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Equilibrium Strategies in an M n /M/1 Queue with Server Breakdowns and Delayed Repairs

Author

Listed:
  • Yingying Pan

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China)

  • Jingchuan Zhang

    (Alibaba Business School, Hangzhou Normal University, Hangzhou 311121, China)

  • Zaiming Liu

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China)

Abstract

Ophthalmic units use sophisticated equipment to enable accurate diagnosis of refractive errors. This equipment is subject to two types of breakdowns. One is simple breakdown which can be repaired in-house, and the other is complex breakdown which requires repair by the original equipment manufacturer (OEM) and results in a delay time between the server breakdown and the start of the repair. In this paper, we model this scenario as an M n /M/1 queuing system with two types of breakdowns and delayed repairs due to complex breakdowns, where the delay time and repair times for simple and complex breakdowns are generally distributed. We obtain the steady-state probabilities and provide the recursive formulas for the Laplace–Stieltjes transforms (LSTs) of conditional residual delay time and repair times given the system state. For the fully observable case, we derive the equilibrium joining strategies of customers who decide to join or balk based on their observation of the system state. Moreover, two numerical experiments are conducted to explore the equilibrium joining probabilities.

Suggested Citation

  • Yingying Pan & Jingchuan Zhang & Zaiming Liu, 2024. "Equilibrium Strategies in an M n /M/1 Queue with Server Breakdowns and Delayed Repairs," Mathematics, MDPI, vol. 12(23), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3695-:d:1529242
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    References listed on IDEAS

    as
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    2. Shone, Rob & Knight, Vincent A. & Williams, Janet E., 2013. "Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior," European Journal of Operational Research, Elsevier, vol. 227(1), pages 133-141.
    3. Hassin, Refael, 1986. "Consumer Information in Markets with Random Product Quality: The Case of Queues and Balking," Econometrica, Econometric Society, vol. 54(5), pages 1185-1195, September.
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    Full references (including those not matched with items on IDEAS)

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