Stochastic scheduling with preemptive-repeat machine breakdowns to minimize the expected weighted flow time

We study a stochastic scheduling problem with a single machine subject to random breakdowns. We address the preemptive-repeat model; this is, if a breakdown occurs during the processing of a job, the work done on this job is completely lost and the job has to be processed from the beginning when the machine resumes its work. The objective is to complete all jobs so that the the expected weighted flow time is minimized. Limited results have been published in the literature on this problem, all with the assumption that the machine uptimes are exponentially distributed. This article generalizes the study to allow that (1) the uptimes and downtimes of the machine follow general probability distributions, (2) the breakdown patterns of the machine may be affected by the job being processed and are thus job dependent; (3) the processing times of the jobs are random variables following arbitrary distributions, and (4) after a breakdown, the processing time of a job may either remain a same but unknown amount, or be resampled according to its probability distribution. We derive the necessary and sufficient condition that ensures the problem with the flow-time criterion to be well posed under the preemptive-repeat breakdown model. We then develop an index policy that is optimal for the problem. Several important situations are further considered and their optimal solutions are obtained.