Towards the 5/6-Density Conjecture of Pinwheel Scheduling

TL;DR

This paper advances understanding of the Pinwheel Scheduling problem by confirming the 5/6-Conjecture for instances with up to 12 tasks and providing a set of schedules that solve all small instances with up to 5 tasks.

Contribution

The paper introduces a novel structural approach using Pareto surfaces to efficiently analyze and solve Pinwheel Scheduling instances, confirming the conjecture for small cases.

Findings

Confirmed the 5/6-Conjecture for instances with up to 12 tasks.

Identified 23 schedules that solve all schedulable instances with up to 5 tasks.

Developed an efficient algorithm leveraging structural insights for Pareto surface computation.

Abstract

Pinwheel Scheduling aims to find a perpetual schedule for unit-length tasks on a single machine subject to given maximal time spans (a.k.a. frequencies) between any two consecutive executions of the same task. The density of a Pinwheel Scheduling instance is the sum of the inverses of these task frequencies; the 5/6-Conjecture (Chan and Chin, 1993) states that any Pinwheel Scheduling instance with density at most 5/6 is schedulable. We formalize the notion of Pareto surfaces for Pinwheel Scheduling and exploit novel structural insights to engineer an efficient algorithm for computing them. This allows us to (1) confirm the 5/6-Conjecture for all Pinwheel Scheduling instances with at most 12 tasks and (2) to prove that a given list of only 23 schedules solves all schedulable Pinwheel Scheduling instances with at most 5 tasks.