correspodig to a viewer beig able to catch up ot oly to the viewer at positio V li but also to the first possible viewer, i.e., viewer at positio V fi. Fig. 6b depicts the evet of a partial hit i o, which illustrates a viewer who is ot able to catch up with the viewer at positio V fi before the movie eds. Sice the movie is started every l miutes, viewers at the same relative positio i differet partitios have a phase differece which is a multiple of l, e.g., viewers at positio V l ad V li have a phase differece of i l miutes. Let us deote the shortest ad logest duratio a viewer must fast forward to have a hit i o by i jump l ad i jump f, respectively. To make our derivatio cosistet with that of P hit w, we express these two quatities i terms of V f ad V c as follows: i jump l i jump f ˆ V li V c ˆ il V f V c B ; ˆ V fi V c ˆ il V f V c : 7 8 The probability of a hit o at the ith partitio ahead, deoted by P hit i o, is show below. Case. The ability to catch up with both viewers at positio V li ad V fi (a complete hit i o ). I this case, a viewer at positio V c is able to catch up with viewers at both, positio V li ad V fi i the ith partitio ahead: P c hit i o jf F ; V c; V f ˆ Z i jumpf i jump l f x dx; 9 where the upper ad lower limit of the itegral are derived based o (9), (7), ad (8). Case 2. The ability to catch up with a viewer at positio V li but ot all at the ith partitio (a partial hit i o ). I boudary cases, a viewer at positio V c may ot be able to catch up with all viewers betwee positio V li ad V fi. However, as log as the viewer is able to catch up with a viewer at positio V li, we will classify this evet as a partial hit i o. As before, we defie V t i to be the positio of the last viewer i the ith partitio with which a viewer at positio V c ca catch up. Costraiig V c V ti V c l ad

10 0 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 V t ˆ V ti il, results i V t ˆ l Vc il. The, the probability of a partial hit i this case is: P p hit i o jff; V c; V f ˆ Z Vti V c i jump l f x dx ˆ Z il Vt Vc i jump l f x dx ˆ Z l Vc i jump l f x dx: 20 Ucoditioig o V f. Whe ucoditioig o V f, we cosider a complete hit i o ad a partial hiti o separately. For a complete hit i o, we ucoditio o V f usig cases a ad b as i the derivatio for evet hit w. For evet hit i o, V t ˆ l Vc il P hit i o jf F; V c ˆ P 2 hit i o jff; V c ˆ Z Vc B : V f ˆV c P c hit i o jff; V c; V f P V f dv f ; 2 Z Vt V f ˆV c P c hit i o jff; V c; V f P V f dv f : 22 Observe that a viewer at positio V c ca have a complete hit o, as well as a partial hit o, depedig o the value of V f. Give a viewer at positio V c, (22) gives the probability of a complete hit o for V c V f V t. But, V t < V f < V c B ca also result i a partial hit o ; this is show i (23) below. Equatio (24) correspods to those viewers who ca oly have a partial hit o for all values of V f, where the largest V f with respect to V c is defied as V t 0 ˆ l Vc il B. P 3 hit i o jf F; V c ˆ P 4 hit i o jff; V c ˆ Z Vc B V f ˆV t P p hit i o jf F; V c; V f P V f dv f ; 23 Z Vt 0 V f ˆV c P p hit i o jf F; V c; V f P V f dv f : 24 Ucoditioig o V c. The rage of V c is calculated usig the same techique as i the hit w case to yield the followig four equatios: Z l B P hit i o jff ˆ il P 2 hit i o jff ˆ Z l il 0 l B P 3 hit i o jf F ˆ Z l il il l B il Z il B l P 4 hit i o jf F ˆ l il P hit i o jff; V c P V c dv c ; P 2 hit i o jf F; V c P V c dv c ; P 3 hit i o jff; V c P V c dv c ; P 4 hit i o jf F; V c P V c dv c : Hece, the probability of resumig at the ith partitio ahead for a FF request, P hit i ojf F, is calculated by summig (25), (26), (27), ad (28): P hit i X4 ojff ˆ P a hit i ojf F : 29 aˆ The umber of partitios to which a viewer at positio V c would be able to jump ahead, resultig i hit o, depeds o which partitio this viewer is before the iitiatio of the FF, e.g., it would be impossible for a viewer i the last partitio of the movie to jump ahead to aother partitio. Also, the existece of V t limits the umber of partitios to which a viewer ca jump ahead. To fid the rage of i, we refer back to (25), which idicates that the upper limit o V c, amely, l B il, must be greater tha or equal to 0. Therefore, we have to satisfy the coditio l B il 0. Rearragig the equatio, we have: l w l i : 30 l 4..3 Fast-Forwardig to the Ed of a Movie Recall that our goal is to maximize the probability that the resources allocated for a VCR request will be released upo resume to ormal playback. Cosider the case where a viewer is at positio V c. The, the logest FF duratio is give by V t V c. As the pdf of a FF duratio is defied i the iterval 0; lš, there is a ozero probability that a viewer issuig a FF request will fast-forward to the ed of the movie. I this case, the resources allocated to him/her i Phase ca also be released. We defie the probability of fast forwardig beyod the ed of a movie to be P ed ; it is give by the followig equatio: P ed ˆ ˆ Z l Z l Z l 0 0 Z l Z l 0 l V c f x dx l dv c l V c f x dx l dv c: 3 Fially, P hitjff, the probability of a hit give that the VCR request is a FF operatio is: P hitjf F ˆ P hit w jff X l b l w l c iˆ P hit i ojf F P ed : 32 The first term of the RHS correspods to the probability of a hit withi the same partitio, i.e., the partitio i which the FF request was iitiated, ad the secod term correspods to the total probability of hit i the ith partitio ahead, where i. We assume that the probability of a hit i partitio i, where i is less tha the curret partitio is defied to be zero. The last term correspods to the probability of a fast-forward to the ed of a movie. All three terms sum up to the probability of releasig the I/O resource upo resume from a FF request. For VCR requests like rewid ad pause, we derive P hitjrw ad P hitjp AU i a maer similar to the derivatio of P hitjff. Due to limitatio of space, we do ot repeat the derivatios here. The detailed derivatios ca be foud i [6].

