Discrete-time queues with correlated arrivals and constant service times

Abstract

A discrete-time single-server finite-capacity queue with correlated arrivals and constant service times of arbitrary length is investigated in this paper. Cells are generated by a bursty on/off source, with geometrically distributed lengths of the on-periods and the off-periods. The performance of the system is evaluated by means of an analytical technique, based on generating functions, whose comutational complexity does not depend on the buffer space. As a result of the analysis, closed-form expressions are obtained for the cell loss ratio, the steady-state probability generating functions of the queue length, the unfinished work and the cell delay and the joint probability generating function of two consecutive interdeparture times at the output of the queue. Some numerical examples illustrate the results. Scope and purpose Discrete-time queueing models are suitable for the performance evaluation of Asynchronous Transfer Mode (ATM) multiplexers and switches. In these models, the time axis is divided into fixed-length slots and the service of a customer must start and end at slot boundaries. Most analytical studies of discrete-time queues assume constant service times equal to one slot, an infinite buffer capacity and/or uncorrelated arrival process. The present paper is an attempt to explore whether analytical techniques are still applicable in absence of these restricting assumptions. Specifically, we focus on a discrete-time single-server queueing system with constant service times of arbitrary length, a finite storage capacity and a simple non-independent arrival model. The analysis method is based on an extensive use of generating functions, an approach which has traditionally been reserved for infinite-capacity queues. We show in this paper that generating functions can also be very useful in the finite-capacity case.