A semi-Markov queue with exponential service times

  • J. H. A. de Smit
  • G. J. K. Regterschot

Abstract

In de Smit (1985) a general model for the single server semi-Markov queue was studied. Its solution was reduced to the Wiener-Hopf factorization of a matrix function. For the general case a formal factorization can be given in which the factors have probabilistic interpretations but cannot be calculated explicitly (see Arjas (1972)). Explicit factorizations can only be given in special cases. The present paper deals with the case in which customers arrive according to a Markov renewal process, while an arriving customer requires an exponential service time whose mean depends on the state of the Markov renewal process at his arrival. For this special case we shall obtain an explicit factorization. The model is formally described as follows.

Keywords

Queue Length Imaginary Axis Implicit Function Theorem Busy Period Interarrival Time 
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References

  1. Arjas, E. (1972). On the use of a fundamental identity in the theory of semi-Markov queues. Adv.Appl.Prob. 4, 271–284.MathSciNetMATHCrossRefGoogle Scholar
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  4. de Smit, J.H.A. (1985). The single server semi-Markov queue. To appear in Stochastic Process.Appl.Google Scholar
  5. Latouche, G. (1983). An exponential semi-Markov process with applications to queueing theory. Report Laboratoire d’Informatique Théorique, Université Libre de Bruxelles.Google Scholar
  6. Regterschot, G.J.K. and J.H.A. de Smit (1985). The queue M|G|1 with Markov modulated arrivals and services. To appear in Math.Oper.Res.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • J. H. A. de Smit
    • 1
  • G. J. K. Regterschot
    • 1
  1. 1.Twente University of TechnologyEnschedeThe Netherlands

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