Scheduling Parallel Jobs Online with Convex and Concave Parallelizability

Online scheduling of parallelizable jobs has received a significant amount of attention recently. Scalable algorithms are known-that is, algorithms that are (1 + epsilon)-speed O(1)-competitive for any fixed epsilon > 0. Previous research has focused on the case where each job's parallelizability can be expressed as a concave speedup curve. However, there are cases where a job's speedup curve can be convex. Considering convex speedup curves has received attention in the offline setting, but, to date, there are no positive results in the online model. In this work, we consider scheduling jobs with convex or concave speedup curves for the first time in the online setting. We give a new algorithm that is (1 + epsilon)-speed O(1)-competitive. There are strong lower bounds on the competitive ratio if the algorithm is not given resource augmentation over the adversary, and thus this is essentially the best positive result one can show for this setting.