Performance Analysis of Resource Selection Schemes for a Large Scale Video-on-demand System

Transcription

1 Performance Analysis of Resorce Selection Schemes for a Large Scale Video-on-demand System Jn Go, Member, IEEE, Eric W. M. Wong, Senior Member, IEEE, Sammy Chan, Member, IEEE, Peter Taylor, Moshe Zkerman, Fellow, IEEE, and Kit-Sang Tang, Member, IEEE Abstract The designers of a large scale video-on-demand system face an optimization problem of deciding how to assign movies to mltiple disks (servers) sch that the reqest blocking probability is minimized sbect to capacity constraints. To solve this problem, it is essential to develop scalable and accrate analytical means to evalate the blocking performance of the system for a given file assignment. The performance analysis is made more complicated by the fact that the reqest blocking probability depends also on how disks are selected to serve ser reqests for mlti-copy movies. In this paper, we analyze several efficient resorce selection schemes. Nmerical reslts demonstrate that or analysis is scalable and sfficiently accrate to spport the task of file assignment optimization in sch a system. Index Terms Blocking probability, fixed-point approximation, resorce selection, video-on-demand. I. INTRODUCTION Research test-beds of video-on-demand (VOD) services [] have been prevalent for many years [2]. There have been several service providers who have attempted to capitalize on this commercial opportnity. To compete with the traditional video rental bsiness, it is important for a VOD system to provide a large poplation of end sers with pleasrable ondemand access to a large variety of movie contents copled with fll VCR-like interactive capabilities [3]. De to the large storage space and I/O bandwidth reqired in storing and delivering movie contents, a large scale VOD system needs to manage a large clster of on-line disks to store the large nmber of movies. This paper was presented in part at the 8th International Teletraffic Congress, Berlin, September The work described in this paper was partially spported by grants from City University of Hong Kong (Proect No and Proect No ) and partially spported by the Astralian Research Concil (ARC). J. Go is with the School of Compter Science and Engineering, The University of New Soth Wales, Astralia ( go@cse.nsw.ed.a). M. Zkerman is with the ARC Special Research Centre for Ultra- Broadband Information Networks (CUBIN), Department of Electrical and Electronic Engineering, The University of Melborne, Astralia ( m.zkerman@ee.nimelb.ed.a). P. Taylor is with CUBIN, the ARC Centre of Excellence for the Mathematics and Statistics of Complex Systems (MASCOS) and the Department of Mathematics and Statistics, The University of Melborne, Astralia ( P.Taylor@ms.nimelb.ed.a). E. W. M. Wong, S. Chan and K. S. Tang are with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China ( {eeewong, eeschan, eekstang}@city.ed.hk). Part of the work presented in this paper was done while J. Go and M. Zkerman were visiting the Department of Electronic Engineering, City University of Hong Kong. Given the significant asymmetry in access demand and file-size of different movies, and de to the limitation of striping techniqes [4], a certain nmber of movies with high access demand or of small file-size are sally replicated over mltiple disks, or logical disks [4] if striping is sed, in the VOD system. In this way, stream resorces from mltiple disks (servers) can be managed to serve ser reqests for poplar movies, while the disk storage space is more efficiently tilized. Since a ser reqest for a movie can only be connected to a disk where a file-copy of the movie is stored, the reqest blocking probability essentially depends on how disks are selected by the VOD schedler to serve ser reqests for mlti-copy movies [5] and how movies are assigned to disks [6]. The designers of a large scale VOD system ths face an optimization problem of deciding how to assigovies to mltiple disks so that the reqest blocking probability is minimized sbect to capacity constraints. To solve this problem, it is essential to develop scalable and accrate analytical means to evalate the reqest blocking probability of a given