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Multi-Path Bound for DAG Tasks

Qingqiang He, Nan Guan, Shuai Zhao, Mingsong Lv

TL;DR

This work addresses bounding the response time $R$ of DAG tasks on identical multi-core platforms. It introduces a generalized multi-path bound that removes the constraint that the first generalized path must be the longest, and provides an optimal computation method by reducing the generalized-path-list problem to a minimum-cost flow instance. The authors prove that the proposed bound dominates Graham's bound and all existing multi-path bounds, and moreover is self-sustainable. Empirical results show the approach yields tighter bounds and significantly improved schedulability compared with state-of-the-art methods, highlighting its practical impact for real-time DAG scheduling on multi-core systems.

Abstract

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.

Multi-Path Bound for DAG Tasks

TL;DR

This work addresses bounding the response time of DAG tasks on identical multi-core platforms. It introduces a generalized multi-path bound that removes the constraint that the first generalized path must be the longest, and provides an optimal computation method by reducing the generalized-path-list problem to a minimum-cost flow instance. The authors prove that the proposed bound dominates Graham's bound and all existing multi-path bounds, and moreover is self-sustainable. Empirical results show the approach yields tighter bounds and significantly improved schedulability compared with state-of-the-art methods, highlighting its practical impact for real-time DAG scheduling on multi-core systems.

Abstract

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.
Paper Structure (16 sections, 16 theorems, 35 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 16 sections, 16 theorems, 35 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. System Model
  4. Task Model
  5. Scheduling Model
  6. New Multi-Path Bound
  7. Computation of the Bound
  8. The Problem
  9. The Optimal Algorithm
  10. The Computation
  11. The Dominance Among Multi-Path Bounds
  12. The Sustainability of Multi-Path Bounds
  13. Evaluation
  14. Response Time Bound of DAG Tasks
  15. Schedulability of DAG Task Sets
  16. ...and 1 more sections

Key Result

Lemma 1

$(\lambda_i)_0^k$, $k \in [0, m-1]$, is a generalized path list. $\varepsilon$ is a regular execution sequence regarding $(\lambda_i)_0^k$. $\lambda^+$ is the restricted critical path of $(\lambda_i)_0^k$ in $\varepsilon$. $\lambda^+_\varepsilon$ is the projection of $\lambda^+$ in $\varepsilon$. Th

Figures (7)

  • Figure 1: (a) A DAG task example. (b) An execution sequence of Fig. \ref{['fig:dag']}.
  • Figure 2: (a) A DAG example for Section \ref{['sec:problem']}. (b) A flow network constructed from the DAG in Fig. \ref{['fig:ill_nested']}.
  • Figure 3: (a) A DAG example for Theorem \ref{['thm:our_date']}. (b) A DAG example for Theorem \ref{['thm:our_chen']}.
  • Figure 4: A hierarchy of multi-path bounds. Solid lines with arrows indicate dominance. Dashed lines indicate nondominance.
  • Figure 5: Normalized bound with changing the parallelism factor. (a) $m=4$. (b) $m=8$. (c) $m=12$.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Example 1
  • Definition 1: Generalized Path List
  • Example 2
  • Lemma 1: Corresponding to Lemma 4 of he2022bounding
  • Lemma 2: Corresponding to Lemma 8 of he2022bounding
  • Lemma 3: Corresponding to Lemma 9 of he2022bounding
  • Lemma 4: Corresponding to Lemma 10 of he2022bounding
  • Theorem 1
  • Example 3
  • Theorem 2
  • ...and 20 more