11 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE The Expected Hit Probability P hit Wheever a viewer issues a VCR request, there is some probability that the request is of FF, RW, or PAU type; we deote these probabilities by P F F, P RW, ad P P AU, respectively. Note that the values of these probabilities ca be determied by measurig user behavior usig statistical techiques. Give these probabilities, we ca obtai the probability of a hit of a resume from a VCR request to ormal operatio, P hit, which ca be expressed as follows: P hit ˆ P hitjf F P FF P hitjrw P RW P hitjpau P P AU : 33 The quatity, P hit, is a fuctio of seve system parameters. Specifically, P hit ˆ l; B; ; w; R F F ; R PB ; R RW : Our goal is to determie the proper values of B (the total size of buffer space eeded for ormal playback) ad (the umber of I/O streams eeded for ormal playback) such that P hit is greater tha or equal to P, which is a give desig parameter. To determie the values of B ad, we use the followig two costraits: (C) w ˆ l B, (C2) P P hit. The first costrait, C, correspods to the maximum waitig time that a viewer might experiece before the movie restarts. The secod costrait, C2, esures that the average probability of a hit coforms to a specified probability P. Substitutig the system parameters ito the two costraits will yield two equatios i two ukows, amely, B ad. By solvig these two equatios umerically, we ca determie the size of the system buffer, B, ad the umber of I/O streams,, which eed to be preallocated for playback of a popular movie i order to satisfy the performace (or quality of service) requiremets of P ad w. Although the above formulatio oly deals with a sigle movie, we will show how to apply our model to hadle multiple movies i solvig the system sizig problem i Sectio Model Verificatio I this sectio, we verify our mathematical model usig simulatio. We first describe the system parameters used i this sectio. The arrivals of viewer requests for a popular movie are modeled as a Poisso process with a rate. As VCR requests are usually of relatively short duratio as simple cases for illustratio purposes, we use a expoetial distributio ad a skewed gamma distributio to represet the duratio of the VCR fuctios. I the followig examples, the rates of fast forward ad rewid, R F F ad R RW, are three times that of ormal playback, R PB, ad the movie legth l is equal to 20 miutes. I the first three experimets, we simulate a system with two types of requests, ormal playback ad oe of the VCR fuctios, amely, either fast forward with viewig (FF), or rewid with viewig (RW), or pause (PAU). I the last experimets, the requests ca be of type ormal playback or ay of the three VCR fuctios, where the probability of a request for a VCR fuctio r is equal to P r, ad r ca be FF, RW, or PAU. I these experimets, we vary both the arrival rate of requests ad the mix of FF, RW, ad PAU. Also, i each experimet, we vary the maximum waitig time, w. The probability of a hit is plotted as a fuctio of the umber of partitios,, where each curve correspods to a specific value of the maximum waitig time, w. The results obtaied from our mathematical model ad from simulatios are plotted i Figs. 7a, 7b, ad 7c. These figures illustrate both the theoretical ad the simulatio results where the oly type of a VCR request issued is either FF, RW, ad PAU, respectively. Results for a mix of all three VCR requests plus requests for ormal display are depicted i Fig. 7d. These figures illustrate that the theoretical results, for the most part, closely match the simulatio results, which verifies the accuracy of our mathematical model. 5 RESOURCE ALLOCATION FOR VCR MODE I this sectio, we preset the computatio of resources eeded to hadle VCR requests so as to satisfy the performace level defied by (6) ad (7). Notice that is defied by v v =m, ad it deotes the fractio of time ay oe of the reserved I/O resources is busy. Therefore, we aim to decrease the value of. Sice ) the total VCR arrival rate to the system, v, is beyod our cotrol (uless we wat to lower the quality of service) ad 2) maximizig the umber of I/O resources for VCR fuctioality, m, is ot favorable because it implies a more expesive system; we are goig to reduce by miimizig v. If we examie v i (2), we see that v is comprised of two compoets. The first compoet caot be reduced because it deotes the mea time a viewer will sped i the fast forward, rewid, or pause operatio wheever the viewer issues a VCR request, which depeds solely o the users' behavior. For the secod compoet, it deotes the expected time a viewer speds holdig oto additioal resources, allocated to him whe he resumes to ormal playback before he ca rejoi a existig partitios, if ormal playback is expected to resume immediately. Notice that this amout of time greatly depeds o how the system hadles the falle out viewers. So, i devisig methods for how to hadle VCR requests, we aim to implemet a method which miimizes the value of hold. I the remaiig sectio, we will preset two schemes for hadlig VCR requests, both of which esure that the resumptio to ormal playback from a VCR operatio is immediate ad show the derivatios of the correspodig values of hold. Scheme (No mergig). If a viewer caot resume at ay of the existig partitios after a VCR request, i.e., resume with a miss, the system will ot do aythig to facilitate the release of resources. I other words, for these viewers to resume to ormal playback without ay delay, they will have to cotiue utilizig I/O resources (allocated durig the VCR phase) after the resume, util they fiish viewig the movie. Assume that each poit of the movie has a equal probability of beig the poit of resume, we obtai the value of hold by calculatig the expected time to hold o to the VCR resource whe the viewer resumes with a miss. Thus, we have hold ˆ l=2 ad this value also serves as a upper boud o hold.

12 2 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 7. Comparig simulatio ad theoretical results for ormal playback ad various VCR requests wherei the iterarrival times are expoetial ad = ˆ 2 miutes; duratio of VCR requests is draw from a skewed gamma distributio with a mea = 8 miutes ( ˆ 2; ˆ 4). (a) Oly fastforward VCR request, (b) oly rewid VCR requests, (c) oly pause VCR requests, ad (d) all types of VCR requests with P F F ˆ 0:2; P RW ad Fig. 8. Catch-up through adaptive piggybackig whe the viewer resumes with a miss. Scheme 2 (Mergig through piggybackig ad bufferig). This scheme is to reduce the value of hold by applyig the techiques of adaptive piggybackig ad bufferig. The adaptive piggybackig approach [], [0] alters display rates of requests i progress, for the purpose of mergig their respective I/O streams ito a sigle stream, which ca serve the etire group of merged requests. After two viewers are ªmerged,º oe of them ca release his/her resources. Thus, the idea is similar to that of batchig except the groupig is doe dyamically while the displays are i progress so o latecy is experieced by the user. Usig adaptive piggybackig, we will alter the display rate of the falle out viewer such that this viewer ca evetually catch up with either the partitio ahead or the partitio behid. I particular, the viewer will either slow dow to eable a catch-up by the partitio behid or speed up to catch up with the partitio ahead, depedig o which partitio he is closer to. I this case, the probability of releasig the allocated VCR resources before the viewer fiishes the movie ca be icreased, thereby decreasig the value of hold. Fig. 8 illustrates two scearios where the viewer resumig with a miss at positio V c of the movie ad cosumes the cotet of the movie at altered rates as compared to the viewers i the partitios, who are cosumig movie cotet at the ormal rate. Let us deote the ormal rate of cosumptio by