file assignment. A. Related work For the sake of a manageable performance analysis, Little and Venkatesh [6] considered what we call a single random trial (SRT) resorce selection scheme. Following SRT, when a ser reqest for a mlti-copy movie arrives, the VOD schedler randomly selects one of the disks storing a file-copy of the reqested movie. If the disk is flly bsy (i.e. the stream capacity of the disk is sed p), the reqest is simply blocked withot frther attempting any other disk that keeps a filecopy of the reqested movie. SRT is a very simple scheme for which exact blocking probability reslts are easily obtained. Specifically, with the assmption of Poisson reqest arrivals (indicated by M), general distribtion of channel holding times (identified by G), and letting k denote the disk stream capacity, each disk is conveniently modelled as an M/G/k/k qeeing system sing Kendall s notation [7]. SRT is inherently inefficient in tilizing system resorces given the existence of mlti-copy movies. Two efficient resorce selection schemes of reasonable operating cost were stdied by simlation in [5]. In both schemes, a ser reqest is blocked if and only if all disks (in an exhastive sense) storing the reqested movie file are fond to be flly bsy. The

2 2 repeated random trials (RRT) scheme is a natral extension of SRT where the VOD schedler contines with repeated random trials ntil all the disks are attempted. If it is feasible to monitor channel occpancies, a more efficient least bsy fit (LBF) scheme can be implemented so that a ser reqest for a mlti-copy movie is always directed to the least bsy disk where a file-copy of the reqested movie is placed. Withot mch concern for cost and complexity of real-time stream schedling in a large scale VOD system, Tsao et al. [8] considered a stream repacking scheme which reschedles the crrently playing streams among disks along a migration path, if this allows an extra ser reqest (potentially blocked by LBF) to be accepted. Variations of the stream repacking scheme were also proposed to be initiated at earlier stages to achieve dynamic load balancing [9], [0]. Althogh in theory the disk resorces can be more efficiently tilized by means of stream repacking, it is not clear how realistically these complicated schemes can be implemented in a large scale VOD system. B. Or contribtion In this paper, we shall demonstrate by nmerical reslts that a file assignment that achieves a smaller blocking probability in the SRT system does not necessarily lead to a smaller blocking probability in systems where exhastive resorce selection schemes are sed. Accordingly, an optimal file assignment soltion established in the SRT system is likely to ndertilize system resorces if we shall test it in systems where exhastive resorce selection schemes are sed. This stifies the importance of or analysis in this paper to spport file assignment optimization in a more realistic VOD system sing these more efficient resorce selection schemes. The maor contribtion of this paper is to provide an analytical framework for analysis of a large scale VOD system sing exhastive resorce selection schemes. We derive scalable and accrate analytical formlas to evalate the reqest blocking probability of a given file assignment. Considering the significant performance gain obtainable de to file assignment optimization [6], or analysis in this paper is key to the design and dimensioning of a large scale VOD system. Specifically, the analytical reslts mst be accrate enogh to differentiate the qality of varios file assignments in terms of reqest blocking probability. In addition, they mst be fast enogh to be embedded in the file assignment optimizatioodle to find ot the optimal or near-optimal blocking probability of the system for a given configration of disks [], [2]. The optimizatioodle can then be embedded in the dimensioning process to find ot the minimm nmber of disks needed for the system to meet the grade of service reqirement. With the complicated interactions between ser reqests for mlti-copy movies and selections of disk resorces to serve these reqests, the entire system is modelled as a mltidimensional Markov chain with the dimension eqal to the nmber of disks. Althogh this