Some reliability measures and maintenance policies for a coherent system composed of different types of components

Abstract

Consider an n-components coherent system monitored at one or two inspection times, and some information about the system and its components is obtained. Under these conditions, some variants of mean residual lifetimes can be defined. Also, the dual concept of the residual lifetime, i.e., inactivity time is defined for a failed system under different conditions. This article is concerned with the study of mean residual lives and mean inactivity times for a coherent system made of multiple types of dependent components. The dependency structure is modeled by a survival copula. The notion of survival signature is employed to represent the system’s reliability function and subsequently its mean residual lives and mean inactivity times under different events at the monitoring time. These dynamic measures are used frequently to study the reliability characteristics of a system. Also, they provide helpful tools for designing the optimal maintenance policies to preserving the system from sudden and costly failures. Here, we extend some maintenance strategies for a coherent system consists of multiple dependent components. Some illustrative examples are provided.

Appendix

1.1 Proof of Theorem 2.2

We have

$$\begin{aligned} A_{\mathbf {l},\mathbf {j}}^{\mathbf {n}}(s,t)&= Pr\left( T_{1}^{(1)}>s, \ldots , T_{l_1}^{(1)}>s, t<T_{l_1+1}^{(1)}\le s, \ldots , t<T_{j_1}^{(1)}\le s, T_{j_1+1}^{(1)}\right. \\&\quad \left. \le t, \ldots , T_{n_1}^{(1)}\le t, \ldots , T_{1}^{(K)}>s, \ldots , T_{l_K}^{(K)}>s, t<T_{l_K+1}^{(K)}\right. \\&\quad \left. \le s, \ldots , t<T_{j_K}^{(K)}\le s, T_{j_K+1}^{(K)}\le t, \ldots , T_{n_K}^{(K)}\le t \right) . \end{aligned}$$

Let

$$\begin{aligned} M&=\{ T_{1}^{(i)}>s, \ldots , T_{l_i}^{(i)}>s, T_{l_i+1}^{(i)}>t, \ldots , T_{j_i}^{(i)}>t, \text {for } i=1,\ldots ,K\},\\ N^c&=\{ T_{l_i+1}^{(i)}\le s, \ldots , T_{j_i}^{(i)}\le s, \text {for } i=1,\ldots ,K \},\\ L^c&=\{T_{j_i+1}^{(i)}\le t, \ldots , T_{n_i}^{(i)}\le t, \text {for } i=1,\ldots ,K \}. \end{aligned}$$

Hence, we can write

$$\begin{aligned} A_{\mathbf {l},\mathbf {j}}^{\mathbf {n}}(s,t)&=Pr(M\cap N^c \cap L^c )\nonumber \\&=Pr(M)-Pr(M\cap N)-Pr(M\cap L)+Pr(M \cap N \cap L). \end{aligned}$$

(18)

For the event M we have

$$\begin{aligned}&Pr(M)= \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1},\underbrace{1}_{n_1-j_1}, \ldots , \underbrace{\bar{F}_K(s)}_{l_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K},\underbrace{1}_{n_K-j_K}\right) . \end{aligned}$$

The event N can be written as the following

$$\begin{aligned} N=\cup _{i=1}^K \cup _{k_i=l_i+1}^{j_i} \{T_{k_i}^{(i)}>s\}, \end{aligned}$$

since there are $\sum _{i=1}^K (j_i-l_i)$ events in N, for convenience we let

$$\begin{aligned} N\equiv \cup _{q=1}^{\sum _{i=1}^K (j_i-l_i)} \{E_q\}, \end{aligned}$$

