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Number 4 - December 2015
Volume 25 - 2015
Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to the origin
Alexander Zeifman, Anna Korotysheva, Yacov Satin, Victor Korolev, Sergey Shorgin, Rostislav Razumchik
Abstract
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.
Keywords
inhomogeneous birth and death processes, ergodicity bounds, perturbation bounds