Speed scaling is a power management technique that involves dynamically changing the speed of a processor. We study policies for setting the speed of the processor for both of the goals of minimizing the energy used and the maximum temperature attained. The theoretical study of speed scaling policies to manage energy was initiated in a seminal paper by Yao et al. [1995], and we adopt their setting. We assume that the power required to run at speed s is P ( s ) = s α for some constant α > 1. We assume a collection of tasks, each with a release time, a deadline, and an arbitrary amount of work that must be done between the release time and the deadline. Yao et al. [1995] gave an offline greedy algorithm YDS to compute the minimum energy schedule. They further proposed two online algorithms Average Rate (AVR) and Optimal Available (OA), and showed that AVR is 2 α − 1 α α -competitive with respect to energy. We provide a tight α α bound on the competitive ratio of OA with respect to energy. We initiate the study of speed scaling to manage temperature. We assume that the environment has a fixed ambient temperature and that the device cools according to Newton's law of cooling. We observe that the maximum temperature can be approximated within a factor of two by the maximum energy used over any interval of length 1/ b , where b is the cooling parameter of the device. We define a speed scaling policy to be cooling-oblivious if it is simultaneously constant-competitive with respect to temperature for all cooling parameters. We then observe that cooling-oblivious algorithms are also constant-competitive with respect to energy, maximum speed and maximum power. We show that YDS is a cooling-oblivious algorithm. In contrast, we show that the online algorithms OA and AVR are not cooling-oblivious. We then propose a new online algorithm that we call BKP. We show that BKP is cooling-oblivious. We further show that BKP is e -competitive with respect to the maximum speed, and that no deterministic online algorithm can have a better competitive ratio. BKP also has a lower competitive ratio for energy than OA for α ≥5. Finally, we show that the optimal temperature schedule can be computed offline in polynomial-time using the Ellipsoid algorithm.

We introduce resource augmentation as a method for analyzing online scheduling problems. In resource augmentation analysis the on-line scheduler is given more resources, say faster processors or more processors, than the adversary. We apply this analysis to two well-known on-line scheduling problems, the classic uniprocessor CPU scheduling problem 1͉r i , pmtn͉͚ F i , and the best-effort firm real-time scheduling problem 1͉r i , pmtn͉͚ w i (1 Ϫ U i ). It is known that there are no constant competitive nonclairvoyant on-line algorithms for these problems. We show that there are simple on-line scheduling algorithms for these problems that are constant competitive if the online scheduler is equipped with a slightly faster processor than the adversary. Thus, a moderate increase in processor speed effectively gives the on-line scheduler the power of clairvoyance. Furthermore, the on-line scheduler can be constant competitive on all inputs that are not closely correlated with processor speed. We also show that the performance of an on-line scheduler in best-effort real time scheduling can be significantly improved if the system is designed in such a way that the laxity of every job is proportional to its length.

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Abstract. Intel's SpeedStep and AMD's PowerNOW technologies allow the Windows XP operating system to dynamically change the speed of the processor to prolong battery life. In this setting, the operating system must not only have a job selection policy to determine which job to run, but also a speed scaling policy to determine the speed at which the job will be run. We give an online speed scaling algorithm that is O(1)-competitive for the objective of weighted flow time plus energy. This algorithm also allows us to efficiently construct an O(1)-approximate schedule for minimizing weighted flow time subject to an energy constraint.

"What matters most to the computer designers at Google is not speed, but power, low power, because data centers can consume as much electricity as a city." -Dr. Eric Schmidt, CEO of Google [12]. AbstractAll of the theoretical speed scaling research to date has assumed that the power function, which expresses the power consumption P as a function of the processor speed s, is of the form P = s α , where α > 1 is some constant. Motivated in part by technological advances, we initiate a study of speed scaling with arbitrary power functions. We consider the problem of minimizing the total flow plus energy. Our main result is a (3+ǫ)-competitive algorithm for this problem, that holds for essentially any power function. We also give a (2+ǫ)-competitive algorithm for the objective of fractional weighted flow plus energy. Even for power functions of the form s α , it was not previously known how to obtain competitiveness independent of α for these problems. We also introduce a model of allowable speeds that generalizes all known models in the literature.

We survey recent research that has appeared in the theoretical computer science literature on algorithmic problems related to power management. We will try to highlight some open problem that we feel are interesting. This survey places more concentration on lines of research of the authors: managing power using the techniques of speed scaling and power-down which are also currently the dominant techniques in practice.

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