Priorities at the end of service

Consider a //1 queue with <i style="box-sizing: border-box; color: rgb(51, 51, 51); font-family: &quot;Open Sans&quot;, sans-serif; font-size: 16px;">N</i> customer priority classes, all with the same service requirement distribution. The special feature of the model is that the assignment of a job is done <i style="box-sizing: border-box; color: rgb(51, 51, 51); font-family: &quot;Open Sans&quot;, sans-serif; font-size: 16px;">at the end</i> of the service period – it is assigned to a customer of the highest priority class then present. Two variations of this scheme are studied: (i) the stoppable server case, in which the server only works when there are customers, and (ii) the unstoppable case, in which the server always works but scraps a job when at its completion there is no customer to receive it. For both variants, we determine the probability generating function of the steady-state joint queue length distribution, as well as the Laplace-Stieltjes transform of the sojourn time distribution of each class. This is done in detail for <i style="box-sizing: border-box; color: rgb(51, 51, 51); font-family: &quot;Open Sans&quot;, sans-serif; font-size: 16px;">N</i> = 2 customer classes and more globally for general <i style="box-sizing: border-box; color: rgb(51, 51, 51); font-family: &quot;Open Sans&quot;, sans-serif; font-size: 16px;">N</i>.<p/>