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The Mn/Gn/1 queue with vacations and exhaustive service

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  • Oz, Binyamin
  • Adan, Ivo
  • Haviv, Moshe

Abstract

We consider the Mn/Gn/1 queue with vacations and exhaustive service in which the server takes (repeated) vacations whenever it becomes idle, the service time distribution is queue length dependent, and the arrival rate varies both with the queue length and with the status of the server, being busy or on vacation. Using a rate balance principle, we derive recursive formulas for the conditional distribution of residual service or vacation time given the number of the customers in the system and the status of the server. We also derive a closed-form expression for the steady-state distribution as a function of the probability of an empty system. As an application of the above, we provide a recursive computation method for Nash equilibrium joining strategies to the observable M/G/1 queue with vacations.

Suggested Citation

  • Oz, Binyamin & Adan, Ivo & Haviv, Moshe, 2019. "The Mn/Gn/1 queue with vacations and exhaustive service," European Journal of Operational Research, Elsevier, vol. 277(3), pages 945-952.
    • Handle: RePEc:eee:ejores:v:277:y:2019:i:3:p:945-952
    DOI: 10.1016/j.ejor.2019.03.016
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    References listed on IDEAS

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    1. Refael Hassin & Moshe Haviv, 2002. "Nash Equilibrium and Subgame Perfection in Observable Queues," Annals of Operations Research, Springer, vol. 113(1), pages 15-26, July.
    2. Boxma, O. J., 1984. "Joint distribution of sojourn time and queue length in the M/G/1 queue with (in) finite capacity," European Journal of Operational Research, Elsevier, vol. 16(2), pages 246-256, May.
    3. Demetrios Fakinos, 1982. "Technical Note—The Expected Remaining Service Time in a Single Server Queue," Operations Research, INFORMS, vol. 30(5), pages 1014-1018, October.
    4. Binyamin Oz & Ivo Adan & Moshe Haviv, 2017. "A rate balance principle and its application to queueing models," Queueing Systems: Theory and Applications, Springer, vol. 87(1), pages 95-111, October.
    5. Kerner, Yoav, 2011. "Equilibrium joining probabilities for an M/G/1 queue," Games and Economic Behavior, Elsevier, vol. 71(2), pages 521-526, March.
    6. 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.
    7. Athanasia Manou & Antonis Economou & Fikri Karaesmen, 2014. "Strategic Customers in a Transportation Station: When Is It Optimal to Wait?," Operations Research, INFORMS, vol. 62(4), pages 910-925, August.
    8. 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.
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    Cited by:

    1. Hanukov, Gabi, 2022. "Improving efficiency of service systems by performing a part of the service without the customer's presence," European Journal of Operational Research, Elsevier, vol. 302(2), pages 606-620.

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