, Volume 50, Issue 4, pp 582–603

Computational analysis of bulk service queue with Markovian arrival process: MAP/R(a, b)/1 queue

Theoretical Article

DOI: 10.1007/s12597-013-0128-3

Cite this article as:
Singh, G., Gupta, U.C. & Chaudhry, M.L. OPSEARCH (2013) 50: 582. doi:10.1007/s12597-013-0128-3


In recent years, queueing models with Markovian arrival process have attracted interest among researchers due to their applications in telecommunications. Such models are generally dealt with matrix-analytic method which appears to be powerful analytically despite the fact that it has numerical difficulties. However, analyzing such queues with the method of roots is always a neglected part since it was assumed that such models are difficult to analyze using roots. In this paper, we consider a bulk service queue with Markovian arrival process and analyze it using the method of roots and present a simple closed-form analysis for evaluating queue-length distribution at a post-departure epoch in terms of roots of the characteristic equation associated with the MAP/R \(^{(a,b)}\)/1 queue, where R represents the class of distributions whose Laplace–Stieltjes transforms are rational functions. We also obtain queue-length distributions at arbitrary epochs. Numerical aspects have been tested for a variety of arrival and service-time (including matrix-exponential (ME)) distributions and a sample of numerical outputs is presented. We hope that the proposed method should be useful for practitioners of queueing theory.


Batch service General bulk service rule Markovian arrival process (MAP) Matrix-exponential (ME) Phase-type (PH) Queueing Queue-length Rational Laplace–Stieltjes transform Roots 

Copyright information

© Operational Research Society of India 2013

Authors and Affiliations

  • Gagandeep Singh
    • 1
  • U. C. Gupta
    • 1
  • M. L. Chaudhry
    • 2
  1. 1.Department of MathematicsIndian Institute of TechnologyKharagpurIndia
  2. 2.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada

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