  This paper studies the response time bound of a DAG (directed acyclic graph)
task. Recently, the idea of using multiple paths to bound the response time of
a DAG task, instead of using a single longest path in previous results, was
proposed and leads to the so-called multi-path bound. Multi-path bounds can
greatly reduce the response time bound and significantly improve the
schedulability of DAG tasks. This paper derives a new multi-path bound and
proposes an optimal algorithm to compute this bound. We further present a
systematic analysis on the dominance and the sustainability of three existing
multi-path bounds and the proposed multi-path bound. Our bound theoretically
dominates and empirically outperforms all existing multi-path bounds. What's
more, the proposed bound is the only multi-path bound that is proved to be
self-sustainable.
