Asymptotics for infinite server queues with fast/slow Markov switching and fat tailed service times

TL;DR

This paper analyzes the asymptotic behavior of infinite server queues with Markov switching, fat-tailed service times, and varying arrival rates, identifying different regimes with distinct limiting processes.

Contribution

It introduces a comprehensive analysis of infinite server queues with fat-tailed service times under different scaling regimes, revealing new limit behaviors.

Findings

Identifies two main regimes: 'fast arrivals' and 'equilibrium' with convergence to limiting processes.

In the 'slow arrivals' regime, establishes convergence of the first two moments of the process.

Provides a unified framework for understanding queue dynamics with heavy-tailed services and Markov switching.

Abstract

We study a general $k$ dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index $\alpha\in (0,1)$. When the arrival rate is sped up by a factor $n^\gamma$, the transition probabilities of the underlying Markov chain are divided by $n^\gamma$ and the service times are divided by $n$, we identify two regimes (''fast arrivals'', when $\gamma>\alpha$, and ''equilibrium'', when $\gamma=\alpha$) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third ''slow arrivals'' regime, $\gamma<\alpha$, we show the convergence of the two first joint moments of the rescaled process.