A Case Study of Load Sharing Based on Popularity in Distributed VoD Systems

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1 IEEE TRASACTIOS O MULTIMEDIA, VOL. 8, O. 6, DECEMBER A Case Study of Load Sharing Based on Popularity in Distributed VoD Systems Sonia González, Angeles avarro, Juan López, and Emilio L. Zapata Abstract In our research, we consider a distributed video-on-demand (VoD) system in which only the most popular videos are replicated in all the servers, whereas the rest of them are distributed through the system following some allocation scheme. In this paper, we present an algorithm to efficiently share the load in such a system and an analytical model that captures the performance of this algorithm, which we validate through simulations. One novelty in our work is that our analytical model lets us relate popularity and partial replication of some of the videos and to predict the user waiting time. We exploit such relationships to assist the system designer to select the size of the servers and network, the optimal number of servers to maintain short waiting time and to predict when the network encounters bottleneck. Index Terms Distributed system, load sharing, partial replication, performance modeling, popularity, video-on-demand. I. ITRODUCTIO A promising way to contain the costs in a video-on-demand (VoD) system, is to link up several small VoD servers to a (limited capacity) network and allow servers with idle retrieval bandwidth to help out servers that are temporarily overloaded (load sharing). The main metric of performance in these systems is the waiting time (Twait) which can be defined as the average of the periods which elapse between the arrival of a video request and the time when the service is initiated; the goal is to minimize that parameter. In Fig. 1, we show the generic architecture of such a distributed VoD system. Our system consists of serv servers, which are connected through an ATM switch. We choose the ATM technology because has emerged as the front-runner for implementing VoD systems [1]. The bandwidth of the ATM switch is Bswitch and each link that connects one server with the switch has a Blink bandwidth in each direction. Each server has term clients attached, and can retrieve stream streams concurrently. In addition, the program provider offers M videos which can be requested by any client of the system. The VoD system that we describe is a distributed system of loosely coupled servers where some objects (videos) are replicated, and eventually a video can be retrieved from alternative servers, depending on their respective load. The advantages of such system are analyzed in [2]. Our research focuses on minimizing the waiting time via exploiting load sharing in a distributed loosely coupled VoD system in which the videos are not replicated in all the servers. Load sharing works as follows: if the video requested by a client attached to a server is stored in this server and there is enough bandwidth in the server to attend to it, then a local service is started. We assume that the clients are willing to wait until the service is obtained (no defection). Thus, when the server is overloaded, the requests are queued instead of blocked. In this case, Manuscript received February 21, 2003; revised January 15, This work was supported in part by the European Union under Contract 1FD , and by the Ministry of Education of Spain under Contract TIC The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jan-Ming Ho. The authors are with the Department of Computer Architecture, University of Malaga, E Malaga, Spain ( sonia@ac.uma.es; angeles@ac.uma.es; juan@ac.uma.es; ezapata@ac.uma.es). Digital Object Identifier /TMM the server initiates a dialog with the other servers which store that video. If some of these servers have idle retrieval capacity, then one of them services the request, but now remotely. This is what we call a remote service. We note that a client attached to a server, which asks for a video that is not stored in that server, also generates a remote service. ote that a remote service consumes more resources than a local service [3]. In a related work, Tay and Pang [2] proposed an algorithm called GWQ (Global Wait Queue) for minimizing the waiting time using load sharing in a similar distributed VoD system. These authors also developed in [2] a framework to address load sharing in distributed architectures. Using this framework, they presented an analytical model that captured the performance of the GWQ algorithm quite accurately. Another approach, from the networking industry point of view, could be to deliver the video streaming in the distributed VoD system, through a Content Distribution etwork (CD) [4], in which application-specific servers or caches (or proxies) at various points in the network, handle the distribution of specific content types, as well as redirect the user requests to the nearest CD server with idle retrieval bandwidth. Although the GWQ algorithm and CD infrastructure focus on load sharing, they consider that all the objects are replicated in all the servers. By contrast, in this paper we explore the possibility of reducing the necessity of storage capacity without degrading the system performance by not replicating all the content. In our research, we propose a new load sharing algorithm that we call Popularity and Partial Replication Load Sharing, PRLS from now on, in which we introduce two new features: i) the videos are distinguishable and, on each server, they are required according