R PB. Fig. 8a illustrates that the falle out viewer will speed up the cosumptio at a rate of f R PB i order to catch up with the last possible viewer, V l, i the partitio ahead. Fig. 8b illustrates that the viewer will be cosumig the movie cotet at a slower rate of f R PB i order to eable a catch-up by the first possible viewer, V f, i the partitio behid. I each case, the elapsed movie time, C t, util a joi of the falle out viewer with either of the partitios ca be accomplished through adaptive piggybackig is f, where is the iitial distace betwee V c ad the target viewer. Sice the calculatio of hold depeds o differet values of C t for viewers resumig at differet positios of the movie, it would be desirable to further reduce this quatity. Thus, i additio to adaptive piggybackig, we ca also make use of the bufferig techique to reduce the catch-up time. The motivatio here is that a viewer resumig with a miss ªearº to a ed of a existig partitio, ca reduce his catch-up time by ªbridgigº the gap betwee him ad the partitio through the use of additioal buffer space, i.e., to merge with that partitio earlier. After the merge, the allocated I/O stream ca be released sice the partitio ad

13 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 3 the additioal buffer space will ow be treated as oe exteded partitio. This additioal buffer space ca be gradually released by cotiuig to apply adaptive piggybackig util the viewer ca joi i the origial (i.e., rather tha the exteded) partitio. Furthermore, viewers resumig with a miss ªfarº from the eds of the partitios ca perform adaptive piggybackig util they are ªclose eoughº to the partitio to utilize additioal buffer space i order to perform a early merge (as suggested above). I this case, the catch-up time, or C t, ca be further reduced ad by usig bufferig, the allocated I/O streams ca be released as soo as possible upo resume, thereby ehacig the availability of I/O streams for other VCR requests. The combied use of bufferig ad adaptive piggybackig techiques is accomplished i the followig maer. We defie a threshold of k miutes (ahead of behid a partitio) withi which the use of bufferig will eable a merge with the partitio. Also, adaptive piggybackig will be used simultaeously to further facilitate the joiig with the partitio. Thus, we divide the hadlig of resume with a miss ito two cases, amely, resume withi the threshold k ad resume beyod that threshold. Case A: Resumig withi the threshold ( k). Whe the viewer resumes with a miss at a positio which is withi the k miute threshold ahead or behid a partitio, we will use miutes of buffer space to joi the viewer with the existig partitio. Sice that viewer is ow cosumig movie frames that have either ot bee fetched yet ito the buffer (i the case of a resume withi k miutes ahead of a partitio) or have bee already replaced (i the case of a resume withi k miutes behid a partitio), we will have to cotiue utilizig the I/O stream util the partitio ad the additioal buffer space ca be merged together as a sigle exteded partitio. At this poit, the merge is said to be completed ad the I/O stream allocated to the viewer ca be released. Meawhile, we will ) cotiue performig adaptive piggybackig so the viewer ca cotiue to catch up with the origial partitio ad 2) release the additioal buffer space gradually. If the viewer resumes at a poit R that is miutes i frot of a partitio, he will cotiue to use the allocated I/O stream to fetch the movie frames, which will be retaied i the additioal buffer space. The maximum size of the additioal buffer space is determied by sice the first viewer i the partitio behid takes miutes to reach poit R. Similarly, the maximum size of additioal buffer space ad the time eeded to eable a merge for a viewer resumig behid a partitio is give by = f miutes as the viewer is adoptig a slightly faster rate with respect to the ormal playback rate. Give a viewer at positio V c of the movie, the average time to release the I/O stream for a resume ahead of the partitio ad a resume behid the partitio is give by R k ˆ0 d l B ad R k d ˆ0 f l B, respectively, where l B is the legth of the gap betwee partitios. Therefore, to calculate the average time to release the I/O stream, we further ucoditio by takig the probability of resumig at ay positio of the movie to be equal to =l. Here, we separate the calculatio ito two cases. The first case correspods to a viewer resumig at a positio which is uits behid a partitio, that is, this viewer will catch up with V l i the partitio ahead. Therefore, the average catch-up time is give by: T k ˆ l B l f " Z l k f k Z k Z l Z # l Vc 2 ddv c V cˆ0 ˆ0 V cˆl ddv c : ˆ0 k f k 34 The first term correspods to the average time to catch up with the partitio ahead whe resumig at ay positio of the movie from the 0th positio to the l k f k th positio. The secod term is eeded for hadlig boudary case of resumig at ay positio withi the rage of the l k f k th positio to the ed of the movie. Therefore, the distace of the resumig viewer from the partitio ahead oly rages from 0 to l V c 2 miutes. The secod case correspods to a viewer resumig at a positio of uits ahead of a partitio. This viewer will slow dow the cosumptio to eable a catch-up by V f i the partitio behid. The average time to release the I/O stream is calculated i a similar maer ad is give by: T 2k ˆ l B l Z l k Z k Z l Z l Vc 35 ddv c ddv c : V cˆ0 ˆ0 V cˆl k ˆ0 Case B: Resumig beyod the threshold ( > k). Whe the viewer resumes at a positio which is more tha k miutes away from a partitio, the viewer will perform adaptive piggybackig [], [0] util the separatio distace reduces to k miutes. At this poit, we ca hadle the situatio as i previous cases. For resumig at a positio beyod the threshold, we cosider the average time for releasig the I/O stream through the followig three cases: Case : Catch up with V l i the partitio ahead. The viewer resumig at positio V c of the movie will speed up the cosumptio of movie frames at a rate of f R PB to catch up with the last viewer i the partitio ahead. The time elapsed before a merge ca be accomplished is k =f, which is the time eeded to decrease the distace betwee the viewer ad the partitio to k miutes. At this poit, k= f more miutes are eeded for the merge to complete. This sceario is illustrated i Fig. 9. Therefore, the total time, C t, eeded to release the I/O stream is give by k =f k= f. However, C t must ot be greater tha the time for V l to fiish the movie; otherwise, it simply meas that the viewer caot release the allocated I/O resource before fiishig the movie. I other words, C t must ot be greater tha l V l. I this case, the followig iequality must be satisfied:

