  We consider a Markovian single server queue in which customers are
preemptively scheduled by exogenously assigned priority levels. The novelty in
our model is that the priority levels are randomly assigned from a continuous
probability measure rather than a discrete one. Because the priority levels are
drawn from a continuum, the queue is modeled by a measure-valued stochastic
process. We analyze the steady state behavior of this process and provide
several results. We derive a measure that describes the average distribution of
customer priority levels in the system; we provide a formula for the expected
sojourn time of a customer as a function of his priority level; and we provide
a formula for the expected waiting time of a customer as a function of his
priority level. We interpret these quantitative results and give a qualitative
understanding of how the priority levels affect individual customers as well as
how they affect the system as a whole. The theoretical analysis is verified by
simulation. We also discuss some directions of future work.
