Scheduling stochastic jobs with increasing hazard rate on identical parallel machines

We consider a discrete time model of m identical machines operating in parallel to complete a collection of jobs. The processing times of the jobs are independent, identically distributed discrete random variables having increasing hazard rate. The jobs may have received different amounts of prior processing. A reward β^t, 0 < β < 1, is acquired when a job is finished at time t. We show that a non-preemptive SEPT (Shortest Expected Processing Time) strategy maximizes the expected total reward among all possible policies. We show that SEPT strategy is still optimal for several generalizations of this problem scenario.