Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks

A double quasi-birth-and-death (QBD) process is the QBD process whose background process is a homogeneous birth-and-death process, which is a synonym of a skip-free random walk in the two-dimensional positive quadrant with homogeneous reflecting transitions at each boundary face. It is also a special case of a 0-partially homogenous chain introduced by Borovkov and Mogul'skii. Our main interest is in the tail decay behavior of the stationary distribution of the double QBD process in the coordinate directions and that of its marginal distributions. In particular, our problem is to get their rough and exact asymptotics from primitive modeling data. We first solve this problem using the matrix analytic method. We then revisit the problem for the 0-partially homogenous chain, refining existing results. We exemplify the decay rates for Jackson networks and their modifications.