Asymptotically optimal policy for stochastic job shop scheduling problem to minimize makespan

This paper studies the large-scale stochastic job shop scheduling problem with general number of similar jobs, where the processing times of the same step are independently drawn from a known probability distribution, and the objective is to minimize the makespan. For the stochastic problem, we introduce the fluid relaxation of its deterministic counterpart, and define a fluid schedule for the fluid relaxation. By tracking the fluid schedule, a policy is proposed for the stochastic job shop scheduling problem. The expected value of the gap between the solution produced by the policy and the optimal solution is proved to be O (1), which indicates the policy is asymptotically optimal in expectation.