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

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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)

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.

Suggested Citation

  • M. L. Chaudhry & Gagandeep Singh & U. C. Gupta, 2013. "A Simple and Complete Computational Analysis of MAP/R/1 Queue Using Roots," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 563-582, September.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:3:d:10.1007_s11009-011-9266-3
    DOI: 10.1007/s11009-011-9266-3
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    References listed on IDEAS

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    1. Ho Woo Lee & Jong Min Moon & Jong Keun Park & Byung Kyu Kim, 2003. "A spectral approach to compute the mean performance measures of the queue with low-order BMAP input," International Journal of Stochastic Analysis, Hindawi, vol. 16, pages 1-12, January.
    2. Mohan L. Chaudhry & Carl M. Harris & William G. Marchal, 1990. "Robustness of Rootfinding in Single-Server Queueing Models," INFORMS Journal on Computing, INFORMS, vol. 2(3), pages 273-286, August.
    3. Sadrac K. Matendo, 1994. "Some performance measures for vacation models with a batch Markovian arrival process," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-14, January.
    4. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
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    Cited by:

    1. S. Pradhan & U. C. Gupta, 2019. "Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process," Annals of Operations Research, Springer, vol. 277(2), pages 161-196, June.
    2. Nitin Kumar & U. C. Gupta, 2020. "A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1293-1324, September.
    3. S. K. Samanta, 2020. "Waiting-time analysis of D-$${ BMAP}{/}G{/}1$$BMAP/G/1 queueing system," Annals of Operations Research, Springer, vol. 284(1), pages 401-413, January.
    4. Miaomiao Yu & Yinghui Tang, 2018. "Analysis of the Sojourn Time Distribution for M/GL/1 Queue with Bulk-Service of Exactly Size L," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1503-1514, December.
    5. M. L. Chaudhry & A. D. Banik & A. Pacheco, 2017. "A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞," Annals of Operations Research, Springer, vol. 252(1), pages 135-173, May.
    6. H. Bruneel & W. Rogiest & J. Walraevens & S. Wittevrongel, 2015. "Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(3), pages 285-315, December.
    7. Michiel Muynck & Herwig Bruneel & Sabine Wittevrongel, 2020. "Analysis of a queue with general service demands and correlated service capacities," Annals of Operations Research, Springer, vol. 293(1), pages 73-99, October.

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