chain can be solved theoretically, brte force soltions to the state eqations are comptationally infeasible for a large scale VOD system de to the crse of dimensionality. We investigate in this paper if the fixed-point approximation (FPA) method [3], [4] can be applied to a large scale VOD system and if the approximation has reasonable accracy and comptational efficiency. The se of FPA has been widespread in the literatre of telecommnications modelling [5], [6]. Specifically, it has been applied to the analysis of large circit-switched networks with random alternate roting and with state dependent roting (see e.g. [7] [20]). We shall see that in the context of a VOD system with the existence of mlti-copy movies, FPA works by decopling the whole system of J disks into J independent sbsystems and treating each sch sbsystem as an M/G/k/k qeeing system. This is done by making the assmptions that ) the occpancy processes at the different disks are statistically independent, and 2) the reqest arrival process seen by each disk is Poisson. Since the blocking probability of an M/G/k/k system is insensitive to the distribtion of channel holding times [2] and depends only on its mean, each disk can then be conveniently modelled as an M/M/k/k qeeing system with exponentially distribted service times. An additional contribtion of this paper is to propose an efficient variation, namely global random trial (GRT), of RRT. The implementation of GRT reqires that the VOD schedler collect the information of flly bsy or not flly bsy from a disk at each time epoch when the disk changes between the two states. Taking advantage of sch state information, GRT handles the reqest for a mlti-copy movie by randomly selecting one disk from those that store a file-copy of the reqested movie and that are not flly bsy. Clearly, GRT and RRT are eqivalent in terms of blocking probabilities since they both block the reqest if and only if all the servers are bsy. Nevertheless, GRT is more efficient in a sense that it avoids mltiple redndant attempts that may be made by RRT pon the arrival of the reqest, which wold otherwise introdce some additional setp delay for processing the reqest. In this paper, we prove that the FPA soltions for RRT and GRT are eqivalent, so that we can se either of the two analytical soltions in practice, as none of them is more accrate in approximating the exact blocking performance of the VOD system. C. Organization In Section II, we describe the system model. In Section III, we present FPA soltions for RRT, GRT and LBF. We prove that the FPA soltions of RRT and GRT are eqivalent. In Section IV, we give nmerical reslts to demonstrate the sfficient accracy of the approximate soltions in spporting the task of file assignment optimization for the system. Finally, we provide conclding remarks in Section V. II. THE SYSTEM MODEL For the reader s convenience, we provide in Table I a list of maor symbols that we shall define and se in this paper. We consider a large scale VOD system with a set D of J disks, labelled, 2,..., J, and a set F of M distinct movies, marked, 2,..., M. The storage space of disk is C nits. The file-size of movie m is L m nits. (For example, one nit of storage space or file-size cold be one Gbyte.)

3 3 TABLE I SUMMARY OF MAJOR SYMBOLS Symbol Definition D Set of disks in the system J Nmber of disks in the system F Set of distinct movies in the system M Nmber of distinct movies in the system C Storage space of disk N Stream capacity of disk L m File-size of movie m Nmber of file-copies of movie m Ω m Set of disks storing a file-copy of movie m Φ Set of distinct movies placed on disk λ Total reqest arrival rate p m Poplarity of movie m /µ m Mean channel holding time for movie m A. Exact soltion of SRT III. PERFORMANCE ANALYSIS The analysis of SRT in a system of heterogeneos disks was discssed in []. For the prpose of performance comparison with other schemes, we derive it again in this section. Since the reqest arrival process of each movie in F is Poisson, if a reqest for a mlti-copy movie m is randomly directed to one of the disks in Ω m, with no sbseqent possible retrials attempted, the reqest arrival process of movie m is simply decomposed into independent