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Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity

Author

Listed:
  • Anatoly Nazarov

    (Institute of Applied Mathematics and Computer Science, Tomsk State University, 634050 Tomsk, Russia)

  • Alexander Dudin

    (Faculty of Applied Mathematics and Informatics, Belarusian State University, 220030 Minsk, Belarus)

  • Alexander Moiseev

    (Institute of Applied Mathematics and Computer Science, Tomsk State University, 634050 Tomsk, Russia)

Abstract

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period ( P S S P ). Distribution of duration of P S S P , its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation.

Suggested Citation

  • Anatoly Nazarov & Alexander Dudin & Alexander Moiseev, 2022. "Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity," Mathematics, MDPI, vol. 10(15), pages 1-12, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2661-:d:874564
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    References listed on IDEAS

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    1. Anatoly Nazarov & Tuan Phung-Duc & Svetlana Paul & Olga Lizyura, 2022. "Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls," Mathematics, MDPI, vol. 10(6), pages 1-14, March.
    2. Moiseev, Alexander & Nazarov, Anatoly, 2016. "Queueing network MAP−(GI/∞)K with high-rate arrivals," European Journal of Operational Research, Elsevier, vol. 254(1), pages 161-168.
    3. Anatoly Nazarov & Alexander Moiseev & Svetlana Moiseeva, 2021. "Mathematical Model of Call Center in the Form of Multi-Server Queueing System," Mathematics, MDPI, vol. 9(22), pages 1-13, November.
    4. Ward Whitt, 1992. "Understanding the Efficiency of Multi-Server Service Systems," Management Science, INFORMS, vol. 38(5), pages 708-723, May.
    5. M. Ramalhoto, 1999. "The infinite server queue and heuristic approximations to the multi-server queue with and without retrials," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 333-350, December.
    6. T. Collings & C. Stoneman, 1976. "The M / M /∞ Queue with Varying Arrival and Departure Rates," Operations Research, INFORMS, vol. 24(4), pages 760-773, August.
    7. Woollcott Smith, 1972. "Technical Note—The Infinitely-Many-Server Queue with Semi-Markovian Arrivals and Customer-Dependent Exponential Service Times," Operations Research, INFORMS, vol. 20(4), pages 907-912, August.
    8. Yiran Liu & Harsha Honnappa & Samy Tindel & Nung Kwan Yip, 2021. "Infinite server queues in a random fast oscillatory environment," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 145-179, June.
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    Cited by:

    1. Anatoly Nazarov & Ekaterina Fedorova & Olga Lizyura & Radmir Salimzyanov, 2023. "Asymptotic Diffusion Method for Retrial Queues with State-Dependent Service Rate," Mathematics, MDPI, vol. 11(14), pages 1-10, July.

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