We suggest four new measures of importance for repairable multistate systems based on the classical Birnbaum measure. Periodic component life cycles and general semi-Markov processes are considered. Similar to the Birnbaum measure, the proposed measures are generic in the sense that they only depend on the probabilistic properties of the components and the system structure. The multistate system model encodes physical properties of the components and the system directly into the structure function. As a result, calculating importance is easy, especially in the asymptotic case. Moreover, the proposed measures are composite measures, combining importance for all component states into a unified quantity. This simplifies ranking of the components with respect to importance. The proposed measures can be characterized with respect to two features: forward-looking versus backward-looking and with respect to how criticality is measured. Forward-looking importance measures focus on the next component states, while backward-looking importance measures focus on the previous component states. Two approaches to measuring criticality are considered: probability of criticality versus expected impact. Examples show that the different importance measures may result in unequal rankings.

中文翻译：

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具有可修复组件的多状态系统的 BIRNBAUM 关键性和重要性措施

基于经典的伯恩鲍姆测量，我们建议了四种对可修复多态系统具有重要意义的新测量。考虑了周期性组件生命周期和一般的半马尔可夫过程。与 Birnbaum 测度类似，所提出的测度是通用的，因为它们仅取决于组件的概率属性和系统结构。多态系统模型将组件和系统的物理特性直接编码到结构函数中。因此，计算重要性很容易，尤其是在渐近的情况下。此外，所提出的措施是复合措施，将所有组件状态的重要性合并为一个统一的量。这简化了组件在重要性方面的排序。提议的措施可以从以下两个特点来描述：前瞻与后瞻，以及衡量重要性的方式。前瞻重要性度量侧重于下一个组件状态，而后瞻重要性度量侧重于先前的组件状态。考虑了两种衡量关键性的方法：关键性概率与预期影响。例子表明，不同的重要性度量可能会导致排名不等。