Static Routing in Stochastic Scheduling: Performance Guarantees and Asymptotic Optimality

AbstractScheduling problems have a deep and well-developed literature in both operations research and computer science. Stochastic scheduling problems with many jobs are notoriously difficult, and optimal policies may require constant tracking of the elapsed time of all jobs in process. When the processors (or “machines”) are identical, a policy that simply schedules jobs in fixed order of job weight to expected processing time—the WSEPT rule—is asymptotically optimal when the number of jobs is large. Much less understood is the case with specialized (or “unrelated”) machines (i.e., each job’s processing distribution may vary across machines). In “Static Routing in Stochastic Scheduling: Performance Guarantees and Asymptotic Optimality,” Balseiro, Brown, and Chen study stochastic scheduling with unrelated machines. The authors study a simple static routing policy that (i) assigns jobs to machines up front and (ii) schedules the jobs on each machine in the WSEPT order. This static routing policy depends only on job processing times through their expected values and is easy to compute with a single convex optimization problem. The authors explicitly characterize the performance loss of this static routing policy relative to an optimal scheduling policy; this result implies that this static routing policy is asymptotically optimal in the regime of many jobs.We study the problem of scheduling a set of J jobs on M machines with stochastic job processing times when no preemptions are allowed and with a weighted sum of expected completion times objective. Our model allows for “unrelated” machines: the distributions of processing times may vary across both jobs and machines. We study static routing policies, which assign (or “route”) each job to a particular machine at the start of the problem and then sequence jobs on each machine according to the weighted shortest expected processing time rule. We discuss how to obtain a good routing of jobs to machines by solving a convex quadratic optimization problem that has J×M variables and depends only on the job processing distributions through their expected values. Our main result is an additive performance bound on the suboptimality of this static routing policy relative to an optimal adaptive, nonanticipative scheduling policy. This result implies that such static routing policies are asymptotically optimal as the number of jobs grows large. In the special case of “uniformly related” machines—that is, machines differing only in their speeds—we obtain a similar but slightly sharper result for a static routing policy that routes jobs to machines proportionally to machine speeds. We also study the impact that dependence in processing times across jobs can have on the suboptimality of the static routing policy. The main novelty in our work is deriving lower bounds on the performance of an optimal adaptive, nonanticipative scheduling policy; we do this through the use of an information relaxation in which all processing times are revealed before scheduling jobs and a penalty that appropriately compensates for this additional information.The online appendices are available at https://doi.org/10.1287/opre.2018.1749.