The Gated Infinite-Server Queue: Uniform Service Times

Abstract

Customers, in a Poisson stream at rate $\lambda $, enter an infinite-server queue. Customer service times are independent and uniformly distributed on $[ 0,s ],\, s > 0$. Gated service is performed in stages as follows. A stage begins with all customers transferred from the queue to the servers. The servers then begin serving these customers, all simultaneously. The stage ends when the service of all customers is complete. At this point, the next stage begins if the queue is nonempty. If the queue is empty, the servers just remain idle, awaiting the next arrival, at which time the next stage begins. This paper develops asymptotics for the equilibrium distribution of the number served in a stage in the light and heavy traffic regimes $\lambda \to 0$ and $\lambda \to \infty $, respectively. The results are obtained by the analysis of a Fredholm integral equation of the second kind. For computational purposes, the integral equation is transformed into an infinite system of linear algebraic equations. The effect of truncating the system to a finite size is then examined.