Performane Benhmarks for an Interative Video-on-Demand System. Guo,P.G.Taylor,E.W.M.Wong,S.Chan,M.Zukerman andk.s.tang ARC Speial Researh Centre for Ultra-Broadband Information Networks (CUBIN) Department of Eletrial and Eletroni Engineering, The University of Melbourne, Australia Email: j.guo@ee.mu.oz.au; m.zukerman@ee.mu.oz.au ARC Speial Researh Centre for Ultra-Broadband Information Networks (CUBIN) and Department of Mathematis and Statistis, The University of Melbourne, Australia Email: p.taylor@ms.unimelb.edu.au Department of Eletroni Engineering, City University of Hong Kong, Hong Kong Email: eeshan@ityu.edu.hk; ewong@ee.ityu.edu.hk; kstang@ee.ityu.edu.hk Abstrat Little and Venkatesh onjetured that, for an interative VoD system with a single random trial resoure seletion sheme, the bloking probability of a user s request is minimized when the overall movie traffi load is spread uniformly on eah disk in the system. In this paper, we generalize this onjeture to the situation where there an be repeated random trials or where a least busy fit resoure seletion sheme is used. We support our onjeture with a simulation of a realisti system, and propose a metri following the idea of our onjeture to assess the goodness of movie assignment in the system. I. INTRODUCTION To ompete with the prevalent video rental business, an interative Video-on-Demand (VoD) system is envisaged to provide users with pleasurable on-demand aess to a large variety of movie ontents oupled with full VCR-like interative funtions [4]. For this purpose, a stringent requirement is usually imposed suh that a single video stream is allotted to eah user, so that the same movie an be viewed simultaneously by many users with random time offsets and independent temporal ontrol ativities. Moreover, the experiene of the video rental business reveals a highly skewed demand distribution among various movies. At any given time, some movies are observed to be muh more popular than others. This, in most ases, entails the alloation of multiple replias for these popular movies so as to inrease the number of onurrent streams that an be supported by the system to satisfy the requests for these hot movies. Beause the storage and transmission of digitally enoded movie data generally onsumes enormous system resoures, and the storage apaity as well as the sustained I/O bandwidth of ommerially available disks (or disk arrays, e.g. RAID) are limited, a VoD system is expeted to manage a large farm of on-line disks (or disk arrays) to aommodate the large library of movie files and yield adequate onurrent video streams suh that ertain Grade of Servie (GoS) and reliability requirements an be met. Without loss of generality, we shall heneforth all any substantive storage devie in a VoD system, be it a onventional hard disk or a single RAID disk array, a disk. Due to the long-lived nature of movie onnetions between users and video servers, interative VoD systems are modelled as loss systems [8], [10], [7]. That is, a user request to aess a partiular movie file is bloked if the system finds that all stream apaity on the set of disks, where a opy of the file is stored, is used up. We see that suh a loss system gives rise to an interesting performane optimization problem: Given a number of disks with limited apaity in both storage and I/O bandwidth, and a range of movie files with vast asymmetry of aess demand, the hallenge is to find an optimal assignment of the movie files suh that the steady state bloking probability of the system is minimized while meeting the system resoure onstraints. The importane of file assignment problems, subjet to optimization of various performane goals, is well established (see [5] and referenes therein). In the ontext of interative VoD systems, due to the presene of multiple-opy files and possible implementation of resoure seletion shemes [7], the exat analytial solution for the bloking probability is generally intratable, exept for rare ases of some simple (but ineffiient) resoure seletion shemes [7]. Considering that file assignment problems with expliit performane goals are in their own right NP hard [5], it is therefore extremely unlikely that the performane optimization problem in the field of interative VoD systems an be takled within a reasonable omputing time. Thus, we have to rely on heuristi approahes to searh for near-optimal solutions. To failitate a heuristi searh, it would be useful if a lower bound on the bloking probability of the system ould be estimated a priori. Suh a bound an be a useful performane benhmark if its derivation does not rely on a speifi assignment of the movie files. For a homogeneous interative VoD system (to be desribed in Setion II), Little and Venkatesh [8] 2189 0-7803-8533-0/04/$20.00 ( 2004 IEEE
