A first passage time problem for spectrally positive Lévy processes and its application to a dynamic priority queue

We study a first passage time problem for a class of spectrally positive Levy processes. By considering the special case where the Levy process is a compound Poisson process with negative drift, we obtain the Laplace-Stieltjes transform of the steady-state waiting time distribution of low-priority customers in a two-class M / G I / 1 queue operating under a dynamic non-preemptive priority discipline. This allows us to observe how the waiting time of customers is affected as the policy parameter varies.