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

ABSTRACT

In this paper, we analyze a sub-class of two-dimensional homogeneous nearest neighbor (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 Neuts, M.F., Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach; The Johns Hopkins University Press: Baltimore, 1981. [Google Scholar]^] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions.^[13 Fayolle, G.; Iasnogorodski, R.; Malyshev, V., Random Walks in the Quarter Plane; Springer-Verlag: New York, 1999. [Google Scholar]^] 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 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 R.