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Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

Author

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  • Alexander Zeifman

    (Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
    Institute of Informatics Problems of the Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 119333 Moscow, Russia
    Vologda Research Center of the Russian Academy of Sciences, 160014 Vologda, Russia)

  • Victor Korolev

    (Institute of Informatics Problems of the Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 119333 Moscow, Russia
    Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia
    Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China)

  • Yacov Satin

    (Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia)

Abstract

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.

Suggested Citation

  • Alexander Zeifman & Victor Korolev & Yacov Satin, 2020. "Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains," Mathematics, MDPI, vol. 8(2), pages 1-25, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:253-:d:320838
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    References listed on IDEAS

    as
    1. Zeifman, A. & Satin, Y. & Kiseleva, K. & Korolev, V. & Panfilova, T., 2019. "On limiting characteristics for a non-stationary two-processor heterogeneous system," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 48-65.
    2. Zeifman, A. I. & Isaacson, Dean L., 1994. "On strong ergodicity for nonhomogeneous continuous-time Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 263-273, April.
    3. Zeifman, A.I. & Korolev, V.Yu. & Satin, Ya.A. & Kiseleva, K.M., 2018. "Lower bounds for the rate of convergence for continuous-time inhomogeneous Markov chains with a finite state space," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 84-90.
    4. Zeifman, A.I., 1995. "Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 157-173, September.
    5. Erik Doorn, 2011. "Rate of convergence to stationarity of the system M/M/N/N+R," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 336-350, December.
    6. Di Crescenzo, A. & Giorno, V. & Nobile, A.G. & Ricciardi, L.M., 2008. "A note on birth-death processes with catastrophes," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2248-2257, October.
    7. Zeifman, A.I. & Korolev, V.Yu., 2014. "On perturbation bounds for continuous-time Markov chains," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 66-72.
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    Cited by:

    1. Yacov Satin & Rostislav Razumchik & Ilya Usov & Alexander Zeifman, 2023. "Numerical Computation of Distributions in Finite-State Inhomogeneous Continuous Time Markov Chains, Based on Ergodicity Bounds and Piecewise Constant Approximation," Mathematics, MDPI, vol. 11(20), pages 1-12, October.
    2. 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.
    3. Ilya Usov & Yacov Satin & Alexander Zeifman & Victor Korolev, 2022. "Ergodicity Bounds and Limiting Characteristics for a Modified Prendiville Model," Mathematics, MDPI, vol. 10(23), pages 1-14, November.

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