to their popularity; and ii) only the most popular videos are replicated in all the servers. This is what we call partial replication. In our system, video popularity is determined by the users behavior. The frequency of requests for a video over the entire system video popularity lets us rank the videos; based on this ranking, we follow a simple policy to allocate the videos in the system: the most requested videos are stored (replicated) on all the servers, whereas the rest of the videos are distributed among the servers following some allocation scheme. The portion of videos replicated in all the servers is what we call percentage of replication.we think that the above mentioned new features allow us to develop a more realistic algorithm for load sharing in a distributed VoD system. There has been some research on the issue of partial replication of the most popular videos in a distributed multimedia system architecture [5] [7]. A major difference between those studies and our approach is that they do not consider load sharing when a server is overloaded. These approaches consider server stream capacity and storage capacity as constraints in their formulation. However, as they do not take into account load sharing, they do not consider the network capacity constraint, and this is a major issue in our work. In this paper, we present an analytical model that captures the performance of our load sharing algorithm. The importance of this model is that it helps us to gain insight into the behavior of the algorithm and the system, and to understand where the bottlenecks are. Another application of the model is that can help us to estimate the suitable percentage of replication in the system that guarantees a reasonably small user waiting time. In addition, the analytical model can help us to select the size of the network, the optimal number of servers that maintains short waiting time, and to predict when the network encounters bottleneck. The paper is organized as follows: in Section II we describe in detail how the videos are allocated and distributed among the servers in our system. In Section III we describe briefly our load sharing algorithm. ext, in Section IV we derive an analytical model to capture the performance of our algorithm. The following section proceeds to present several experiments to validate our model and to highlight the influence of video popularity and percentage of replication on the user /$ IEEE

2 1300 IEEE TRASACTIOS O MULTIMEDIA, VOL. 8, O. 6, DECEMBER 2006 waiting time. Finally, in Section VI we conclude with a review of our contribution. II. DISTRIBUTIO OF VIDEOS: POPULARITY AD PARTIAL REPLICATIO Any of the M videos provided by the program provider can be required in any server. Basically, the popularity of a video in the system is defined as the frequency of requests for this video issued for all the customers in the system. To keep our proposal sufficiently general, we assume that the popularity of the videos in the system follows a Zipf law distribution [8], which models video popularity quite well as was noted in [9]. The Zipf distribution takes two parameters: M (the cardinality, in our case the number of videos) and (the degree of skew, which we call popularity degree). The probability p j of requesting the jth video in our system, is given by p j = c=j, for each j 2f1;...;Mg, M where c =1= k=1 (1=k ) is a normalization constant. We note that the probabilities assigned by the Zipf distribution decrease with the indexes in the sequence of videos, that is p1 >p2 >... >p M. Since we are going to study the behavior of a load sharing algorithm in a distributed architecture, we focus on local or metropolitan area networks, which is where more benefits from a load sharing feature can be obtained [2]. In this context, it seems reasonable to assume similar user preferences in all the servers, and for this reason we work within global video popularities. In any case, the load sharing algorithm that we propose in the next section can handle different user behavior and video popularities on each server. We focus our study on one load period [3], for which the estimated number of requests per unit time and popularity of videos are known and do not fluctuate. Regarding the video distribution, we assume that the most popular videos (hot videos) are going to be allocated in all the servers. The M videos offered by the program provider are ordered with regard to their popularity. Using the percentage of replication (that is, the portion of videos replicated in all the servers) we establish the number s of videos to be replicated, select the s first videos in the ordered sequence and store them on each server. ext, the remaining M 0 s videos are distributed among the servers following some allocation scheme. The interested reader can find in [3] a discussion about several allocation strategies. One example of an allocation strategy could be simply to place the M 0 s videos among the servers according to a round-robin scheme. Specifically, this is the allocation scheme that we have selected for our simulations in Section V. We selected this one because it is the most challenging worst case of video distribution for our system, as we demonstrated in [3], although our algorithm and our analytical model support another strategies. The videos allocated on a server are what we call the local videos of the server. Let s be the number of videos replicated in all the servers, and let Ki represent the set of the other videos allocated in server i (8i 