We consider Markovian multi-server queues with two types of impatient customers: high-and low-priority ones. The first type of customers has a non-preemptive priority over the other type. After entering the queue, a customer will wait a random length of time for service to begin. If service has not begun by this time he or she will abandon and be lost. We consider two cases where the discipline of service within each customer type is FCFS or LCFS. For each type of customers, we focus on various performance measures related to queueing delays: unconditional waiting times, and conditional waiting times given service and given abandonment. The analysis we develop holds also for a priority queue with mixed policies, i.e., FCFS for the first type and LCFS for the second one, and vice versa. We explicitly derive the Laplace-Stieltjes transforms of the defined random variables. In addition we show how to extend the analysis to more than two customer types. Finally we compare FCFS and LCFS and gain insights through numerical experiments.