Poisson processes, each of which has rate y (m) = λp m /. For disk, D, the total reqest arrival rate de to the sperposition of the reqest arrival processes of all movie files in Φ is given by y = m Φ y (m). () For all D, we assme that max m F L m C, so that each disk can store a nmber of movie files. We also assme m F L m < D C, so that the system has spare disk storage space to place mltiple file-copies for certaiovies in F. For a feasible file assignment, movie m has copies, J. Those copies are allocated on separate disks, which constitte the set Ω m. The set of movie files placed on disk is denoted Φ, and satisfies the storage space constraint, i.e. m Φ L m C. If the independent video streams emanating from disk are considered to be approximately statistically eqivalent, disk may spport p to N concrrent streams (logical channels). We shall assme that the access link to the J disks has enogh bandwidth capacity sch that it will not impose frther constraint on the nmber of concrrent video streams that can be spported by the J disks. In a statistical sense, making a reqest for a movie in a VOD system is similar to making a call in telephony, where the Poisson assmption is widely accepted. This Poisson assmption was recently stified in [22], where Costa et al. observed that inter-arrival times of ser reqests in streaming mltimedia systems are exponentially distribted. We therefore assme that the aggregate arrivals of reqests for all movies follow a Poisson process with rate λ reqests per time nit. The reqest arrival processes of different movies are mtally independent Poisson processes. The ser holding time of a video stream for movie m is arbitrarily distribted with mean /µ m time nits. The demand rate for movie m creates its poplarity profile p m, defined as the relative probability of movie m being reqested by a ser, and M m= p m =. The reqest arrival rate of movie m is given by λp m. We ignore the setp delay in this paper becase it has negligible effect on the blocking probability. Notice that relative to the large ser holding time of a video stream (e.g. for a two-hor featre length movie), the setp delay for processing a ser reqest in a VOD system is of the order of milliseconds [23]. The mean channel holding time /ˆµ for all movie files in Φ is obtained by = y (m). (2) ˆµ y µ m m Φ Let ξ = ( ) ξ (N), ξ (N2) 2,..., ξ (N J ) J denote the vector of stationary probabilities that disk, D, is in state N, or in other words, it has all N logical channels occpied. In the case of SRT, ξ (N ) is simply given by the Erlang B Formla [7] ξ (N) (3) ( ) def y = E, N = ( y ˆµ ) N /N! ˆµ N i=0 ( y ˆµ ) i /i!. (4) For m F, the reqest blocking probability of movie m is calclated by B m =. Ω m ξ (N) We compte the mean reqest blocking probability of mlticopy movies by m F,n B = p m> mb m m F,n p, (5) m> m and the mean reqest blocking probability of single-copy movies by m F,n ˆB = p m= mb m m F,n p, (6) m= m and the mean reqest blocking probability of all movies from B = m F p m B m. (7)

4 4 B. FPA soltion of RRT If sbseqent random trials are repeated ntil all disks in the set Ω m are attempted, the reqest arrival process of a mlti-copy movie m directed to disk, Ω m, is composed of two processes: ) a Poisson process made of first choice reqests to disk with rate λp m /, and 2) a process made of reqests overflowed from disks in Ω m other than disk, which is normally non-poisson. To ease the bookkeeping of these overflowed reqests, we define Ψ(Ω m {}, x) as the set of all possible permtations of arranging x disks ot of Ω m {} (all disks in Ω m except disk ), for x =, 2,...,. By S Ψ(Ω m {}, x) and S = {s, s 2,..., s x }, we say that S is one sch possible permtation, and s, s 2,..., s x are the ordered disks enmerated in S. Assme that a reqest for any movie m that has been denied at disk is independent of other reqests for movie m that have been denied at other disks in the set Ω m, the rate of overflowed reqests for movie m originally blocked by disk s, and sbseqently blocked by s 2, s 3,..., s x, and finally offered to disk is calclated by λp m x ξ (N s i ) s i i. i= Taking into accont all the possible permtations from Ψ(Ω m {}, x) and all x, the aggregate rate of overflowed