Hardness Results for Static Priority Real-Time Scheduling

Real-time systems are often modeled as a collection of tasks, describing the structure of the processor's workload. In the literature, task-models of different expressiveness have been developed, ranging from the traditional periodic task model to highly expressive graph-based models. For dynamic priority schedulers, it has been shown that the schedulability problem can be solved efficiently, even for graph-based models. However, the situation is less clear for the case of static priority schedulers. It has been believed that the problem can be solved in pseudo-polynomial time for the generalized multiframe model (GMF). The GMF model constitutes a compromise in expressiveness by allowing cycling through a static list of behaviors, but disallowing branching. Further, the problem complexity for more expressive models has been unknown so far. In this paper, we show that previous results claiming that a precise and efficient test exists are wrong, giving a counterexample. We prove that the schedulability problem for GMF models (and thus also all more expressive models) using static priority schedulers is in fact coNP-hard in the strong sense. Our result thus establishes the fundamental hardness of analyzing static priority real-time scheduling, in contrast to its dynamic priority counterpart of pseudo-polynomial complexity.