On Recurrence of the Infinite Server Queue

TL;DR

This paper investigates the recurrence and stability properties of the infinite-server queue through the maximum-dater recursion $X_{n+1}=\max(X_n - t_{n+1}, s_{n+1})$ under stationary ergodic inputs and contrasts with the single-server setting. It extends the Loynes construction to obtain a stationary version and develops a Markov-chain recurrence framework to classify transience, Harris recurrence, and positive Harris recurrence for $GI/GI/\infty$, including tail behavior of the stationary maximum-dater $\widetilde{X}_\infty$. The authors provide explicit criteria and regenerative-structure arguments, with further sharp tail asymptotics: for exponential service tails $\bar F(x)\sim\delta e^{-\mu x}$, $P(\widetilde{X}_\infty>x)\sim\delta e^{-\mu x}\sum_{n\ge0} E e^{-\mu T_n}$, and for Pareto tails $\bar F(x)\sim \delta x^{-\alpha}$, $P(\widetilde{X}_\infty>x)\sim\frac{\delta}{E t_1 (\alpha-1)} x^{1-\alpha}$. These results highlight how stability and tail behavior depend on service-time tails and compare the infinite-server case to GI/GI/1.

Abstract

This paper concerns the recurrence structure of the infinite server queue, as viewed through the prism of the maximum dater sequence, namely the time to drain the current work in the system as seen at arrival epochs. Despite the importance of this model in queueing theory, we are aware of no complete analysis of the stability behavior of this model, especially in settings in which either or both the inter arrival and service time distributions have infinite mean. In this paper, we fully develop the analog of the Loynes construction of the stationary version in the context of stationary ergodic inputs, extending earlier work of E.Altman (2005), and then classify the Markov chain when the inputs are independent and identically distributed. This allows us to classify the chain, according to transience, recurrence in the sense of Harris, and positive recurrence in the sense of Harris. We further go on to develop tail asymptotics for the stationary distribution of the maximum dater sequence, when the service times have tails that are asymptotically exponential or Pareto, and we contrast the stability theory for the infinite server queue relative to that for the single server queue.

Theorems & Definitions (19)

• Proposition 1
• proof
• Theorem 1
• proof
• Remark 1
• Proposition 2
• proof
• Proposition 3
• Theorem 2
• proof