Computational analysis of bulk service queue with Markovian arrival process: MAP/R (a, b) /1 queue
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Computational analysis of bulk service queue with Markovian arrival process: MAP/R(a,b) /1 queue Gagandeep Singh · U. C. Gupta · M. L. Chaudhry
Accepted: 27 February 2013 © Operational Research Society of India 2013
Abstract 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 matrixexponential (ME)) distributions and a sample of numerical outputs is presented. We hope that the proposed method should be useful for practitioners of queueing theory.
G. Singh () · U. C. Gupta Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India e-mail: [email protected]; [email protected] U.C. Gupta e-mail: [email protected] M. L. Chaudhry Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston, ON, Canada K7K 7B4 e-mail: [email protected]
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Keywords Batch service · General bulk service rule · Markovian arrival process (MAP) · Matrix-exponential (ME) · Phase-type (PH) · Queueing · Queue-length · Rational Laplace–Stieltjes transform · Roots 1 Introduction Bulk service queues have received considerable attention due to their wide applications in several areas including computer-communication, telecommunication, transportation, and manufacturing systems. In the past few decades, the analytic and numerical aspects of the M/G(a,b) /1 queue (both finite and infinite buffer) and their applications have been discussed by several authors, see, e.g., Neuts [31, 32], Chaudhry et al. [17], Chaudhry and Gupta [15]. Discrete time queues viz., GeoX /Geoc /1 and GeoX /Dc /1 have been analyzed by Claeys et al. [21, 22]. Bulk service queues with batch-size dependent service have recently been analyzed by BarLev et al. [9], Chaudhry and Gai [14], and more recently by Claeys et al. [23] who discuss the queue GoeX /G(a,b) /1 with batch-size dependent service. Banerjee and Gupta (a,b) [7] also analyze the finite buffer queue M/Gr /1/N with batch-size dependent service. They obtain the joint distribution of the number of customers in queue and the number in se
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