On partially homogeneous nearest-neighbour random walks in the quarter plane and their application in the analysis of two-dimensional queues with limited state-dependency

TL;DR

This paper analyzes the stationary behavior of two-dimensional partially homogeneous nearest-neighbour random walks, applying advanced mathematical techniques to model queueing systems with state-dependent parameters, motivated by wireless network applications.

Contribution

It introduces a novel approach to analyze such random walks using matrix functional equations and boundary value problems, specifically tailored for queueing models with state-dependent features.

Findings

Stationary distribution can be obtained by solving a finite linear system.

The method involves matrix functional equations and Riemann-Hilbert boundary value problems.

Numerical implementation demonstrates practical applicability.

Abstract

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that its stationary behaviour is investigated by solving a finite system of linear equations, two matrix functional 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 flow level performance of wireless networks that give rise in queueing models with scalable service capacity, as well as in queue-based random access protocols, where the network's parameters are functions of the queue lengths. A simple numerical illustration, along with some details on the numerical implementation are also presented.