Time-Dependent Queues

Abstract

A single server queue is considered having exponentially distributed inter-arrival and service times with slowly changing time-dependent rates $\lambda (\varepsilon t)$ and $\mu (\varepsilon t)$. The parameter $\varepsilon $ is the ratio of an average inter-arrival time to the time over which the rates change appreciably, so it is small. Therefore an asymptotic solution, valid for $\varepsilon $ small, is constructed for the time-dependent queue length probability distribution. It consists of five typical parts corresponding to five typical time periods. They are the initial period, the period of light traffic when ${\lambda / \mu } < 1$, the saturation transition period when ${\lambda / \mu }$ increases through unity, the oversaturation period when ${\lambda / \mu }$ starts out greater than unity and then decreases below unity, and the transition period at the end of oversaturation, when the queue returns to the light traffic condition. By combining the solutions for these five intervals, the solution for any queue with slowly varying rates can be obtained. Some of these parts were found previously by G. F. Newell [1], [2], and some of the formal results have been shown to be asymptotic by W. A. Massey [3].