reqests for movie m from disks in Ω m except disk is given by λp m n m x= S Ψ(Ω m {},x) x i= ξ (N s i ) s i i. The reqest arrival rate of disk de to movie m is therefore given by y (m) = λp [ n m x m ξ (Ns i ) ] s + i, (8) i x= S Ψ(Ω m {},x) i= and the total reqest arrival rate of disk de to all movie files in the set Φ is given by (). The mean channel holding time /ˆµ for all movie files in Φ is obtained by (2). FPA works by treating the reqest arrival process seen by disk as if it were Poisson, even if it incldes a de facto non-poisson overflow process. Correspondingly, we allow calclating the blocking probability of disk in the RRT system sing (4). For compatibility, we reqire that the blocking probabilities so calclated be the same as those sed to calclate the redced load in (8). Ths (), (2), (3), (4) and (8) constitte a set of fixed-point eqations of the form ξ = e( ξ ). (9) These fixed-point eqations can often be solved efficiently by the sccessive sbstittioethod to be presented in Section III-E. Solving (9) for ξ (N ) of disk, and sing the fact that for a reqest of movie m to be blocked, it wold need to be denied at all disks in Ω m, we therefore have B m =. (0) Ω m ξ (N) B, ˆB, and B are then compted by (5), (6) and (7) respectively. C. FPA soltion of GRT According to this scheme, when a reqest for a mlti-copy movie m arrives, the VOD schedler filters ot all disks in the set Ω m that are flly bsy, and randomly selects one of the remaining not flly bsy disks to serve the reqest. The implementation of GRT reqires that the VOD schedler collect the information of flly bsy or not flly bsy from a disk at each time epoch when the disk changes between the two states. Provided that disk is in Ω m and has one or more free channels available pon the arrival of the reqest, it is sefl to define Υ(Ω m {}, x) as the set of all possible combinations of choosing x disks ot of Ω m {}. Note that for RRT, Ψ( ) is a set of permtations, while here Υ( ) is a set of combinations. Then, pon the arrival of the reqest for movie m, the probability that, among the other disks in Ω m except disk, h disks also have one or more free channels available, and the remaining h disks are all flly bsy is given by S Υ(Ω m {},h ) ( ) ξ (N) v Ω m {} S ξ (Nv) v. Therefore, the reqest arrival rate of disk de to movie m is given by y (m) = λp m h h= v Ω m {} S S Υ(Ω m {},h ) ξ (Nv) v, ( ) ξ (N ) () and the total reqest arrival rate y of disk de to all movie files in the set Φ is given by (). The mean channel holding time /ˆµ for all movie files in Φ is obtained by (2). The system of eqations (), (2), (3), (4) and () comprises a set of fixed-point eqations, again of the form (9). Once ξ (N) of disk is obtained by solving (9), by the disk independence assmption and sing the fact that for a reqest of movie m to be blocked, all channels on the disks in Ω m mst be crrently occpied, B m is given by (0). B, ˆB, and B are then compted by (5), (6) and (7) respectively. It is clear that GRT and RRT are eqivalent in terms of blocking probabilities since they both block the reqests if and only if all the servers are bsy. Theorem below establishes the eqivalency of GRT and RRT in terms of the redced load to each disk based on FPA. We first need a combinatorial lemma. Lemma : For integers n and k with n > k, k ( ) k ( ) i k!(n k )! =. i n k + i n! i=0 The Lemma follows from (.4) in ([24], page 6). A closer examination of the right hand side (RHS) of (8) reveals that the pain of enmerating all possible permtations of arranging x disks ot of Ω m {} can be mch alleviated by taking advantage of the fact that the prodct of the probabilities is always the same, regardless of the particlar

