Relaxation Time for the Discrete D/G/1 Queue

For the discrete D/G/1 queue, the stationary waiting time 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. We therefore seek for some means to characterize the speed at which these series converge. Such a characterization is related to the notion of relaxation time in queueing theory, a generic term for the time required for a transient system to reach its stationary regime. 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 may serve as a stopping criterium for the number of convolutions to calculate 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.