A queue with preemptive resume priorities between customers from n different classes, is studied. The customers arrive in batches of arbitrary size according to a Poisson process and the service times are arbitrarily distributed with a different distribution for each class. The sever finally takes multiple vacations each time the system becomes empty. For this model, the Laplace transforms of the joint distributions of the system states and the elapsed service times of the customer in service and the customers in limbo are obtained, both in a transient state and in the steady state. Similar results are also derived for the model without vacations, and the mean performance measures for both models are extracted. These measures are finally compared with the corresponding measures for models with nonpreemptive priority and useful relationships are obtained which provide insight into the impact of the various strategies on the performance measures.
