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A Simple and Complete Computational Analysis of MAP/R/1 Queue Using Roots

M. L. Chaudhry (), Gagandeep Singh () and U. C. Gupta ()
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M. L. Chaudhry: Royal Military College of Canada
Gagandeep Singh: Indian Institute of Technology
U. C. Gupta: Indian Institute of Technology

Methodology and Computing in Applied Probability, 2013, vol. 15, issue 3, 563-582

Abstract: Abstract In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at three epochs of time (arbitrary, pre-arrival, and post-departure) and queueing-time distribution (virtual and actual) of the MAP/R/1 queue, where R represents the class of distributions whose Laplace–Stieltjes transforms are rational functions. Our analysis is based on roots of the associated characteristic equations of the (i) vector-generating function of system-length distribution and (ii) Laplace–Stieltjes transform of the virtual queueing-time distribution. The proposed method for evaluating boundary probabilities is an alternative to the matrix-analytic method as well as spectral method. 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. The method is analytically quite simple and easy to implement. It is hoped that the results obtained would prove to be beneficial to both theoreticians and practitioners.

Keywords: Queueing; Markovian arrival process (MAP); Queueing-time; Roots; System-length; Rational Laplace–Stieltjes transform; Matrix-exponential (ME); Phase-type (PH); Primary 60K25; Secondary 90B25 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s11009-011-9266-3

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