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Inform. Primen., 2014, Volume 8, Issue 1, Pages 106–117 (Mi ia303)  

This article is cited in 1 scientific paper (total in 1 paper)

General bounds for nonstationary continuous-time Markov chains

A. I. Zeifmanabc, V. Yu. Korolevdc, A. V. Korotyshevab, S. Ya. Shorginc

a ISEDT, Russian Academy of Sciences, Vologda, Russian Federation
b Vologda State University, 15 Lenin Str., Vologda 160000, Russian Federation
c Institute of of Informatics Problems, Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
d Department of MathematicalStatistics, Faculty ofComputational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation

Abstract: A general approach for obtaining perturbation bounds of nonstationary continuous-time Markov chains is considered. The suggested approach deals with a special weighted norms related to total variation. The method is based on the notion of a logarithmic norm of a linear operator function and respective bounds for the Cauchy operator of a differential equation. Special transformations of the reduced intensity matrix of the process are applied. The statements are proved which provide estimates of perturbation of probability characteristics for the case of absence of ergodicity in uniform operator topology. Birth–death–catastrophe queueing models and queueing systems with batch arrivals and group services are also considered in the paper. Some classes of such systems are studied, and bounds of perturbations are obtained. Particularly, such bounds are given for the $M_t/M_t/S$ queueing system with possible catastrophes and a simple model of a queueing system with batch arrivals and group services is analyzed. Moreover, approximations of limiting characteristics are considered for the queueing model.

Keywords: nonstationary continuous-time chains and models; nonstationary Markov chains; perturbation bounds; special norms; queueing models.

DOI: https://doi.org/10.14357/19922264140111

Full text: PDF file (312 kB)
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Received: 27.08.2013

Citation: A. I. Zeifman, V. Yu. Korolev, A. V. Korotysheva, S. Ya. Shorgin, “General bounds for nonstationary continuous-time Markov chains”, Inform. Primen., 8:1 (2014), 106–117

Citation in format AMSBIB
\Bibitem{ZeiKorKor14}
\by A.~I.~Zeifman, V.~Yu.~Korolev, A.~V.~Korotysheva, S.~Ya.~Shorgin
\paper General bounds for~nonstationary continuous-time Markov chains
\jour Inform. Primen.
\yr 2014
\vol 8
\issue 1
\pages 106--117
\mathnet{http://mi.mathnet.ru/ia303}
\crossref{https://doi.org/10.14357/19922264140111}
\elib{http://elibrary.ru/item.asp?id=21337623}


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    This publication is cited in the following articles:
    1. A. I. Zeifman, A. V. Korotysheva, K. M. Kiseleva, V. Yu. Korolev, S. Ya. Shorgin, “Ob otsenkakh skorosti skhodimosti i ustoichivosti dlya nekotorykh modelei massovogo obsluzhivaniya”, Inform. i ee primen., 8:3 (2014), 19–27  mathnet  crossref  elib
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