Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms

TL;DR

This paper develops algorithms to determine when the invariant measure of a homogeneous random walk in the quarter-plane is a sum of geometric terms, and provides approximation schemes with error bounds for cases when it is not.

Contribution

It introduces an algorithm to verify if the invariant measure is a sum of geometric terms and offers explicit forms and approximation methods for other cases.

Findings

Algorithm to check geometric sum invariant measures

Explicit form of invariant measure when it is a sum of geometric terms

Error bounds for performance measures in non-geometric cases

Abstract

We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in [2,3]. Based on these results, we first develop an algorithm to check whether the invariant measure of a given random walk is a sum of geometric terms. We also provide the explicit form of the invariant measure if it is a sum of geometric terms. Secondly, for random walks of which the invariant measure is not a sum of geometric terms, we provide an approximation scheme to obtain error bounds for the performance measures. Finally, some numerical examples are provided.