5 5 disk order in each of the permtations. The RHS of (8) can ths be expressed as [ n m x!( x )! ] λp m + ξ (N).! x= S Υ(Ω m {},x) (2) Now the problem redces to showing that (2) and the RHS of () are the same. Theorem : Let P A () = n m + x= ξ (N ), and P B () = h= h S Υ(Ω m {},x) S Υ(Ω m {},h ) v Ω m {} S ξ (N v) v. x!( x )!! ( ) ξ (N) (3) (4) Then P A () = P B (). Proof: To facilitate the proof, we shall rewrite (3) as n(r)!( n(r) )! P A () = ξ (N ),! and (4) as P B () = R Ω m {} S Ω m {} ξ (N) n(s) ( v Ω m {} S ξ (Nv) v R ), (5) where we se n(r) to denote the nmber of elements in the set R. Expanding the RHS of (5), we get P B () = n(s) = S Ω m {} [ + ξ (N) v v 2 Ω m {} S S Ω m {} v Ω m {} S n(s) ξ (N v ) v ξ (Nv ) v ξ (Nv 2 ) v 2 ξ (N) ξ (N )... ξ (N) S R Ω m {}( ) n(r) n(s) R = ξ (N) ( ) n(r) n(s) n(s) R Ω m {} R S R n(r) ( ) = ξ (N ) n(r) ( ) i i n(r) + i R Ω m {} R = P A (), i=0 where we enmerate the sbsets S and make the sbstittion i = n(r) n(s) to get from the third line to the forth line and the final eqivalence follows from the Lemma. ] Showing that P A () = P B () is not only an interesting mathematical challenge, bt it also provides confidence in the correctness of or soltions. As both RRT and GRT will eventally find a free disk if one is available, and also since we have shown they are eqivalent, we shall treat them as the same and henceforth consider only GRT in or analysis. D. FPA soltion of LBF As discssed, nder LBF, when a reqest for movie m arrives, it will be directed to the least bsy disk in the set Ω m. In the case where there is more than one least bsy disk, the reqest will be randomly dispatched to one of them. The implementation of LBF ths reqires that the VOD schedler pdate its channel occpancy information of a disk at each time epoch when ( a video session is terminated ) from the disk. Let ξ = ξ (0), ξ (),..., ξ (N) denote the vector of stationary probabilities that disk is in state i, i = 0,, 2,..., N, and ξ ( = ξ, ξ 2,..., ξ ) J. Given that disk is in Ω m, let s again define Υ(Ω m {}, x) as the set of all possible combinations of choosing x disks ot of Ω m {}. For movie m, when disk is in state i, the probability that, among the other disks containing movie m, h disks also have i channels occpied, and the remaining h disks have more than i channels occpied is P (h, i) = S Υ(Ω m {},h ) h= ξ (i) h N v v Ω m {} S k=i+ ξ (k) v, (6) for i = 0,,..., N and h =, 2,...,. Note that if =, we simply set P (h, i) =. Hence, when disk is in state i, its movie m reqest arrival rate is ( ) y (i) λpm (m) = P (h, i), (7) and its total reqest arrival rate de to all movie files is = (m). (8) ( y (i) y (i) m Φ Let y = y (0), y (),..., y (N ) and y = ) ( y, y 2,..., y J. Ths, (6) to (8) define a fnction f( ) that can be sed to obtain y from ξ: y = f( ξ). (9) On the other hand, let s model the state transition process of disk as a birth-death process with the birth rate y (i), i = 0,,..., N, and the death rate iˆµ (i), i =, 2,..., N, where ˆµ (i) = y (i ) m Φ y (i ) (m) ) µ m. (20) From the steady-state eqations of a birth-death process (see e.g. [7], page 3), we have ξ (i) = N! i! N k=i y (k) ˆµ (k+) ξ (N ). (2)

6 6 By normalization, we obtain N i=0 N! i! N k=i y (k) ˆµ (k+) ξ (N ) + ξ (N ) =. (22) Therefore, for disk, =, 2,..., J, (20) to (22) define a fnction g( ) that can be sed to obtain ξ from y: ξ = g( y). (23) The system of eqations (9) and (23) composes the following fixed-point eqations: ξ = g(f( ξ)). (24) By the disk independence assmption and having obtained ξ by solving (24), B m is given by (0). B, ˆB, and B are then compted by (5), (6) and (7) respectively. E. Comments on comptational effort The fixed-point eqations (9) and (24) can often be solved efficiently by the following sccessive sbstittioethod. Let T (z) denote the transformation given by the RHS of (9) or (24), where z is the vector ξ in (9) or the vector ξ in (24). We start the sccessive sbstittion procedre with a certain initial vector z 0, and then iteratively compte z k+ = T (z k ), for k = 0,,..., ntil z k+ is sfficiently close to z k. Since the transformation T (z) is a continos mapping J N+ = in from the compact set [0, ] J in (9) or [0, ] (24) to itself, by Brower s fixed-point theorem [25], a fixed point exists sch that z = T (z), ensring that the fixedpoint soltions for (9) and (24) exist. Althogh we can not establish the niqeness of the soltions, for varios nmerical experiments we have condcted, the iterations have always stabilized at the same fixed point regardless of z 0. It may appear that the task of enmerating the combination set Υ( ) involved in the comptation of () and (6) is cmbersome. However, the size of Υ( ) merely depends on the nmber of copies a movie has in the system. It is