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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  and connecting it with the result of Fayolle et al. which is based on Lyapunov functions . Furthermore, we consider the sub-class of random walks with equilibriumarXiv:1603.04842v4 fatcat:sal54ywcurfozfla4stpkxebgi