On Multi-Processor Speed Scaling with Migration

We investigate a very basic problem in dynamic speed scaling where a sequence of jobs, each specified by an arrival time, a deadline and a processing volume, has to be processed so as to minimize energy consumption. Previous work has focused mostly on the setting where a single variable-speed processor is available. In this paper we study multiprocessor environments with m parallel variable-speed processors assuming that job migration is allowed, i.e. whenever a job is preempted it may be moved to a different processor. We first study the offline problem and show that optimal schedules can be computed efficiently in polynomial time. In contrast to a previously known strategy, our algorithm does not resort to linear programming. We develop a fully combinatorial algorithm that relies on repeated maximum flow computations. The approach might be useful to solve other problems in dynamic speed scaling. For the online problem, we extend two algorithms Optimal Available and Average Rate proposed by Yao et al. [16] for the single processor setting. We prove that Optimal Available is alpha(alpha)-competitive, as in the single processor case. Here alpha > 1 is the exponent of the power consumption function. While it is straightforward to extend Optimal Available to parallel processing environments, the competitive analysis becomes considerably more involved. For Average Rate we show a competitiveness of (3 alpha)(alpha)/2 + 2(alpha).