2014-09-01

# 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.

Additional Metadata | |
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Infinite-server queues, Markov modulation, central limit theorem, deviation matrices | |

Life Sciences (theme 5) | |

Cambridge U.P. | |

Probability in the Engineering and Informational Sciences | |

Coarse grained stochastic methods for biochemical reactions | |

Organisation | Life Sciences and Health |

Blom, J.G, de Turck, K, & Mandjes, M.R.H. (2014). Analysis of Markov-modulated infinite-server queues in the central-limit regime.
Probability in the Engineering and Informational Sciences. |