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Analysis of Vacation Fluid M / M /1 Queue in Multi-Phase Random Environment

Author

Listed:
  • Sherif I. Ammar

    (Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
    Department of Mathematics and Computer Science, Faculty of Science, Menofia University, Shibin El Kom 32511, Egypt)

  • Yousef F. Alharbi

    (Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia)

  • Yiqiang Q. Zhao

    (School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada)

Abstract

An M / M /1 fluid queue with various vacations is studied in the context of a multi-phase random environment. When the system is in operation ( i = 1, 2, …, n ), it behaves according to the M / M /1 fluid queue model. However, in any other situation, the system is on vacation, so this leads it to transition into the vacation phase ( i = 0). This transition occurs only when there is no data in the system. If the system returns from a vacation and finds it still empty of jobs, it will initiate a new vacation and continue in this pattern until jobs become available in the system, at which point it resumes working. When the vacation phase ends, the probability of the system transitioning to the operational phase is denoted as q i ( i = 1, 2, …, n ). Subsequently, we derive the stationary probability and analyze the buffer content in relation to the modified Bessel function of the first kind. We utilize the generating function approach and the Laplace–Stieltjes transform to achieve this, enabling us to accomplish our objectives. We provide numerical results to elucidate the overall behavior of the system under consideration.

Suggested Citation

  • Sherif I. Ammar & Yousef F. Alharbi & Yiqiang Q. Zhao, 2023. "Analysis of Vacation Fluid M / M /1 Queue in Multi-Phase Random Environment," Mathematics, MDPI, vol. 11(21), pages 1-14, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4444-:d:1268252
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    References listed on IDEAS

    as
    1. Jianjun Li & Liwei Liu, 2016. "On the Discrete-Time Queue with Vacations in Random Environment," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-9, June.
    2. Jinping Liu & Xiuli Xu & Shuo Wang & Dequan Yue, 2021. "Equilibrium analysis of the fluid model with two types of parallel customers and breakdowns," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(24), pages 5792-5805, November.
    3. R. Bekker & M. Mandjes, 2009. "A fluid model for a relay node in an ad hoc network: the case of heavy-tailed input," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(2), pages 357-384, October.
    4. Jianjun Li & Liwei Liu & Tao Jiang, 2018. "Analysis of an M/G/1 queue with vacations and multiple phases of operation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(1), pages 51-72, February.
    5. K.V. Vijayashree & A. Anjuka, 2018. "Stationary analysis of a fluid queue driven by an M/M/ 1 /N queue with disaster and subsequent repair," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 31(4), pages 461-477.
    6. Kolinjivadi Viswanathan Vijayashree & Atlimuthu Anjuka, 2016. "Fluid Queue Driven by an Queue Subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption," Advances in Operations Research, Hindawi, vol. 2016, pages 1-11, June.
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