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Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment

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  • Nurehemaiti Yiming

    (College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China)

Abstract

In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C 0 —semigroup theory, we prove that the system has a unique positive time-dependent solution (TDS) that satisfies the probability condition. As our second result, we prove that the TDS of the system strongly converges to its steady-state solution (SSS) if the service rates of the servers are constants. For this asymptotic behavior, we analyze the spectrum of the system operator associated with the system. Additionally, the stability of the semigroup generated by the system operator is also discussed.

Suggested Citation

  • Nurehemaiti Yiming, 2024. "Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment," Mathematics, MDPI, vol. 12(6), pages 1-26, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:6:p:805-:d:1353960
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    References listed on IDEAS

    as
    1. Ghosh, Souvik & Hassin, Refael, 2021. "Inefficiency in stochastic queueing systems with strategic customers," European Journal of Operational Research, Elsevier, vol. 295(1), pages 1-11.
    2. Kailash C. Madan, 1999. "An M/G/1 queue with optional deterministic server vacations," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3-4), pages 83-95.
    3. Qi‐Ming He & James H. Bookbinder & Qishu Cai, 2020. "Optimal policies for stochastic clearing systems with time‐dependent delay penalties," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(7), pages 487-502, October.
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