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Martingale relations for the M[+45 degree rule]GI[+45 degree rule]1 queue with Markov modulated Poisson input

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  • Baccelli, François
  • Makowski, Armand M.

Abstract

This paper is concerned with single server queueing systems with renewal service process and Poisson arrivals modulated by a finite-state Markov chain. Exponential martingales are associated with a chain embedded at service completion epochs in the stochastic process describing the joint evolution of the number of customers in the queue and the state of the environment. The analysis of these martingales leads to a new and unified treatment of various known results concerning the stability condition and the steady state statistics, as well as to several new properties. Noteworthy among them are a conservation law that relates the duration of the busy period to the state of the environment at the end of the busy period, and some absolute continuity properties with respect to other queues of the same type.

Suggested Citation

  • Baccelli, François & Makowski, Armand M., 1991. "Martingale relations for the M[+45 degree rule]GI[+45 degree rule]1 queue with Markov modulated Poisson input," Stochastic Processes and their Applications, Elsevier, vol. 38(1), pages 99-133, June.
    • Handle: RePEc:eee:spapps:v:38:y:1991:i:1:p:99-133
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