Customer delay in very large multi-queue single-server systems

The objective of this work is the modeling and analysis of multi-queue single-server systems consisting of many queues. The size of such systems imposes difficulties in either applying numerical procedures or in using simulations. We address the problem of whether a very large multi-queue single-server system (polling system), consisting of 100 or more queues, can be modeled by a significantly smaller system without considerably distorting its performance. In particular we study systems in which the service discipline is either exhaustive or gated and the service times in the different queues are identically distributed. We consider the delay incurred by an arbitrary customer in the system as the performance measure of interest. The main result of this paper is that in this framework a polling system consisting of several queues can approximate the behavior of a very large system fairly accurately. However, an approximation by a system consisting of a single queue (with vacation periods) will yield a fairly poor approximation. We propose an algorithm for transforming the original system (called System A) into the approximate system (called System B). We discuss the errors introduced by this transformation and provide bounds for the error in the estimates of the mean customer delay. Numerical results show that System B is good for predicting the tail probabilities of System A as well.