Relaxation Time for the Discrete D/G/1 Queue

Abstract

When queueing models are used for performance analysis of some stochastic system, it is usually assumed that the system is in steady-state. Whether or not this is a realistic assumption depends on the speed at which the system tends to its steady-state. A characterization of this speed is known in the queueing literature as relaxation time. The discrete D/G/1 queue has a wide range of applications. We derive relaxation time asymptotics for the discrete D/G/1 queue in a purely analytical way, mostly relying on the saddle point method. We present a simple and useful approximate upper bound which is sharp in case the load on the system is not very high. A sharpening of this upper bound, which involves the complementary error function, is then developed and this covers both the cases of low and high loads. For the discrete D/G/1 queue, the stationary waiting time distribution can be expressed in terms of infinite series that follow from Spitzer's identity. These series involve convolutions of the probability distribution of a discrete random variable, which makes them suitable for computation. For practical purposes, though, the infinite series should be truncated. The relaxation time asymptotics can be applied to determine an appropriate truncation level based on a sharp estimate of the error caused by truncating.