Non-clairvoyant Speed Scaling for Weighted Flow Time
Lecture Notes in Computer Science
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 8α 2 -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 ) 2 -competitive when using a processor that, given the power P (s), can run at speed (1 + )s for some > 0. Without such speedup, no non-clairvoyant algorithm can be O(1)-competitive for an arbitrary power function  . 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    . 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 )-competitive algorithm using a processor with (1 + )-speedup. Flow and energy. The past few years have witnessed several interesting results on online scheduling for optimizing the tradeoff between energy usage and flow time (e.g., [1,        ). The flow time (or simply the flow) of a job is the time elapsed since the job is released until it is completed. Assuming jobs are equally important, it is natural to find a schedule that minimizes the total flow time (also known as minimizing the total/average response time). In general, jobs have varying importance or weights, and it is more meaningful to minimize the total weighted flow time. In the speed scaling model, minimizing total flow time and energy usage are orthogonal objectives. To understand their tradeoff, Albers and Fujiwara  initiated the study of minimizing a linear combination of flow and energy. The intuition is that, from an economic viewpoint, users are willing to pay a certain (say, ρ) units of energy to reduce one unit of flow time. By changing the units of time and energy, one can further assume ρ = 1 and thus wants to minimize flow plus energy, or in general, weighted flow plus energy. Clairvoyant scheduling for flow plus energy. Most of the previous work on minimizing flow plus energy focused on the online setting where the size of a job is known at its release time. This is known as the clairvoyant setting. Bansal, Pruhs and Stein  were the first to consider jobs with arbitrary weights, and they give an online algorithm BPS that is O(( α ln α ) 2 )-competitive for minimizing weighted flow plus energy. Very recently, Bansal, Chan and Pruhs  improved the analysis of BPS, which implies that BPS is O( α ln α )-competitive. The BPS algorithm scales the speed as a function of the fraction of unfinished work and thus it keeps changing the speed continuously over time.