  In this paper, we consider a discrete-time preemptive priority queue with
different service rates for two classes of customers, one with high-priority
and the other with low-priority. This model corresponds to the classical
preemptive priority queueing system with two classes of independent Poisson
customers and a single exponential server. Due to the possibility of customers'
arriving and departing at the same time in a discrete-time queue, the model
considered in this paper is more complicated. In this model, we focus on the
characterization of exact tail asymptotics for the joint stationary
distribution of the queue length of the two classes of customers, for the two
boundary distributions and for the two marginal distributions, respectively. By
using generating functions and kernel method, we get an explicit expression of
exact tail asymptotics along the low-priority queue direction, as well as along
the high-priority queue direction.