important to observe in (0) the dramatic performance gain obtainable by a movie m if its nmber of replicas increases from to +. Conseqently, it is not necessary in practice to place a large nmber of copies for any movie to achieve significant redction in its reqest blocking probability. Not only does this fact dramatically save storage space, bt it garantees that the size of the combination set is small, as we can set the maximal nmber of replicas of any movie to be not too large. IV. NUMERICAL EXAMPLES We consider a VOD system that manages 20 disks and provides sers with on-demand access to a library of 200 distinct movies. Each disk has a storage space of nits, and spports p to 30 concrrent video streams. The movie file-size is randomly generated, ranging from as small as 0.76 nits to as large as.29 nits. It reqires an overall storage space of nits to allocate a file-copy for each movie. Conseqently, the system has an extra storage space of 8.68 nits to accommodate mltiple file-copies for certaiovies. Let the poplarity profiles of the 200 movies be distribted so that p m = m ζ / 200 k= k ζ, for m =, 2,..., 200, where the parameter ζ determines the skewness of the distribtion. This distribtion is known as a Zipf-like distribtion, since when ζ = it is the Zipf distribtion [26]. It was fond that sch a distribtion with ζ = 0.27 statistically matches client access freqencies to varios movie titles observed from the video rental bsiness [27]. We assme that the channel holding time for movie m, taking into consideration ser interactive behavior [28], follows a lognormal distribtion. Withot loss of generality, we assme that the vale of the mean channel holding time for movie m is eqal to the vale of its filesize. For the prpose of or simlation, we set the standard deviation of channel holding time for movie m to be eqivalent to its mean, and set one time nit to be one hndred mintes. A. Approximation validation In the first example, we investigate the accracy of or analysis for GRT and LBF by means of a discrete event simlation stdy. For this prpose, we se one feasible file assignment that allocates two file-copies for 4 movies and one file-copy for all other movies. We validate the analytical reslts with respect to its mean reqest blocking probability of mlticopy movies, single-copy movies, and all movies in F. The analytical formlas and the simlation tests are implemented in C programming langage. In a typical rn of the simlation process, each of the one hndred million random events represents either the arrival of a reqest or the termination of a video session for a movie in F. We obtain B m by conting the nmber of its reqest arrivals and the nmber of its reqest losses. B, ˆB, and B are then compted by (5), (6) and (7) respectively. The simlation estimates are presented in Fig. with λ ranging from 420 to 520 at increments of 0. By repeating the simlation test with six independent rns, we have kept the radii of the 95% confidence intervals ([29], page 273) between 0.09% and 0.37% of the observed mean at each data point for B, between 0.09% and 0.37% for ˆB, and between 0.26% and 3.47% for B. We see that the approximate reslts for the mean reqest blocking probability of all movies are very close to those from the simlation. The approximation is also sccessfl in predicting the reslts of single-copy movies. Moreover, the reslts of mlti-copy movies are very good nder GRT. Unfortnately, the reslts of mlti-copy movies are less accrate nder LBF. This is mainly de to the fact that the FPA soltion of LBF involves state-dependent elements. Conseqently, the two assmptions made by FPA in Section I-B tend to introdce more errors to the approximate soltion of LBF than that of GRT. Nevertheless, this is less of a practical problem since in practice it is more realistic that we measre the performance of a VOD system sing either the mean reqest blocking probability of all movies or the worst blocking performance of the system (sally experienced by reqests for singlecopy movies). We will se B as the performance metric in all following examples. As a byprodct of the simlation stdy, we demonstrate the inefficiency of RRT in handling reqests for mlti-copy

