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

TL;DR

This work tackles probabilistic timing analysis for parallel real-time tasks modeled as $p$-DAGs, where execution uncertainty yields a distribution of response times rather than a single bound. It introduces a non-enumeration approach that identifies the exact set of longest paths $\Lambda^*$ using a lower bound $\Delta(\tau)$ and a minimal-longest-path graph $\mathcal{G}^\diamond$, thereby avoiding exhaustive enumeration. For each $\lambda_h \in \Lambda^*$, it derives $P(\lambda_h)$ with bounded cross-path dependencies and combines these with worst-case interferences to construct the full probabilistic timing distribution, with formal correctness guarantees. Empirical results show the method scales to large $p$-DAGs, achieving up to six orders of magnitude reduction in computation time and average deviations around $1.04\%$, while enabling more resource-efficient system designs compared to enumeration-based approaches. The approach complements probabilistic WCET analyses and supports design decisions under ISO-26262-style probabilistic timing constraints.

Abstract

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.

Figures (5)

• Figure 1: A $p$-DAG with two probabilistic structures and its cumulative probability distribution in response time (numbers in black: worst-case execution time; numbers in blue: execution probability).
• Figure 2: The illustrative examples used in Sec. \ref{['sec:identify']}.
• Figure 3: Computation of $P(\lambda_h)$ for every $\lambda_h \in \Lambda^*$ ordered in non-increasing path length (blue: probabilities that are examined; red: the probability under computation; grey: probabilities to be examined).
• Figure 4: The deviation in percentage between the proposed analysis and Ueter2021 under varied $psr$, $p$ and $|\Theta|$.
• Figure 5: Comparison of the computation cost under varied $|\Theta|$.

Theorems & Definitions (14)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2
• proof