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Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience

Author

Listed:
  • Yacov Satin

    (Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia)

  • Rostislav Razumchik

    (Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., 119133 Moscow, Russia
    Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., 117198 Moscow, Russia)

  • Ivan Kovalev

    (Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia)

  • Alexander Zeifman

    (Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, Russia
    Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., 119133 Moscow, Russia
    Vologda Research Center of the Russian Academy of Sciences, 556A Gorky Str., 160014 Vologda, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia)

Abstract

We consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no other customers are waiting. As a result, customers in the queue may become impatient and leave it. Under this setting and with certain restrictions on the intensity functions, the quantity of interest, the total number of customers in the system, is the level-dependent birth-and-death process (BPD). In this paper, for the first time in the literature, explicit upper bounds for the distance between two probability distributions of this BDP are obtained. Using the obtained ergodicity bounds in combination with the sensitivity bounds, we assess the stability of BDP under perturbations. Truncation bounds are also given, which allow for numerical solutions with guaranteed truncation errors. Finally, we provide numerical results to support the findings.

Suggested Citation

  • Yacov Satin & Rostislav Razumchik & Ivan Kovalev & Alexander Zeifman, 2023. "Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:1979-:d:1129969
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    References listed on IDEAS

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    2. Zeifman, A.I. & Razumchik, R.V. & Satin, Y.A. & Kovalev, I.A., 2021. "Ergodicity bounds for the Markovian queue with time-varying transition intensities, batch arrivals and one queue skipping policy," Applied Mathematics and Computation, Elsevier, vol. 395(C).
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