, 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 serviceGeneral bulk service ruleMarkovian arrival process (MAP)Matrix-exponential (ME)Phase-type (PH)QueueingQueue-lengthRational Laplace–Stieltjes transformRoots

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