On the stochastic and dependence properties of the three-state systems

Abstract

Suppose that a system has three states up, partial performance and down. We assume that for a random time $T_1$ the system is in state up, then it moves to state partial performance for time $T_2$ and then the system fails and goes to state down. We also denote the lifetime of the system by $T$, which is clearly $T=T_1+T_2$. In this paper, several stochastic comparisons are made between $T$, $T_1$ and $T_2$ and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided.

Appendix

Computation of signature in Example 1:

We show the computation regarding part (a). The number of possible orderings of the links failures is $5!=120$. One can easily verify that in $24$ situations out of $120$, the first and second failure links cause the states of the network change from $K=2$ to $K=1$ and $K=1$ to $K=0$, respectively; that is, $n_{1,2}=24$. Similarly, it can be seen that $n_{1,3}=56, \ n_{1,4}=16, \ n_{2,3}=16, \ n_{2,4}=8.$ Hence, based on the fact that $s_{i,j}=\frac{n_{i,j}}{n!},$ the nonzero elements of the signature matrix of this network are given, respectively, as $s_{1,2}=\frac{24}{120}, \ s_{1,3}=\frac{56}{120}, \ s_{1,4}=\frac{16}{120}, \ s_{2,3}=\frac{16}{120}, \ s_{2,4}=\frac{8}{120}.$ The computation for part (b) can be done similarly and hence we omit the details.