Many modern real-time parallel applications can be modeled as a directed acyclic graph (DAG) task. Recent studies show that the worst-case response time (WCRT) bound of a DAG task can be significantly reduced when the execution order of the vertices is determined by the priority assigned to each vertex of the DAG. How to obtain the optimal vertex priority assignment, and how far from the best-known WCRT bound of a DAG task to the minimum WCRT bound are still open problems. In this article, we aim to construct the optimal vertex priority assignment and derive the minimum WCRT bound for the DAG task. We encode the priority assignment problem into an integer linear programming (ILP) formulation. To solve the ILP model efficiently, we do not involve all variables or constraints. Instead, we solve the ILP model iteratively, i.e., we initially solve the ILP model with only a few primary variables and constraints, and then at each iteration, we increment the ILP model with the variables and constraints which are more likely to derive the optimal priority assignment. Experimental work shows that our method is capable of solving the ILP model optimally without involving too many variables or constraints, e.g., for instances with 50 vertices, we find the optimal priority assignment by involving 12.67% variables on average and within several minutes on average.

Capacity augmentation bound (CAB) is a widely used quantitative metric in theoretical analysis for directed acyclic graph (DAG) parallel real-time tasks, which reveals the key factors the schedulability of DAG tasks heavily depending on: the normalized utilization (the ratio of the total utilization to the core numbers) and the tensity (the maximum ratio of task's longest path length to task's deadline). However, CAB requires both factors of a schedulable task system to be capped by the same threshold. A task system with a normalized utilization slightly larger than that threshold but very small tensity, or very smaller normalized utilization but slightly larger than that threshold has good chance to be scheduled are both denied by CAB. To this end, we propose a new concept called capacity augmentation function (CAF) to better characterize the schedulability of parallel real-time tasks, which provides a more loose and different threshold for both factors. In particular, we derive a CAF-based linear-time schedulability test for real-time constrained-deadline DAG tasks under global EDF, which entirely dominates the state-of-the-art CAB-based test for constrained-deadline settings. Finally, we conduct experiments to compare the acceptance ratio of our CAF-based test with the existing schedulability tests also having linear-time complexity. The results show that CAF-based test significantly outperforms the existing linear-time schedulability test under different parameter settings.

Capacity augmentation bound is a widely used quantitative metric in theoretical studies of schedulability analysis for directed acyclic graph (DAG) parallel real-time tasks, which not only quantifies the suboptimality of the scheduling algorithms, but also serves as a simple linear-time schedulability test. Earlier studies on capacity augmentation bounds of the sporadic DAG task model were either restricted to a single DAG task or a set of tasks with implicit deadlines. In this paper, we consider parallel tasks with constrained deadlines under global earliest deadline first policy. We first show that it is impossible to obtain a constant bound for our problem setting, and derive both lower and upper bounds of the capacity augmentation bound as a function with respect to the maximum ratio of task period to deadline. Our upper bound is at most 1.47 times larger than the optimal one. We conduct experiments to compare the acceptance ratio of our capacity augmentation bound with the existing schedulability test also having linear-time complexity. The results show that our capacity augmentation bound significantly outperforms the existing linear-time schedulability test under different parameter settings.