Discrete-Time Multiserver Buffer Systems with Correlated Arrivals and Geometric Service Times Peixia Gao, Sabine Wittevrongel, Herwig Bruneel SMACS * Research Group, Department of Telecommunications and Information Processing, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium. E-mail: {pg, sw, hb}@telin.rug.ac.be Keywords: multiserver queueing system, correlated arrivals, geometric service times, system contents, de- lay Abstract In this paper, we investigate the behavior of a discrete- time multiserver buffer system with infinite buffer size. Packets arrive in the system according to a two-state correlated arrival process. The service times of the packets are assumed to be independent and identically distributed according to a geometric distribution. We present an analytical technique, based on the use of generating functions, for the analysis of the system. Explicit expressions are obtained for the mean values, the variances and the tail distributions of the system contents and the packet delay. The influence of the various model parameters on the behavior of the sys- tem is shown by means of some numerical examples. 1. INTRODUCTION Discrete-time buffer systems with different kinds of correlated packet arrival processes have been studied by many researchers during the last decades, given the bursty traffic characteristics in digital communication networks. In a discrete-time model, time is assumed to be slotted and the service or transmission of the pack- ets is mostly assumed to be deterministic at the rate of one packet per slot, in accordance with the constant packet length used in e.g. ATM-based networks. Only a limited number of papers in literature consider more general service-time distributions, even though there are fundamentally theoretic as well as practical rea- sons to do this in view of the complicated and irregular service mechanisms in the internet. In [1]-[3], discrete- time buffer systems with (various types of) correlated arrivals and constant service times of arbitrary length ([1]), geometrically distributed service times ([2]) or a * SMACS: Stochastic Modeling and Analysis of Communica- tion Systems general service-time distribution ([3]) are analyzed, un- der the assumption that there is only one single server. Discrete-time multiserver systems with either constant service times of multiple slots ([4]) or geometric service times ([5], [6]) have been studied as well, but only for the case of an uncorrelated packet arrival process. The assumptions considered in [7] and [8] are uncorrelated arrivals, a single server and geometric ([7]) or general ([8]) service times. In this paper, we investigate the behavior of a dis- crete time buffer system with multiple servers, corre- lated arrivals and geometric service times. From the above survey, the paper can be seen as a generaliza- tion of [5] and [6] to the case of correlated arrivals or as an extension of [2] to the case of multiple servers. We present an analytical technique, based on the use of generating functions, for the analysis of the system behavior. The presented technique leads to explicit expressions for the mean values, the variances and the tail distributions of both the system contents and the packet delay, which can easily be evaluated numeri- cally. The outline of the paper is as follows. In Section 2, the system under study is described. In Section 3, the probability generating function (pgf) of the system contents is derived, and from this pgf, the mean value, the variance and the tail distribution of the system contents are obtained. In Section 4, the characteristics of the packet delay are analyzed by means of a gen- eral relationship between system contents and delay. In Section 5, some numerical examples are given to il- lustrate the impact of the various model parameters on the system behavior. Finally, the paper is concluded in Section 6. 2. SYSTEM DESCRIPTION We consider a discrete-time buffer system with c (c ≥ 1) servers (output channels) and an infinite buffer size. The time axis is assumed to be divided into fixed-