Markov-modulated infinite-server queues with general service times

  • Authors:

  • J. Blom;O. Kella;M. Mandjes;H. Thorsdottir

  • Affiliations:

  • CWI, Amsterdam, The Netherlands;Department of Statistics, The Hebrew University of Jerusalem, Jerusalem, Israel;CWI, Amsterdam, The Netherlands and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands;CWI, Amsterdam, The Netherlands and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 2014

Quantified Score

Hi-index 0.00

Visualization

Abstract

This paper analyzes several aspects of the Markov-modulated infinite-server queue. In the system considered (i)聽particles arrive according to a Poisson process with rate $$\lambda _i$$驴i when an external Markov process ("background process") is in state $$i$$i, (ii)聽service times are drawn from a distribution with distribution function $$F_i(\cdot )$$Fi(路) when the state of the background process (as seen at arrival) is $$i$$i, (iii)聽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 $$t\ge 0$$t驴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 $$N$$N, and the transition times by a factor $$N^{1+\varepsilon }$$N1+驴 (for some $$\varepsilon 0$$驴0). Under this scaling it turns out that the number of customers at time $$t\ge 0$$t驴0 obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.