In this paper, the multi-state survival signature is first redefined for multi-state coherent or mixed systems with independent and identically distributed (i.i.d.) multi-state components. With the assumption of independence of component lifetimes at different state levels, transformation formulas of multi-state survival signatures of different sizes are established through the use of equivalent systems and a generalized triangle rule for order statistics from several independent and non-identical distributions. The results obtained facilitate stochastic comparisons of multi-state coherent or mixed systems with different numbers of i.i.d. multi-state components. Specific examples are finally presented to illustrate the transformation formulas established here, and also their use in comparing systems of different sizes.

在本文中，首先为具有独立同分布（iid）多状态分量的多状态相干或混合系统重新定义了多状态生存特征。假设不同状态级别的组件寿命独立，通过使用等效系统和广义三角法则，建立了不同大小的多状态生存特征的变换公式，对多个独立且不相同的分布进行阶次统计。获得的结果有助于对具有不同数量iid的多状态相干或混合系统进行随机比较多状态组件。最后给出了具体的例子来说明这里建立的变换公式，以及它们在比较不同规模的系统中的用途。