2f1;...; serverg).wedefine the first parameter of interest in our research: F i, the probability of requesting a local video in server i. F i can be computed as, s F i = p j + p k : (1) j=1 8k2K From F i we can compute the average probability of requesting a local video from a server, F, as, F =( i=1 F i)=server. Obviously, the average probability of requesting a non-local video is 1 0 F. For instance, when all videos are replicated in all the servers then we say that percentage of replication is 100% it follows that F =1and 1 0 F = 0. Fig. 1. Distributed VoD system (Color version available online at III. LOAD SHARIG I DISTRIBUTED VoD SYSTEMS The main objective of the load sharing algorithms in a distributed VoD system is to minimize the waiting time for a service to begin by allowing servers with idle retrieval bandwidth to help out servers that are temporarily overloaded. While conventional distributed systems require that load sharing algorithms provide load balance [10], this is a less important consideration for VoD systems because remote services consume network bandwidth in the links of both local and remote servers as well as in the ATM switch. Consequently, for VoD systems, remote services due to load sharing should be generated only when some server is fully loaded. However, since our system maintains only replication of the hot videos, a user can request a video that is not locally stored. In this case, a remote service is generated as well. We propose a load sharing algorithm entitled PRLS that takes into account these previous considerations. Summarizing, the main characteristics of our algorithm are (again more details can be found in [3]): 1) The first priority of every server is to service the requests to the local videos. 2) A server generates a remote request if it receives a request to a local video and is fully loaded, that is, it does not have enough resources to attend to this request. In addition, a request to a nonlocal video generates a remote request too. IV. AALYTICAL MODEL We must note here that we have based our model on the three steps framework presented in [2], which was developed for the GWQ algorithm. Our contribution is that we have incorporated the partial replication feature in the model, so we are now able to quantify the impact that the percentage of replication will have on system performance. Another effect that our model captures is the influence of the popularity degree. A. Model otation We assume, as has been done in most prior studies [2], that the inter-arrival time of requests to a VoD server follows an exponential distribution and that the service time of a video follows another exponential distribution. For our system, a server can be modeled as a M=M= stream= term= term queue, i.e., a closed system with term terminals, and a queue with stream servers and term buffer slots for pending requests. We are interested in the case where several multimedia servers are connected to an ATM switch (Fig. 1), and they share their load and

3 IEEE TRASACTIOS O MULTIMEDIA, VOL. 8, O. 6, DECEMBER service requests to non-local videos following our PRLS algorithm sketched in Section III. We assume that the servers are homogeneous, i.e., they have the same term clients attached, the same stream, and similar exponential distributions to model the arrival of requests (T sleep ) and the service time (T active ). On the other hand, the videos are allocated on each server following the procedure explained in Section II. One important parameter in our system is the server utilization [11]. The server utilization, for a server i can be approximately calculated (being the throughput requests per unit time [11]) as, i = 1 Tactive stream : (2) B. Model for the PRLS Algorithm Our model is based on two key observations established in [2]: 1) the server utilization is common for any server (i.e., 8r 2 f1;...; serverg, r = ). This fact lets us estimate the server utilization for any load sharing algorithm simply by calculating for an isolated server. Thus, our model is derived from the perspective of one of the servers; and 2) the effect of load sharing on a server s performance is similar to that of adding more capacity to that server. In addition, based on several simulations of our algorithm, we have added a new point: 3) the fact that not all the videos are replicated in the system can be captured by subtracting some capacity from the network. This is due to the obligatory nature of some remote services (the non-local video requests) that obviously subtract resources for load sharing. The incorporation of these three observations lets us organize the analytical model into three steps (as in [2]): Step 1: M=M= stream = term = term is the analytical model of an isolated server. For this model, we compute the throughput and server utilization i (2). More details in [11], [12]. The goal of this step is to calculate the arrival rate of obligatory remote service requests, r. In the PRLS algorithm we know that a request becomes a candidate for obligatory remote service when the request is for a non-local video. Thus, an estimation of r can be, r = 1 (1 0 F ) (3) Step 2: M=M=n is the model for computing the total idle remote capacity for load sharing. We choose this model because we assume that there is no defection in the system [9], i.e., the users wait until they are served. From the perspective of a server, the total remote idle capacity can be viewed as an additional capacity of n streams. This n is determined by the link and switch bandwidths, as well as