We first consider online speed scaling algorithms to minimize the energy used subject to the constraint that every job finishes by its deadline. We assume that the power required to run at speed ¡ is ¢¤ £. We provide a tight bound on the competitive ratio of the previously pro-posed Optimal Available algorithm. This improves the best known competitive ratio by a factor � � of. We then introduce a new online algorithm, and show that this algorithm’s competitive ratio is at � £ �� � £ �¨����¥�¥����� � most. This competitive ratio is significantly better and is � ������� approximately for large �. Our result is essentially tight for large �. In particular, as � approaches infinity, we show that any algorithm must have competitive ratio �� � (up to lower order terms). We then turn to the problem of dynamic speed scaling to minimize the maximum temperature that the device ever reaches, again subject to the constraint that all jobs finish by their deadlines. We assume that the device cools according to Fourier’s law. We show how to solve this problem in polynomial time, within any error bound, using the Ellipsoid algorithm. 1.

We survey recent research that has appeared in the theoretical computer science literature on algorithmic

Abstract Power-aware operating systems ensure that the system temperature does not exceed a threshold by utilizing system-throttling. In this technique, the system load (or alternatively, the clock speed) is scaled when the temperature hits this threshold. At other times, the system operates at maximum load. In this paper, we show that such simple system-throttling rules are in fact the best one can achieve under certain assumptions. We show that maintaining a constant operating speed (and thus temperature) always does more work than operating in alternating periods of cooling and heating. As a result, for certain settings and for a reasonable temperature model, we prove that system-throttling is the most effective temperature aware-scheduling. Naturally, these assumptions do not always hold; we also discuss the scenario when some of our assumptions are relaxed, and argue why one needs more complex scheduling algorithms in this case. 1 Introduction Energy and temperature management of processor systems is an increasingly important problem as the power consumption of these chips rises drastically with every new generation. At the same time, the rate of technological improvements in cooling systems has not been keeping pace [4]. Naturally, this has resulted in a large body of work that attempts to incorporate energy and temperature considerations into processor scheduling levels. Some of these are incorporated at the system level, where the on-chip architecture scales the speed of the processor if it is getting too hot. With advances in processor technology, this is also possible at the operating system level since most modern day processors have interfaces that allow the user to control its speed in real-time. The current processors by Intel, AMD, and IBM now allow a mechanism called dynamic voltage scaling (DVS) to control the clock speed of the processor [15] by varying the supply voltage. Most operating systems now also include commands which allow the user to access this interface (e.g., cpufreq in Linux).

Abstract. Power-aware operating systems/processor controllers ensure that the system temperature does not exceed a threshold by utilizing system-throttling, where the clock speed is scaled to an equilibrium load. We denote this as the Constant policy, and compare against Zig-Zag policies that alternate between phases of cooling and heating. In this paper, we characterize and calculate the best possible Zig-Zag policy, and argue that simple system-throttling rules are often optimal. In reality, however, the system design often forces us to implement Zig-Zag policies. In particular, we consider the case where the processor can operate only at a few discrete states; thus it is required to alternate between cooling and heating phases. In such a setting, we develop an algorithm that outperforms all other Zig-Zag policies, and present computational experiments emphasizing the performance of our algorithm. 1

Abstract. 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