We study a mutually enriching connection between response time analysis in real-time systems and the mixing set problem. Thereby generalizing over known results we present a new approach to the computation of response times in fixed-priority uniprocessor real-time scheduling. We even allow that the tasks are delayed by some period-constrained release jitter. By studying a dual problem formulation of the decision problem as an integer linear program we show that worst-case response times can be computed by algorithmically exploiting a conditional reduction to an instance of the mixing set problem. In the important case of harmonic periods our new technique admits a near-quadratic algorithm to the exact computation of worst-case response times. We show that generally, a smaller utilization leads to more efficient algorithms even in fixed-priority scheduling. Worst-case response times can be understood as least fixed points to non-trivial fixed point equations and as such, our approach may also be used to solve suitable fixed point problems. Furthermore, we show that our technique can be reversed to solve the mixing set problem by computing worst-case response times to associated real-time scheduling task systems. Finally, we also apply our optimization technique to solve 4-block integer programs with simple objective functions.

翻译：我们研究了实时系统中响应时间分析与混合集问题之间相互丰富的联系。通过推广已知结果，我们提出了一种计算固定优先级单处理器实时调度中响应时间的新方法。我们甚至允许任务受到周期性约束的释放抖动延迟。通过将决策问题作为整数线性规划进行对偶问题形式化研究，我们证明最坏情况响应时间可以通过算法性地利用条件归约到混合集问题实例来计算。在谐波周期的重要情况下，我们的新技术允许用近二次算法精确计算最坏情况响应时间。我们证明，一般而言，即使是在固定优先级调度中，较小的利用率也能导致更高效的算法。最坏情况响应时间可以理解为非平凡不动点方程的最小不动点，因此，我们的方法也可用于求解合适的不动点问题。此外，我们证明我们的技术可以反向应用，通过计算相关实时调度任务系统的响应时间来解决混合集问题。最后，我们还应用我们的优化技术来求解具有简单目标函数的4块整数规划。