Scheduling parallelizable tasks to minimize average response time

Abstract

A parallelizable (or malleable) task is one which can be run on an arbitrary number of processors, with a task execution time that depends on the number of processors allotted to it. Consider a system of M independent parallelizable tasks which are to be scheduled without preemption on a parallel computer consisting of P identical processors. For each task, the execution time is a known function of the number of processors allotted to it. The goal is to find (1) for each task i, an allotment of processors βt, and (2) overall, a non-preemptive schedule assigning the tasks to the processors which minimizes the average response time of the tasks. Equivalently, we can minimize the flow time, which is the sum of the completion times of each of the tasks. In this paper we tackle the problem of finding a schedule with minimum average response time in the special case where each task in the system has sublinear speedup. This natural restriction on the task execution time means simply that the efficiency of a task decreases or remains constant as the number of processors allotted to it increases. The scheduling problem with sublinear speedups has been shown to be NP-complete in the strong sense. We therefore focus on finding a polynomial time algorithm whose solution comes within a fixed multiplicative constant of optimal. In particular, we give an algorithm which finds a schedule having a response time that is within 2 times that of the optimal schedule and which runs in O(M(M2 + P)) time.