Optimal reward-based scheduling for periodic real-time tasks

Reward-based scheduling refers to the problem in which there is a reward associated with the execution of a task. In our framework, each real-time task comprises a mandatory and an optional part. The mandatory part must complete before the task's deadline, while a nondecreasing reward function is associated with the execution of the optional part, which can be interrupted at any time. Imprecise computation and Increased-Reward-with-Increased-Service models fall within the scope of this-framework. In this paper, we address the reward-based scheduling problem for periodic tasks. An optimal schedule is one where mandatory-parts complete in a timely manner and the weighted average reward is maximized. For linear and concave reward functions, which are most common, we 1) show the existence of an optimal schedule where the optional service time of a task is constant at every instance and 2) show how to efficiently compute this service time. We also prove the optimality of Rate Monotonic Scheduling (with harmonic periods), Earliest Deadline First, and Least Laxity First policies for the case of uniprocessors when used with the optimal service times we computed. Moreover, we extend our result by showing that any policy which can fully utilize all the processors is also optimal for the multiprocessor periodic reward-based scheduling. To show-that our optimal solution is pushing the limits of reward-based scheduling, we further prove that, when the reward functions are convex, the problem becomes NP-Hard. Our static optimal solution, besides providing considerable reward improvements over the previous suboptimal strategies, also has a major practical benefit. Run-time overhead is eliminated and existing scheduling disciplines may be used without modification with the computed optimal service times.