the utilization and number of remote servers, and the probability of requesting a nonlocal video. We now describe, briefly, how n is calculated. The interested reader can find the details of the complete discussion in [3]. Server i sees (on average) a total idle capacity, n idle (i), that is the sum of the idle capacity of the other servers r 6= i, n idle (i) = r=1;r6=i (1 0 r ) 1 stream j=1 1 0 F j server From observation 1) we know that server utilization is common for any server (i.e., 8r 2f1;...; server g, r = ). On the other hand, we are interested in the average idle capacity which sees a server, n idle, so we can approximate the probability of requesting a non-local video in a server 1 0 F j by the average probability of requesting a non-local video, i.e., 1 0 F. Thus, we can compute n idle as, n idle =(10 ) 1 stream 1 F 1 (server 0 1) (4) However, only part of this idle capacity is usable for load sharing, basically because we have to take into account the available link and switch bandwidths. The number of streams that the link connected to server i has available for offering load sharing, as well as the number of streams that the switch has available for load sharing, can be computed as, n link (i) = n switch (i) = B link (1 0 F j ) B stream j=1;j6=i B switch B stream 1 server j=1;j6=i (1 0 F j ) Again, we are interested in the average link capacity, n link, and in the average switch capacity, n switch, so we can approximate 1 0 F j by 1 0 F. Thus, n link and n switch can be calculated as, n B link link = 1[10(10F )1( server 01)] B stream (5) B switch n switch = 1[10(10F )1( server 01)] B stream 1 server (6) One important point here is that (4), (5) and (6) depend on F ; thus, the influence of the replication can be measured in our system. Clearly, we can see that when the percentage of replication increases, then s increases and F increases (1), and therefore n idle, n link and n switch grow. In other words, more replication means that less requests will be obligatorily serviced remotely, and for this reason there will be more resources available for load sharing. Another effect that is captured in our model is the influence of the popularity degree. A greater popularity degree means that the hot videos are more frequently requested. For this reason, F increases again. As we explained before, then n idle, n link and n switch grow. That is, a greater popularity degree means more requests to the replicated videos (which are local videos) and therefore less obligatory remote services, and for this reason there will be more resources available for load sharing. Once we have computed n idle, n link and n switch, we can deduce the average number of streams that are available remotely, n, as, n = min(n idle ;n link ;n switch ) (7) Having calculated n, we consider an M=M=n queue with an arrival rate given by r [from (3)] and an average service time T active. For this queue we compute m, the average number of active streams in this queue. This m is our estimation for the average number of requests that are serviced remotely due to the non-local video requests. The expression to calculate m is, m = r 1 Tactive = 1 (1 0 F ) 1 Tactive (8) Step 3: M=M= stream + v= term = term is now our model for load sharing, as we established in observation 2). In fact, through simulations, we have verified that the performance of our PRLS algorithm can be closely approximated with a M=M= stream + v=term= term queue; i.e., each server behaves as if a (virtual) capacity of v streams were added. ow, we have to compute v. This v is the average number of remote resources available for load sharing, and can be estimated as v = n 0 m. This subtraction is what we had determined in observation 3). It is the difference between the average number of streams available remotely (n) minus the average number of requests that are obligatorily serviced remotely (m). In other words, v represents the remote capacity that is not occupied by obligatory services, and therefore that is available for load sharing. Once we have computed v, from the equations of

4 1302 IEEE TRASACTIOS O MULTIMEDIA, VOL. 8, O. 6, DECEMBER 2006 the M=M= stream + v= term= term queue [12] we compute the average waiting time in our system (T wait ). C. Estimation of the System Capacities One important feature in the design of the system, is the storage capacity, S, that a server must offer. Let L j be the length of video j, s the number of videos replicated in all the servers, and K i the set of the non-replicated videos allocated on server i. The minimum storage capacity required in server i, S i, can be computed as, s S>S i = L j 1 B stream + j=1 8k2K L k 1 B stream (9) The actual storage size of the servers, for a given allocation scheme, will fix the maximum percentage of replication, and obviously the number of replicated videos s. Other parameters that we could estimate are the suitable stream capacity of the servers ( stream ) and the size of the network (B link and B switch ) in order to avoid any of these elements becoming bottlenecks in the system. One of the first restrictions, to prevent our system collapsing, is that the average number of requests that are obligatorily serviced remotely (i.e. m, given by (8)) must be bounded, and must be lower than the average number of streams available remotely (i.e., n). That is, n>m. From a server s point of view, from (7) we can define the smallest stream capacity that a server must offer in order to prevent a server becoming the bottleneck in the system, stream. min In this