A pseudo-linear time algorithm for the optimal discrete speed minimizing energy consumption

Abstract

We consider the classical problem of minimizing off-line the total energy consumption required to execute a set of n real-time jobs on a single processor with a finite number of available speeds. Each real-time job is defined by its release time, size, and deadline (all bounded integers). The goal is to find a processor speed schedule, such that no job misses its deadline and the energy consumption is minimal. We propose a pseudo-linear time algorithm that checks the schedulability of the given set of n jobs and computes an optimal speed schedule. The time complexity of our algorithm is in O(n), to be compared with O(n log(n)) for the best known solution. Besides the complexity gain, the main interest of our algorithm is that it is based on a completely different idea: instead of computing the critical intervals, it sweeps the set of jobs and uses a dynamic programming approach to compute an optimal speed schedule. Our linear time algorithm is still valid (with some changes) when arbitrary (non-convex) power functions and when switching costs are taken into account.

Energy Optimal Activation of Processors for the Execution of a Single Task with Unknown Size

“…Other relevant related scenarios consider the case where the given task is composed of a stream of sporadic jobs all with known sizes, arrival times and deadlines. Here, the problem of designing an algorithm for the optimal speed profile, or the number of active processors to use at any point in time, has been solved in a series of papers, with improving complexity [8], [9], [13]. In this online clairvoyant case, where the size and deadline of each job are revealed at the arrival time of the task, the problem can be formulated as a Markov Decision process and has been solved in [7] resp.…”

“…Then, dτ (w) dw = 1 s(w) (8) and integrating both sides and using that τ (0) = 0 and that τ (W max ) ≤ D, we obtain (7).…”