Optimal harmonic period assignment: complexity results and approximation algorithms

Abstract

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 831 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.

Optimal Harmonic Period Assignment With Constrained Number of Distinct Period Values

“…The third research area includes papers which are focused on harmonic period assignment with period ranges. In recent research [9], authors determined that two classical harmonic period optimization problems labeled UHPA (utilizationmaximizing harmonic period assignment) and CHPA (costminimizing harmonic period assignment) are in the NP-hard complexity class. Additionally, they devised approximation algorithms for the relaxed version of the CHPA problem, i.e., the constraints on period ranges are removed.…”

“…We show that the already studied UHPA problem is Turing reducible to the UDHPA problem. Additionally, using the complexity results for the UHPA problem from [9], we determine that the UDHPA problem is at least weakly NP-hard by reduction from the well-known partition sum problem (ii). We devise an optimal and heuristic algorithms for the UDHPA problem, provide time-complexity analysis, and show the effectiveness of the approach with extensive evaluation on a large number of synthetically generated instances, which correspond to real-world motivational scenarios (iii).…”

“…In this paper, system S is represented as a set T of n periodic tasks with no initial offset, i.e., a synchronous task set [24]. We use a task model which is common for the period assignment with period ranges proposed in [9], [10], and [11]. Therefore, task τ i is represented as a tuple τ i = {C i , I i }, where C i is the worst-case execution time (WCET) of τ i , and I i is the period range of allowed period values for τ i .…”

“…First off, we describe existing harmonic period assignment problems which are analyzed and discussed in the literature [9], [10], [11]. Inputs of these problems are tasks described with worst-case execution times C i and period ranges…”

“…As it can be seen, actual periods T i are allowed to be in range I i and have to be in harmonic relation. It is worth noting that in the related literature (e.g., [9]) period values T i are not restricted to the integer values. However, the period ratios of the two consecutive integers Ti Tj are required to be integers.…”