14 4 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 9. Speed up to catch up V l util the distace betwee the two is reduced to k. l V c k f f f : 36 Case 2: Catch up with V f i the partitio behid. The viewer resumig at positio V c of the movie will slow dow the cosumptio of movie cotet at a rate of f R PB to facilitate the catch-up by the first viewer, V f, i the partitio behid, as illustrated i Fig. 0. Usig the same calculatios as i the above case, we ca see that the time elapsed before a merge ca be accomplished ad must ot be greater tha the time for V f to fiish the movie, i.e., f k k l V c : However, if the time take to merge with the partitio behid is greater tha that take to fiish the movie by V c, we caot beefit from performig adaptive piggybackig. Therefore, the followig iequality has to be satisfied istead: l V c k f: 37 f Case 3: Caot catch up with either of partitios. I this case, viewer V c caot perform a merge before he fiishes the movie. The, the time elapsed before the I/O resources ca be released ad is give by l V c. The average time to release a resources, give that the viewer is at positio V c of the movie, is computed usig (38). The first term below correspods to the average time a viewer takes to catch up with the partitio ahead while the secod term below correspods to the average time a viewer takes to eable a catch-up by the first possible viewer i the partitio behid. I this equatio, k deotes the largest distace from V l i the partitio ahead where a catch-up to the partitio ahead ca be accomplished before the ed of the movie ad k 2 deotes the largest distace from V f i the partitio behid where a catch-up from behid is still possible. The last term of the equatio correspods to the average time spet to release a I/O resource whe a viewer caot perform adaptive piggybackig i order to eable a catch-up with either of the partitios: Fig. 0. Slow dow to eable a catch-up by V f util the distace betwee the two is reduced to k. Z k k k! d ˆk f f l B Z k2! d k k ˆk f l B l B l V c k! k 2 : l B 38 To fid the average time to release resources at ay positio of the movie, we ucoditio o V c i the three cases as stated above, which correspods to the three terms give i (38). The rage of V c is determied by the rage of. Whe the viewer resumes at a distace of l B 2 from either of the partitios, i.e., half way betwee two partitios, it takes the same amout of time for him to joi either the partitio ahead or the oe behid. I each of the three cases, we determie the rage of V c by settig the iequality: l B 2 to equality for Cases ad 2 usig (36) ad (37), respectively. For Case : V c l k f f For Case 2: V c l k f f l B ˆ : 2 l B ˆ 2 : f Let us cosider the ucoditioig of V c i (38) term by term. Case. For V c, this correspods to the viewer resumig at the maximum distace of l B 2 from the ed of the partitio where a merge with the partitio ahead ca still be accomplished before the ed of the movie. However, for V c >, the maximum distace from the ed of the partitio is give by (36). Thus, we have the followig: T ˆ " Z l B l Z l k f k V cˆ0 Z Vc k ˆk Z l B 2 k k ddv c ˆk f f 3 f k k 7 ddv c 5: f f f 4 Case 2. The equatio give below is derived i the same maer as i Case with referece to (37) ad the rage of V c defied i (40):

15 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 5 TABLE 2 Parameters of the Two Popular Movies for Simulatio T 2 ˆ " Z 2 l B l V cˆ0 Z l k Z l Vc k f V cˆ 2 ˆk Z l B 2 ˆk Šf k k ddv c f k k ddv c #: f 42 with referece to the secod term i (34) ad (35). Therefore, the average time to release the I/O stream, hold, i this scheme, is give by the followig equatio: hold ˆ T k T 2k T T 2 T 3 : 44 Case 3. The followig equatio is calculated with respect to (34), (35), (4), ad (42). Makig use of the upper boud of i each equatio ad ucoditio o the correspodig values of V c, the average time to release the I/O stream whe adaptive piggybackig caot be performed to facilitate the joiig of partitios is give by the followig equatio: T 3 ˆ l B l 2 0 Z 2 l B l V c l B V c k V cˆ 2 f f A dv c Z l k f k 0 l B V c k f f l V V cˆ 2 f l V c k! f dv c f Z l k V cˆl f k k l V c l B l V c 2 l V c k f dv c f l V c l B l V c l V c dv c : 2 Z l V cˆl k 43 Thus, the average time to release the I/O stream for Case 3 is give by (43). The first two terms are derived with referece to (4) ad (42); the last two terms are derived 5. Verificatio I the previous sectio, we have derived the equatios for calculatig the value of hold for Scheme 2. These equatios help to calculate the overall time the viewers will hold o to the I/O resources before they ca joi a existig partitio. I this sectio, we will verify our derivatio usig simulatio. We carry out our simulatio usig two popular movies, ad 2, with movie legth l beig 60 ad 90 miutes, respectively. We also set the maximum time that the viewers ca wait for the start of the movies, w to be 0:5 ad 0.25 miute, respectively. The duratio of VCR requests of movie ad 2 are draw from a expoetial distributio with a mea equal to five miutes ad two miutes, respectively. Let the performace requiremet of P of each movie be equal to 0:5. We apply our mathematical model to calculate the amout of buffer ad I/O resources for ormal playback; these are summarized i Table 2. The arrival rate of viewers ito system,, is 5=mi ad the simulatio is ru for 50,000 viewers. The average time for a falle out viewer to hold o to the allocated resources, ( hold ), is calculated for differet values of the threshold k. The respective values of hold obtaied by simulatio ad through our derivatio of both movies are plotted i Fig.. From Fig., we ca see that the calculated values of hold match closely the simulatio results, which verifies our derivatio. Yet, the values of hold obtaied from simulatio are smaller tha the calculated couterparts, which ca be explaied as follows: I our derivatio, we try to calculate the time take for the viewers to joi a existig partitio. However, i the simulatio, there also exist other falle out Fig.. Simulated ad calculated results of hold for differet values of the threshold k. (a) Movie ad (b) movie 2.

16 6 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 2. Calculatig the resources for ormal playback. (a) The relatio betwee the probability of hit ad umber of I/O streams required for ormal playback. (b) The relatio betwee the cost of resources for ormal playback ad the probaility of a hit. viewers tryig to piggyback o a existig partitio ad they may have already formed a exteded partitio. These exteded partitios reduce the time for other falle out viewers to joi a existig partitio, which explais why our derivatio slightly overestimates the values of hold as compared with the simulated values. 6 APPLICATIONS TO SYSTEM SIZING Let us illustrate how we apply the mathematical model described i Sectios 4 ad 5 to makig system sizig decisios. I determiig the amout of resources required for a VOD system, we divide the system resources ito two categories. The first category of resources is for hadlig ormal playback ad is calculated by applyig the mathematical model described i Sectio 4, while the secod category of resources is for servicig VCR requests ad is calculated by the model described i Sectio 5. Our goal is to optimize the performace of a VOD system by reducig the demad of system resources while providig a acceptable level of quality of service to the viewers. The total system cost is determied by the amout of buffer space ad I/O streams (both for servicig ormal ad VCR playbacks) eeded to esure a required level of performace. Thus, the overall cost of resources will be a fuctio of the costs for ) memory buffers per miute of a movie deoted by C b ad 2) a I/O stream deoted by C. Although the two categories of resources serve differet purposes, the resources allocated for ormal playback will directly affect the amout of resources required for hadlig VCR requests while the amout of resources for servicig VCR requests will affect the QoS. Thus, i calculatig the system resources required for ormal playback, we also cosider the behavior of viewers i issuig VCR requests. This is achieved by esurig a miimum probability of a hit, P : Wheever a viewer resumes from a VCR request, we ca esure that the VCR resources will ot be cosumed for ormal playback with a probability greater tha or equal to P. Therefore, if we ca desig our system such that the probability of a hit upo resume from a VCR request is high, the the amout of resources which eed to be reserved for servig VCR fuctios ca be reduced. If the probability of hit is very small, it is ulikely that the viewer ca release the I/O stream upo resume util he fiishes viewig the whole movie. However, to esure a high probability of a hit requires more resources to be reserved for ormal playback, which also implies a higher system cost. Therefore, it is very importat to determie the optimal probability of a hit which will yield a system with the lowest overall cost, but oe that still satisfies the required performace ad quality of service levels. I the remaiig sectio, we will show how to determie the cost of resources for ormal playback as well as those for servicig VCR requests. 