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.

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