Long time behavior of a stochastically modulated infinite server queue

TL;DR

This paper analyzes the long-term behavior of an infinite server queue with stochastic environment modulation, providing explicit descriptions of the limiting distribution and a sampling scheme, including an example of system transience.

Contribution

It offers a novel explicit characterization of the queue's limiting distribution under semi-Markovian modulation and introduces a sampling scheme with convergence diagnostics.

Findings

Explicit limiting measure described via affine stochastic recurrence

Sampling scheme with convergence diagnostics proposed

Example demonstrating system transience under stochastic environment

Abstract

We consider an infinite server queue where the arrival and the service rates are both modulated by a stochastic environment governed by an $S$-valued stochastic process $X$ that is ergodic with a limiting measure $\pi\in \mathcal{P}(S)$. Under certain conditions when $X$ is semi-Markovian and satisfies the renewal regenerative property, long-term behavior of the total counts of people in the queue (denoted by $Y:=(Y_{t}:t\ge 0)$) becomes explicit and the limiting measure of $Y$ can be described through a well-studied affine stochastic recurrence equation (SRE) $X\stackrel{d}{=}CX+D,\,\, X\perp\!\!\!\perp (C, D)$. We propose a sampling scheme from that limiting measure with explicit convergence diagnostics. Additionally, one example is presented where the stochastic environment makes the system transient, in absence of a `no-feedback' assumption.