We study a stochastic scheduling problem with a single machine
subject to random breakdowns. We address the
preemptive-repeat model; that 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.