Light-Tailed Behavior in QBD Processes with Countably Many Phases

Generally speaking, analysis of tail asymptotics in two-dimensional queueing systems is very challenging. Earlier work based on complex analysis led to determinations of exact forms of tail asymptotics. Ideas of large deviations, a powerful tool for characterizing light-tailed decay rates or analysis of rough tail asymptotics, have been utilized recently to develop probabilistic methods to do exact tail asymptotic analysis. Another promising approach to do tail asymptotics analysis, both exact and rough, is the matrix-analytic method. In this article, we combine the matrix-analytic method with techniques from probability and analysis to characterize tail asymptotics in a QBD process with infinitely many phases. The main results include conditions on: (1) exact geometric decay; (2) light-tailed behavior without an exact geometric decay, which in general is not the focus of the large deviations method; and (3) upper and lower bounds for stationary probabilities. We apply the main results to two two-dimensional queueing systems, including a polling system and a gated random-order server queue to characterize their light-tailed behavior of the queue length processes.