Functional central limit theorems for Markov-modulated infinite-server systems

TL;DR

This paper establishes a functional central limit theorem for Markov-modulated M/M/∞ queues, revealing different asymptotic behaviors depending on the scaling of background process rates, and explicitly characterizing the limiting Gaussian process.

Contribution

It provides the first detailed functional limit theorem for the correlation structure of Markov-modulated infinite-server queues under various scaling regimes.

Findings

Different behaviors for >1 and <1 regimes

Explicit identification of the Ornstein-Uhlenbeck type limiting process

Alignment with known mean, variance, and covariance results

Abstract

In this paper we study the Markov-modulated M/M/$\infty$ queue, with a focus on the correlation structure of the number of jobs in the system. The main results describe the system's asymptotic behavior under a particular scaling of the model parameters in terms of a functional central limit theorem. More specifically, relying on the martingale central limit theorem, this result is established, covering the situation in which the arrival rates are sped up by a factor $N$ and the transition rates of the background process by $N^\alpha$, for some $\alpha>0$. The results reveal an interesting dichotomy, with crucially different behavior for $\alpha>1$ and $\alpha<1$, respectively. The limiting Gaussian process, which is of the Ornstein-Uhlenbeck type, is explicitly identified, and it is shown to be in accordance with explicit results on the mean, variances and covariances of the number of jobs in the system.