case, a server must have an idle stream capacity such as n idle > n min idle = min(n link ;n switch ). From (4) and from the previous observation, the smallest number of streams available in the server must be bigger than m, if we want to avoid collapse. That is, stream >stream min 1 (1 0 F ) 1 T active = (1 0 ) 1 F 1 ( server 0 1) (10) If we study the network, from (7), we can define the smallest link resources required to prevent the link being the bottleneck as n link >n min link = min(n idle ;n switch ), as well as the smallest switch resources required to prevent the switch being the bottleneck as n switch >n min switch = min(n idle ;n link ). Besides this, that smallest number of streams available remotely through the link (or the switch) must be bigger than m, if we do not want the link (or the switch) to collapse the system. In other words, B link >Blink min 1(10F)1T = 10(10F)1( 01) 1 Bstream; if C<1 1; if C 1 B switch >Bswitch min 1(10F)1T 10(10F)1( 01) = 1 server 1 B stream ; if C<1 1; if C 1 (11) (12) being C =(10F ) 1 ( server 0 1). We should remark here that C (both in (11) and (12)) could be bigger than 1. In such cases, a negative value for Blink min (or Bswitch) min could be obtained, but this result would not make physical sense. What happens is that the number of requests to remote services overwhelm the available resources. We express this situation with an asymptotic behavior of B link (and B switch ), which is represented with the term 1 in the second part of (11) and (12). If that bound is violated, then the link (or the switch) is the bottleneck in the network design. In this case, the waiting time could be reduced by increasing the link (or the switch) bandwidth, without changing any other parameter. V. SIMULATIOS In the simulations we assume the following parameters: the number of clients attached to each server is term = 90, whereas the number of streams that can be retrieved concurrently on each server is stream = 50 (as in [2]); each client generates requests with inter-arrival times that obey an exponential distribution with a mean time of T sleep = 7200 seconds; the service times are also exponentially distributed with a mean time of T active = 7200 seconds; the total number of videos offered by the system provider is M = 100; the distribution of the videos among the servers is carried out as described in Section II. As mentioned in that Section, this video distribution scheme is the most pessimistic scenario from the load sharing point of view, and helps us to simplify our analysis and understand the impact on system performance that partial replication and popularity degree may involve, as well as to identify where the bottlenecks are. We assumed standard values for the different bandwidths: B switch = 1 Gbps, B link = 155 Mbps and B stream = 15 Mbps. For each simulation experiment, the number of videos serviced by each server was at least The initial 1000 samples were discarded to achieve a statistical stationary state. In order to examine the influence of the percentage of replication, we conducted a first set of experiments where we set =1(this corresponds to a so-called pure Zipf distribution) and we varied the percentage of replication through 100%, 75% and 50%. In the second set of experiments, we established =1:8 (this give us a highly skewed distribution, where popular videos are the most frequently demanded). Again, we varied the percentage of replication through 100%, 75% and 50%. The goal of this new set of experiments is to measure the impact of the popularity degree on system performance. A. Accuracy of the Analytical Model Fig. 2 represents the performance of our PRLS algorithm, obtained by simulations the circular marks. The x-axis represents the number of servers in the system, serv, and the y-axis gives the average waiting times in seconds. In addition, we represent the results derived from the analytical model the triangular marks. From Fig. 2 we notice that, for less than 6 servers, the waiting times of the analytical model are slightly smaller than the measured ones. We think that this is because our model over-estimates the number of available streams when we compute (4). On the other hand, for more than 10 servers, the waiting times of the analytical model are now a bit higher than the simulated ones. The reason is because the analytical model assumes a worst-case scenario when it computes the number of obligatory remote services, those due to non-local video requests. Both issues are discussed in more detail in [3]. In any case, we think that the results show that our analytical model makes an accurate estimation of the PRLS algorithm performance. B. Effect of Replication and Popularity in the Waiting Time We now study the influence of the percentage of replication and the popularity degree on client waiting times. In both set of experiments, the measured waiting times increase when the percentage of replication decreases, especially for =1. This is due to the rise of obligatory remote services because less replication means that a high number of requests will be to non-local videos, particularly when = 1. For example, the waiting time for 10 servers increases when we fix the popularity degree, and the percentage of replication changes through 100%, 75% to 50%. This can be noted in Fig. 2(a) (c) when =1. However, in Fig. 2(d) (f) when =1:8, the growth of times is very small. One observation from these figures is that the percentage of replication has a more important effect when the popularity degree is small.