hence using inclusion–exclusion formula we get

$$\begin{aligned}&Pr(M \cap N) =Pr\left( \cup _{q=1}^{\sum _{i=1}^K (j_i-l_i)} (M \cap E_q)\right) \\&\quad =\sum _{q=1}^{\sum _{i=1}^K (j_i-l_i)}(-1)^{q+1}\underset{r_1+\ldots +r_L=q}{\sum _{r_1=0}^{j_1-l_1}\ldots \sum _{r_K=0}^{j_K-l_K}} \left( {\begin{array}{c}j_1-l_1\\ r_1\end{array}}\right) \\&\qquad \ldots \left( {\begin{array}{c}j_K-l_K\\ r_K\end{array}}\right) \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1+r_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1-r_1},\underbrace{1}_{n_1-j_1},\right. \\&\qquad \left. \ldots , \underbrace{\bar{F}_K(s)}_{l_K+r_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K-r_K},\underbrace{1}_{n_K-j_K}\right) . \end{aligned}$$

Similarly, for the event L we have

$$\begin{aligned} L=\cup _{i=1}^K \cup _{k_i=j_i+1}^{n_i} \{T_{k_i}^{(i)}>t\}, \end{aligned}$$

as there are $n-\sum _{i=1}^K j_i$ events in L, for simplicity we denote the L by the following

$$\begin{aligned} L=\cup _{q=1}^{n-\sum _{i=1}^K j_i} \{D_q\}, \end{aligned}$$

subsequently, by inclusion-exclusion formula, we have

$$\begin{aligned}&Pr(M \cap L) =Pr\left( \cup _{q=1}^{n-\sum _{i=1}^K j_i} (M \cap D_q)\right) \\&\quad =\sum _{q=1}^{n-\sum _{i=1}^K j_i}(-1)^{q+1}\underset{r_1+\ldots +r_L=q}{\sum _{r_1=0}^{n_1-j_1}\ldots \sum _{r_K=0}^{n_K-j_K}} \left( {\begin{array}{c}n_1-j_1\\ r_1\end{array}}\right) \\&\qquad \ldots \left( {\begin{array}{c}n_K-j_K\\ r_K\end{array}}\right) \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1+r_1},\underbrace{1}_{n_1-j_1-r_1},\right. \\&\qquad \left. \ldots \underbrace{\bar{F}_K(s)}_{l_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K+r_K},\underbrace{1}_{n_K-j_K-r_K}\right) . \end{aligned}$$

Finally, using $E_q$’s and $D_q$’s we get the following

$$\begin{aligned}&Pr(M \cap N \cap L)=Pr\left( (M\cap L) \cap \cup _{q=1}^{\sum _{i=1}^K (j_i-l_i)} \{E_q\} \right) \\&\quad =Pr(\cup _{q=1}^{\sum _{i=1}^K (j_i-l_i)} ((M\cap L) \cap E_q))\\&\quad =\sum _{q=1}^{\sum _{i=1}^K (j_i-l_i)}(-1)^{q+1}\underset{r_1+\ldots +r_L=q}{\sum _{r_1=0}^{j_1-l_1}\ldots \sum _{r_K=0}^{j_K-l_K}} \left( {\begin{array}{c}j_1-l_1\\ r_1\end{array}}\right) \ldots \left( {\begin{array}{c}j_K-l_K\\ r_K\end{array}}\right) \\&\qquad \times \sum _{v=1}^{n-\sum _{i=1}^K j_i}(-1)^{v+1}\underset{y_1+\ldots +y_L=v}{\sum _{y_1=0}^{n_1-j_1}\ldots \sum _{y_K=0}^{n_K-j_K}} \left( {\begin{array}{c}n_1-j_1\\ y_1\end{array}}\right) \\&\quad \ldots \left( {\begin{array}{c}n_K-j_K\\ y_K\end{array}}\right) \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1+r_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1-r_1+y_1},\underbrace{1}_{n_1-j_1-y_1},\right. \\&\qquad \left. \ldots , \underbrace{\bar{F}_K(s)}_{l_K+r_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K-r_K+y_K},\underbrace{1}_{n_K-j_K-y_K}\right) . \end{aligned}$$

