## 3 Methodology
### Component Swapping in the Original System
For the system shown in Figure 4, the survival signature of the original system without swapping \(\Phi(l_{1},l_{2})\), and the system with swapping \(\Phi^{w}(l_{1},l_{2})\) are presented in Table 3. It can be noted that the values of \(\Phi(1,1)\) and \(\Phi(1,2)\) are changed from \(1/2\) to \(1\), which means the increase in the probability of system's reliability. Using the failure times of components of different types and cumulative distribution function (CDF) \(F_{k}(t)\), the system's reliability can be calculated by  
\[R(t)=\sum_{l_{1}=0}^{m_{1}}\cdots\sum_{l_{K}=0}^{m_{K}}\left[\Phi(l_{1},\cdots,l_{K})\prod_{k=1}^{K}\left(\binom{m_{k}}{l_{k}}[F_{k}(t)]^{m_{k}-l_{k}}[1-F_{ k}(t)]^{l_{k}}\right)\right] \tag{1}\]  
For a situation that some components can be swapped upon the failure of a component, the system structure function given the defined swap in place is represented as \(\phi^{w}(\underline{x})\). Two swapping rules are made in this study: (i) once a component fails another good component will take over its role in the system; (ii) swaps of components take neglectable time and do not affect the functioning of the component that changes its role in the system nor its remaining useful life [19]. Therefore, the survival signature given the defined swapping regime in place can be expressed as follows  
\[\Phi^{w}(l_{1},\cdots,l_{K})=\left(\prod_{k=1}^{K}\binom{m_{k}}{l_{k}}^{-1} \right)\times\sum_{\underline{x}\in S_{l_{1},\cdots,l_{K}}}\phi^{w}(\underline {x}) \tag{2}\]  
Figure 4: Illustration of component swapping. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{l_{1}}\) & \(\mathbf{l_{2}}\) & \(\mathbf{\Phi(l_{1},l_{2})}\) & \(\mathbf{\Phi^{w}(l_{1},l_{2})}\) \\ \hline
0 & 1 & 0 & 0 \\
0 & 2 & 0 & 0 \\
1 & 2 & 1/2 & 1 \\
2 & 2 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Survival signature of the system in Figure 4 with and without swapping. Then the system reliability with the specified swapping regime in place \(R^{w}(t)\) can be obtained as follows  
\[R^{w}(t)=\sum_{l_{1}=0}^{m_{1}}\cdots\sum_{l_{K}=0}^{m_{K}}\left[\Phi^{w}(l_{1}, \cdots,l_{K})\prod_{k=1}^{K}\left(\binom{m_{k}}{l_{k}}[F_{k}(t)]^{m_{k}-l_{k}}[1- F_{k}(t)]^{l_{k}}\right)\right] \tag{3}\]  
where all swapping strategies are reflected in the survival signature, and effects of component swapping on the system can be seen from the structure function. The improvement of system reliability due to the swapping regime compared to the situation without swaps is given from Equations (1) and (3)  
\[\Delta R_{s}(t)=\sum_{l_{1}=0}^{m_{1}}\cdots\sum_{l_{K}=0}^{m_{K}}\left\{[ \Phi^{w}(l_{1},\cdots,l_{K})-\Phi(l_{1},\cdots,l_{K})]\cdot\prod_{k=1}^{K} \left(\binom{m_{k}}{l_{k}}[F_{k}(t)]^{m_{k}-l_{k}}[1-F_{k}(t)]^{l_{k}}\right)\right\} \tag{4}\]  
The example shown in Figure 4 is used to illustrate the effects of component swapping on system reliability. Let CDFs of components of each type be \(F_{1}(t)=1-e^{-(t/3)^{4}}\) and \(F_{2}(t)=1-e^{-(t/4)^{4}}\), and the system reliability curves with and without swapping are shown in Figure 5. The results of this figure clearly present that component swapping can greatly improve the reliability of the entire system.