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 .