A Quadratic-Time Response Time Upper Bound with a Tightness Property

The response time analysis (RTA) is one of the fundamental tools used to guarantee the schedulability of sets of real-time tasks scheduled by Fixed Priorities. Also, several analysis methods inspired by RTA have been successfully developed to address more sophisticated execution platforms (distributed systems, multiprocessor) and application models (DAGs). The major issue with RTA is its time complexity, which is NP-hard. Such a complexity shows up when the task set has high utilization and RTA needs to check all jobs until the first idle instant. In this paper, we propose a continuous upper bound to the response time with quadratic time complexity in the number of tasks. Such an upper bound is demonstrated to be tighter than previously proposed ones with linear time complexity. In addition, with two tasks only, we prove that the proposed bound is the tightest continuous function upper bounding the exact response time of sets of tasks with full utilization. Whether or not this property holds with more than two tasks is still an open problem.