Optimal Rules for Single Machine Scheduling with Stochastic Breakdowns

This paper studies the problem of scheduling a set of jobs on a single machine subject to stochastic breakdowns, where jobs have to be restarted if preemptions occur because of breakdowns. The breakdown process of the machine is independent of the jobs processed on the machine. The processing times required to complete the jobs are constants if no breakdown occurs. The machine uptimes are independently and identically distributed (i.i.d.) and are subject to a uniform distribution. It is proved that the Longest Processing Time first (LPT) rule minimizes the expected makespan. For the large-scale problem, it is also showed that the Shortest Processing Time first (SPT) rule is optimal to minimize the expected total completion times of all jobs.