A novel real-time calculus for arbitrary job patterns and deadlines

In schedulability analysis, the timing of all tasks of a real-time system is verified by finding the worst-case behavior. Two well-studied methods in this field are the demand bound test and the real-time calculus. However, the former is only applicable to specific task models whereas the latter does not formalize concrete task models of complex job patterns. This work presents a new approach to formally describe and analyze the worst-case of any complex job pattern. The approach consists of a task model that reduces all kinds of job patterns to a vector space of jobs. Furthermore, the worst-case analysis searches for local maxima to find the worst-case of any job pattern by differentiating any cumulative function of the real-time calculus. Therefore, the analysis in this work implies an algorithm to compute request as well as demand bounds by construction. This formal approach allows the integration of mathematical results from real-time calculus into real-time scheduling theory. In fact, this is, to our knowledge, the first method to compute a demand bound function with arbitrary deadlines by a min-plus deconvolution in the real-time calculus. This now allows the analysis of complex task models as the generalized multiframe model in the real-time calculus.