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Analytically elegant and computationally efficient results in terms of roots for the $$GI^{X}/M/c$$ G I X / M / c queueing system

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  • Mohan L. Chaudhry

    (Royal Military College of Canada)

  • James J. Kim

    (Royal Canadian Air Force (RCAF))

Abstract

An elegant and simple solution to determine the distributions of queue length at different epochs and the waiting time for the model $$GI^{X}/M/c$$ G I X / M / c is presented. In the past, the model $$GI^{X}/M/c$$ G I X / M / c has been extensively analyzed using various techniques by many authors. The purpose of this paper is to present a simple and effective derivation of the analytic solution for pre-arrival epoch probabilities as a linear combination of specific geometric terms (except for the boundary probabilities when the number of servers is greater than the maximum batch size) involving the roots of the underlying characteristic equation. The solution is then leveraged to compute the waiting-time distributions of both first and arbitrary customers of an incoming batch. Numerical examples with various arrival patterns and batch size distributions are also presented. The method that is being proposed here not only gives an alternate solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

Suggested Citation

  • Mohan L. Chaudhry & James J. Kim, 2016. "Analytically elegant and computationally efficient results in terms of roots for the $$GI^{X}/M/c$$ G I X / M / c queueing system," Queueing Systems: Theory and Applications, Springer, vol. 82(1), pages 237-257, February.
    • Handle: RePEc:spr:queues:v:82:y:2016:i:1:d:10.1007_s11134-015-9469-3
    DOI: 10.1007/s11134-015-9469-3
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    References listed on IDEAS

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    1. 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.
    2. P. J. Burke, 1975. "Technical Note—Delays in Single-Server Queues with Batch Input," Operations Research, INFORMS, vol. 23(4), pages 830-833, August.
    3. Ronald W. Wolff, 1982. "Poisson Arrivals See Time Averages," Operations Research, INFORMS, vol. 30(2), pages 223-231, April.
    4. Laoucine Kerbache & G. M. Gontijo & G. S. Atuncar & F.R.B. Cruz, 2011. "Performance Evaluation and Dimensioning of GIX/M/c/N Systems Through Kernel Estimation," Post-Print hal-00796342, HAL.
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    Cited by:

    1. J. J. Kim & M. L. Chaudhry & V. Goswami & A. D. Banik, 2021. "A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 273-289, March.
    2. James J. Kim & Douglas G. Down & Mohan Chaudhry & Abhijit Datta Banik, 2022. "Difference Equations Approach for Multi-Server Queueing Models with Removable Servers," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1297-1321, September.
    3. 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.

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