Scheduling Stochastic Jobs with a Two-Point Distribution on Two Parallel Machines

We analyze the optimal preemptive sequencing of n jobs on two machines to minimize expected total flow time. The running times of the jobs are independent samples from the distribution Pr ( X = 1) = p , Pr ( X = κ + 1) = 1 − p . We verify that the shortest-expected-remaining-processing-time (SERPT) policy, which is optimal for independent and identically distributed (i.i.d.) running times with a monotone hazard-rate distribution, is not optimal for this distribution. However, we prove that if p ≥ 1/κ, then the number of decisions where SERPT and an optimal policy disagree is bounded by a constant independent of n . For p < 1/k, we prove that the expected number of such decisions has a similar bound. In addition, bounds on the expected increase in flow times under SERPT are derived; these bounds are also independent of n .