On G-networks and resource allocation in multimedia systems

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Abstract

Consider a multimedia-on-demand server who transmits video documents through a high-speed network, to geographically distributed clients. The server accumulates requests for specific documents in separate queues. The queues need to share the transmission medium in some fashion, typically in Round-Robin (RR) mode. We describe the resulting performance modeling problem, and develop an approximate representation using queuing networks. Our analytic model enables the efficient implementation of a new scheduling scheme, that we call the Local Round-Robin (LRR). We show that LRR yields significant improvement in system performance, compared to the original RR.

Introduction

Multimedia servers are quite different from those of conventional computer file systems, due to their strict timing requirements. In the common application of multimedia-on-demand service a large database of video documents is stored at a centralized server; the documents are transferred through high-speed communication network to geographically distributed clients [3], [5], [15], [17], [18]. Subscribers can choose both the program they wish to view and the time they wish to view it [18], [22]. The server is required to provide service within a small latency and to guarantee a sustained and almost constant transfer rate of the data. Consequently, a major bottleneck at the server is disk bandwidth, measured as the number of video streams that can be simultaneously delivered by the server.

Modern communication networks, such as ATM, are equipped with multicast facility [13], [14], [23], i.e., the same data can be sent to multiple clients without causing any extra overhead at the server. Multicasting can reduce the number of streams required by the server to support a given number of clients: if two requests for the same document are separated by a small time interval, then by delaying the service of the first request, a single stream can be used to satisfy both [1] (see Fig. 1). Thus, time-sharing of a given server capacity is implemented by batching requests for the same video document in one queue, and by serving all the requests waiting in the queue simultaneously. This mode of operation distinguishes multimedia systems from the traditional computer/communication systems, that can sometimes be modeled as standard queuing networks [6], [7], [11]. Various decision rules can be used for selecting the queue that will be serviced when a stream becomes available. We call these rules batching schemes.

In the present paper we develop a queuing model that enables to analyze and compare the performance of multimedia servers under Round-Robin (RR) based batching schemes (see in Section 3). Our analysis addresses the general multimedia-on-demand scenario, where waiting users may lose patience and withdraw their requests. This leads to a loss in potential revenue for the system. The formal framework that we use is based on the G-network model proposed in [9], [10]. The G-network captures the two important features of batch scheduling and the impatience of users.

In the sequel we also explore some optimization problems that arise in the management of a multimedia server with a given stream capacity. In particular, we show that using our queuing model, the problem of allocating a fixed server capacity to individual queues so as to maximize the system throughput (or minimize the average wait time) can be solved efficiently, in time that is polynomial in the server capacity and the number of documents.

We mention briefly some of the past work in this area. In [2] a multimedia system is viewed as a special case of an on-line batching system, where a single server gives simultaneous service to a batch of clients; the length of this service is independent of the size of the batch, and no assumptions are made on the arrival times of requests, or the frequency of viewing requests for the various documents. The model was used for the study of several decision rules for adaptive video-on-demand, in which a batching scheme is combined with scheduling algorithms, that decide whether to accept or reject user requests in a movie vending environment.

In [19] the problem of finding an efficient schedule for a multimedia-on-demand system was formulated as a stochastic optimization problem, and functional equations were derived, for defining the optimal scheme. It was shown, that the computational complexity as well as the space required to solve these equations can be sizable, and the authors proceed with experimental study of heuristic solutions.

In [4] a Markovian model was used for representing a multimedia system. The authors show that even when simplifying assumptions are used, this conventional method is prohibitive, due to the large state space of the corresponding Markov chain. In [20] a server of the walking type model was proposed for studying the performance of multimedia applications; however, the analysis is valid only for the special case where the users patience interval is of infinite length.

The rest of the paper is organized as follows. In Section 2.1 we present the system model and our performance measures. In Section 2.2 we introduce the G-network formalism. Section 2.3 gives some definitions and the notation used in our study of the discrete resource allocation problem, that arises in multimedia systems. In Section 3 we analyze the multimedia server performance under the RR scheme.

In Section 4 we introduce the Local Round-Robin (LRR) scheme, that assigns a fixed server capacity to each of the queues (based on the relative frequency of requests for the corresponding document). We show that efficient allocation of the server capacity to the queues can be found using our analytic model and the relation of the LRR scheme to the discrete resource allocation (DRA) problem.

Section 5 describes the results of a simulation study of the RR and the LRR, and a comparison to the results we obtained using our analytic model. We summarize in Section 6 with a discussion of possible directions for future work.

Section snippets

Modeling and mathematical preliminaries

In this Section we first describe our system model and the performance measures that are of interest. Then we introduce the G-network formalism and the resource allocation problem, that will be discussed in the following sections.

Consider a database of M video documents. User requests (to view specific documents) arrive as Poisson process with rate Λ; a user chooses to view document i with probability pi,0<pi<1, where ∑Mi=1pi=1. The probabilities p1,…,pM reflect the relative popularity of each

Round-Robin scheduling of the multimedia server

Consider the implementation of the RR scheme in the queuing system of Fig. 2. Let Li=L denote the length of a service of queue i, 1⩽iM,1 and denote by D the inspection frequency parameter of the system. The queues are inspected at time instances that are integral multiples of D. Indeed, by tuning D, the exhaustive service

Multimedia scheduling and the resource allocation problem

In this section we study a scheduling approach that is based on the assignment of a fixed portion of the server capacity to each of the queues. Thus, the ith queue has a set of I/O streams that are used only for servicing requests for the ith document. The resulting queuing system is depicted in Fig. 4.

For a server capacity of N streams, the LRR scheme allocates Ni streams to document i, 1⩽iM. The LRR schedules each of the queues by the RR algorithm, as given in Fig. 3, with M=1, and N=Ni for

Numerical results

We give below the numerical results obtained in a simulation study of the multimedia server system. Our results validate the mathematical model used in Section 3, and evaluate its relative efficiency in the implementation of the LRR scheme.

We simulated a multimedia system with a database of M=100 documents; user requests arrived by Poisson process. The rpvp̄ formed a Zipf's distribution, i.e., pi=1/(iHM), where HM, the Mth Harmonic number, is the normalization constant. The Zipf's distribution

Discussion

We have studied the performance of two service schemes for multimedia systems, based on a RR schedule, namely, the original RR and a new scheme called LRR. The usage of the G-network formalism enabled us to derive analytic results for the steady state distribution of such systems under these schemes, and to obtain efficient partition of the server capacity to queues, for the implementation of the LRR. Our numerical study showed that LRR can improve significantly the performance of the system,

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