( m , k )-firm real-time tasks require meeting the deadline of at least m jobs out of any k consecutive jobs. When compared to hard real-time tasks, ( m , k )$-firm tasks open up the possibility of tighter resource-dimensioning in implementations. Firmness analysis verifies the satisfaction of ( m , k )-firmness conditions. Scheduling policies under which a set of periodic tasks runs on a resource influence the number of deadline missed jobs. Therefore, the nature of the firmness analysis problem depends on scheduling policies. In this work, we present Firmness Analysis (FAn) methods for three common scheduling policies—synchronous and asynchronous Static Priority Preemptive (SPP) policies and Time Division Multiple Access (TDMA). We first introduce the Balloon and Rake problem—the problem of striking the maximum number of balloons in a balloon line with a rake. We show that the common core of firmness analysis problems can be abstracted as the Balloon and Rake problem. Next, we prove that the Finite Point method is a solution to the Balloon and Rake problem. We illustrate how existing FAn methods for the TDMA and asynchronous SPP policies can be adapted to use the same solution framework for the Balloon and Rake problem. Using the solution of the Balloon and Rake problem, we adapt the existing FAn methods to synchronous SPP scheduling policies. The scalability of the FAn methods is compared with that of a timed-automata approach, a brute-force approach, and a Mixed Integer Linear Programing method. The FAn methods scale substantially better to firmness analysis problem instances with a large k and a high number of tasks.

中文翻译：

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实时任务的稳固性分析

(米,ķ)-企业实时任务要求至少在截止日期前完成米任何工作ķ连续工作。与硬实时任务相比，（米,ķ)$-firm 任务为实现中更严格的资源维度提供了可能性。牢固度分析验证满足 (米,ķ)-坚固性条件。在资源上运行一组定期任务的调度策略会影响最后期限错过的作业的数量。因此，稳固性分析问题的性质取决于调度策略。在这项工作中，我们提出了三种常见调度策略的稳固性分析 (FAn) 方法——同步和异步静态优先级抢占 (SPP) 策略和时分多址 (TDMA)。我们首先介绍气球和耙子问题——用耙子在气球线上击中最大数量的气球的问题。我们表明，稳固性分析问题的共同核心可以抽象为 Balloon 和 Rake 问题。接下来，我们证明有限点法是气球和耙子问题的解决方案。我们说明了用于 TDMA 和异步 SPP 策略的现有 FAn 方法如何适用于对 Balloon 和 Rake 问题使用相同的解决方案框架。使用 Balloon 和 Rake 问题的解决方案，我们将现有的 FAn 方法调整为同步 SPP 调度策略。将 FAn 方法的可扩展性与时间自动机方法、蛮力方法和混合整数线性规划方法的可扩展性进行了比较。FAn 方法可以更好地扩展到具有大的坚固性分析问题实例 蛮力方法和混合整数线性规划方法。FAn 方法可以更好地扩展到具有大的坚固性分析问题实例 蛮力方法和混合整数线性规划方法。FAn 方法可以更好地扩展到具有大的坚固性分析问题实例ķ和大量的任务。