In this paper two classes of multi-state systems are proposed and the concept of modules of those systems is clarified from mathematical aspects. The purpose of this paper is to investigate some properties of modules on basis of the minimal elements which determine the structure of systems. Furthermore, we discuss a criterion, what we call test for modularity", for deciding whether or not a subset of components is a module, and develop three modules theorem" which is well-known in the theory of binary-state coherent systems.

In this paper we derive several results on the dependent 2-unit system with single repair facility. The dependency means that it is allowed for the two units to fail at the same time. First, for the dependent 2-unit warm standby system, we obtain the following results: Laplace transform of the system reliability by using the conditional transform, limit theorems for the system reliability and mean time to the system failure (MTSF), distribution of the number of repairs completed during (0, t] and Laplace transform of the system reliability and MTSF in the case that the repair facility is capable of only n repairs. Second we discuss on the effects of repair policies to the availability of the dependent 2-unit hot standby system.