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

TL;DR

This paper derives a subgeometric convergence formula for the stationary distribution of finite-level M/G/1-type Markov chains as the level tends to infinity, using a block-decomposition approach to the Poisson equation.

Contribution

It introduces a fundamental deviation matrix approach that simplifies analyzing the convergence of stationary distributions in finite-to-infinite level limits.

Findings

Established a subgeometric convergence formula for the stationary distribution.

Provided a difference formula linking finite and infinite-level stationary distributions.

Utilized a block-decomposition-friendly solution to the Poisson equation.

Abstract

This paper studies the subgeometric convergence of the stationary distribution in taking the infinite-level limit of a finite-level M/G/1-type Markov chain, that is, in letting the upper boundary level go to infinity. This study is performed through the fundamental deviation matrix, which is a block-decomposition-friendly solution for the Poisson equation of the deviation matrix. The fundamental deviation matrix yields 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 deriving the main result of this paper: a subgeometric convergence formula for the infinite-level limit of the stationary distribution of the finite-level chain.