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Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

Part of: Markov processes Special processes

Published online by Cambridge University Press:  03 December 2020

Masatoshi Kimura  and
Tetsuya Takine
Masatoshi Kimura*
Affiliation:
Osaka University
Tetsuya Takine*
Affiliation:
Osaka University
*
*Postal address: Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, Suita 565-0871, Japan.
*Postal address: Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, Suita 565-0871, Japan.
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Abstract

This paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

Keywords

Markov chainscountably infinite state spaceconditional stationary distributionlinear inequalitiesconvex polytopesaugmented truncation approximation

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K25: Queueing theory
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1249 - 1283
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Copyright
© Applied Probability Trust 2020

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