Federated Scheduling Admits No Constant Speedup Factors for Constrained-Deadline DAG Task Systems

In the federated scheduling approaches in multiprocessor systems, a task either 1) is restricted to execute sequentially on a single processor or 2) has exclusive access to the assigned processors. There have been several positive results to conduct good federated scheduling policies, which have constant speedup factors with respect to any optimal federated scheduling algorithm. This paper answers an open question: "For constrained-deadline task systems with directed acyclic graph (DAG) dependency structures, do federated scheduling policies have a constant speedup factor with respect to any optimal scheduling algorithm?" The answer is "No!" This paper presents an example, which demonstrates that any federated scheduling algorithm has a speedup factor of at least $Ω(\min\{M, N\})$ with respect to any optimal scheduling algorithm, where $N$ is the number of tasks and $M$ is the number of processors.