Optimization for Real-Time Systems with Non-convex Power Versus Speed Models

Until now, the great majority of research in low-power systems has assumed a convex power model. However, recently, due to the confluence of emerging technological and architectural trends, standard convex models have been invalidated for the proper specification of power models with different execution speeds. For example, the use of a shutdown energy minimization strategy to eliminate leakage power in multiprocessor systems results in a nonconvex trade-off between power and speed. Non-convexity renders the majority of previous power management schemes, algorithms, and even basic theorems invalid. For instance, the main premise that one has to run continuously using a single speed in order to minimize energy consumption for constant computation requirements is not valid anymore. We study techniques for energy minimization where the power versus speed curve has a non-convex shape. We first identify and quantify sources of nonconvexity. Minimizing energy when the power-speed model is non-convex is an NP-complete problem, even in the canonical and simple case where a task is to execute a specified amount of computation without dependencies, in a given amount of time. We address this problem using a non-linear function minimization based approach and demonstrate that on average the new solution saves at least 40% more energy on industrial processors than techniques that follow the convexity paradigm. Then we address common real-time task scenarios where the power-speed model is non-convex. Specifically, we introduce a heuristic for scheduling tasks onto a multiprocessor system with a non-trivial start-up cost and compare its performance to our mixed integer linear programming (MIP) formulation. We experimentally compare our neighbors heuristic with the wellknown average rate algorithm, and find that it results in a 106% improvement while being only 14% worse than the optimal MIP solution.