Stochastic Scheduling on Unrelated Machines

Two important characteristics encountered in many real-world scheduling problems are heterogeneous processors and a certain degree of uncertainty about the sizes of jobs. In this paper we address both, and study for the first time a scheduling problem that combines the classical unrelated machine scheduling model with stochastic processing times of jobs. Here, the processing time of job j on machine i is governed by random variable P-i,P-j, and its realization becomes known only upon job completion. With w(j), being the given weight of job j, we study the objective to minimize the expected total weighted completion time E[Sigma(j) w(j)C(j)], where C-j is the completion time of job j. By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee (3 + Delta)/2 + epsilon. Here, epsilon > 0 is arbitrarily small, and Delta is an upper bound on the squared coefficient of variation of the processing times. When jobs also have individual release dates our bound is (2 + Delta) + epsilon. We also show that the dependence of the performance guarantees on A is tight. Via Delta = 0, currently best known bounds for deterministic scheduling on unrelated machines are contained as special case.