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Mathematical Methods of Operations Research

, Volume 83, Issue 3, pp 351–372 | Cite as

Functional central limit theorems for Markov-modulated infinite-server systems

  • J. Blom
  • K. De TurckEmail author
  • M. Mandjes
Article

Abstract

In this paper we study the Markov-modulated M/M/\(\infty \) queue, with a focus on the correlation structure of the number of jobs in the system. The main results describe the system’s asymptotic behavior under a particular scaling of the model parameters in terms of a functional central limit theorem. More specifically, relying on the martingale central limit theorem, this result is established, covering the situation in which the arrival rates are sped up by a factor N and the transition rates of the background process by \(N^\alpha \), for some \(\alpha >0\). The results reveal an interesting dichotomy, with crucially different behavior for \(\alpha >1\) and \(\alpha <1\), respectively. The limiting Gaussian process, which is of the Ornstein–Uhlenbeck type, is explicitly identified, and it is shown to be in accordance with explicit results on the mean, variances and covariances of the number of jobs in the system.

Keywords

Queues Infinite-server systems Markov modulation  Central limit theorems 

Notes

Acknowledgments

The authors thank Peter Spreij (Korteweg-de Vries Institute for Mathematics, University of Amsterdam) for useful suggestions.

References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.CWIAmsterdamThe Netherlands
  3. 3.Laboratoire Signaux et Systèmes (L2S, CNRS UMR8506), École CentraleSupélecUniversité Paris SaclayGif-sur-YvetteFrance
  4. 4.EURANDOMEindhoven University of TechnologyEindhovenThe Netherlands
  5. 5.IBIS, Faculty of Economics and BusinessUniversity of AmsterdamAmsterdamThe Netherlands

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