k2Q: A Quadratic-Form Response Time and Schedulability Analysis Framework for Utilization-Based Analysis

In this paper, we present a general response-time analysis and schedulability-test framework, called k2Q (k to Q). It provides automatic constructions of closed-form quadratic bounds or utilization bounds for a wide range of applications in real-time systems under fixed-priority scheduling. The key of the framework is a $k$-point schedulability test or a $k$-point response time analysis that is based on the utilizations and the execution times of $k-1$ higher-priority tasks. The natural condition of k2Q is a quadratic form for testing the schedulability or analyzing the response time. The response time analysis and the schedulability analysis provided by the framework can be viewed as a "blackbox" interface that can result in sufficient utilization-based analysis. Since the framework is independent from the task and platform models, it can be applied to a wide range of applications. We show the generality of k2Q by applying it to several different task models. k2Q produces better uniprocessor and/or multiprocessor schedulability tests not only for the traditional sporadic task model, but also more expressive task models such as the generalized multi-frame task model and the acyclic task model. Another interesting contribution is that in the past, exponential-time schedulability tests were typically not recommended and most of time ignored due to high complexity. We have successfully shown that exponential-time schedulability tests may lead to good polynomial-time tests (almost automatically) by using the k2Q framework.