Exploiting the Uncertainty of the Longest Paths: Response Time Analysis for Probabilistic DAG Tasks

Parallel real-time systems (e.g., autonomous driving systems) often contain functionalities with complex dependencies and execution uncertainties, leading to significant timing variability which can be represented as a probabilistic distribution. However, existing timing analysis either produces a single conservative bound or suffers from severe scalability issues due to the exhaustive enumeration of every execution scenario. This causes significant difficulties in leveraging the probabilistic timing behaviours, resulting in sub-optimal design solutions. Modelling the system as a probabilistic directed acyclic graph (p-DAG), this paper presents a probabilistic response time analysis based on the longest paths of the p-DAG across all execution scenarios, enhancing the capability of the analysis by eliminating the need for enumeration. We first identify every longest path based on the structure of p-DAG and compute the probability of its occurrence. Then, the worst-case interfering workload is computed for each longest path, forming a complete probabilistic response time distribution with correctness guarantees. Experiments show that compared to the enumeration-based approach, the proposed analysis effectively scales to large p-DAGs with computation cost reduced by six orders of magnitude while maintaining a low deviation (1.04% on average and below 5% for most p-DAGs), empowering system design solutions with improved resource efficiency.