An improved greedy algorithm for stochastic online scheduling on unrelated machines

Most practical scheduling applications involve some uncertainty about the arriving times and lengths of the jobs. Stochastic online scheduling is a well-established model capturing this. Here the arrivals occur online, while the processing times are random. For this model, Gupta, Moseley, Uetz, and Xie recently devised an efficient policy for non-preemptive scheduling on unrelated machines with the objective to minimize the expected total weighted completion time. We improve upon this policy by adroitly combining greedy job assignment with ${\alpha }_j$-point scheduling on each machine. In this way we obtain a $(3+\sqrt{5})(2+\Delta )$-competitive deterministic and an $(8+4\Delta )$-competitive randomized stochastic online scheduling policy, where $\Delta$ is an upper bound on the squared coefficients of variation of the processing times. We also give constant performance guarantees for these policies within the class of all fixed-assignment policies. The ${\alpha }_j$-point scheduling on a single machine can be enhanced when the upper bound $\Delta$ is known a priori or the processing times are known to be $\delta$-NBUE for some $\delta \ge 1$. This implies improved competitive ratios for unrelated machines but may also be of independent interest.