M/M/1 queue in two alternating environments and its heavy traffic approximation

TL;DR

This paper studies an M/M/1 queue with two alternating environments governed by a Markov chain, deriving steady-state and transient distributions, and introduces a heavy-traffic approximation leading to a PDE-based continuous process.

Contribution

It provides a novel analysis of an M/M/1 queue in switching environments, including steady-state, transient distributions, and a heavy-traffic approximation with PDE characterization.

Findings

Steady-state distribution expressed as a mixture of two geometric distributions.

Transient distribution and busy period analysis in special cases.

Heavy-traffic approximation results with PDE for the limiting process.

Abstract

We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of the process in terms of a generalized mixture of two geometric distributions. In the special case when only one kind of switch is allowed, we analyze the transient distribution, and investigate the busy period problem. The analysis is also performed by means of a suitable heavy-traffic approximation which leads to a continuous random process. Its distribution satisfies a partial differential equation with randomly alternating infinitesimal moments. For the approximating process we determine the steady-state distribution, the transient distribution and a first-passage-time density.