This paper presents a scheduling model for reducing CPU energy consumption in computer systems, particularly for battery-powered devices. The model focuses on job scheduling on a single processor with variable speed, where energy usage is a convex function of the processor speed. The goal is to minimize energy consumption by scheduling jobs between their arrival and deadline times. The authors propose an offline algorithm that computes a minimum-energy schedule for any set of jobs, assuming the power function is convex. They then analyze online algorithms, particularly the Average Rate heuristic (AVR), which uses a constant factor times the minimum energy required. The analysis involves bounding the largest eigenvalue in matrices of a special type. The paper shows that for the power function $ P(s) = s^2 $, the competitive ratio of the AVR heuristic is between 4 and 8. For general power functions $ P(s) = s^p $ where $ p \geq 2 $, the competitive ratio is between $ p^p $ and $ 2^{p-1}p^p $. The analysis of the AVR heuristic is based on reducing the problem to canonical forms, which allows for the use of matrix eigenvalues to bound the competitive ratio. The paper concludes that the AVR heuristic has a constant competitive ratio for the power function $ P(s) = s^p $, and that the true competitive ratio is conjectured to be $ p^p $. The authors also mention that simulations suggest the Optimal-Available heuristic has a competitive ratio of 4 for $ p = 2 $. The paper highlights the importance of energy efficiency in portable computing devices and suggests further research into scheduling algorithms and average-case performance.This paper presents a scheduling model for reducing CPU energy consumption in computer systems, particularly for battery-powered devices. The model focuses on job scheduling on a single processor with variable speed, where energy usage is a convex function of the processor speed. The goal is to minimize energy consumption by scheduling jobs between their arrival and deadline times. The authors propose an offline algorithm that computes a minimum-energy schedule for any set of jobs, assuming the power function is convex. They then analyze online algorithms, particularly the Average Rate heuristic (AVR), which uses a constant factor times the minimum energy required. The analysis involves bounding the largest eigenvalue in matrices of a special type. The paper shows that for the power function $ P(s) = s^2 $, the competitive ratio of the AVR heuristic is between 4 and 8. For general power functions $ P(s) = s^p $ where $ p \geq 2 $, the competitive ratio is between $ p^p $ and $ 2^{p-1}p^p $. The analysis of the AVR heuristic is based on reducing the problem to canonical forms, which allows for the use of matrix eigenvalues to bound the competitive ratio. The paper concludes that the AVR heuristic has a constant competitive ratio for the power function $ P(s) = s^p $, and that the true competitive ratio is conjectured to be $ p^p $. The authors also mention that simulations suggest the Optimal-Available heuristic has a competitive ratio of 4 for $ p = 2 $. The paper highlights the importance of energy efficiency in portable computing devices and suggests further research into scheduling algorithms and average-case performance.