7 7 0 GRT 0 LBF TABLE III SIGNALLING OVERHEAD OF GRT IN COMPARISON WITH LBF Blocking Probability Blocking Probability 0 2 All, Simlation All, Analysis λ 0 GRT Fig.. Single, Simlation 0 4 Single, Analysis Mlti, Simlation Mlti, Analysis λ Approximation validation. Blocking Probability Blocking Probability TABLE II 0 2 All, Simlation All, Analysis λ 0 LBF Single, Simlation 0 4 Single, Analysis Mlti, Simlation Mlti, Analysis λ INTENSITY OF REDUNDANT ATTEMPTS MADE BY RRT λ Ratio (%) TABLE IV FILE ASSIGNMENT QUALITY COMPARISON Cases SRT B GRT B LBF B Analysis Order Simlation Analysis Order Simlation Analysis Order λ (q a q r )/q r (%) movies. Let q a cont the nmber of random attempts for mlticopy movies in RRT. Let q r cont the nmber of reqests for mlti-copy movies. The performance metric sed in Table II indicates the intensity of redndant attempts made by RRT. Even thogh there are only 4 mlti-copy movies in this example and all of them have merely two file-copies, the overhead de to the rond trip signalling involved in the redndant attempts is qite significant. This is especially tre nder high traffic loads. We also compare the signalling overhead between the implementation of GRT and that of LBF. As we have discssed, both schemes reqire that the VOD schedler collect certain state information from the disks. Reslts in Table III demonstrate that LBF entails considerably larger signalling overhead than GRT. For this particlar example, the signalling overhead of GRT is less than 0% of that of LBF even at high traffic loads. B. Applications of the models In the second example, we investigate if the proposed analytical soltions are applicable to the task of file assignment optimization in a VOD system. From the large space of feasible file assignments for the test system, we frther present the reslts of nine file assignments for comparison in Table IV with the one stdied in the first example. We consider these different cases becase their blocking probabilities are close to each other. In this way, we can see if the analytical models are sfficiently accrate to establish the qality in terms of reqest blocking probability for a file assignment. In addition to both simlation and analytical reslts for GRT and LBF, we also report the analytical reslts for SRT in Table IV for comparison. Table V gives the average CPU time reqired to obtain the corresponding simlation and analytical reslts on a 2.4 GHz Pentim 4 machine. We observe that ) for both GRT and LBF, the qality of each file assignment established by analysis is consistent with that confirmed by simlation, bt the CPU time reqired by analysis is drastically redced, and 2) a file assignment that achieves a smaller blocking probability in the SRT system does not necessarily lead to a smaller blocking probability in systems where GRT and LBF are sed. In fact, for this particlar example, the best file assignment established in the SRT system trns ot to be the worst in the LBF system. This stifies the importance of or analysis in this paper to spport the task of file assignment optimization in the VOD system sing the more efficient resorce selection schemes. V. CONCLUSIONS In this paper, we have stdied three exhastive resorce selection schemes, namely RRT, GRT and LBF, for a large scale VOD system. These schemes tilize system resorces efficiently given the existence of mlti-copy movies. We have provided an analytical framework for analysis of the VOD system sing these efficient resorce selection schemes. The proposed analytical framework relies on the well-known FPA method, so that fast and sfficiently accrate soltions can be derived to facilitate the design and dimensioning of the VOD system. We have rigorosly proved the eqivalence between the FPA soltions of RRT and GRT. This allows s to se either of the two analytical soltions in practice, as none of them is more accrate in approximating the exact blocking performance of the VOD system. The reslts from this paper are directly sed in [2], where they spport an optimization program aiming to obtain the optimal or nearoptimal assignment of movie files that minimizes the reqest blocking probability of the VOD system given a configration of mltiple disks.

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