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Estimating the Rate of Convergence of the PH/M/1 Model by Reducing to Quasi-Birth-Death Processes

Author

Listed:
  • Ilya Usov

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

  • Yacov Satin

    (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 Sciences and Control” of the Russian Academy of Sciences, 44-2 Vavilova Str., 119333 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 are studying the quasi-birth-death process and the property of weak ergodicity. Using the C-matrix method, we derive estimates for the rate of convergence to the limiting regime for the general case of the P H / M / 1 model, as well as the particular case when m = 3 . We provide a numerical example for the case m = 3 , and construct graphs showing the probability of an empty queue and the probability of p 1 ( t ) .

Suggested Citation

  • Ilya Usov & Yacov Satin & Alexander Zeifman, 2023. "Estimating the Rate of Convergence of the PH/M/1 Model by Reducing to Quasi-Birth-Death Processes," Mathematics, MDPI, vol. 11(6), pages 1-11, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1494-:d:1101092
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    References listed on IDEAS

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    1. 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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