Note that if $q=0$ then we get $Pr(M \cap N \cap L)=-Pr(M \cap L)$ and $Pr(M\cap N)=-Pr(M)$. By replacing the obtained relations in (18), the following result is gained

$$\begin{aligned}&A_{\mathbf {l},\mathbf {j}}^{\mathbf {n}}(s,t)\\&\quad =-\sum _{q=0}^{\sum _{i=1}^K (j_i-l_i)}(-1)^{q+1}\underset{r_1+\ldots +r_L=q}{\sum _{r_1=0}^{j_1-l_1}\ldots \sum _{r_K=0}^{j_K-l_K}} \left( {\begin{array}{c}j_1-l_1\\ r_1\end{array}}\right) \\&\quad \ldots \left( {\begin{array}{c}j_K-l_K\\ r_K\end{array}}\right) \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1+r_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1-r_1},\underbrace{1}_{n_1-j_1}, \ldots ,\right. \\&\qquad \left. , \underbrace{\bar{F}_K(s)}_{l_K+r_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K-r_K},\underbrace{1}_{n_K-j_K}\right) +\sum _{q=0}^{\sum _{i=1}^K (j_i-l_i)}(-1)^{q+1}\underset{r_1+\ldots +r_L=q}{\sum _{r_1=0}^{j_1-l_1}\ldots \sum _{r_K=0}^{j_K-l_K}} \left( {\begin{array}{c}j_1-l_1\\ r_1\end{array}}\right) \\&\quad \ldots \left( {\begin{array}{c}j_K-l_K\\ r_K\end{array}}\right) \times \sum _{v=1}^{n-\sum _{i=1}^K j_i}(-1)^{v+1}\underset{y_1+\ldots +y_L=v}{\sum _{y_1=0}^{n_1-j_1}\ldots \sum _{y_K=0}^{n_K-j_K}} \left( {\begin{array}{c}n_1-j_1\\ y_1\end{array}}\right) \\&\quad \ldots \left( {\begin{array}{c}n_K-j_K\\ y_K\end{array}}\right) \hat{C}\left( \underbrace{\bar{F}_i(s)}_{l_i+r_i},\underbrace{\bar{F}_i(t)}_{j_i-l_i-r_i+y_i},\underbrace{1}_{n_i-j_i-y_i}, \ldots \right. \\&\qquad \left. , \underbrace{\bar{F}_i(s)}_{l_i+r_i},\underbrace{\bar{F}_i(t)}_{j_i-l_i-r_i+y_i},\underbrace{1}_{n_i-j_i-y_i}\right) \\&\quad =\sum _{q=0}^{\sum _{i=1}^K (j_i-l_i)}(-1)^{q+1}\underset{r_1+\ldots +r_L=q}{\sum _{r_1=0}^{j_1-l_1}\ldots \sum _{r_K=0}^{j_K-l_K}} \left( {\begin{array}{c}j_1-l_1\\ r_1\end{array}}\right) \ldots \left( {\begin{array}{c}j_K-l_K\\ r_K\end{array}}\right) \\&\qquad \times \sum _{v=0}^{n-\sum _{i=1}^K j_i}(-1)^{v+1}\underset{y_1+\ldots +y_L=v}{\sum _{y_1=0}^{n_1-j_1}\ldots \sum _{y_K=0}^{n_K-j_K}} \left( {\begin{array}{c}n_1-j_1\\ y_1\end{array}}\right) \\&\quad \ldots \left( {\begin{array}{c}n_K-j_K\\ y_K\end{array}}\right) \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1+r_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1-r_1+y_1},\underbrace{1}_{n_1-j_1-y_1}, \right. \\&\quad \left. \ldots ,\underbrace{\bar{F}_K(s)}_{l_K+r_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K-r_K+y_K},\underbrace{1}_{n_K-j_K-y_K}\right) \\&\quad ={\sum _{r_1=0}^{j_1-l_1}\ldots \sum _{r_K=0}^{j_K-l_K}}(-1)^{r_1+\ldots +r_L}\left( {\begin{array}{c}j_1-l_1\\ r_1\end{array}}\right) \\&\quad \ldots \left( {\begin{array}{c}j_K-l_K\\ r_K\end{array}}\right) {\sum _{y_1=0}^{n_1-j_1}\ldots \sum _{y_K=0}^{n_K-j_K}} (-1)^{y_1+\ldots +y_L}\left( {\begin{array}{c}n_1-j_1\\ y_1\end{array}}\right) \ldots \left( {\begin{array}{c}n_K-j_K\\ y_K\end{array}}\right) \\&\qquad \times \hat{C}\left( \underbrace{\bar{F}_1(s)}_{l_1+r_1},\underbrace{\bar{F}_1(t)}_{j_1-l_1-r_1+y_1},\underbrace{1}_{n_1-j_1-y_1}, \ldots , \underbrace{\bar{F}_K(s)}_{l_K+r_K},\underbrace{\bar{F}_K(t)}_{j_K-l_K-r_K+y_K},\underbrace{1}_{n_K-j_K-y_K}\right) . \end{aligned}$$

