We study classical deadline-based preemptive scheduling of tasks in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each task is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of tasks in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. [16], its exact computational complexity has been repeatedly posed as an open question (see e.g. [2,8,15]). The currently best known upper and lower bounds are a 4/3-approximation algorithm and NP-hardness due to [2] and [2,17], respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules.
我们重新审视了非抢占式的速度缩放问题，其中一组作业必须在一个或一组并行的速度可伸缩的处理器（S）的发布日期和截止日期之间执行，以使能量消耗最小化。我们采用在[Yao et al.，FOCS 1995]，根据该方法，耗散的功率是处理器速度的凸函数。直觉上，处理器的速度越高，能耗就越高。对于单处理器的情况下，我们通过提供一个$(1+ε)^α\tilde{B}_α$ -近似算法，其中$\tilde{B}_α$是贝尔数的推广，提高了最知名的近似算法。对于多处理机的情况下，我们提出了一个近似算法的比率$\tilde{B}_α((1+ε)(1+\frac{w_{\max}}{w_{\min}}))^α$提高了$(\frac{5}{2})^{α-1}(\frac{w_{\max}}{w_{\min}})^α$的最佳已知结果的一个因素。请注意，我们的结果适用于完全异构的环境，而以前已知的结果只适用于更受限制的情况下，具有相同的功率函数的并行处理器。
University of Twente（特文特大学）；Sharif University of Technology（谢里夫理工大学）；Max Planck Institut für Informatik（马克斯·普朗克信息学研究所）；Universität des Saarlandes（萨尔大学）
