Optimal harmonic period assignment: complexity results and approximation algorithms

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. As a result, all instances of a task are interfered by the same amount of workload. This property decreases the jitters that 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 and the task set is required to be feasible. We study two variants of this problem. In the first one, the objective is to maximize the system utilization, while in the second one, the goal is to minimize the total weighted sum of the periods. First, we assume that an interval is determined a priori for each task from which its period can be selected. We show that both variants of the problem are (at least) weakly NP-hard. This is shown by reducing the NP-complete number partitioning problem to the mentioned harmonic period assignment problems. Afterwards, we consider a variant of the second problem in which the periods are not restricted to a special interval. We present two approximation algorithms with polynomial-time complexity for this problem and show that the maximum relative error of these algorithms is bounded by a factor of 1.125. Our evaluations show that, on the average, results of the approximation algorithms are very close to an optimal solution.