Pinwheel scheduling with two distinct numbers

Abstract

“The Pinwheel” is a real-time scheduling problem based on a problem in scheduling satellite ground stations but which also addresses scheduling preventive maintenance. Given a multiset of positive integers A = {a ₁, a ₂, ..., a _n }, a schedule S for A is an infinite sequence over {1, 2, ..., n} such that any subsequence of length a _i (1 ≤ i ≤ n) contains at least one i. Schedules can always be made cyclic; that is, a segment can be found that can be repeated indefinitely to form an infinite schedule. Interesting questions include determining whether schedules exist, determining the minimum cyclic schedule length, and creating an online scheduler. The “density” of an instance is defined as $d = \sum\nolimits_{i = 1}^n {1/a} _i$. It has been shown that any instance with d > 1.0 cannot be scheduled. In the present paper we limit ourselves to instances in which A contains elements having only two distinct values. We prove that all such instances with d ≤ 1.0 can be scheduled, using a scheduling strategy based on balancing. The schedule so created is not always of minimum length, however. We use a related but more complicated method to create a minimum-length cyclic schedule, and prove its correctness. The former is computationally easier to obtain but not necessarily minimal. The latter, although still obtainable in polynomial time, requires significantly more computation. In addition, we show how to use either method to produce a fast online scheduler. Thus, we have solved completely the three major problems for this class of instances.

This work was supported in part by U.S. Office of Naval Research Grant No. N00014-86-K-0763 and National Science Foundation Grant No. CCR-8711579.