  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. We further extend the proposed approach
to multi-DAG task systems. Our schedulability test theoretically dominates
federated scheduling and outperforms the state-of-the-art by a considerable
margin.
