Speed scaling with an arbitrary power function

This article initiates a theoretical investigation into online scheduling problems with speed scaling where the allowable speeds may be discrete, and the power function may be arbitrary, and develops algorithmic analysis techniques for this setting. We show that a natural algorithm, which uses Shortest Remaining Processing Time for scheduling and sets the power to be one more than the number of unfinished jobs, is 3-competitive for the objective of total flow time plus energy. We also show that another natural algorithm, which uses Highest Density First for scheduling and sets the power to be the fractional weight of the unfinished jobs, is a 2-competitive algorithm for the objective of fractional weighted flow time plus energy.