6. Cost of Resources for Normal Playback For a movie of legth l ad a give maximum waitig time, w, we ca calculate the amout of resources eeded for ormal playback as follows: Usig our mathematical model, we calculate the correspodig hit probability for each possible umber of I/O streams eeded for ormal playback, (where ˆ ; 2;... l w ). The respective buffer size, B, for each ca be obtaied by usig (0). We ca the determie the cost of resources for ormal playback, C P B, for each B; pair, by solvig C b B C, which o rearragig gives C PB ˆ C 'B, where ' ˆ Cb C ad ca be viewed as a price ratio of oe miute of memory buffer to oe I/O stream. For each possible B; pair, there is a correspodig probability of a hit, P, ad cost, C P B. We express each possible cofiguratio of resources for ormal playback as a PB-tuple, which is i the form of B; ; P ; C P B. 6.. Experimet A movie is of legth, 60 miutes, ad the maximum time a viewer is willig to wait before a movie starts is 30 secods. Assume that the time a viewer speds i a VCR fuctio follows a expoetial distributio with the mea equal to miute of movie time. By applyig our mathematical model, the respective probability of a hit is plotted i Fig. 2a. We also assume that a 2G-SCSI disk is used i this experimet ad that the cost of each disk is aroud $700 ad that its trasfer rate is 5 MB/sec. The data rate of a MPEG-2 stream is assumed to be 4 Mbps, ad the cost of MB of mai memory is $24. We calculate C b ad C as follows: C b ˆ 60s4Mbps 8 $24 ˆ $720 US$700 5MBytes=sec8=4Mbps ad C ˆ ˆ $70. Give the above equatios, we ca see that the amout of buffer space required to store oe miute of a MPEG-2 movie is approximately 0 times as expesive as oe I/O stream, i.e., ' 0. The cost for ormal playback is plotted i Fig. 2b. From Fig. 2a, we ca see that decreasig the

17 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 7 Fig. 3. Cost of resources for servicig VCR requests. umber of partitios will icrease the probability of a hit. More buffer space is utilized to compesate for the saved I/O streams, which meas that a larger portio of the movie will reside i the buffer space thereby icreasig the probability of a hit. However, we ca see from Fig. 2b that the cost of resources for ormal playback also icreases with the probability of a hit. So, it may seem that adoptig a lower probability of hit will yield a lower system cost. We will show, i the ext experimet, that this is ot true because we are tryig to reduce the overall system cost. I determiig how much resources eed to be allocated for ormal playback, we have to cosider the overall system cost to obtai the required P. 6.2 Cost of Resources for VCR Fuctios As metioed i Sectio 3.3, the provisio of VCR fuctioality should esure the satisfactio of a certai performace level, give by (6), (7), ad (8). As i the ormal playback case, we express each possible cofiguratio of resources for servicig VCR requests as a VCR-tuple, which is i the form of B v ; m; v ; V CR ; P ; C v, where C v is the cost of resources eeded for servicig VCR requests Experimet 2 Cosider the system i Experimet. The system will make use of adaptive piggybackig oly to facilitate the joiig of falle out viewers with existig partitios. I other words, we will adopt Scheme 2 but set the threshold value k ˆ 0. We also assume that the average waitig time for servicig ay VCR request is less tha three secods. This performace level is chose because the waitig time is a measure that is oticeable to the viewers as compared to the queue legth ad probability of waitig. The arrival rate of VCR requests, v is assumed to be 200/mi. For a movie of 60 miutes, the expected umber of viewers i the system is 900. Therefore, each viewer will, o average, issue a VCR request oce every 4:5 mi. We will show later i this sectio the effect of alterig VCR arrival rates. Fig. 3 shows the relatioship betwee the cost of resources for servicig VCR requests ad the probability of a hit. From Fig. 3, we ca see that the cost of resources decreases with the icrease of probability of a hit, which is opposite to the effect observed i Fig. 2b. A lower probability of a hit will icrease the average time a viewer speds i holdig o to the additioal resources, which implies that more resources are eeded to be reserved for servicig VCR requests. Therefore, i makig system sizig decisios, we have to Fig. 4. Total cost of the system. cosider both the cost of resources for ormal playback as well as the cost of resources for servicig VCR requests (the overall system cost) before we ca determie how much resources are to be allocated for ormal playback ad for servicig VCR requests. 6.3 Overall System Cost The overall system cost is calculated by summig up the cost of ) resources for ormal playback ad 2) resources for servicig VCR requests. Let us describe how to determie the optimal system cofiguratio by cosiderig the overall system cost. The overall system costs for a system usig Scheme ad Scheme 2 are compared. I additio, we also show that our proposed mathematical model, together with Scheme 2 adopted for servicig VCR requests, will yield a lower cost as compared with systems usig pure I/O ad pure buffer space resources. Last, we describe how to choose the right value of a hit probability, P ad of the threshold k. Assume that we wat to determie the amout of system resources which eed to be allocated for a sigle movie, we have to calculate the overall system cost i the followig maer: Step. Geerate all the PB-tuples by determiig all the possible cofiguratios for ormal playback by applyig the model preseted i Sectio 4 ad calculate the respective hit probabilities ad cost. Step 2. Obtai hold for each of the cofiguratios geerated i Step usig (44). With referece to the hit probability i each PB-tuple obtaied i Step, we ca v by (2). The, we ca determie a correspodig VCR-tuple by applyig the M=M=m queue model. Step 3. For each pair of PB-tuple ad VCR-tuple, obtai the overall system cost, C, by summig up C PB i the PBtuple ad C v i the VCR-tuple. Step 4. The optimal choice of system resources allocatio is the tuple pair with the lowest cost Experimet 3 Cosider the system i Experimet ad 2. The total cost of the system is plotted agaist the umber of I/O streams eeded for ormal playback ad is show i Fig. 4. I this figure, we ca see that the overall cost of the system decreases iitially for the smaller umber of I/O streams ad icreases toward the larger umber of I/O streams. The system yields the lowest cost whe the umber of I/O streams for ormal playback lies withi the rage of With referece to Fig. 2a, the probability of a hit

