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October, 1975 Limit Theorems for a $GI/G/\infty$ Queue
Norman Kaplan
Ann. Probab. 3(5): 780-789 (October, 1975). DOI: 10.1214/aop/1176996265

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.

Citation

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Norman Kaplan. "Limit Theorems for a $GI/G/\infty$ Queue." Ann. Probab. 3 (5) 780 - 789, October, 1975. https://doi.org/10.1214/aop/1176996265

Information

Published: October, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0322.60081
MathSciNet: MR413306
Digital Object Identifier: 10.1214/aop/1176996265

Subjects:
Primary: 60K25

Keywords: $GI/G/\infty$ queue , cluster process , point process , regeneration point

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 5 • October, 1975
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