Pinwheel Scheduling with Real Periods

TL;DR

This paper extends the pinwheel scheduling problem to real periods, proving validity for sequences with three distinct real periods and density at most 5/6, and conjectures the original conjecture for real periods.

Contribution

It proves the existence of valid schedules for sequences with three real periods and density at most 5/6, extending previous discrete period results.

Findings

Valid schedules exist for sequences with three real periods and density ≤ 5/6.

Supports the conjecture that all sequences with density ≤ 5/6 have valid schedules.

Extends pinwheel scheduling theory to real-valued periods.

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 $\lceil l a_{i} \rceil$ consecutive 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.