PACE Optimal Formula and Supporting Proofs

This supplemental material presents proofs for the mathematical claims in the paper, “PACE: A New Approach to Dynamic Voltage Scaling.” In §2, we describe the model of the processor we will use in our theorems and their proofs. We also demonstrate the existence, given this model, of a function S with certain interesting properties. This function will later play an important role in the computation of the optimal speed schedule for any distribution of task work, and these properties will let us prove its optimality. The optimal speed after completing w units of work is S[F (w)/K], where F (w) is the probability the task will take more than w cycles of work and K is a scaling parameter used to meet a given performance target. In §3, we will present some notation and lemmas we will need in subsequent theorems and their proofs. The main result here is the continuous improvement lemma, which shows us how to demonstrate that a function h(w) satisfies ∫ b