In this paper, by considering two multi-state semi-coherent systems sharing some components, we define the multi-state joint signature. Then, some properties of this multi-state joint signature are discussed in detail, and some preservation results are established in terms of stochastic ordering. A south-east shift order for comparing the multi-state joint signature matrices is then discussed. Finally, some numerical examples are presented for illustrating all the theoretical results established here.

中文翻译：

---

基于多状态联合签名的相依多状态半相干系统

在本文中，通过考虑共享一些组件的两个多状态半相干系统，我们定义了多状态联合签名。然后，详细讨论了这种多状态联合签名的一些性质，并根据随机排序建立了一些保存结果。然后讨论用于比较多状态联合签名矩阵的东南移位顺序。最后，给出了一些数值例子来说明这里建立的所有理论结果。