Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process

We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) \(\{(\varvec{X}_n,J_n)\}\) on \(\mathbb {Z}_+^2\times S_0\), where \(\varvec{X}_n=(X_{1,n},X_{2,n})\) is the level state, \(J_n\) the phase state (background state) and \(S_0\) a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in an arbitrary direction. This result is an extension of the corresponding result for a certain two-dimensional reflecting random work without background processes, obtained by using the large deviation techniques and the singularity analysis methods. We also present a condition ensuring the sequence of the stationary probabilities geometrically decays without power terms, asymptotically. Asymptotic properties of the stationary tail distribution in the coordinate directions in a 2d-QBD process have already been studied in the literature. The results of this paper are also important complements to those results.