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Many problems can be modeled as single-server queues with impatient customers. An example is that of the transmission of voice packets over a packet-switched network. If the voice packets do not reach their destination within a certain time interval of their transmission, they are useless to the receiver and considered lost. It is therefore desirable to schedule the customers such that the fraction of customers served within their respective deadlines is maximized. For this measure of performance, it is shown that the shortest time to extinction (STE) policy is optimal for a class of continuous and discrete time nonpreemptive M/G/1 queues that do not allow unforced idle times. When unforced idle times are allowed, the best policies belong to the class of shortest time to extinction with inserted idle time (STEI) policies. An STEI policy requires that the customer closest to his or her deadline be scheduled whenever it …