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Accelerating Probabilistic Response-Time Analysis: Revised Critical Instant and Optimized Convolution

Hiroto Takahashi, Atsushi Yano, Takuya Azumi

TL;DR

<3-5 sentence high-level summary>

Abstract

Accurate estimation of the Worst-Case Deadline Failure Probability (WCDFP) has attracted growing attention as a means to provide safety assurances in complex systems such as robotic platforms and autonomous vehicles. WCDFP quantifies the likelihood of deadline misses under the most pessimistic operating conditions, and safe estimation is essential for dependable real-time applications. However, achieving high accuracy in WCDFP estimation often incurs significant computational cost. Recent studies have revealed that the classical assumption of the critical instant, the activation pattern traditionally considered to trigger the worst-case behavior, can lead to underestimation of WCDFP in probabilistic settings. This observation motivates the use of a revised critical instant formulation that more faithfully captures the true worst-case scenario. This paper investigates convolution-based methods for WCDFP estimation under this revised setting and proposes an optimization technique that accelerates convolution by improving the merge order. Extensive experiments with diverse execution-time distributions demonstrate that the proposed optimized Aggregate Convolution reduces computation time by up to an order of magnitude compared to Sequential Convolution, while retaining accurate and safe-sided WCDFP estimates. These results highlight the potential of the approach to provide both efficiency and reliability in probabilistic timing analysis for safety-critical real-time applications.

Accelerating Probabilistic Response-Time Analysis: Revised Critical Instant and Optimized Convolution

TL;DR

<3-5 sentence high-level summary>

Abstract

Accurate estimation of the Worst-Case Deadline Failure Probability (WCDFP) has attracted growing attention as a means to provide safety assurances in complex systems such as robotic platforms and autonomous vehicles. WCDFP quantifies the likelihood of deadline misses under the most pessimistic operating conditions, and safe estimation is essential for dependable real-time applications. However, achieving high accuracy in WCDFP estimation often incurs significant computational cost. Recent studies have revealed that the classical assumption of the critical instant, the activation pattern traditionally considered to trigger the worst-case behavior, can lead to underestimation of WCDFP in probabilistic settings. This observation motivates the use of a revised critical instant formulation that more faithfully captures the true worst-case scenario. This paper investigates convolution-based methods for WCDFP estimation under this revised setting and proposes an optimization technique that accelerates convolution by improving the merge order. Extensive experiments with diverse execution-time distributions demonstrate that the proposed optimized Aggregate Convolution reduces computation time by up to an order of magnitude compared to Sequential Convolution, while retaining accurate and safe-sided WCDFP estimates. These results highlight the potential of the approach to provide both efficiency and reliability in probabilistic timing analysis for safety-critical real-time applications.

Paper Structure

This paper contains 23 sections, 2 theorems, 17 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Probabilistic Notation and Symbols
  4. Task Set Assumptions
  5. WCDFP and a Revised Upper Bound
  6. Definitions and Notations
  7. Upper Bound Considering the Critical Instant
  8. Foundations for Efficient Convolution
  9. Mathematical Preliminaries for Convolution
  10. Efficient Convolution Techniques
  11. Single Convolution using Fast Fourier Transform
  12. Exponentiation by Squaring in Convolutions
  13. Accelerated Aggregate Convolution with Revised Critical Instant
  14. Optimizing Convolution Merge Order
  15. Worst-Case Cost of Sequential Merging
  16. ...and 8 more sections

Key Result

Corollary 1

Given a fully preemptive fixed-priority scheduler, a set of constrained-deadline sporadic tasks, and under the assumption that incomplete jobs are aborted at their deadline, the following inequality holds: Here, $S_{k,t}$ is defined as: where $\mathbb{P}(S_{k,t} > t)$ represents the probability that the processor demand of task $\tau_k$ and higher-priority tasks exceeds the time $t$.

Figures (5)

  • Figure 1: Visualization of the job arrival sequence, showing jobs released at various time instants for different tasks, along with their respective priorities
  • Figure 2: Processor load indicator $S_{k,t}$ for task $\tau_k$, including carry-in and preemption effects from higher-priority tasks
  • Figure 3: Comparison of WCDFP estimation accuracy and execution time across MC, SC, AC (Imp.), and BE. Points are positioned relative to baseline values at $x=1.0$ and $y=1.0$, with quadrant counts displayed in the corners
  • Figure 4: Comparison of WCDFP values computed using MC, SC, AC (Imp.), and BE. Each point represents a taskset, with counts of task sets above and below the $y = x$ line displayed in the top-left and bottom-right corners, respectively
  • Figure 5: Comparison of execution times for WCDFP estimation across MC, SC, AC (Orig.), AC (Imp.), and BE. The boxplots summarize execution time distributions, with each method corresponding to a box

Theorems & Definitions (18)

  • Definition 1: Random Variable
  • Definition 2: Cumulative Distribution Function (CDF)
  • Definition 3: Partial Order of Random Variables
  • Definition 4: Workload
  • Definition 5: Total Workload
  • Definition 6: Service Time
  • Definition 7: Carry-in
  • Definition 8: Total Carry-in of Task $\tau_i$
  • Definition 9: Aborted Execution Time
  • Definition 10: Total Aborted Execution Time of Task $\tau_i$
  • ...and 8 more