Sufficient conditions for a geometric tail in a QBD process with many countable levels and phases

In this paper, we present sufficient conditions, under which the stationary probability vector of a QBD process with both infinite levels and phases decays geometrically, characterized by the convergence norm n and the 1/eta-left-invariant vector x of the rate matrix R. We also present a method to compute eta and x based on spectral Properties of the censored matrix of a matrix function constructed with the repeating blocks of the transition matrix of the QBD process. What makes this method attractive is its simplicity; finding eta reduces to determining the zeros of a polynomial. We demonstrate the application of our method through a few interesting examples