Static Routing in Stochastic Scheduling: Performance Guarantees and Asymptotic Optimality

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 the weighted sum of expected completion time 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 WSEPT (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 only depends on the job processing distributions through their expected values. Our main result is an additive performance bound on the sub-optimality of this static routing policy relative to an optimal adaptive, non-anticipative 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 differ 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 sub-optimality of the static routing policy. The main novelty in our work is deriving lower bounds on the performance of an optimal adaptive, non-anticipative 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. Subject classifications: Stochastic scheduling, unrelated machines, dynamic programming, information relaxation duality, asymptotic optimality.