Optimal Scheduling on Parallel Processors with Precedence Constraints and General Costs

We consider preemptive and nonpreemptive scheduling of partially ordered tasks on parallel processors, where the precedence relations have an interval-order, an in-forest, or a uniform out-forest structure. Processing times of tasks are random variables with an increasing in likelihood ratio distribution in the nonpreemptive case and an exponential distribution in the preemptive case. We consider a general cost that is a function of time and of the uncompleted tasks and show that the most successors (MS) policy stochastically minimizes the cost function when it satisfies certain agreeability conditions. A consequence is that the MS policy stochastically minimizes makespan, weighted flowtime, and the weighted number of late jobs.