have onjetured that the bloking probability of the system is minimal when the movie files are alloated suh that eah disk in the system has an equal probability of being aessed. While they did not expliitly state it, we shall see that they were thinking only in the ontext of a simple resoure seletion sheme, under whih a request for a multiple-opy movie file is randomly direted to only one of the disks where a opy of that file is stored, and there is no exhaustive attempt made to find an available disk to satisfy the request. Our important ontribution in this paper is to provide an extension of the onjeture to the situation where more ompliated and effiient resoure seletion shemes are implemented, suh that bloked requests for a multiple-opy movie file an retry at other disks where that partiular file is stored. Let a file that has opies be alled a Type file. Let be the total number of disks in the system. We amend the onjeture by suggesting that the optimal performane of the system may be reahed only if the aggregate traffi of eah type of movie file is evenly loaded on all ( ombination groups of disks. This is the soalled Combination Load Balaning (CLB) algorithm initially proposed in [6]. Although the proof of suh a onjeture is an open researh issue and beyond the sope of this paper, we attempt its justifiation in Setion III and Setion IV by simulations for various resoure seletion shemes. In this paper, we do not onsider the use of striping tehniques to resolve the load balaning problem. This approah has inherent limitations whih undermine its viability in real interative VoD systems [2], [9]. II. THE VOD SYSTEM MODEL Consider an interative VoD system with a set D of disks labelled 1, 2,..., and a set F of M distint movie files marked 1, 2,..., M. Eah disk supports a number of onurrent video streams and stores a range of movie files. Assume that file m has n m opies, and those opies are alloated on n m separate disks, whih onstitutes the set Ω m. For onveniene, we all a file that has opies a Type file. The set of movie files plaed on disk j is denoted Φ j. Further assume that all disks in the system are alike. If the independent video streams emanating from eah disk in the system are approximated to be statistially equivalent, eah disk may serve up to N onurrent logial hannels. Note that the terms stream and hannel are interhangeable in this paper. Assume that the aggregate arrivals of requests for all movie files follow a Poisson proess with rate λ. In a statistial sense, making a request for a movie in a VoD system is similar to making a all in telephony, where the Poisson assumption is widely aepted. Sine the insensitivity of bloking probability to the holding time is ommon in many loss systems, we assume in our ontext that the onnetion times of movie files are exponentially distributed with mean 1/µ. Without loss of generality, the mean onnetion time of movie files is normalized to 1. Assume that the request arrival proesses of different movie files are mutually independent Poisson proesses. The demand rate for file m reates its popularity profile p m, defined as the likelihood of file m being requested by a user, and M m=1 p m = 1. The popularity profiles of the movie files in the system are updated on a daily basis to apture the variability of users demand rate [8]. Therefore, in a single day, the request arrival rate of file m is given by λp m, for m = 1, 2,..., M. The aggregate rate of requests for all Type files is then obtained by λˆp, where ˆp = m F,n p m= m. Let the popularity profile p m of movie file m follow a Zipflike distribution, so that m ζ p m = M, (1) k=1 k ζ for m =1, 2,..., M, where the parameter ζ determines the skewness of the distribution. Though we do not restrit the popularity distribution of the movie files in the VoD system to be governed by (1), it was found that suh a distribution with ζ = 0.271 statistially mathes the lient aess frequenies to various movies observed from the video store rental data [3]. III. DISK LOAD BALANCING AND COMBINATION LOAD BALANCING For a homogeneous interative VoD system desribed in the previous setion, Little and Venkatesh made the following onjeture ([8], page 284): The probability that a ustomer s request results in a suessful onnetion is maximal when the movies are distributed suh that eah disk has a uniform probability of being aessed. While they did not expliitly state it, we shall see that they were thinking only in the ontext of a simple resoure seletion sheme, under whih a request for a multiple-opy movie file is randomly direted to only one of the disks where a opy of that file is stored, and there is no exhaustive attempt made to find an available disk