On a job resequencing issue in parallel processor stochastic scheduling

Abstract

In flexible assembly systems, it is often necessary to coordinate jobs and materials so that specific jobs are matched with specific materials. This requires that jobs depart from upstream parallel workstations in some predetermined order. One way to satisfy this requirement is to temporarily hold the serviced jobs getting out of order at a resequencing buffer and to release them to downstream workstations as soon as all their predecessors are serviced. In this paper we consider the problem of scheduling a fixed number of non-preemptive jobs on two IHR non-identical processors with the resequencing requirement. We prove that the individually optimal policy, in which each job minimizes its own expected departure time subject to the constraint that available processors are offered to jobs in their departure order, is of a threshold type. The policy is independent of job weights and the jobs residing at the resequencing buffer and possesses the monotonicity property which states that a job will never utilize a processor in the future once it has declined the processor. Most importantly, we prove that the individually optimal policy has the stability property; namely: if at any time a job deviated from the individually optimal policy, then the departure time of every job, including its own, would be prolonged. As a direct consequence of this property, the individually optimal policy is socially optimal in the sense that it minimizes the expected total weighted departure time of the system as a whole. We identify situations under which the individually optimal policy also minimizes the expected makespan of the system.