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Numerical Computation of Distributions in Finite-State Inhomogeneous Continuous Time Markov Chains, Based on Ergodicity Bounds and Piecewise Constant Approximation

Author

Listed:
  • Yacov Satin

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

  • Rostislav Razumchik

    (Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119133 Moscow, Russia)

  • Ilya Usov

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

  • Alexander Zeifman

    (Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
    Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119133 Moscow, Russia
    Vologda Research Center, Russian Academy of Sciences, 160014 Vologda, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow State University, 119991 Moscow, Russia)

Abstract

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4265-:d:1258668
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    References listed on IDEAS

    as
    1. Linda Green & Peter Kolesar, 1991. "The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals," Management Science, INFORMS, vol. 37(1), pages 84-97, January.
    2. 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.
    3. Ward Whitt & Wei You, 2019. "Time-Varying Robust Queueing," Operations Research, INFORMS, vol. 67(6), pages 1766-1782, November.
    4. M. Arns & P. Buchholz & A. Panchenko, 2010. "On the Numerical Analysis of Inhomogeneous Continuous-Time Markov Chains," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 416-432, August.
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