A stochastic fluid model approach to the stationary distribution of the maximal priority process

We consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process (Formula presented.), which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process (Formula presented.) to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of (Formula presented.). Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples.