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On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process

Author

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  • Alexander Zeifman

    (Department of Applied Mathematics, Vologda State University, IPI FRC CSC RAS, VolSC RAS, 160000 Vologda, Russia)

  • Yacov Satin

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

  • Ksenia Kiseleva

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

  • Victor Korolev

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China)

Abstract

We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this case, a non-Markov process is obtained, in which the transitions to neighboring states are possible in small periods of time. For this one-dimensional process, by modifying the method previously developed by the authors of the note, estimates of the rate of convergence in weakly ergodic and null-ergodic cases are obtained. The simplest example of a two-dimensional process of this type is considered.

Suggested Citation

  • Alexander Zeifman & Yacov Satin & Ksenia Kiseleva & Victor Korolev, 2019. "On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process," Mathematics, MDPI, vol. 7(5), pages 1-10, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:477-:d:234437
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    References listed on IDEAS

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    1. G.Sh. Tsitsiashvili & M.A. Osipova & N.V. Koliev & D. Baum, 2002. "A Product Theorem for Markov Chains with Application to PF-Queueing Networks," Annals of Operations Research, Springer, vol. 113(1), pages 141-154, July.
    2. 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.
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