Discrete multistate coherent systems and the three modules theorem

Systems of n components, in which the system and its components either fail or function, and where the system is not degraded by performance improvement of any of its components, are known as binary coherent systems (BCS's). Sometimes there are subsets of components which function collectively as a single unit. In this case such subsets could be replaced by single units, consequently reducing the system, without affecting the system performance. These subsets are called modular sets and play a significant role in system reliability analysis.

Complex systems are quite common in the present high-technology age; for example, nuclear power systems and space technology systems, among others. These complex systems require a large amount of computation, and interaction among their components can make the exact computation of their reliabilities virtually impossible. The analyst may attempt to attack these problems by incorporating modules into the system. These modules facilitate the analysis of the system, since they reduce its general configuration, and provide the analyst with a good methodology for obtaining bounds for the system reliability.

For BCS's there is an important result known as the Three Modules Theorem (Birnbaum and Esary (1965)). This theorem asserts that if two modular sets have at least one common component, then we can obtain a larger modular set (by combining the two original sets) or three smaller modular sets (by decomposing the two original sets into disjoint sets). While BCS's have been extended to multistate coherent systems (MCS's) since the late 1970's, the Three Modules Theorem has undergone no generalization. In our paper this theorem is generalized to a specific class of multistate coherent systems.

We also provide bounds for the reliability of a specific class of a MCS, by extending the (binary) results given by Shanthikumar (1986).