On the analysis of partially homogeneous nearest-neighbour random walks in the quarter plane

TL;DR

This paper analyzes the stationary behavior of two-dimensional partially homogeneous nearest-neighbour random walks in the quarter plane, using linear equations and Riemann-Hilbert boundary value problems, with applications in queueing and access systems.

Contribution

It introduces a method to analyze such random walks by solving linear systems and functional equations, extending existing techniques to state-dependent transition probabilities.

Findings

Stationary distribution can be obtained via linear equations and boundary value problems.

Application to queue-aware multiple access systems demonstrates practical relevance.

Numerical example illustrates the approach's effectiveness.

Abstract

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks in the quarter plane are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that its stationary behavior is investigated by solving a finite system of linear equations, and a functional equation with the aid of the theory of Riemann(-Hilbert) boundary value problems. This work is strongly motivated by emerging applications in multiple access systems as well as in the study of a general class of queueing systems with state dependent parameters. A simple numerical illustration providing useful information about a queue-aware multiple access system is also presented.