Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to the origin

Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, time-dependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided. © 2015 Alexander Zeifman et al., published by De Gruyter Open 2015.

Zeifman A.1, 2, 4 , Korotysheva A.1, 4 , Satin Y.1, 4 , Korolev V.4, 3 , Shorgin S.4 , Razumchik R. 4, 5
Walter de Gruyter GmbH
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  • 1 Department of Applied Mathematics, Vologda State University, S. Orlova, 6, Vologda, Russian Federation
  • 2 Institute of Socio-Economic Development of Territories, Russian Academy of Sciences, Gorkogo Str., 56A, Vologda, Russian Federation
  • 3 Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, Moscow, Russian Federation
  • 4 Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russian Federation
  • 5 Department of Applied Informatics and Probability Theory, Peoples' Friendship University of Russia, Miklukho-Maklaya str., 6, Moscow, 117198, Russian Federation
Ключевые слова
ergodicity bounds; inhomogeneous birth and death processes; perturbation bounds
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