  In traditional priority queues, we assume that every customer upon arrival
has a fixed, class-dependent priority, and that a customer may not commence
service if a customer with a higher priority is present in the queue. However,
in situations where a performance target in terms of the tails of the
class-dependent waiting time distributions has to be met, such models of
priority queueing may not be satisfactory. In fact, there could be situations
where high priority classes easily meet their performance target for the
maximum waiting time, while lower classes do not.
  Here, we are interested in the stationary distribution at the times of
commencement of service of this maximum priority process. Until now, there has
been no explicit expression for this distribution. We construct a mapping of
the maximum priority process to a tandem fluid queue, which enables us to find
expressions for this stationary distribution. We derive the results for the
stationary distribution of the maximum priority process at the times of the
commencement of service.
