We consider the MAP, M/G1, G2/1 queue with preemptive resume priority, where low priority customers arrive to the system according to a Markovian arrival process (MAP) and high priority customers according to a Poisson process. The service time density function of low (respectively: high) priority customers is g1(x) (respectively: g2(x)). We use the supplementary variable method with Extended Laplace Transforms to obtain the joint transform of the number of customers in each priority queue, as well as the remaining service time for the customer in service in the steady state. We also derive the probability generating function for the number of customers of low (respectively, high) priority in the system just after the service completion epochs for customers of low (respectively, high) priority.
