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A subgeometric convergence formula for finite-level M/G/1-type Markov chains: via a block-decomposition-friendly solution to the Poisson equation of the deviation matrix

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 December 2023

Hiroyuki Masuyama Open the ORCID record for Hiroyuki Masuyama [Opens in a new window] ,
Yosuke Katsumata  and
Tatsuaki Kimura
Hiroyuki Masuyama*
Affiliation:
Tokyo Metropolitan University
Yosuke Katsumata*
Affiliation:
Kyoto University
Tatsuaki Kimura*
Affiliation:
Osaka University
*
*Postal address: Graduate School of Management, Tokyo Metropolitan University, Tokyo 192–0397, Japan. Email address: masuyama@tmu.ac.jp
**Postal address: Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606–8501, Japan. Email address: katsumata@sys.i.kyoto-u.ac.jp
***Postal address: Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, Suita 565–0871, Japan. Email address: kimura@comm.eng.osaka-u.ac.jp
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Abstract

The purpose of this study is to present a subgeometric convergence formula for the stationary distribution of the finite-level M/G/1-type Markov chain when taking its infinite-level limit, where the upper boundary level goes to infinity. This study is carried out using the fundamental deviation matrix, which is a block-decomposition-friendly solution to the Poisson equation of the deviation matrix. The fundamental deviation matrix provides a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial role in the derivation of the main result of this paper, and the main result is used, for example, to derive an asymptotic formula for the loss probability in the MAP/GI/1/N queue.

Keywords

Augmented truncation approximationMarkovian arrival processloss probabilitysecond-order long-tailedsquare-root-insensitivesubexponential

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K25: Queueing theory
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 2 , June 2024 , pp. 389 - 429
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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