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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 / ∞

Author

Listed:
  • M. L. Chaudhry

    (Royal Military College of Canada)

  • A. D. Banik

    (Indian Institute of Technology
    Universidade de Lisboa)

  • A. Pacheco

    (Universidade de Lisboa)

Abstract

We consider a batch arrival infinite-buffer single-server queue with generally distributed inter-batch arrival times with arrivals occurring in batches of random sizes. The service process is correlated and its structure is governed by a Markovian service process in continuous time. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at a pre-arrival epoch. We also obtain the steady-state probability distribution at an arbitrary epoch using the classical argument based on Markov renewal theory. Some important performance measures such as the average number of customers in the system and the mean sojourn time have also been obtained. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:252:y:2017:i:1:d:10.1007_s10479-015-2026-y
    DOI: 10.1007/s10479-015-2026-y
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    References listed on IDEAS

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    1. John F. Shortle & Percy H. Brill & Martin J. Fischer & Donald Gross & Denise M. B. Masi, 2004. "An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue," INFORMS Journal on Computing, INFORMS, vol. 16(2), pages 152-161, May.
    2. António Pacheco & Helena Ribeiro, 2008. "Consecutive customer losses in oscillating GI X /M//n systems with state dependent services rates," Annals of Operations Research, Springer, vol. 162(1), pages 143-158, September.
    3. 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.
    4. A. Banik & U. Gupta, 2007. "Analyzing the finite buffer batch arrival queue under Markovian service process: GI X /MSP/1/N," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(1), pages 146-160, July.
    5. 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.
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    Cited by:

    1. Souvik Ghosh & A. D. Banik, 2018. "Computing conditional sojourn time of a randomly chosen tagged customer in a $$\textit{BMAP/MSP/}1$$ BMAP / MSP / 1 queue under random order service discipline," Annals of Operations Research, Springer, vol. 261(1), pages 185-206, February.
    2. Abhijit Datta Banik & Souvik Ghosh & M. L. Chaudhry, 2020. "On the optimal control of loss probability and profit in a GI/C-BMSP/1/N queueing system," OPSEARCH, Springer;Operational Research Society of India, vol. 57(1), pages 144-162, March.

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