Improved Bounds for Speed Scaling in Devices Obeying the Cube-Root Rule

Speed scaling is a power management technique that involves dynamically changing the speed of a processor. This gives rise to dualobjective scheduling problems, where the operating system both wants to conserve energy and optimize some Quality of Service (QoS) measure of the resulting schedule. In the most investigated speed scaling problem in the literature, the QoS constraint is deadline feasibility, and the objective is to minimize the energy used. The standard assumption is that the power consumption is the speed to some constant power α. We give the first non-trivial lower bound, namely e^α−1/α, on the competitive ratio for this problem.

For CMOS based processors, and many other types of devices, α = 3, that is, they satisfy the cube-root rule. Thus the most interesting case is when α = 3.When α = 3, the algorithm with the best known competitive ratio is Optimal Available (OA), which is 27-competitive.We introduce a new algorithm qOA, and show that qOA is 6.7-competitive when α = 3. So when the cube-root rule holds, our results reduce the range for the optimal competitive ratio from [1.8, 27] to [2.4, 6.7].We also analyze qOA for general α and give almost matching upper and lower bounds.