Pinwheel Scheduling with Real Periods
Hiroshi Fujiwara, Kota Miyagi, Katsuhisa Ouchi
TL;DR
This work extends pinwheel scheduling to real-valued periods and proves that any instance with exactly three distinct periods and density at most $\tfrac{5}{6}$ is schedulable, via a detailed geometric case analysis in the frequency space and explicit constructive schedules. It builds on Kawamura's real-period framework and prior two-value results, employing monotonicity lemmas and a seven-case partition to cover all possibilities. The paper also discusses two-value schedulability, Pareto-surface perspectives, and the emergence of instance-sensitive versus instance-insensitive schedules, highlighting a path toward broader generalization. It concludes with a roadmap for extending the results to more than three distinct periods and for developing a general real-valued framework, noting substantial challenges ahead.
Abstract
For a sequence of tasks, each with a positive integer period, the pinwheel scheduling problem involves finding a valid schedule in the sense that the schedule performs one task per day and each task is performed at least once every consecutive days of its period. It had been conjectured by Chan and Chin in 1993 that there exists a valid schedule for any sequence of tasks with density, the sum of the reciprocals of each period, at most $\frac{5}{6}$. Recently, Kawamura settled this conjecture affirmatively. In this paper we consider an extended version with real periods proposed by Kawamura, in which a valid schedule must perform each task $i$ having a real period~$a_{i}$ at least $l$ times in any consecutive $\lceil l a_{i} \rceil$ days for all positive integer $l$. We show that any sequence of tasks such that the periods take three distinct real values and the density is at most $\frac{5}{6}$ admits a valid schedule. We hereby conjecture that the conjecture of Chan and Chin is true also for real periods.