Long time behavior of a stochastically modulated infinite server queue

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) $XCX+D,X\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp (C,D)$. 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.