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Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls

Author

Listed:
  • Anatoly Nazarov

    (Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 36 Lenina Ave., 634050 Tomsk, Russia)

  • Tuan Phung-Duc

    (Department of Policy and Planning Sciences, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Ibaraki, Japan)

  • Svetlana Paul

    (Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 36 Lenina Ave., 634050 Tomsk, Russia)

  • Olga Lizyura

    (Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 36 Lenina Ave., 634050 Tomsk, Russia)

Abstract

This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of N − 1 ( N ≥ 2 ) categories occur according to N − 1 independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:6:p:948-:d:772342
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    References listed on IDEAS

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    1. Hiroyuki Sakurai & Tuan Phung-Duc, 2016. "Scaling limits for single server retrial queues with two-way communication," Annals of Operations Research, Springer, vol. 247(1), pages 229-256, December.
    2. R. Lillo, 1996. "A G/M/1-queue with exponential retrial," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(1), pages 99-120, June.
    3. J. Artalejo, 1999. "A classified bibliography of research on retrial queues: Progress in 1990–1999," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 187-211, December.
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    Cited by:

    1. 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.
    2. Gabi Hanukov & Uri Yechiali, 2024. "Orbit while in service," Operational Research, Springer, vol. 24(2), pages 1-32, June.

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