2014-03-01

# Markov-modulated infinite-server queues with general service times

## Publication

### Publication

*Queueing Systems , Volume 76 - Issue 4 p. 403- 424*

This paper analyzes several aspects of the Markov-modulated infinite-server queue. In the system considered (i) particles arrive according to a Poisson process with rate $\lambda_i$ when an external Markov process (“background process”) is in state $i$ , (ii) service times are drawn from a distribution with distribution function $F_i (·)$ when the state of the background process (as seen at arrival) is $i$ , (iii) there are infinitely many servers.We start by setting up explicit formulas for the mean and variance of the number of particles in the system at time $t ≥ 0$, given the system started empty. The special case of exponential service times is studied in detail, resulting in a recursive scheme to compute the moments of the number of particles at an exponentially distributed time, as well as their steady-state counterparts. Then we consider an asymptotic regime in which the arrival rates are sped up by a factor $N$, and the transition times by a factor $N^{1+\vareps}$ (for some $\vareps > 0$). Under this scaling it turns out that the number of customers at time $t ≥ 0$ obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.

Additional Metadata | |
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Keywords | Markov-modulated Poisson process, General service times, Queues, Infinite-server systems, Markov modulation, Laplace transforms, Fluid and diffusion scaling |

MSC | Queueing theory (msc 60K25), Processes in random environments (msc 60K37), Central limit and other weak theorems (msc 60F05) |

THEME | Life Sciences (theme 5) |

Publisher | Springer |

Persistent URL | dx.doi.org/10.1007/s11134-013-9368-4 |

Journal | Queueing Systems |

Citation |
Blom, J.G, Kella, O, Mandjes, M.R.H, & Thorsdottir, H. (2014). Markov-modulated infinite-server queues with general service times
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Queueing Systems, 76(4), 403–424. doi:10.1007/s11134-013-9368-4 |