5 IEEE TRASACTIOS O MULTIMEDIA, VOL. 8, O. 6, DECEMBER Fig. 2. Performance results for the PRLS algorithm and the analytical model. =1: (a), (b), (c). =1:8: (d), (e), (f). The x-axes represent the number of servers, and the y-axes are the waiting times in seconds. Another observation is that when the popularity degree is high, the percentage of replication has a very small impact on the system performance. In Fig. 2(b), we note that for a number of servers between 8 and 14 the waiting time increases by an average factor of 2. However, in Fig. 2(c), for a number of servers between 8 and 14 the waiting time increases by an average factor of more than 10. This is due to the increase in the number of obligatory remote services. In fact, our model predicts that there is a point at which the number of requests that must be serviced remotely (m) needs more than the number of available network streams (n). This point (which is the bottleneck) is reached when the number of servers is higher than server =12for 75% of replication [Fig. 2(b)], and higher than server =6for 50% of replication [Fig. 2(c)]. This point is not reached for =1:8. The point at which the servers or the network become the bottleneck can be estimated in our model. As we indicated in Section IV, (10), (11) and (12) let us compute the values for stream, min Blink min and Bswitch min in order to avoid the servers or the network becoming a bottleneck. The interested reader can find in [3] details in how to use those equations to guide the system designer in the selection of the appropriate servers and network sizes. Another interesting issue is to find the number of servers for which the system achieves the minimum waiting time (see again Fig. 2). For 100% of replication and any, the minimum is achieved with 6 8 servers. However, for =1, when the percentage of replication decreases, the minimum is achieved when the number of servers tends to 4. On the other hand, for =1:8 when the percentage of replication decreases, the minimum is reached when the number of servers tends to 8. Other application of the analytical model could be to estimate the appropriate percentage of replication that gives us the minimum waiting time. Knowing the system parameters ( server, stream, S i, B link, B switch ) and video parameters (p j, L j), we could formulate an optimization problem whose objective function is the minimization of the average waiting time, and whose constraints are given by (9) (the disk storage limitation), and (10) (12) (the minimum stream capacities). This is an interesting issue that could be explored in future work, but we consider it is beyond the scope of this paper. VI. COCLUSIOS In this work, we have presented a new load sharing algorithm for VoD distributed systems that considers issues such as videos being requested according to their popularity and which are partially replicated on the servers. To capture the key characteristics of this algorithm, we have developed an analytical model and proved through simulations that it accurately represents the performance of our algorithm. Using the analytical model and simulation experiments, we have analyzed the influence of the percentage of replication and the popularity degree on client waiting times. One important application of the analytical model is that it lets us estimate for different percentages of replication and popularity degrees, some basic characteristics of the behavior of our PRLS algorithm. These estimations would let us select the size of the servers and the network, the optimal number of servers to maintain short waiting time and to predict when the network encounters bottleneck.

6 1304 IEEE TRASACTIOS O MULTIMEDIA, VOL. 8, O. 6, DECEMBER 2006 ACKOWLEDGMET The authors would like to thank the anonymous referees for their helpful and insightful suggestions. REFERECES [1] T. Little and D. Venkatesh, Prospects for interactive video-on-demand, IEEE Multimedia, vol. 1, no. 3, pp , [2] Y. Tay and H. Pang, Load sharing in distributed multimedia-on-demand systems, IEEE Trans. Knowl. Data Eng., vol. 12, no. 3, pp , Jun [3] S. Gonzalez, A. avarro, J. Lopez, and E. L. Zapata, A Case Study of Load Sharing Based on Popularity in Distributed VoD Systems Dept. Comput. Architecture, Univ. Malaga, Spain, Tech. Rep. UMA-DAC-05/06, [4] I. Lazar and W. Terrill, Exploring content delivery networking, IT Professional, vol. 3, no. 4, pp , [5] T. Little and D. Venkatesh, Popularity-based assignment of movies to storage devices in a video-on-demand system, Multimedia Syst., vol. 2, pp , [6] A. Mourad, Issues in the design of a storage server for video-on-demand, Multimedia Syst., vol. 4, pp , [7] Y. Leung and R. Hou, Assignment of Movies to Heterogeneous Video Servers Dept. Comput. Sci., Hong Kong Baptist University, Kowloon Tong, Hong Kong, Tech. Rep. COMP , Mar [8] A. Chervenak, Tertiary Storage: An Evaluation of ew Applications, Ph.D. dissertation, Univ. California, Berkeley, Dec. 1994, Tech. Rep. UDB/CSD 94/847. [9] C. Aggarwal, J. Wolf, and P. Yu, The maximum factor queue length batching scheme for video-on-demand systems, IEEE Trans. Comput., vol. 50, no. 2, pp , [10] J. K.-Y. g, S. Xiong, and H. Shen, A multi-server video-on-demand system with arbitrary rate playback support, J. Syst. Softw., no. 51, pp , [11] A. Allen, Probability, Statistics and Queueing Theory: With Computer Sciences Applications, 2nd ed. ew York: Academic, [12] D. Gross and C. Harris, Fundamentals of Queueing Theory, 3rd ed. ew York: Wiley, 1998.