Pinwheel Scheduling: Achievable Densities

Abstract

A pinwheel schedule for a vector v= (v ₁ , v ₂ , . . ., v _n ) of positive integers 2 ≤ v ₁ ≤ v ₂ ≤ ⋅s ≤ v _n is an infinite symbol sequence {S _j : j ∈ Z} with each symbol drawn from [n] = {1,2, . . ., n } such that each i ∈ [n] occurs at least once in every v _i consecutive terms (S _j+1 , S _j+2 , . ., S _j+vi ) . The density of v is d(v) = 1/v ₁ + 1/v ₂ + ⋅s + 1/v _n . If v has a pinwheel schedule, it is schedulable . It is known that v(2,3,m) with m ≥ 6 and density d(v) = 5/6 + 1/m is unschedulable, and Chan and Chin [2] conjecture that every v with d(v) ≤ 5/6 is schedulable. They prove also that every v with d(v) ≤ 7/10 is schedulable.

We show that every v with d(v) ≤ 3/4 is schedulable, and that every v with v ₁ =2 and d(v) ≤ 5/6 is schedulable. The paper also considers the m -pinwheel scheduling problem for v , where each i ∈ [n] is to occur at least m times in every mv _i consecutive terms (S _j+1 , . ., S _j+mvi ) , and shows that there are unschedulable vectors with d(v) =1- 1/[(m+1)(m+2)] + ɛ for any ɛ > 0 .