The expected competitive ratio for weighted completion time scheduling

Abstract

A set of n independent jobs is to be scheduled without preemption on m identical parallel machines. For each job j, a so called diffuse adversary chooses the distribution Fj of the random processing time P j from a certain class of distributions Fj. The scheduler is given the expectation μj = E[Pj], but the actual duration is not known in advance. A positive weight Wj is associated with each job j and all jobs are ready for execution at time zero. The objective is to minimise the expected competitive ratio maxF∈F E [∑wjcj/OPT], Cjdenotes the completion time of job j and OPT the offline optimum value. The scheduler determines a list of jobs, which is then scheduled in non-preemptive static list policy. We show a general bound on the expected competitive ratio for list scheduling algorithms, which holds for a class of so called new-better-than-used processing time distributions. This class includes, among others the exponential distribution. Our bound depends on the probability of any pair of jobs being in the wrong order in the list of an arbitrary list scheduling algorithm, compared to an optimum list. As a special case, we show that the so called WSEPT algorithm achieves E [WSEPT/OPT] ≤ 3-1/m for exponential distributed processing times. © Springer-Verlag 2004.