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On Truncation of the Matrix-Geometric Stationary Distributions

Author

Listed:
  • Valeriy A. Naumov

    (Service Innovation Research Institute, 00120 Helsinki, Finland)

  • Yuliya V. Gaidamaka

    (Department of Applied Informatics and Probability, Peoples’ Friendship University of Russia (RUDN University), Miklukho-Maklaya St. 6, Moscow 117198, Russia
    Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilov St. 44-2, Moscow 119333, Russia)

  • Konstantin E. Samouylov

    (Department of Applied Informatics and Probability, Peoples’ Friendship University of Russia (RUDN University), Miklukho-Maklaya St. 6, Moscow 117198, Russia
    Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilov St. 44-2, Moscow 119333, Russia)

Abstract

In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed.

Suggested Citation

  • Valeriy A. Naumov & Yuliya V. Gaidamaka & Konstantin E. Samouylov, 2019. "On Truncation of the Matrix-Geometric Stationary Distributions," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:798-:d:262871
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    References listed on IDEAS

    as
    1. Debjit Roy, 2016. "Semi-open queuing networks: a review of stochastic models, solution methods and new research areas," International Journal of Production Research, Taylor & Francis Journals, vol. 54(6), pages 1735-1752, March.
    2. B. Avi-Itzhak & D. P. Heyman, 1973. "Approximate Queuing Models for Multiprogramming Computer Systems," Operations Research, INFORMS, vol. 21(6), pages 1212-1230, December.
    3. Jing Jia & Sunderesh S. Heragu, 2009. "Solving Semi-Open Queuing Networks," Operations Research, INFORMS, vol. 57(2), pages 391-401, April.
    4. Marcel F. Neuts, 1982. "Explicit Steady-State Solutions to Some Elementary Queueing Models," Operations Research, INFORMS, vol. 30(3), pages 480-489, June.
    5. Dhingra, Vibhuti & Kumawat, Govind Lal & Roy, Debjit & Koster, René de, 2018. "Solving semi-open queuing networks with time-varying arrivals: An application in container terminal landside operations," European Journal of Operational Research, Elsevier, vol. 267(3), pages 855-876.
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