2.1. DVFS and optimization problems
When dealing with energy consumption, the most usual optimization function consists in minimizing the energy consumption, while ensuring a deadline on the execution time (i.e., a real-time constraint), as discussed in the following papers.
In [14], Okuma et al. demonstrate that voltage scaling is far more effective than the shutdown approach, which simply stops the power supply when the system is inactive. Their target processor employs just a few discretely variable voltages. De Langen and Juurlink [26] discuss leakage-aware scheduling heuristics that investigate both dynamic voltage scaling (DVS) and processor shutdown, since static power consumption due to leakage current is expected to increase significantly. Chen et al. [27] consider parallel sparse applications, and they show that when scheduling applications modeled by a directed acyclic graph with a well-identified critical path, it is possible to lower the voltage during non-critical execution of tasks, with no impact on the execution time. Similarly, Wang et al. [11] study the slack time for non-critical jobs, they extend their execution time and thus reduce the energy consumption without increasing the total execution time. Kim et al. [22] provide power-aware scheduling algorithms for bag-of-tasks applications with deadline constraints, based on dynamic voltage scaling. Their goal is to minimize power consumption as well as to meet the deadlines specified by application users.
For real-time embedded systems, slack reclamation techniques are used. Lee and Sakurai [17] show how to exploit slack time arising from workload variation, thanks to a software feedback control of supply voltage. Prathipati [12] discusses techniques to take advantage of run-time variations in the execution time of tasks; it determines the minimum voltage under which each task can be executed, while guaranteeing the deadlines of each task. Then, experiments are conducted on the Intel StrongArm SA-1100 processor, which has eleven different frequencies, and the Intel PXA250 XScale embedded processor with four frequencies. In [28], the goal of Xu et al. is to schedule a set of independent tasks, given a worst case execution cycle (WCEC) for each task, and a global deadline, while accounting for time and energy penalties when the processor frequency is changing. The frequency of the processor can be lowered when some slack is obtained dynamically, typically when a task runs faster than its WCEC. Yang and Lin [23] discuss algorithms with preemption, using DVS techniques; substantial energy can be saved using these algorithms, which succeed to claim the static and dynamic slack time, with little overhead.
Since an increasing number of systems are powered by batteries, maximizing battery life also is an important optimization problem. Battery-efficient systems can be obtained with similar techniques of dynamic voltage and frequency scaling, as described by Lahiri et al. in [18]. Another optimization criterion is the energy-delay product, since it accounts for a trade-off between performance and energy consumption, as for instance discussed by Gonzalez and Horowitz in [29]. We do not discuss further these latter optimization problems, since our goal is to minimize the energy consumption, with a fixed deadline.
In this paper, the application is a task graph (directed acyclic graph), and we assume that the mapping, i.e., an ordered list of tasks to execute on each processor, is given. Hence, our problem is closely related to slack reclamation techniques, but instead on focusing on non-critical tasks as for instance in [11], we consider the problem as a whole. Our contribution is