Generalized Inequality-based Approach for Probabilistic WCET Estimation
Hayate Toba, Atsushi Yano, Takuya Azumi
TL;DR
This work tackles the challenge of obtaining safe and tight probabilistic WCET bounds without the model-uncertainty inherent in EVT methods. By embedding saturating functions into Chebyshev’s inequality, specifically $f(X)=(\arctan(X/d))^{k}$ and $f(X)=(\tanh(X/d))^{k}$, the approach reduces sensitivity to extreme outliers while preserving a valid upper bound: $P(X\ge b) \le \min_{d,k} \mathbb{E}[f(X)]/f(b)$. Through synthetic distributions and real-world Autoware timing data, the proposed ATAN and TANH variants generally yield tighter bounds than MEMIK, with ATAN showing robust, consistent performance across scenarios. The results suggest that saturating-function Chebyshev bounds can provide safe, practically useful pWCET estimates for heavy-tailed and multimodal workloads in real-time systems, without requiring EVT-tail modeling. Future work includes automated, data-driven selection of the scale parameter $d$ to maximize tightness while guaranteeing safety and reducing the computational burden of the $(d,k)$ search.
Abstract
Estimating the probabilistic Worst-Case Execution Time (pWCET) is essential for ensuring the timing correctness of real-time applications, such as in robot IoT systems and autonomous driving systems. While methods based on Extreme Value Theory (EVT) can provide tight bounds, they suffer from model uncertainty due to the need to decide where the upper tail of the distribution begins. Conversely, inequality-based approaches avoid this issue but can yield pessimistic results for heavy-tailed distributions. This paper proposes a method to reduce such pessimism by incorporating saturating functions (arctangent and hyperbolic tangent) into Chebyshev's inequality, which mitigates the influence of large outliers while preserving mathematical soundness. Evaluations on synthetic and real-world data from the Autoware autonomous driving stack demonstrate that the proposed method achieves safe and tighter bounds for such distributions.