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Discrete-time queue with batch renewal input and random serving capacity rule: $$GI^X/ Geo^Y/1$$ G I X / G e o Y / 1

Author

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  • F. P. Barbhuiya

    (Indian Institute of Technology Kharagpur)

  • U. C. Gupta

    (Indian Institute of Technology Kharagpur)

Abstract

In this paper, we provide a complete analysis of a discrete-time infinite buffer queue in which customers arrive in batches of random size such that the inter-arrival times are arbitrarily distributed. The customers are served in batches by a single server according to the random serving capacity rule, and the service times are geometrically distributed. We model the system via the supplementary variable technique and further use the displacement operator method to solve the non-homogeneous difference equation. The analysis done using these methods results in an explicit expression for the steady-state queue-length distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the underlying characteristic equation. Our approach enables one to estimate the asymptotic distribution at a pre-arrival epoch by a unique largest root of the characteristic equation lying inside the unit circle. With the help of few numerical results, we demonstrate that the methodology developed throughout the work is computationally tractable and is suitable for light-tailed inter-arrival distributions and can also be extended to heavy-tailed inter-arrival distributions. The model considered in this paper generalizes the previous work done in the literature in many ways.

Suggested Citation

  • F. P. Barbhuiya & U. C. Gupta, 2019. "Discrete-time queue with batch renewal input and random serving capacity rule: $$GI^X/ Geo^Y/1$$ G I X / G e o Y / 1," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 347-365, April.
    • Handle: RePEc:spr:queues:v:91:y:2019:i:3:d:10.1007_s11134-019-09600-7
    DOI: 10.1007/s11134-019-09600-7
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    References listed on IDEAS

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    1. Seok Ho Chang & Dae Won Choi, 2006. "Modeling and Performance Analysis of a Finite-Buffer Queue with Batch Arrivals, Batch Services, and Setup Times: The M X /G Y /1/K + B Queue with Setup Times," INFORMS Journal on Computing, INFORMS, vol. 18(2), pages 218-228, May.
    2. Dorit S. Hochbaum & Dan Landy, 1997. "Scheduling Semiconductor Burn-In Operations to Minimize Total Flowtime," Operations Research, INFORMS, vol. 45(6), pages 874-885, December.
    3. Lev Abolnikov & Alexander Dukhovny, 1991. "Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications," International Journal of Stochastic Analysis, Hindawi, vol. 4, pages 1-23, January.
    4. Carl M. Harris & Percy H. Brill & Martin J. Fischer, 2000. "Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis," INFORMS Journal on Computing, INFORMS, vol. 12(4), pages 261-271, November.
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    Cited by:

    1. Nitin Kumar & F. P. Barbhuiya & U. C. Gupta, 2020. "Unified killing mechanism in a single server queue with renewal input," OPSEARCH, Springer;Operational Research Society of India, vol. 57(1), pages 246-259, March.

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