This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.
Some Stochastic Orderings Among (n-1)-out-of-n Systems from Scale Components
