Graph-based task models have been studied to better model and analyze the schedulability of real-time systems. Among them, the digraph task model, with its powerful expressiveness to describe the behavior of a large class of real-time tasks, receives a wide range of interests recently. However, the exact schedulability analysis of digraph tasks on a uni-processor with preemptive static-priority scheduling has been shown to be coNP-hard. Approximate analyses based on the request and interference bound functions ($$\textit{rbf}$$rbf and $$\textit{ibf}$$ibf) have been proposed to improve the analysis efficiency. In this work, we summarize the existing results on these analysis techniques, and seek to further improve their generality, complexity, and accuracy. Specifically, we develop analysis techniques for tasks with arbitrary deadlines. We prove the periodicity of interference bound function such that it can be expressed as a finite aperiodic part and an infinite periodic part, which makes the asymptotic complexity of its calculation independent from the length of the time interval. Moreover, we develop a linear upper bound on $$\textit{ibf}$$ibf that is tighter than that of $$\textit{rbf}$$rbf, to derive a better response time bound.

中文翻译：

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静态优先级调度的有向图实时任务的响应时间分析：泛化、逼近和改进

已经研究了基于图的任务模型以更好地建模和分析实时系统的可调度性。其中，有向图任务模型以其强大的表达能力来描述一大类实时任务的行为，最近受到了广泛的关注。然而，在具有抢占式静态优先级调度的单处理器上对有向图任务进行精确的可调度性分析已被证明是 coNP-hard。已经提出了基于请求和干扰边界函数（$$\textit{rbf}$$rbf 和 $$\textit{ibf}$$ibf）的近似分析来提高分析效率。在这项工作中，我们总结了这些分析技术的现有结果，并寻求进一步提高它们的通用性、复杂性和准确性。具体来说，我们为具有任意期限的任务开发分析技术。我们证明了干涉界函数的周期性，使得它可以表示为有限的非周期性部分和无限的周期性部分，这使得其计算的渐近复杂度与时间间隔的长度无关。此外，我们在 $$\textit{ibf}$$ibf 上开发了一个比 $$\textit{rbf}$$rbf 更严格的线性上限，以获得更好的响应时间界限。