Geometric decay in level-expanding QBD models

Level-expanding quasi-birth-and-death (QBD) processes have been shown to be an efficient modeling tool for studying multi-dimensional systems, especially twodimensional ones. Computationally, it changes the more challenging problem of dealing with algorithms for two-dimensional systems to a less challenging one for block-structured transition matrices of QBD type with a varying finite block size in terms of results from the matrix-analytic method. In this paper, we focus on tail asymptotics in the stationary distribution of a level-expanding QBD process. Specifically, we provide sufficient conditions for geometric tail asymptotics for the level-expanding QBD process, and then apply the result to an interesting twodimensional system, an inventory queue model.