The shape of a DAG: bounding the response time using long paths

In 1969, Graham developed a well-known response time bound for a DAG task using the total workload and the longest path of the DAG, which has been widely applied to solve many scheduling and analysis problems of DAG-based task systems. This paper presents a new response time bound for a DAG task using the total workload and the lengths of multiple long paths of the DAG, instead of the longest path in Graham's bound. Our new bound theoretically dominates and empirically outperforms Graham's bound. Based on the insight of the new bound, we propose a new task model called the multi-path model, which intuitively describes the shape of a DAG task. We further extend the proposed approach to multi-task systems using the new task model under both federated scheduling and global scheduling. Our schedulability test theoretically dominates federated scheduling and significantly outperforms the state-of-the-art.