We propose three novel mathematical optimization formulations that solve the same two-type heterogeneous multiprocessor scheduling problem for a real-time taskset with hard constraints. Our formulations are based on a global scheduling scheme and a fluid model. The first formulation is a mixed-integer nonlinear program, since the scheduling problem is intuitively considered as an assignment problem. However, by changing the scheduling problem to first determine a task workload partition and then to find the execution order of all tasks, the computation time can be significantly reduced. Specifically, the workload partitioning problem can be formulated as a continuous nonlinear program for a system with continuous operating frequency, and as a continuous linear program for a practical system with a discrete speed level set. The latter problem can therefore be solved by an interior point method to any accuracy in polynomial time. The task ordering problem can be solved by an algorithm with a complexity that is linear in the total number of tasks. The work is evaluated against existing global energy/feasibility optimal workload allocation formulations. The results illustrate that our algorithms are both feasibility optimal and energy optimal for both implicit and constrained deadline tasksets. Specifically, our algorithm can achieve up to 40% energy saving for some simulated tasksets with constrained deadlines. The benefit of our formulation compared with existing work is that our algorithms can solve a more general class of scheduling problems due to incorporating a scheduling dynamic model in the formulations and allowing for a time-varying speed profile.

中文翻译：

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两类异构多处理器的节能实时调度

我们提出了三种新颖的数学优化公式，用于解决具有硬约束的实时任务集的相同的两种类型的异构多处理器调度问题。我们的公式基于全局调度方案和流体模型。第一个公式是一个混合整数非线性程序，因为调度问题直观地被认为是一个分配问题。但是，通过改变调度问题，先确定一个任务的工作负载分区，然后再找出所有任务的执行顺序，可以显着减少计算时间。具体来说，对于具有连续工作频率的系统，工作负载划分问题可以表述为连续非线性程序，对于具有离散速度水平集的实际系统，可以表述为连续线性程序。因此，后一个问题可以通过内点法在多项式时间内以任意精度解决。任务排序问题可以通过复杂度与任务总数成线性关系的算法来解决。该工作根据现有的全球能源/可行性最佳工作负载分配公式进行评估。结果表明，我们的算法对于隐式和受限期限任务集都是可行性最优和能量最优的。具体来说，我们的算法可以为一些具有受限期限的模拟任务集实现高达 40% 的节能。与现有工作相比，我们的公式的好处是我们的算法可以解决更一般的调度问题，因为在公式中加入了调度动态模型并允许随时间变化的速度分布。