Markov-modulated infinite-server queues with general service times

J. Blom*, O. Kella, M. Mandjes, H. Thorsdottir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Scopus citations

Abstract

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 λi when an external Markov process ("background process") is in state i, (ii) service times are drawn from a distribution with distribution function Fi(·) 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 N1+ε(for some ε >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.

Original languageAmerican English
Pages (from-to)403-424
Number of pages22
JournalQueueing Systems
Volume76
Issue number4
DOIs
StatePublished - Apr 2014

Keywords

  • Fluid and diffusion scaling
  • General service times
  • Infinite-server systems
  • Laplace transforms
  • Markov modulation
  • Markov-modulated Poisson process
  • Queues

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