A Pinwheel Scheduler for Three Distinct Numbers with a Tight Schedulability Bound

Abstract.

Given a multiset of positive integers $A=\{a_1,a_2,\ldots,a_n\}$ , the pinwheel problem is to find an infinite sequence over $\{1,2,\ldots,n\}$ such that there is at least one symbol i within any subsequence of length a _i . The density of A is defined as $\rho(A)=\sum^n_{i=1} (1/a_i)$ . 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 $\frac{5}{6}\approx 0.83$ density threshold. This scheduler has achieved the tight schedulability bound of this problem.