TELETRAFFIC AND OAT ATRAFFIC in a Period of Change, ITC-13 A. Jensen and V.B. Iversen (Editors) Elsevier Science Publishers B.V. (North-Holland) © lAC, 1991 795 ANALYSIS OF A GRADUAL INPUT MODEL FOR BURSTY TRAFFIC IN ATMt Huanxu Pan·, Hiroyuki Okazaki·· and Issei Kino·· Institute 01 Applied Mathematics, Academia Sinica, Beijing, China· NEC Corporation, Kawasaki, Japan·· Abstract: We consider a model for bursty traffic in ATM. The model is a single server queue with infinite buffer size. The arrival process is characterized by two aspects: arrivals of bursts and arrivals of cells in bursts. It is assumed that bursts arrive according to a Poisson process and cells in a burst arrive successively with a deterministic inter arrival time, say 1. We further make the fluid flow approximations: i) the burst size is assumed to be exponentially distributed though it can only be an integer in the practical case; ii) an arriving cell increases its amount in the buffer gradually at a rate of 1; iii) the server, which represents an output line, releases a cell in the buffer, if any, for transmission in the same gradual way as the cell arrives. This model gives a consideration on the correlations existing in bursty traffic. We present a unique approach for our model. By comparing it with a corresponding bulk arrival model, we find that the number of bursts in the system has the same steady-state distribution as the queue length in an M/M/l queue. Finally, we obtain an exact expression for the Laplace-Stieltjes transform of the steady-state distribution on the buffer content. The result can be used to estimate the delay of cells, and the loss probability in the case of finite buffer size. 1. Introduction In this paper, we consider a model for bursty traffic in an ATM (Asynchronous Transfer Mode) system and present an exact analysis. In an ATM network, information is sent in blocks of small fixed size, i.e., data are broken into cells before being trans- mitted. Therefore the stream of cells from a single source is most likely to be bursty as shown in Fig. 1. We refer to those cells that flow successively in a sequence (clustered) as a burst. Thus the arrival process to a switching node can be characterized by two aspects: arrivals of bursts and arrivals of cells during a burst (which keep a deterministic interarrival time). The compound arrival process of all cells is generally not a renewal process in terms of correlations in interarrival times. Even if the arrival process from each source is renewal, the superposition arrival process from all sources to a switching node can hardly be renewal. This is one major characteristic of ATM traffic and makes the performance analysis difficult. Source i ata Cell Burst _____ ____ Fig. 1 Bursty traffic in ATM. Recently, studies have been conducted on bursty traffic. Many of them fail to give enough considerations on the above-mentioned correlations existing in arrival processes and use ordinary queueing models with renewal input. The results are not fine enough since correlations in arrival pro- cesses can significantly affect queueing performance. A pop- ular way to incorporate correlation into a bursty traffic model is to use the Markov modulated Poisson process (MMPP)[I]. The MMPP provides analytic simplicity and versatility, but fails to give an exact representation on the deterministic na- ture of interarrival times of cells during a burst. Another model for correlated bursty arrival process is the branching Poisson process (BPP)[2]. The BPP possesses the correla- tions of arrivals as well as the deterministic nature of inter- arrival times during a burst. Analyses of both models with the MM PP and BPP arrivals need a considerable amount of numerical calculation and for the BPP only approximate results have been obtained. Other models for bursty traffic can be found in [3] and [4]. In the model considered in this paper, both the correlations in arrivals and the deterministic nature of interarrival times during a burst are well represented. The model is a single server queueing system with infinite buffer size. The arrival process is characterized by arrivals of bursts and arrivals of cells during bursts. We assume that bursts (specifically, head cells of bursts) arrive according to a Poisson process, and the cells in a burst arrive at fixed intervals. We further make the following fluid flow approximations: tThis paper is based on research carried out while H. Pan, one of the authors, was with C&C Research Laboratories, NEC Corporation in Kawasaki, Japan.