to satisfy the request. In situations where other more ompliated resoure seletion shemes are implemented, their onjeture may lead to an underestimation of the optimal performane that the system an reah. For ease of exposition, we support our argument by looking into the performane of a small VoD system with four disks and 12 distint movie files. Eah of the four disks may provide up to 10 onurrent streams. All the 12 files are Type 2 files. Their popularity profiles are produed by (1) with ζ =0.271, and listed in Table I. We set the overall request rate λ = 24. Among many feasible assignments of these 24 files, we onsider three partiular ases as shown in Fig. 1. To differentiate and ompare the level of disk load balaning among these three ases, we define a Disk Load Balaning Index (DLBI), given by DLBI = 1 ( j D m Φ j p m 1 ) 2. n m This formula is a measure of the spread of the traffi distribution among the disks in the set D. After routine omputation, 2190 0-7803-8533-0/04/$20.00 ( 2004 IEEE
(a) Case 1 (b) Case 2 ( Case 3 Fig. 1. Assignment of movie files in the 4-disk VoD system. we find that in both Case 1 and Case 2, DLBI = 0 (i.e., the load is uniformly distributed among the disks in D), while in Case 3, DLBI = 0.0088, whih is not as balaned as in the other two ases. Based on the idea of CLB from [6], we further make up an ideal ase, where we assume that the traffi of all Type files an be evenly distributed among ( ombination groups of disks enumerated in the set D. We then ondut a disrete event simulation to evaluate the bloking probability of the system. When a user request arrives, say for movie file m, we first find out the number of opies n m of file m in the system. If file m is singular (i.e. a single-opy file), we straightforwardly diret the request to its arrying disk provided the disk is available. If file m has multiple replias, we perform a resoure seletion sheme to selet an available disk in the set Ω m to satisfy the request. To be onsistent with [7], we onsider the following three resoure seletion shemes: Single Random Trial (SRT): When a request for movie file m arrives, one of the disks in the set Ω m is hosen at random. If all the possible hannels provided by the seleted disk are busy, the request will be bloked. No further attempt to retrieve a replia of file m from other disks in Ω m is made. Repeated Random Trials (RRT): Under this sheme, the first step is as in SRT. However, if all the possible hannels provided by the seleted disk are busy, we ontinue with repeated random trials among all remaining disks in the set Ω m until we are suessful. If after all disks in the set Ω m have been attempted and we fail to find an available hannel on any of them, the request is bloked. Least Busy Fit (LBF): Based on the latest statistis of all disks in the set Ω m, if all possible hannels are used up, the request is bloked. Otherwise, we diret the request to the least busy disk in Ω m, or the disk with the maximal number of available hannels. In the ase where there are more than one least busy disk, the request will be randomly dispathed to one of them. In the ideal CLB ase, the above three resoure seletion shemes are slightly varied. When a request for a partiular movie file arrives, we merely look at what type of file it is. If it is a Type file, we then find out on whih ombination group of disks the file is stored. Sine in the ase of CLB, we do not require any speifi assignment information of eah individual Type file, but assume that the traffi of all Type files an be evenly distributed among ( ombination groups of disks, eah ombination group therefore has equal likelihood of being aessed. For the purpose of simulation, instead of maintaining a umbersome list of ( ombination groups and then randomly hoosing one of them upon the request for a Type file, we use an equivalent (but more effiient) way of randomly seleting disks out of the set D, and then proeed with the orresponding resoure seletion sheme to proess the request. In a typial run of our simulation test, eah of the fifty million random events represents either an arrival of a movie file request or a termination of a movie file onnetion. We obtain the overall request bloking probability by dividing the total number of request losses by the total number of request arrivals. To guarantee the onfidene in our simulation estimates, we repeat the simulation test with six independent runs, and we keep the radii of the 95% onfidene intervals ([1], page 273) within 1% of the average of the results measured. We see in Table II that, in all ases when either RRT or LBF is implemented, the simulation outome is apparently in ontradition with what Little and Venkatesh have asserted in their onjeture. Although