Optimal scheduling in a queue with differentiated impatient users

We consider a M/M/1 queue in which the average reward for servicing a job is an exponentially decaying function of the job's sojourn time. The maximum reward and mean service times of a job are i.i.d. and chosen from arbitrary distributions. The scheduler is assumed to know the maximum reward, service rate, and age of each job. We prove that the scheduling policy that maximizes average reward serves the customer with the highest product of potential reward and service rate.