A finite compensation procedure for a certain class of two-dimensional random walks

TL;DR

This paper investigates conditions under which a finite compensation procedure can derive explicit invariant measures for a specific class of two-dimensional random walks, motivated by queueing theory applications.

Contribution

It establishes conditions on transition probabilities that enable a finite compensation approach to find invariant measures in certain 2D random walks.

Findings

Identifies key relations among transition probabilities for finite compensation applicability

Provides a thorough discussion on the importance of these conditions

Offers a framework for deriving explicit invariant measures

Abstract

Motivated by queueing applications, we consider a certain class of two-dimensional random walks for which their invariant measure is written as a linear combination of a finite number of product-form terms. In this work, we investigate under which conditions such an elegant solution can be derived by applying a finite compensation procedure. The conditions are formulated in terms of relations among the transition probabilities in the inner area, the boundaries as well as the origin. A thorough discussion on the importance of these conditions is also given.