the disk loading is perfetly balaned in both Case 1 and Case 2, requests for movie files experiene muh less bloking probability in the latter. Moreover, there is an even smaller bloking probability in Case 3 despite its imbalaned traffi distribution. Note that, even though LBF is in general more effiient than RRT, they produe the same bloking probability in Case 1. This is beause the assignment of movie files in Case 1 onstitutes a standard Non-overlapping Load Balaning (NLB) ase that is not sensitive to any exhaustive resoure seletion sheme at all [6]. Interestingly, however, in the irumstanes of SRT, the simulation agrees with what the onjeture of [8] has predited. In an equivalent manner to the way we defined DLBI, we will now propose what we all the Combination Load Balaning Index (CLBI). CLBI is a weighted index over the various file types in the system defined by CLBI = ˆp CLBI ˆp, where CLBI = ( 1 s Υ(D, [ m F,Ω m=s p m ˆp ( ] 2, where Υ(D, is the set of all ombinations of hoosing disks out of the set D, and s Υ(D, is one suh possible ombination. CLBI omputes, in an atual movie file assignment and for Type files, how far the traffi distribution, among the ( ombination groups of disks enumerated in the set D, deviates from the ideal CLB ase. After working out the load of the traffi aumulated on eah of the six ombination groups from the requests for the 12 Type 2 files, as shown in Table III, we see that CLBI = 0.2357 for Case 1, 0.1182 for Case 2, and 0.0134 for Case 3. These 2191 0-7803-8533-0/04/$20.00 ( 2004 IEEE
results learly explain our interesting observations in Table II, and we so arrive at our onjeture of the ondition on the assignment of movie files suh that the optimal performane of the system an be reahed. Conjeture 1: The bloking probability of a user s request is minimized when, for eah, the traffi wishing to aess movie files of Type is uniformly distributed among all ( ) ombination groups enumerated in the set D. The size of a typial interative VoD system that provides on-demand aess to hundreds of distint movies is usually of the order of dozens of disks. In the next setion, we shall justify our onjeture by onsidering a system of this size. IV. PERFORMANCE BENCHMARK USTIFICATION We report the simulation estimates for the bloking probability of a realisti interative VoD system of 20 disks and 200 distint movie files. The radii of the 95% onfidene intervals are again kept below 1% of the average of the measured results. All disks in the system have the same stream apaity of size 30. The popularity distribution of the 200 movie files is again produed by (1) with ζ =0.271. In our example, we speifially alloate four opies for eah of the first three movie files in the system, three opies for files 4 to 25, two opies for files 26 to 50, and one single opy for the remaining 150 movie files. Our ongoing study of optimizing the number of opies for eah movie file and their plaement in the VoD system will be reported elsewhere. Two different assignments of the 200 movie files as well as their multiple replias are speified in Fig. 2(a) for Case 1 and in Fig. 2(b) for Case 2. The results of the simulation are given in Table IV. We intentionally hoose these two ases beause both of them give an index value of 0.00050 for DLBI. However, after an elaborate omputation, it turns out that CLBI equals 0.00352 in Case 1 and 0.00348 in Case 2. This again explains why in Table IV the bloking probabilities experiened by the system are higher in Case 1, when either RRT or LBF is implemented. It is partiularly true for LBF where bloking probabilities in the two ases may differ by up to 5.8% in spite of the small CLBI disrepany. Interestingly, the system one more sees the same bloking probabilities in both ases when SRT takes over. Nevertheless, in all ases, CLB learly justifies its role of being the ondition on the assignment of movie files that the system may reah its performane benhmark. Although the differene in the bloking probability in the two examples might reasonably be expeted to follow from the differene in CLBI, we have not proved that the bloking probability inreases with inreasing CLBI. Suh a result would be appealing, but is beyond the sope of this paper. It is interesting to observe that, the ideal CLB ase (i.e. CLBI = 0) is less likely obtainable in pratie, sine in real VoD systems, the number of Type files is usually muh fewer than the number of ombination groups ( enumerated in the set D. Nevertheless, we see that, suh an ideal ase an be muh useful in effetively estimating a lower bound on the number of disks needed for the system, when a speifi resoure seletion sheme is implemented, to satisfy ertain predefined performane threshold. Relevant results