Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

Abstract

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.

Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes

“…Inequality (12) and the assumptions made in proposition (v) of Lemma 2 finally imply that the following inequality holds, for 0 < ε ≤ ε D given in relation (f) of this proposition,…”

“…A comprehensive bibliography of works in the area can be found in these books and, also, in the research report by Silvestrov, D. and Silvestrov, S. (2015), which is an extended preliminary version of the present paper.…”

“…The assumptions made in proposition (iv) of Lemma 2 and inequality (12) imply that the following inequality holds, for 0 < ε ≤ ε C ,…”

Nonlinearly Perturbed Birth-Death-Type Models

“…The present paper is devoted to studies of asymptotic expansions for stationary and conditional quasi-stationary distributions for perturbed birthdeath-type semi-Markov processes. The algorithms of sequential phase space reduction for perturbed semi-Markov processes combined with techniques of Laurent asymptotic expansions developed in the recent papers by Silvestrov, D. and Silvestrov, S. (2015Silvestrov, S. ( , 2016 are applied to birth-death-type semi-Markov processes. In this model, the proposed algorithms of phase space reduction preserve the birth-death structure for reduced semi-Markov processes.…”

“…The results of the present section are based on the explicit formula (104) for expected return times and the expressions which connect these quantities with stationary and conditional quasi-stationary distributions. We obtain the first and second order asymptotic expansions from these formulas by using operational rules for Laurent asymptotic expansions presented in Silvestrov, D. and Silvestrov, S. (2015Silvestrov, S. ( , 2016. Some of these operational rules which are relevant for this paper can be found in Subsection 6.1.…”