Eulerian polynomials and Quasi-Birth-Death processes with time-varying-periodic rates

A Quasi-Birth-Death (QBD) process is a stochastic process with a two dimensional state space, a level and a phase. An ergodic QBD with time-varying periodic transition rates will tend to an asymptotic periodic solution as time tends to infinity . Such QBDs are also asymptotically geometric. That is, as the level tends to infinity, the probability of the system being in state ( k , j ) (k,j) at time t t within the period tends to an expression of the form f j ( t ) α − k Π j ( k ) f_j(t)\alpha ^{-k}\Pi _j(k) where α \alpha is the smallest root of the determinant of a generating function related to the generating function for the unbounded (in the level) process, Π j ( k ) \Pi _j(k) is a polynomial in k k , the level, that may depend on j j , the phase of the process, and f j ( t ) f_j(t) is a periodic function of time within the period which may also depend on the phase. These solutions are analogous to steady state solutions for QBDs with constant transition rates. If the time within the period is considered to be part of the state of the process, then they are steady-state solutions. In this paper, we consider the example of a two-priority queueing process with finite buffer for class-2 customers. For this example, we provide explicit results up to an integral in terms of the idle probability of the queue. We also use this asymptotic approach to provide exact solutions (up to an integral equation involving the probability the system is in level zero) for some of the level probabilities.