18 8 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 lies approximately withi the rage of 0:75 0:85. This illustrates that i determiig the proper amout of resources to be allocated for ormal playback, we should also cosider the cost of resources for VCR requests. The allocatio of correspodig resources should yield a system with the lowest cost while satisfyig a predefied performace level. 6.4 Compariso of Differet Schemes Let us ow demostrate the effectiveess of our approach by ruig several experimets to determie the overall system cost. First, we icorporate Scheme ad Scheme 2 ito our ormal playback model, where their total system cost is plotted agaist differet arrival rates of VCR requests while other parameters remai the same as those described i the previous experimets. Secod, we examie two other systems, amely, a system usig oly I/O streams for ormal playback (referred to as pure I/O hereafter) ad a system usig oly buffer space for ormal playback (referred to as pure buffer hereafter). These are the two extreme cases where oly I/O streams or oly memory buffers are used for ormal playback. We aim to show that a proper balace betwee I/O ad memory resource will yield a system with a lower cost Experimet 4 I this experimet, we determie the overall system cost of each scheme for differet arrival rates of VCR requests, ragig from very low arrival rates to very high arrival rates. We assume that the arrival rate of a ormal playback request is 5=mi, so that the expected umber of viewers i the system is give by 5 60 ˆ 900. We plot the overall system cost of each scheme agaist the umber of I/O streams eeded for ormal playback with the average VCR arrival rates v of 5=mi, 00=mi, 200=mi, ad 400=mi i Figs. 5a, 5b, 5c, ad 5d, respectively. From these figures, we ca see that differet arrival rates will yield a system with the lowest cost occurrig at differet positios o the cost curve. For Scheme, except for very low VCR arrival rates, the miimum cost is always located at the smallest value of I/O streams, that is, with the largest amout of buffer space for ormal playback. This ca be explaied as follows: Recall that hold for Scheme is l 2, which is 30 miutes i this case. That is to say, if the mea time for a falle out viewer to hold o to the additioally allocated resource is large, more resources eed to be reserved for servicig VCR requests ad hadlig the falle out viewers. This amout of resources becomes so large that it is more beeficial to allocate more buffer space for ormal playback, such that the probability of a hit ca be icreased. By icreasig the probability of a hit, the mea time a viewer speds i utilizig the additioal resources ca be reduced, ad we ca make sure that the reserved resources are used for servicig VCR requests rather tha servicig the ormal playback of the falle out viewers. The same ratioal applies to Scheme 2 whe the VCR arrival rate is very high. O the cotrary, for low VCR arrival rates i Scheme 2, the lowest cost is located at the largest value of I/ O streams, that is, whe the least amout of buffer space is used for ormal playback. A low arrival rate of VCR requests meas that ot may viewers will be issuig VCR requests. Eve if the viewers fall out of a partitio upo resume, Scheme 2 ca facilitate the joiig of these viewers with existig partitios ad, i tur, the release of allocated resources i a very short time. Therefore, oly a small amout of resources is eeded for servicig VCR requests ad hadlig the falle out viewers. I this case, usig a large amout of buffer space for ormal playback to esure a larger probability of a hit is ot beeficial. Whe we compare Scheme ad Scheme 2, we ca see that Scheme 2 always outperforms Scheme i the sese that the miimum cost for a system usig Scheme 2 is more smaller tha that of a system usig Scheme. To further demostrate the effectiveess of Scheme 2, we plot the percetage of total system cost saved by usig Scheme 2 as compared to usig Scheme. The percetage saved through Scheme 2 for differet arrival rates, as show i Fig. 6 is calculated by comparig the miimum overall cost i each of the two schemes, with referece to Scheme. The percetage saved by Scheme 2 as compared to Scheme rages approximately from 3 percet to as high as 70 percet Experimet 5 I this experimet, we compare the overall system cost of our proposed system with that of pure I/O ad pure buffer. We assume that the arrival rate of ormal playback requests is 5=mi, thus, a total of 900 I/O streams are eeded for ormal playback i the pure I/O case. The rate of fastforward or rewid is assumed to be three times faster tha that of ormal playback. The percetage of system cost saved by Scheme 2 for the case where the total cost is depedet o the VCR requests arrival rate is plotted agaist the differet VCR requests arrival rates ad is show i Fig. 7a, while the idepedet case is plotted i Fig. 7b. The percetage saved is determied by the differece betwee the miimum cost of Scheme 2 ad that of the pure I/O ad the pure buffer schemes. As show i Fig. 7a, the cost curve decreases rapidly as the VCR arrival rate icreases ad early levels off for very high VCR arrival rates. Scheme 2 outperforms the pure I/O method by at least approximately 2 percet ad ca outperform it by as high as 83 percet. Scheme 2 ca also outperform the pure buffer scheme by upto 75 percet for low to moderate VCR arrival rates, although beyod a VCR arrival rate of aroud 220=mi, pure buffer has a lower cost tha Scheme 2, which is demostrated by a egative percetage i the graph. As for the idepedet case illustrated i Fig. 7b, the cost curves also decrease rapidly at first from aroud 80 percet util the VCR arrival rate reaches aroud 200=mi, after which the decrease is slowed dow. I cotrast with the depedet case, both the pure I/O ad the pure buffer method perform better tha Scheme 2 whe the VCR arrival rate icreases beyod a certai poit. The percetage saved becomes egative whe the VCR arrival rate icreases to aroud 200=mi ad 290=mi for pure buffer ad pure I/O, respectively. Furthermore, istead of levelig off as i the depedet case, the egative percetage cotiues to drop for very high VCR arrival rates because whe the VCR arrival rate icreases, more ad more resources are eeded i Scheme 2, which meas the cost also

19 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 9 Fig. 5. Total cost of system for Scheme ad Scheme 2 for differet VCR arrival rates. (a) Average arrival rate of VCR requests ˆ 5=mi. (b) Average arrival rate of VCR requests ˆ 00=mi. (c) Average arrival rate of VCR requests ˆ 200=mi. (d) Average arrival rate of VCR requests ˆ 400=mi. icreases. O the other had, sice the cost of pure I/O ad pure buffer is idepedet of the VCR arrival rate, this meas that their cost is fixed, thereby resultig i the cotiuous decrease i the percetage saved. We also plot the same graphs for two more system settigs, where the movie is of legth 90 mi ad 20 mi; these are show i Figs. 8 ad 9, respectively. From these figures, we ca see

20 20 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 6. Percetage of system cost saved by Scheme 2 as compared to Scheme. that the percetage saved by Scheme 2 becomes more sigificat as the movie legth icreases. The percetage saved by Scheme 2 is more sigificat at lower VCR arrival rates because for systems usig pure I/O or pure buffer method, the cost is domiated by the resources allocated for ormal playback. However, as the VCR arrival rate icreases, the requiremet of VCR I/O resources becomes larger ad larger, util it becomes the domiat factor i the case of Scheme 2. I spite of that, we ca claim that Scheme 2 outperforms the pure I/O ad pure buffer methods i a VOD system. This ca be explaied as follows: Cosider the movie of legth 60 mi ad the rage of VCR arrival rates where the percetage saved by Scheme 2 is positive, i.e., at most 200=mi, where the expected umber of viewers i the system is 900 i this example. That meas that approximately 4 of the viewers are doig VCR fuctios per miute or the viewers are issuig VCR requests oce every 4:5 miutes o the average, ad this is a very high VCR rate for viewig movies. I a VOD system, relatively less time will be spet i doig VCR fuctios as compared with ormal playback. For a movie of legth 60 miutes, we expect Fig. 7. Percetage of system cost saved by Scheme 2 over pure I/O ad pure buffer. (a) VCR arrival rate depedat ad (b) VCR arrival rate idepedet. Fig. 8. Percetage of system cost saved by Scheme 2 as compared to pure I/O ad pure buffer for a movie of legth of 90 mi. (a) VCR arrival rate depedet ad (b) VCR arrival rate idepedet. Fig. 9. Percetage of system cost saved by Scheme 2 over pure I/O ad pure buffer for a movie of legth of 20 mi. (a) VCR arrival rate depedet ad (b) VCR arrival rate idepedet.

