Limit Theorems for a $GI/G/\infty$ Queue

Abstract

The $GI/G/\infty$ queue is studied. For the stable case $(\nu = \text{expected service time} < \infty)$, necessary and sufficient conditions are given for the process to to have a legitimate regeneration point. In the unstable case $(\nu = \infty)$, several limit theorems are established. Let $X(t)$ equal the number of servers busy at time $t$. It is proven that when $\nu = \infty$, \begin{equation*}\tag{i}\frac{X(t)}{\lambda(t)} \Rightarrow 1\end{equation*} and \begin{equation*}\tag{ii}\frac{X(t) - \lambda(t)}{\sqrt{\lambda(t)}} \Rightarrow N(0, 1)\end{equation*} where $\lambda(t)$ is a deterministic function. ($\Rightarrow$ means convergence in distribution). A Poisson type limit result is also proved when the arrival of a customer is a rare event.