Matrix geometric approach for random walks: stability condition and equilibrium distribution

TL;DR

This paper analyzes a specific class of two-dimensional random walks using the matrix geometric approach, providing new stability conditions, spectral properties, and explicit eigenvalue calculations for the equilibrium distribution.

Contribution

It introduces an alternative stability condition derivation, extends existing methods for eigenvalue analysis, and connects multiple approaches for analyzing equilibrium distributions.

Findings

Derived a new stability condition connecting drift and Lyapunov methods

Calculated eigenvalues and eigenvectors of the matrix R for series of product-form distributions

Extended the matrix geometric approach by integrating compensation and boundary value methods

Abstract

In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix $\mathbf{R}$ appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix $\mathbf{R}$.