2014-02-01

# Analysis of Markov-modulated infinite-server queues in the central-limit regime

## Publication

### Publication

This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix $Q\equiv(q_{ij})_{i,j=1}^d$. {Both arrival rates and service rates are depending on the state of the background process.} The main contribution concerns the derivation of central limit theorems for the number of customers in the system at time $t\ge 0$, in the asymptotic regime in which the arrival rates $\lambda_i$ are scaled by a factor $N$, and the transition rates $q_{ij}$ by a factor $N^\alpha$, with $\alpha \in \mathbb R^+$. The specific value of $\alpha$ has a crucial impact on the result: (i)~for $\alpha>1$ the system essentially behaves as an M/M/$\infty$ queue, and in the central limit theorem the centered process has to be normalized by $\sqrt{N}$; (ii)~for $\alpha<1$, the centered process has to be normalized by $N^{{1-}\alpha/2}$, with the deviation matrix appearing in the expression for the variance.

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Cornell University Library | |

arXiv.org e-Print archive | |

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Blom, J.G, de Turck, K, & Mandjes, M.R.H. (2014).
Analysis of Markov-modulated infinite-server queues in the central-limit regime. arXiv.org e-Print archive. Cornell University Library . |