A Random Family of Queueing Systems with a Dynamic Priority Discipline

We consider a family {∑_u, 0 ≤ u ≤ ∞} of single server queueing systems whose customers belong to two priority classes. They arrive in two independent Poisson processes and their service times are independent with general distributions for each class. The system operates under a dynamic priority queue discipline in which the relative priority of a customer increases with his waiting time, and which can be characterized in terms of the urgency number u. The processes of interest are the waiting times of the two classes of customers. First we investigate the behavior of these processes as functions of the urgency number and establish stochastic orderings for them as u increases from 0 to ∞. Next we suppose that the urgency number is a random variable whose distribution has an exponential density for 0 < u < ∞ and atoms at u = 0 and u = ∞. We investigate the transient as well as the steady state behavior of the waiting time processes. Finally we derive limit theorems for these processes in the heavy traffic case.