On the Problem of Finding Optimal Harmonic Periods

Harmonic periods have wide applicability in industrial real-time systems. Rate monotonic (RM) is able to schedule task sets with harmonic periods up to 100% utilization. Also, if there is no release jitter and execution time variation, RM and EDF generate the same schedule for each instance of a task. This property decreases the jitters which happen during sampling and actuation of the tasks, and hence, it increases the quality of service in control systems. In this paper, we consider the problem of optimal period assignment where the periods are constrained to be harmonic. First, we assume that an interval is determined a priori for each task from which its period can be selected. The goal is to assign a (harmonic) period to each task such that the total system utilization is maximized while the task set remains feasible. We show that this problem is (at least) weakly NP-hard. This is shown by reducing the NP-complete number partitioning problem to the mentioned harmonic period assignment problem. Afterwards, we consider a variant of the problem in which the periods are not restricted to a special interval and the objective is to minimize the total weighted sum of the periods with the same feasibility constraint. We present two approximation algorithms for the second problem and show that the maximum error of these algorithms is bounded by a factor of 2. Our evaluations show that, on the average, results of the approximation algorithms are very close to an optimal solution.