Revisit to the tail asymptotics of the double QBD process: Refinement and complete solutions for the coordinate and diagonal directions

We consider a two dimensional skip-free reflecting random walk on a nonnegative integer quadrant. We are interested in the tail asymptotics of its stationary distribution, provided its existence is assumed. We derive exact tail asymptotics for the stationary probabilities on the coordinate axis. This refines the asymptotic results in the literature, and completely solves the tail asymptotic problem on the stationary marginal distributions in the coordinate and diagonal directions. For this, we use the so-called analytic function method in such a way that either generating functions or moment generating functions are suitably chosen. The results are exemplified by a two node network with simultaneous arrivals.