21 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE... 2 Fig. 20. Total system cost for differet values of k. (a) VCR arrival rate ˆ 5=mi. (b) VCR arrival rate ˆ 00=mi. (c) VCR arrival rate ˆ 200=mi. (d) VCR arrival rate ˆ 400=mi. the rage of VCR arrival rates to be far less that 200=mi. I this regio, we ca have a substatial reductio i the cost of resources by usig Scheme 2. I additio, the percetage saved as compared to the pure I/O method is greater tha the percetage saved as compared to the pure buffer method, which meas that the cost of usig pure buffer is lower tha that of usig pure I/O. This is maily due to the fact that the cost of the pure I/O method will deped o the arrival rate of ormal playback requests, whereas that of pure buffer is fixed for ay arrival rate. As both the pure I/O ad the pure buffer method require the same amout of VCR resources (sice they have the same v i the depedet case), the mai differece lies oly i the cost of ormal playback resources. Therefore, the pure I/O method will have a lower cost tha the pure buffer method for smaller values of arrival rates Experimet 6 (Differet Values of Threshold Buffer k) As metioed before, the use of additioal buffers ca further facilitate the joiig of the falle out viewer with existig partitios. I this experimet, we plot the overall system cost agaist differet values of k for differet arrival rates, as depicted i Fig. 20. For differet values of k, we determie the correspodig values of hold ad calculate the lowest cost of supportig such a system. We plot the total cost of the system with differet values of k by varyig the VCR arrival rate. I this figure, we ca see that the miimum poit of the cost curve occurs at differet values of k for differet values of VCR arrival rates. For smaller VCR arrival rates, the miimum cost occurs at the smallest k. Whe VCR arrival rate icreases, the miimum cost shifts to larger values of k. For large VCR arrival rates, the miimum cost occurs at the largest k. This meas that the use of more buffer space is beeficial whe the VCR arrival rate is high. The use of buffer space has reduced the time for the viewers to joi a existig partitio, which implies that fewer I/O streams are eeded for servicig VCR requests. For a system with very high VCR arrival rates, we ca aticipate that a large amout of I/O streams are eeded for hadlig VCR requests. Although the use of additioal buffer space will impose more cost o the VCR resources, this cost ca be compesated by the reductio i I/O streams. This reductio becomes very sigificat whe the VCR arrival rate is high. O the other had, if the VCR arrival rate is small, oly a small amout of additioal I/O streams are eeded. Thus, the use of additioal buffer space caot be compesated for by the cost of the I/O streams saved. Therefore, for systems with small VCR arrival rates, usig oly I/O streams for servicig VCR requests ca yield a lower cost Experimet 7: Differet Values of Memory-I/O Cost Ratio, '. I the previous experimets, we have assumed that the cost of memory buffers per miute of a movie is approximately 0 times that of oe I/O stream eeded to retrieve that movie from the disk subsystem. However, as techological chages occur, the cost of memory may decrease faster (or slower) tha the cost of I/O badwidth; thus, the miimum cost might occur at other positios o the cost curve, as illustrated i Fig. 2. I this experimet, we vary ' as follows: 3; 8; 0, ad. Figs. 2a ad 2b illustrate the situatio where the cost of the memory decreases faster tha the cost of I/O badwidth while Fig. 2d illustrates the

22 22 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 200 Fig. 2. Overall system cost versus the umber of I/O streams for differet values of '. (a) ' ˆ 3, (b) ' ˆ 8, (c) ' ˆ 0, ad (d) ' ˆ. situatio where the cost of I/O badwidth decreases faster tha the cost of memory. As we ca see from the graph, the miimum cost teds to shift to the right as ' icreases, i.e., more I/O streams should be used istead of memory buffers for large values of ', but more memory buffers should be used for smaller values of '. Smaller values of ' idicate that the use of more buffer space ca icrease the probability of a hit ad, cosequetly, causes a reductio i the amout of VCR resources to be reserved. Recall that, i ormal playback, we are usig w miutes of buffer space as a trade-off for oe I/O stream, therefore, the decrease of ' simply meas that we ca trade-off oe I/O stream with a eve lower cost. Therefore, i choosig a optimal system cofiguratio, our model aids system desigers i makig proper system sizig decisios ad, at the same time, allows them to meet the performace requiremets. 7 CONCLUSIONS Determiig the proper amout of resources to be allocated is crucial for optimizig the performace of VOD systems so as to maitai the beefits of data sharig techiques. I this paper, we choose specific combiatios of data sharig techiques (e.g, batchig with static partitioig) which result i a reasoably represetative of data sharig scheme. We the preset our mathematical model so as to illustrate how we ca perform system sizig. The mathematical model we preset ca help desigers to perform efficiet resource maagemet ad system sizig for a large scale VOD system. Although the amout of resources eeded will also deped o the arrival rate of ormal playback requests ad the arrival rate of VCR requests, our mathematical model is geeral eough so that it ca be applied i solvig the system sizig problem. The amout of resources calculated esures that the service to be provided satisfies a give performace level, such as esurig that the maximum waitig time for startig or resumig the movie is withi the viewers' tolerace level. Furthermore, we use our model i determiig the optimal distributio of resources for ormal playback ad servicig of VCR requests, so as to reduce the overall system cost. We show, through experimets, that differet arrival rates of VCR requests will affect the overall costs ad the distributio of resources for servicig ormal playback requests ad VCR requests. Usig the mathematical model we proposed, system desigers ca make proper system sizig ad resource distributio decisios. Our future research work icludes extedig the proposed mathematical model to ecompass other data sharig ad VCR fuctioality schemes. For istace, oe promisig directio seems to be the possibility of icludig, i the model, schemes which utilize cliet-side resources to improve overall system performace, e.g., such as bufferig of video data at the cliet's set-top box for provisio of VCR fuctios (as proposed i [2]). ACKNOWLEDGMENTS The research work of Joh C.S. Lui was supported i part by the RGC Earmarked Grat ad the CUHK Direct Grat. The research work of Leaa Golubchik was supported i part by the US Natioal Sciece Foudatio Grat CCR ad the ANI grat. REFERENCES [] C. Aggarwal, J. Wolf, ad P. Yu, ªOptimal Piggyback Policies for Video-o-Demad Systems,º Proc. ACM SIGMETRICS Cof., pp , May 996. [2] K.C. Almeroth ad M.H. Ammar, ªThe Use of Multicast Delivery to Provide a Scalable ad Iteractive Video-o-Demad Service,º IEEE J. Selected Areas i Comm., vol. 4, o. 6, pp. 0-22, Aug. 996.

