Optimal power-down strategies

We consider the problem of selecting threshold times to transition a device to low-power sleep states during an idle period. The two-state case in which there is a single active and a single sleep state is a continuous version of the ski-rental problem. We consider a generalized version in which there is more than one sleep state, each with its own power consumption rate and transition costs. We give an algorithm that, given a system, produces a deterministic strategy whose competitive ratio is arbitrarily close to optimal. We also give an algorithm to produce the optimal online strategy given a system and a probability distribution that generates the length of the idle period. We also give a simple algorithm that achieves a competitive ratio of 3 + 2root2 approximate to 5.828 for any system.