The bell is ringing in speed-scaled multiprocessor scheduling

Abstract

This paper investigates the problem of scheduling jobs on multiple speed-scaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s#945; per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βBα-approximation algorithm for multiple processors, where Bα is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a βBα-competitive online algorithm for multiple processors. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βBα-approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes. All algorithms are based on the surprising fact that we can remove migration with a blowup of Bα in expectation.