Long time behavior of a stochastically modulated infinite server queue

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.