Occasional deadline misses are acceptable for soft real-time systems. Quantifying probabilistic and deterministic characteristics of deadline misses is therefore essential to ensure that deadline misses indeed happen only occasionally. This is supported by recent research activities on probabilistic worst-case execution time, worst-case deadline failure probability, the maximum number of deadline misses, upper bounds on the deadline miss probability, and the deadline miss rate. This paper focuses on the deadline miss rate of a periodic soft real-time task in the long run. Our model assumes that this soft real-time task has an arbitrary relative deadline and that a job can still be executed after a deadline-miss until a dismiss point. This model generalizes the existing models that either dismiss a job immediately after its deadline miss or never dismiss a job. We provide mathematical notation on the convergence of the deadline miss rate in the long run and essential properties to calculate the deadline miss rate. Specifically, we use a Markov chain to model the execution behavior of a periodic soft real-time task. We present the required ergodicity property to ensure that the deadline miss rate in the long run is described by a stationary distribution.

翻译：对于软实时系统而言，偶发的截止期限错过是可以接受的。因此，量化截止期限错过的概率与确定性特征至关重要，以确保截止期限确实仅偶发地发生。这一需求得到了近期研究活动的支持，这些研究涉及概率性最差情况执行时间、最差情况截止期限失效概率、最大截止期限错过次数、截止期限错过概率的上界以及截止期限错过率。本文聚焦于周期性软实时任务在长期运行中的截止期限错过率。我们的模型假设该软实时任务具有任意相对截止期限，并且作业在截止期限错过之后仍可被执行，直至某个丢弃点。该模型泛化了现有模型，这些模型要么在截止期限错过时立即丢弃作业，要么从不丢弃作业。我们提供了关于长期截止期限错过率收敛性的数学符号，以及用于计算该错过率的基本性质。具体而言，我们采用马尔可夫链对周期性软实时任务的执行行为进行建模，并提出了所需的遍历性性质，以确保长期截止期限错过率可由平稳分布描述。