for the 200-file VoD system subjet to a 1% GoS requirement are presented in Table V. V. MAIN CONCLUSIONS For a homogeneous interative VoD system, a previous onjeture made by Little and Venkatesh in [8] asserts that the bloking probability of the system is minimal when the movie files are distributed suh that eah disk has an equal probability of being aessed. While they did not expliitly state it, it is lear that they were thinking only in the ontext of a simple SRT resoure seletion sheme. In this ontext, our work has supported their onjeture. However, in situations where other more ompliated resoure seletion shemes (like RRT and LBF) are implemented, we have observed that the onjeture of [8] may lead to an underestimation of the optimal performane that the system an reah. Based on our previously proposed CLB algorithm, we have extended the onjeture by suggesting that the performane benhmark of suh a VoD system may only be reahed when the traffi wishing to aess movie files of the same type is uniformly distributed among all ombination groups of disks enumerated in the system for the assoiated file type. Although the ideal CLB ase is less likely obtainable in pratie, we have shown that it an fill the role of effetively estimating the lower bound on disk onsumption in the VoD system subjet to ertain predefined performane threshold. ACKNOWLEDGMENTS The work desribed in this paper was partially supported by grants from City University of Hong Kong (Projet No. 7001224 and Projet No. 7001458) and partially supported by the Australian Researh Counil. It was done while. Guo and M. Zukerman were visiting the Department of Eletroni Engineering, City University of Hong Kong. REFERENCES [1] S. K. Bose, An Introdution to Queueing Systems, Kluwer Aademi/Plenum Publishers, New York, 2002. [2] C. F. Chou, L. Golubhik, and. C. S. Lui, Striping doesn t sale: how to ahieve salability for ontinuous media servers with repliation, in Pro. 20th Int. Conf. Distributed Computing Systems, Taipei, Taiwan, Apr 2000, pp. 64 71. [3] A. Dan, D. Sitaram, and P. Shahabuddin, Sheduling poliies for an on-demand video server with bathing, in Pro. 2nd ACM Int. Conf. Multimedia, San Franiso, CA, USA, 1994, pp. 15 23. [4] D. Deloddere, W. Verbiest, and H. Verhille, Interative video on demand, IEEE Commun. Mag., vol. 32, no. 5, pp. 82 88, May 1994. [5] Lawrene W. Dowdy and Derrell V. Foster, Comparative models of the file assignment problem, ACM Computing Surveys, vol. 14, no. 2, pp. 287 313, 1982. [6]. Guo, P. G. Taylor, M. Zukerman, S. Chan, K. S. Tang, and E. W. M. Wong, On the effiient use of video-on-demand storage faility, in Pro. IEEE ICME 03, Baltimore, MD, USA, ul 2003, vol. 2, pp. 329 332. [7]. Guo, S. Chan, E. W. M. Wong, M. Zukerman, P. G. Taylor, and K. S. Tang, On bloking probability evaluation for video-on-demand systems, in Pro. ITC 18, Berlin, Germany, Sep 2003, vol. 5a, pp. 211 220. [8] T. D. C. Little and D. Venkatesh, Popularity-based assignment of movies to storage devies in a video-on-demand system, Multimedia Syst., vol. 2, pp. 280 287, an 1995. 2192 0-7803-8533-0/04/$20.00 ( 2004 IEEE
(a) Case 1 (b) Case 2 Fig. 2. Assignment of movie files in the 20-disk VoD system. [9] M. Reisslein, K. W. Ross, and S. Shrestha, Striping for interative video: is it worth it?, in Pro. IEEE Int. Conf. Multimedia Computing and Systems, Florene, Italy, un 1999, vol. 2, pp. 635 640. [10] K. S. Tang, K. T. Ko, S. Chan, and E. Wong, Optimal file plaement in VOD system using geneti algorithm, IEEE Trans. Ind. Eletron., vol. 48, no. 5, pp. 891 897, Ot 2001. TABLE I MOVIE POPULARITY DISTRIBUTION IN THE 4-DISK VOD SYSTEM File ID Popularity File ID Popularity 1 0.128 7 0.076 2 0.106 8 0.073 3 0.095 9 0.071 4 0.088 10 0.069 5 0.083 11 0.067 6 0.079 12 0.065 TABLE II BLOCKING PROBABILITY IN THE 4-DISK VOD SYSTEM SRT RRT LBF Case 1 0.04318 0.00980 0.00980 Case 2 0.04319 0.00603 0.00243 Case 3 0.04370 0.00526 0.00206 CLB 0.04317 0.00516 0.00204 TABLE III COMBINATION GROUPING TRAFFIC IN THE 4-DISK VOD SYSTEM Combination Group Case 1 Case 2 Case 3 (D1, D2) 0.5 0 0.154 (D1, D3) 0 0.238 0.193 (D1, D4) 0 0.262 0.155 (D2, D3) 0 0.262 0.161 (D2, D4) 0 0.238 0.164 (D3, D4) 0.5 0 0.173 TABLE IV BENCHMARK USTIFICATION IN THE 20-DISK VOD SYSTEM λ = 440 λ = 470 λ = 500 Case 1 0.02056 0.03455 0.05265 SRT Case 2 0.02056 0.03455 0.05265 CLB 0.02055 0.03454 0.05263 Case 1 0.01427 0.02489 0.03957 RRT Case 2 0.01426 0.02489 0.03955 CLB 0.01417 0.02478 0.03935 Case 1 0.00273 0.00751 0.01744 LBF Case 2 0.00258 0.00711 0.01669 CLB 0.00185 0.00572 0.01459 TABLE V LOWER BOUND ON DISK CONSUMPTION OF THE 200 MOVIE FILES λ = 440 λ = 470 λ = 500 CLB-SRT 22 24 25 CLB-RRT 21 23 24 CLB-LBF 18 20 21 2193 0-7803-8533-0/04/$20.00 ( 2004 IEEE