Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

TL;DR

This paper derives exact asymptotic formulas for the stationary distribution of a discrete-time two-dimensional QBD process with a finite background process, providing insights into its long-term behavior.

Contribution

It introduces a method to obtain precise asymptotic formulas for the stationary distribution of 2D-QBD processes under certain conditions.

Findings

Derived exact asymptotic formulas for stationary distribution

Applicable to processes with finite background modulation

Provides theoretical foundation for analyzing 2D-QBD processes

Abstract

We consider a discrete-time two-dimensional process $\{(L_{1,n},L_{2,n})\}$ on $\mathbb{Z}_+^2$ with a supplemental process $\{J_n\}$ on a finite set, where individual processes $\{L_{1,n}\}$ and $\{L_{2,n}\}$ are both skip free. We assume that the joint process $\{Y_n\}=\{(L_{1,n},L_{2,n},J_n)\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{1,n},L_{2,n})\}$ are modulated depending on the state of the background process $\{J_n\}$. This modulation is space homogeneous except for the boundaries of $\mathbb{Z}_+^2$. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process and, under several conditions, obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.