Corollaries to Amdahl's Law for Energy

This paper studies the important interaction between parallelization and energy consumption in a parallelizable application. Given the ratio of serial and parallel portion in an application and the number of processors, we first derive the optimal frequencies allocated to the serial and parallel regions in the application to minimize the total energy consumption, while the execution time is preserved (i.e., speedup = 1). We show that dynamic energy improvement due to parallelization has a function rising faster with the increasing number of processors than the speed improvement function given by the well-known Amdahl's Law. Furthermore, we determine the conditions under which one can obtain both energy and speed improvement, as well as the amount of improvement. The formulas we obtain capture the fundamental relationship between parallelization, speedup, and energy consumption and can be directly utilized in energy aware processor resource management. Our results form a basis for several interesting research directions in the area of power and energy aware parallel processing.