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

## TL;DR

This paper introduces a novel matrix-infinite-product-form solution for upper block-Hessenberg Markov chains that does not require parameter sets or conditions, and features a quasi-algorithmic construction method.

## Contribution

It proposes a new MIP-form solution that simplifies computation and removes the need for certain parameter conditions, with a quasi-algorithmic constructibility feature.

## Key findings

- The new MIP-form solution requires no parameter set or convergence conditions.
- It is constructed via an iterative recursive procedure with finite complexity per iteration.
- The solution is applicable to stationary distribution computation in UBH-MCs.

## 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 Syst., Vol. 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 solutions for the stationary distribution in UBH-MCs.