23 LEUNG ET AL.: USE OF ANALYTICAL PERFORMANCE MODELS FOR SYSTEM SIZING AND RESOURCE ALLOCATION IN INTERACTIVE [3] M-S. Che, D.D. Kadlur, ad P.S. Yu, ªStorage ad Retrieval Methods to Support Fully Iteractive Playout i a Disk-Array- Based Video Server,º Multimedia Systems, pp , 995. [4] A. Da, D.M. Dias, R. Mukherjee, D. Sitaram, ad R. Tewari, ªBufferig ad Cachig i Large-Scale Video Servers,º Proc. IEEE CompCo, 995. [5] A. Da, D. Sitaram, ªBuffer Maagemet Policy for a O- Demad Video Server,º IBM Research Report RC 9347, Ja [6] A. Da, D. Sitaram, ad P. Shahabuddi, ªSchedulig Policies for a O-Demad Video Server with Batchig,º Proc. ACM Multimedia Cof., pp , 994. [7] E. de Souza e Silva, H.R. Gail, L. Golubchik, ad J.C.S. Lui, ªAalytical Models for Mixed Workload Multimedia Storage Servers,º J. Performace Evaluatio, vols , os. -4, pp. 85-2, 999. [8] D. Deloddere, W. Verbiest, ad H. Verhille, ªIteractive Video o Demad,º IEEE Comm. Magazie, vol. 32, o. 5, pp , 994. [9] S. Ghadeharizadeh ad R.R. Mutz, ªDesig ad Implemetatio of Scalable Cotiuous Media Servers,º special issue of Parallel Computig J. Parallel Data Servers ad Applicatios, Ja [0] L. Golubchik, J.C.S. Lui, ad R. Mutz, ªAdaptive Piggyback: A Novel Techique for Data sharig i Video-O-Demad Storage Servers,º ACM J. Multimedia Systems, vol. 4, o. 3, pp , Jue 996. [] L. Golubchik, J.C.S. Lui, ad M. Papadopouli, ªDesigig Efficiet Fault Tolerat VOD Storage Server: Techiques, Aalysis ad, Comparisos,º J. Parallel Computig, vol. 24, o., pp , 998. [2] M. Kamath, K. Ramamritham, ad D. Towsley, ªBuffer Maagemet for Cotiuous Media Sharig i Multimedia Database Systems,º techical report, Uiv. of Massachusetts, Feb [3] K.D. Jayata, J.D. Salehi, J.F. Kurose, ad D. Towsley, ªProvidig VCR Capabilities i Large-Scale Video Servers,º Proc. ACM It'l Cof. Multimedia, pp , Oct [4] S.W. Lau ad J.C.S. Lui, ªSchedulig ad Data Layout Policies for a Near-Lie Multimedia Storage Architecture,º The ACM J. Multimedia Systems, vol. 5, o. 5, pp , Sept [5] K.W. Law, J.C.S. Lui, ad L. Golubchik, ªEfficiet Support for Iteractive Services i Multi-Resolutio VOD Systems,º The Very Large Data Base J., vol. 8, o. 2, pp , 999. [6] M.Y.Y. Leug, ªSystem Sizig ad Resource Allocatio for Videoo-Demad Systems,º master's thesis, The Chiese Uiversity of Hog Kog, Jue 997. [7] M.Y.Y. Leug, J.C.S. Lui, ad L. Golubchik, ªBuffer ad I/O Resource Pre-Allocatio for Implemetig Batchig ad Bufferig Techiques for Video-o-Demad Systems,º Proc. It'l Cof. Data Eg., Apr [8] W. Liao ad V.O.K. Li, ªThe Split ad Merge Protocol for Iteractive Video-o-Demad,º IEEE Multimedia, vol. 4, o. 4, pp. 5-62, Oct [9] P.W.K. Lie, J.C.S. Lui, ad L. Golubchik, ªThreshold-Based Dyamic Replicatio i Large-Scale Video-o-Demad Systems,º J. Multimedia Tools ad Applicatios, vol., o., May [20] D.J. Makaroff ad R.T. Ng, ªBuffer Sharig Schemes for Cotiuous-Media Systems,º Techical Report TR-95-0, Dept. Computer Sciece, Uiv. of British Columbia, 995. [2] R.T. Ng ad J. Yag, ªMaximizig Buffer ad Disk Utilizatios for News o-demad,º Proc. 20th Very Large Data Base Cof., pp , 994. [22] D. Rotem ad J.L. Zhao, ªBuffer Maagemet for Video Database Systems,º Proc. It'l Cof. Data Eg., pp , Mar [23] W. Shi ad S. Ghadeharizadeh, ªBuffer Sharig i Video-o- Demad Servers,º ACM Sigmetrics Bull., 997. [24] K.L. Wu ad P.S. Yu, ªCosumptio-Based Buffer Maagemet for Maximizig System Through Puts of News-o-Demad Multimedia Systems,º IBM Research Report RC 2043, Ja [25] P.S. Yu, J.L. Wolf, ad H. Shachai, ªDesig ad Aalysis of A Look-Ahead Schedulig Scheme to Support Pause-Resume for Video-o-Demad Applicatios,º ACM J. Multimedia Systems, vol. 3, o. 4, pp , 995. Mary Y.Y. Leug received the bachelor's degree (with hoors) i computer sciece ad the MPhil degree i computer sciece from the Chiese Uiversity of Hog Kog i 995 ad 997, respectively. Sice the, she has bee workig with Aderse Cosultig o various projects, icludig the digital televisio ad video-o-demad i Lodo from March 998 to July Curretly, she is workig o the iteractive televisio project i Sydey. Joh C.S. Lui received PhD degree i computer sciece from the Uiversity of Califoria at Los Ageles. Durig the summer of 990, he participated i a parallel database project i the IBM Thomas J. Watso Research Ceter. After his graduatio, he joied a team at the IBM Almade Research Laboratory/Sa Jose Laboratory ad participated i the research ad developmet of a parallel I/O architecture ad file system project. He later joied the Departmet of Computer Sciece ad Egieerig at the Chiese Uiversity of Hog Kog. His curret research iterests are i commuicatio etworks, distributed multimedia systems, OS desig issues, parallel I/O ad storage architectures, ad performace evaluatio theory. His persoal iterests iclude geeral readig ad movies. He is a member of the IEEE. Leaa Golubchik received the BS, the MS, ad the PhD degrees from the Computer Sciece Departmet at the Uiversity of Califoria at Los Ageles i 989, 992, ad 995, respectively. She is curretly a assistat professor i the Departmet of Computer Sciece at the Uiversity of Marylad at College Park. From the Fall of 995 util the Summer of 997, she was a assistat professor i the Departmet of Computer Sciece at Columbia Uiversity. Her research iterests iclude computer systems modelig ad performace evaluatio, Iteret-based computig, ad multimedia storage systems. She is a guest coeditor for special issues of the Parallel Computig Joural ad the Iteratioal Joural of Itelliget Systems, a program cochair of SIGMETRICS/Performace 200 ad MIS'99, as well as a program committee member of several cofereces, icludig SIG- METRICS, SIGMOD, ICDCS, ICDE, ad PDIS. Leaa has received several awards, icludig the US Natioal Sciece Foudatio (NSF) CAREER award, the IBM Doctoral Fellowship, ad the US NSF Doctoral Fellowship. She is a member of Tau Beta Pi, the ACM, ad the IEEE.