Characterization and computation of optimal policies for an M/G/1 priority queue

We consider two independent Poisson-streams with different holding charges, each one arriving to a single server who provides independent, identically distributed service times. We assume that the priority corresponds to the highest holding. The service is controlled by an exhaustive policy which activates the server at some arrival time. Bell showed that the only stationary optimal policy could be obtained from a monotone procedure determined by a linear boundary depending on some unknown quantities. In this paper, these quantities and two different explicit solutions for the optimal policy are obtained and this policy can also be characterized by a quadratic boundary.