Priority queues and the stationary distribution calculated using tandem fluid queues

We consider a time-dependent priority queuing model as described in the work of Stanford, Taylor and Ziedins, in which two classes of customers accumulate priority over time at linear and class-dependent rates. Each time the server becomes idle, the customer with the highest accumulated priority commences service. The corresponding maximum priority process is such that for every time $t$ which is not a departure time, it records the least upper bounds of the priorities for both customer classes in the queue. We are interested in the stationary distribution of this process considered at the moments immediately after the commencement of the service. We construct a mapping of the maximum priority process to a tandem fluid queue which enables us to find expressions for this stationary distribution.