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A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations Under Late and Early Arrival System

Author

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  • F. P. Barbhuiya

    (Indian Institute of Technology Kharagpur)

  • U. C. Gupta

    (Indian Institute of Technology Kharagpur)

Abstract

This paper studies a discrete-time batch arrival GI/Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature.

Suggested Citation

  • F. P. Barbhuiya & U. C. Gupta, 2020. "A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations Under Late and Early Arrival System," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 599-624, June.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:2:d:10.1007_s11009-019-09724-6
    DOI: 10.1007/s11009-019-09724-6
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    References listed on IDEAS

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    1. S K Samanta & U C Gupta & R K Sharma, 2007. "Analysis of finite capacity discrete-time GI/Geo/1 queueing system with multiple vacations," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(3), pages 368-377, March.
    2. Naishuo Tian & Zhe George Zhang, 2006. "Vacation Queueing Models Theory and Applications," International Series in Operations Research and Management Science, Springer, number 978-0-387-33723-4, April.
    3. Naishuo Tian & Zhe George Zhang, 2006. "Applications of Vacation Models," International Series in Operations Research & Management Science, in: Vacation Queueing Models Theory and Applications, chapter 0, pages 343-358, Springer.
    Full references (including those not matched with items on IDEAS)

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