Vacations in GI^X/M^Y/1 systems and Riemann boundary value problems

  • Authors:

  • Alexander Dukhovny

  • Affiliations:

  • Department of Mathematics, San Francisco State University, San Francisco, CA 94132, USA E-mail: dukhovny@math.sfsu.edu

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 1997

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Abstract

We consider systems of GI/M/1 type with bulk arrivals, bulk service and exponential server vacations. The generating functions of the steady-state probabilities of the embedded Markov chain are found in terms of Riemann boundary value problems, a necessary and sufficient condition of ergodicity is proved. Explicit formulas are obtained for the case where the generating function of the arrival group size is rational. Resonance between the vacation rate and the system is studied. Complete formulas are given for the cases of single and geometric arrivals.