Tighter Bounds of Speedup Factor of Partitioned EDF for Constrained-Deadline Sporadic Tasks

Abstract

Even though earliest-deadline-first (EDF) is optimal in terms of uniprocessor schedulability, it is co-NP-hard to precisely verify uniprocessor schedulability for constrained-deadline task sets. The most efficient way to solve this problem in polynomial time is via a partially linear approximation of the demand bound function. Such approximation leads to a simple uniprocessor schedulability testing with speedup factor ρ. Such a result further leads to Deadline-Monotonic Partitioned-EDF on multi-processors with speedup factor of 1 + ρ − 1/m (where m is the number of processors). The current state of the art results indicate that ρ is within the range [1.5,14/9]. Especially, it has been a conjecture that ρ = 1.5.This paper improves the range of ρ to (1.5026,1.5380). The improved lower bound disproves the conjecture of lower bound 1.5. A novel technique is to construct an auxiliary function that is larger than the approximate demand bound function but keeps the supremum ρ unchanged. It solves the dilemma that beating the lower bound 1.5 requires extremely large task sets, while the large size makes it difficult to check the schedulability. This technique not only enables us to disprove 1.5 by a task set of only eight tasks, but also sheds light on future work in transferring/downsizing task sets and deriving utilization bound based tests for various workload abstraction models, such as DAG tasks.