A pinwheel scheduler for three distinct numbers with a tight schedulability bound

Given a multiset of positive integers A = {a₁, a₂, . . . , a_n}, the pinwheel problem is to find an infinite sequence over {1, 2, . . . , n} such that there is at least one symbol i within any subsequence of length a_i. The density of A is defined as ρ (A) = ∑_i=1ⁿ(1/a_i). In this paper we limit ourselves to instances composed of three distinct integers. The best scheduler [5] published previously can schedule all instances with a density of less than 0.77. A new and fast scheduling scheme based on spectrum partitioning is presented in this paper which improves the 0.77 result to a new 5/6 ≈ 0.83 density threshold. This scheduler has achieved the tight schedulability bound of this problem.