Perturbation analysis of an M/M/1 queue in a diffusion random environment

TL;DR

This paper analyzes an M/M/1 queue with a server rate influenced by an Ornstein-Uhlenbeck process, establishing differential systems, proving solution uniqueness, and performing perturbation analysis to validate a reduced service rate approximation.

Contribution

It develops a differential system for the queue's joint distribution, proves solution uniqueness under bounded perturbations, and validates a simplified approximation through perturbation analysis.

Findings

Differential system for the queue and environment is established.

Solution uniqueness is proven under certain conditions.

Perturbation analysis confirms the reduced service rate approximation.

Abstract

We study in this paper an $M/M/1$ queue whose server rate depends upon the state of an independent Ornstein-Uhlenbeck diffusion process $(X(t))$ so that its value at time $t$ is $\mu \phi(X(t))$, where $\phi(x)$ is some bounded function and $\mu>0$. We first establish the differential system for the conditional probability density functions of the couple $(L(t),X(t))$ in the stationary regime, where $L(t)$ is the number of customers in the system at time $t$. By assuming that $\phi(x)$ is defined by $\phi(x) = 1-\varepsilon ((x\wedge a/\varepsilon)\vee(-b/\varepsilon))$ for some positive real numbers $a$, $b$ and $\varepsilon$, we show that the above differential system has a unique solution under some condition on $a$ and $b$. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when $\phi$ is replaced with $\Phi(x)=1-\varepsilon x$ for sufficiently small $\varepsilon$. We finally perform a perturbation analysis of this latter solution for small $\varepsilon$. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to $\mu(1-\eps\E(X(t)))$.