Minimizing expected makespans of multi-priority classes of jobs on uniform processors

Consider a scheduling problem where several classes of non-preemptive jobs with different priorities are to be processed on two processors with different processing speeds. Of the jobs waiting to be processed, jobs with the highest priority have the first option of being dispatched to the available processors. Let processing times have increasing hazard rate distributions which depend on the processor and job class. It is shown that for each job class there exists a simple threshold policy which minimizes the expected makespan (the completion time of the last job) of the class, among all non-preemptive policies. This research extends the result of Coffman, Flatto, Garey and Weber to include several priority job classes and more general processing time distributions. The proof is presented in a more intuitive and simpler manner.