Energy-Efficient Online Scheduling with Deadlines

Whether viewed as an environmental, financial, or convenience concern, efficient management of power resources is an important problem. In this paper, we explore the problem of scheduling tasks on a single variable-speed processor. Our work differs from previous results in two major ways. First, we consider a model where not all tasks need to be completed, and where the goal is to maximize the difference between the benefit of completed tasks and the cost of energy (previous work assumed that all tasks must be completed). Second, we permit a wide range of functions relating task completion time to energy (previous work assumed a polynomial relationship). We begin by exploring multiple speed packet scheduling, and we develop 2-competitive algorithm where tasks are unit-sized and indivisible. This extends to a fractional version where benefit can be obtained for partially-completed tasks, and also extends to permit arbitrary nonnegative relationships between task value and completion time. The proof introduces a novel version of online maximum-weight matching which may be of independent interest. We then consider the problem of processor scheduling with preemption. We develop a randomized poly-logarithmic competitive algorithm by showing how to effectively “guess” a speed close to that which the optimal solution will use. We also prove a number of lower bounds, indicating that our result cannot be significantly improved and that no deterministic algorithm can be better than polynomially-competitive. We also consider the case where all tasks must be completed by their deadlines and the goal is to minimize energy, improving upon the best previous competitive result (as well as extending to arbitrary convex functions). Finally, we consider a problem variant where speedup affects distinct tasks differently, and provide a logarithmic-speedup competitive result and matching lower bounds.