Non-clairvoyant speed scaling for weighted flow time

We study online job scheduling on a processor that can vary its speed dynamically to manage its power. We attempt to extend the recent success in analyzing total unweighted flow time plus energy to total weighted flow time plus energy. We first consider the non-clairvoyant setting where the size of a job is only known when the job finishes. We show an online algorithm WLAPS that is 8α²-competitive for weighted flow time plus energy under the traditional power model, which assumes the power P(s) to run the processor at speed s to be s^α for some α > 1. More interestingly, for any arbitrary power function P(s), WLAPS remains competitive when given a more energy-efficient processor; precisely, WLAPS is 16(1 + 1/ε) ²-competitive when using a processor that, given the power P(s), can run at speed (1 + ε)s for some ε > 0. Without such speedup, no non-clairvoyant algorithm can be O(1)-competitive for an arbitrary power function [8]. For the clairvoyant setting (where the size of a job is known at release time), previous results on minimizing weighted flow time plus energy rely on scaling the speed continuously over time [5-7]. The analysis of WLAPS has inspired us to devise a clairvoyant algorithm LLB which can transform any continuous speed scaling algorithm to one that scales the speed at discrete times only. Under an arbitrary power function, LLB can give an 4(1 + 1/ε)-competitive algorithm using a processor with (1 + ε)-speedup.