Analyzing M/M/1 Queues with Perturbations in the Arrival Process

Abstract

This paper investigates when the M/M/1 model can be used to predict accurately the operating characteristics of queues with arrival processes that are slightly different from the Poisson process assumed in the model. The arrival processes considered here are perturbed Poisson processes. The perturbations are deviations from the exponential distribution of the inter-arrival times or from the assumption of independence between successive inter-arrival times. An estimate is derived for the difference between the expected numbers in perturbed and M/M/1 queueing systems with the same traffic intensity. The results, for example, indicate that the M/M/1 model can predict the performance of the queue when the arrival process is perturbed by inserting a few short inter-arrival times, an occasional batch arrival or small dependencies between successive inter-arrival times. In contrast, the M/M/1 is not a good model when the arrival process is perturbed by inserting a few long inter-arrival times.