Discrete time GI/Geom/1 queueing system with priority

We study a discrete time single server system with generic distribution of the number of arrivals in a time slot, geometric distribution of the service time and two classes of customers. The customers can be served only when each time slot begins. The customers of the second class can be served only if the customers of the first class are absent. The model is motivated by the description of the congestion in the information networks. We give a complete description of this systems: we are able to compute the expected waiting time of the customers of the two classes, and solving a boundary value problem we are able to write the probability distribution of the length of the queue. To obtain these results, we use judiciously the standard technique of the generating function of the probability distribution. Although the boundary value problem is relatively easy, it is has to be pointed out that the generating function is not simply the product of two independent generating functions for the two classes.