1.2 Proof of relation (8)

$$\begin{aligned}&Pr(W_{t_1}, t_1<T<t_2)\\&\quad ={\sum _{l_1=0}^{n_1}\ldots \sum _{l_K=0}^{n_K}}~~{\sum _{j_1=l_1}^{n_1}\ldots \sum _{j_K=l_K}^{n_K}} Pr(W_{t_1}, t_1<T<t_2, C_1(t_2)\\&\quad =l_1, \ldots , C_K(t_2)=l_K, C_1(t_1)=j_1, \ldots , C_K(t_1)=j_K)\\&\quad ={\sum _{l_1=0}^{n_1}\ldots \sum _{l_K=0}^{n_K}}~~\underset{(j_1,\ldots ,j_K) \in D_{W_{t_1}}}{\sum _{j_1=l_1}^{n_1}\ldots \sum _{j_K=l_K}^{n_K}} Pr(t_1<T<t_2| C_1(t_2)=l_1, \ldots , C_K(t_2)\\&\quad =l_K, C_1(t_1)=j_1, \ldots , C_K(t_1)=j_K)\\&\qquad \times Pr(C_1(t_2)=l_1, \ldots , C_K(t_2)=l_K, C_1(t_1)=j_1, \ldots , C_K(t_1)=j_K)\\&\quad ={\sum _{l_1=0}^{n_1}\ldots \sum _{l_K=0}^{n_K}}~~\underset{(j_1,\ldots ,j_K) \in D_{W_{t_1}}}{\sum _{j_1=l_1}^{n_1}\ldots \sum _{j_K=l_K}^{n_K}} \left[ Pr(T>t_1| C_1(t_2)=l_1, \ldots , C_K(t_2)=l_K, C_1(t_1)\right. \\&\quad =j_1, \ldots , C_K(t_1)=j_K)\\&\quad \left. -Pr(T>t_2| C_1(t_2)=l_1, \ldots , C_K(t_2)=l_K, C_1(t_1)=j_1, \ldots , C_K(t_1)=j_K)\right] \\&\quad \times Pr(C_1(t_2)=l_1, \ldots , C_K(t_2)=l_K, C_1(t_1)=j_1, \ldots , C_K(t_1)=j_K)\\&\quad ={\sum _{l_1=0}^{n_1}\ldots \sum _{l_K=0}^{n_K}}~~ \underset{(j_1,\ldots ,j_K) \in D_{W_{t_1}}}{\sum _{j_1=l_1}^{n_1}\ldots \sum _{j_K=l_K}^{n_K}} [\Phi (j_1,\ldots , j_K)-\Phi (l_1,\ldots , l_K)]PC_{\mathbf {l},\mathbf {j}}^{\mathbf {n}}(t_2,t_1). \end{aligned}$$