Optimal Online Scheduling of Parallel Jobs with Dependencies. Anja Feldmann, Ming-Yang Kao, Jiri Sgall,Shang-Hua Teng; preliminary version appeared as STOC 1993.
Abstract
We study the following general online scheduling problem. Parallel jobs arrive on a parallel machine dynamically according to the dependencies between them. Each job requests a certain number of processors in a specific communication configuration, but its running time is not known until it is completed. We present optimal online algorithms for PRAMs, hypercubes, and one-dimensional meshes. For PRAMs we obtain optimal tradeoffs between the competitive ratio and the largest number of processors requested by any job.
Our results demonstrate that online scheduling with dependencies differs from scheduling without dependencies in several crucial aspects. First, it is essential to use virtualization, i.e., to schedule parallel jobs on fewer processors than requested. Second, the maximal number of processors requested by a job has significant influence on the performance. Third, the geometric structure of the network topology is an even more important factor than in the absence of dependencies.

Postscript