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Packing-Inspired Algorithms for Periodic Scheduling Problems with Harmonic Periods

Josef Grus, Claire Hanen, Zdeněk Hanzálek

TL;DR

$Weaddress non-preemptive periodic scheduling with harmonic periods on a single machine, and establish a bijection to height-divisible 2D packing (HD2D) to leverage packing techniques. The approach combines a CP model using pack constraints with packing-inspired heuristics (notably RG-FF-OPT) and compares against ILP on synthetic, time-triggered communication-like instances. The main contributions are the formal equivalence PSP  HD2D, a CP formulation outperforming ILP on hard cases, and a packing-based heuristic framework that improves solvability for highly utilized PSPs. Overall, the work provides a novel structural perspective and practical algorithms for time-triggered scheduling in industrial and avionics settings, enabling scalable exact and heuristic solutions under high utilization.$

Abstract

We tackle the problem of non-preemptive periodic scheduling with a harmonic set of periods. Problems of this kind arise within domains of periodic manufacturing and maintenance, and also during the design of industrial, automotive, and avionics communication protocols, where efficient scheduling of messages is crucial for the performance of a time-triggered network. We consider the decision variant of the periodic scheduling problem on a single highly-utilized machine. We first prove a bijection between periodic scheduling and a particular (so-called height-divisible) 2D packing of rectangles. We formulate the problem using Constraint Programming and compare it with equivalent state-of-the-art Integer Linear Programming formulation, showing the former's superiority on difficult instances. Furthermore, we develop a packing-inspired first fit heuristic, which we compare with methods described in the literature. We justify our proposed methods on synthetically generated problem instances inspired by the communication of messages on one channel.

Packing-Inspired Algorithms for Periodic Scheduling Problems with Harmonic Periods

TL;DR

Abstract

We tackle the problem of non-preemptive periodic scheduling with a harmonic set of periods. Problems of this kind arise within domains of periodic manufacturing and maintenance, and also during the design of industrial, automotive, and avionics communication protocols, where efficient scheduling of messages is crucial for the performance of a time-triggered network. We consider the decision variant of the periodic scheduling problem on a single highly-utilized machine. We first prove a bijection between periodic scheduling and a particular (so-called height-divisible) 2D packing of rectangles. We formulate the problem using Constraint Programming and compare it with equivalent state-of-the-art Integer Linear Programming formulation, showing the former's superiority on difficult instances. Furthermore, we develop a packing-inspired first fit heuristic, which we compare with methods described in the literature. We justify our proposed methods on synthetically generated problem instances inspired by the communication of messages on one channel.

Paper Structure

This paper contains 23 sections, 5 theorems, 34 equations, 6 figures, 5 tables.

Table of Contents

  1. INTRODUCTION AND RELATED WORK
  2. PROBLEM DEFINITION
  3. 2D PACKING AND PERIODIC SCHEDULING
  4. Height-Divisible 2D Packing Problem
  5. $HD2D$ Packing Instance Associated with a $PSP$ Instance
  6. Flip Operation and Equivalence between $HD2D$ Packing Problem and $PSP$
  7. Proof of Theorem \ref{['theor:flip']}
  8. Canonical Form of a $HD2D$ Packing
  9. CP MODEL
  10. HEURISTICS
  11. Baseline Heuristics
  12. Time-wise First Fit
  13. Spatial First Fit and Best Fit
  14. Longest Processing Time First
  15. Rectangle-Guided First Fit
  16. ...and 8 more sections

Key Result

Lemma 1

A periodic schedule induces a collision between two jobs $J_{i},J_{j}$ with periods $T_{\alpha_{i}},T_{\alpha_{j}}$ such that $\alpha_{i}\le\alpha_{j}$ if and only if the two following conditions hold:

Figures (6)

  • Figure 1: Solutions to $PSP$ and $HD2D$ packing problems, with corresponding jobs and rectangles presented using the same colors. Row indices use the mixed-radix system.
  • Figure 2: Canonical form of the packing shown in Figure \ref{['fig:sample_placement']}. Sub-bin division is highlighted by T-like contours.
  • Figure 3: Bag view of dummy rectangles for height $H_1=6$ constructed for instance from Section \ref{['sec:equivalence']}.
  • Figure 4: Unfinished (a), (b) and finished (c) packings created by different methods on instance from Section \ref{['sec:equivalence']}.
  • Figure 5: Utilization experiment on instance set $S_3$. Colored dots mark final utilizations $U_F$ for each instance. Black dots mark the average value per method.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • proof
  • proof
  • Definition 1
  • Theorem 2
  • ...and 1 more