Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

Abstract

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.