Algebraic Task Models

有很多事件和任务模型用于检查实时系统。这是因为到目前为止，还没有已知的模型来分析实时系统领域中最多样化的应用程序的不同要求。一方面，有硬的最后期限和简单的激励模式的系统，另一方面，软的最后期限，但强烈波动的执行时间通信系统。此外，不同模型之间的正式联系也没有得到很好的理解。一方面是因为每个任务模型都有自己的分析算法，这些算法与该模型紧密相连，另一方面是因为嵌入式实时系统的新应用或平台需要新的复杂的分析方法。在这个项目中，另一种方法是追求与已知的程序：任务建模是减少到几个方法，以及建立在工程和物理。嵌入式实时系统分析的主要步骤，事件和任务建模，确定最坏情况的界限的建立和随后的分析应该正式封装和数学上明确的联系。为此，一方面，要使用事件和任务模型的向量计算，另一方面，要适应从网络演算已知的代数。该项目旨在证明，它是可能的，以获得每一个事件和任务模型中描述的文献中使用三个基本的数学运算。首先，任务，它们的激活模式和净执行时间被建模为向量，并转换为边界进行分析，例如，使用标量积和积分。最坏的情况下，然后构造使用离散区间变换或卷积从最小代数，并进行实时分析的基础上，这个新的界限。对于所有可能的应用程序，此过程应始终相同，因此不再需要特定的模型和与之相关的算法。为了说明这一点，在项目结束时，我们希望用新方法描述过去40年来文献中最重要的模型，并研究它们之间的一般联系。

There exists a bunch of event and task models for examining real-time systems. This follows from the fact that to date there is no known model for the different requirements for the analysis of the most diverse applications in the field of real-time systems. On the one hand there are systems with hard deadlines and simple excitation patterns, on the other hand communication systems with soft deadlines but strongly fluctuating execution times. In addition, the formal connection between the different models is not well understood. On the one hand, because each task model has its own analysis algorithms that are closely linked to this model, and on the other hand, because new applications or platforms of embedded real-time systems require new and complex analysis methods. In this project, an alternative approach is pursued in contrast to the known procedure: The task modeling is to be reduced to a few methods well established in engineering and physics. The main steps for the analysis of embedded real-time systems, event and task modelling, the setting up of bounds with the determination of the worst case and the subsequent analysis should be formally encapsulated and mathematically unambiguously linked. For this purpose, on the one hand, vector calculation for event and task models is to be used, on the other hand, the algebra known from network calculus is to be adapted. The project aims to prove that it is possible to derive every event and task model described in the literature using three basic mathematical operations. First, tasks, their activation patterns and net execution times are modeled as vectors and converted into bounds to be analyzed using, for example, the scalar product and integral calculus. The worst case is then constructed using a discrete interval transformation or convolution from the min-plus algebra, and the real-time analysis is carried out on the basis of this new bound. This procedure should always be the same for all possible applications, so that specific models and the algorithms linked to them are no longer necessary. In order to show this, at the end of the project we want to describe the most important models from the literature of the past 40 years in the new methodology and examine the general connections between them.