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A new matrix-infinite-product-form solution for upper block-Hessenberg Markov chains and its quasi-algorithmic constructibility

Part of: Markov processes

Published online by Cambridge University Press:  29 July 2022

Hiroyuki Masuyama
Hiroyuki Masuyama*
Affiliation:
Tokyo Metropolitan University
*
*Postal address: Graduate School of Management, Tokyo Metropolitan University, Tokyo 192–0397, Japan. Email address: masuyama@tmu.ac.jp
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Abstract

This paper presents a new matrix-infinite-product-form (MIP-form) solution for the stationary distribution in upper block-Hessenberg Markov chains (UBH-MCs). The existing MIP-form solution (Masuyama, Queueing Systems 92, 2019, pp. 173–200) requires a certain parameter set that satisfies both a Foster–Lyapunov drift condition and a convergence condition. In contrast, the new MIP-form solution requires no such parameter sets and no other conditions. The new MIP-form solution also has ‘quasi-algorithmic constructibility’, which is a newly introduced feature of being constructed by iterating infinitely many times a recursive procedure of finite complexity per iteration. This feature is not found in the other existing solutions for the stationary distribution in general UBH-MCs.

Keywords

Upper block-Hessenberg Markov chain (UBH-MC)level-dependent M/G/1-type Markov chainmatrix-infinite-product-form (MIP-form) solutionquasi-algorithmic constructionstationary distribution vector

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J22: Computational methods in Markov chains

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 1 - 28
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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