中文摘要：

偶发实时任务最早截止期优先（earliest deadline first,简称EDF）可调度分析是实时系统领域经典的NP困难问题.现有的伪多项式时间判定算法（pseudo-polynomail time decision algorithm,简称PTDA）均局限于利用率U严格小于1的同步任务系统.对于U≤1的同步系统或更加困难的异步系统,现有PTDA则不再适用.针对以上问题,为同步和异步两类实时系统建立了统一的整数规划模型,其规模并不依赖于利用率U的取值.基于多面体理论证明了模型维数和极大诱导不等式,进而提出了同/异步系统上EDF可调度性分析问题统一的多项式时间线性松弛求解方法.实验结果表明,该方法能够获得较紧的问题解下界,在异步和同步系统中,线性松弛解与最优解之间的平均百分界差gap分别为0.78%和1.27%.另外,随机生成了大量同步和异步系统的算例,用于该算法和传统算法进行性能比较.对于同步算例,实验结果表明,在U〉0.99时,该算法能够对70%的算例给出判定结果,算法性能与QPA算法相比有指数级提升.对于异步算例,实验结果表明,该算法能够对近96%的算例给出可调度性判定.与传统算法相比,该方法将不能判定可调度性的算例比例平均降低了29.27%.对于剩余的4%的算例,该算法将可调度上界的值平均降低了近10~4倍.

英文摘要：

Schedulability test for EDF（earliest deadline first） systems is one of the classical NP-hard problems in the study of real-time system. Current researches mainly focus on the synchronous systems with the utilization U strictly less than 1, which can be decided exactly in pseudo-polynomial time. However, these results cannot be easily extended to the synchronous systems with U≤1 or to the asynchronous systems even with U〈1. In this paper, a unified integer programming formulation, where the associated scale is independent of utilization U, is proposed for the EDF schedulability problems in both of the synchronous and asynchronous systems. The polyhedral structure of the formulation is investigated and a kind of facet inequalities is derived, resulting in a linear relaxation approach with polynomial-time complexity. Numerical results on a large scale randomly generated asynchronous and synchronous instances show that the proposed method can obtain a tight gap（0.78% and 1.27% respectively on average） between the relaxation and the optimal integer solutions. Furthermore, the comparison with the QPA exhibits that the new method is available for 70% synchronous instances and exponentially reduces the calculation time especially in situations when U〉0.99. Finally, experiments on asynchronous systems find that nearly 96% instances can be exactly solved by the method, which is 29.27% lesser than the traditional method. For the rest of the instances, the upper bound of the schedulability test can be sharply reduced. For most instances, the new bound is 104 smaller than the traditional ones.