Queues with time-dependent arrival rates I—the transition through saturation

Abstract

Suppose that the arrival rate λ(t) of customers to a service facility increases with time at a nearly constant rate, dλ(t)/dt = a, so as to pass through the saturation condition, λ(t) = μ = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time Tproportional to α^–2/3 such that if , then the queue distribution stays close to the prevailing equilibrium distribution associated with the λ(t) and μ, evaluated at time t. For |t| = O(T), however, the mean queue length is much less than the equilibrium mean, and is measured in units of some characteristic length L which is proportional to α^–1/3. For , the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units.