The joint signatures of binary-state and multi-state (semi-coherent or mixed) systems with i.i.d. (independent and identically distributed) binary-state components are considered in this work. For the comparison of pairs of binary-state systems of different sizes, transformation formulas of their joint signatures are derived by using the concept of equivalent systems and a generalized triangle rule for order statistics. Similarly, for facilitating the comparison of pairs of multi-state systems of different sizes, transformation formulas of their multi-state joint signatures are also derived. Some examples are finally presented to illustrate and to verify the theoretical results established here.

在这项工作中，考虑了具有iid（独立同分布）二进制状态组件的二进制状态和多状态（半相干或混合）系统的联合签名。为了比较不同大小的二元态系统对，利用等价系统的概念和阶数统计的广义三角形规则，推导了它们的联合签名的变换公式。同理，为了便于比较不同大小的多态系统对，还导出了它们的多态联合签名的变换公式。最后给出了一些例子来说明和验证这里建立的理论结果。