## 4 Reliability Analysis of Redundant Systems  
In this section, a survival signature-based system reliability model is established for the OWT. Reliability analysis of the entire system with considerations of components swapping is conducted. Calculations of multiple swapping cases are performed to explore the effects of component swapping of different types on the system reliability. The reliability block diagram of swapping components of three types is given in Figure 9. Considering the subsystem in Figure 9, which has twelve components of three types (K = 3), components 5, 6, 28, 31, 42, 43 and 44 are of type 1 (\(T_{1}\)), components 29, 45 and 46 are of type 2 (\(T_{2}\)), and components 30 and 47 are of type 3 (\(T_{3}\)). Four swapping cases of the OWT system are defined. The case without swapping is treated as case 0. In case 1, all components of three types can be swapped, and swapping can only happen among the same type components. In cases 2, 3 and 4, only components of type 1, type 2, and type 3 can be swapped when needed to keep the system functioning, respectively. It is very time-consuming and difficult to calculate the survival signature of all cases. For this reason, an \(R\) package created by [PERSON] is used to obtain the survival signature of each case in this study [28]. The survival signature of the subsystem with swapping components is derived and presented in Table 4, where \(\Phi_{0}\), \(\Phi_{1}^{w}\), \(\Phi_{2}^{w}\), \(\Phi_{3}^{w}\) and \(\Phi_{4}^{w}\) represent structure functions of Case 0, Case 1, Case 2, Case 3, and Case 4, respectively. According to the approaches proposed in Section 3.1, the reliability function of the subsystem shown in Figure 9 with component swapping is derived as follows  
\[R_{RS}^{w}(t)=\sum_{l_{1}=0}^{m_{1}=7}\sum_{l_{2}=0}^{m_{2}=3}\sum_{l_{3}=0}^{ m_{3}=2}\Phi_{i}^{w}(l_{1},l_{2},l_{3})\prod_{k=1}^{K=3}\binom{m_{k}}{l_{k}} \left[F_{k}(t)\right]^{m_{k}-l_{k}}[1-F_{k}(t)]^{l_{k}} \tag{25}\]  
where \(i=0\), \(1\), \(2\), \(3\) and \(4\). For load-sharing redundancy subsystems, alike events \(E_{5}\) and \(E_{6}\), the corresponding reliability function can be obtained from Equation (16)  
\[R_{E_{k}}^{\text{ls}}(t)= e^{-(\lambda_{1}\cdot t)^{\gamma_{1}}}\cdot e^{-(\lambda_{2} \cdot t)^{\gamma_{2}}}+\int_{0}^{t}\gamma_{1}\lambda_{1}^{\gamma_{1}}t_{1}^{ \gamma_{1}-1}\cdot e^{-\left\{\left(\lambda_{1}\cdot t_{1}\right)^{\gamma_{1} }+\left[\lambda_{2}^  
where \(k=5,6\), \(\lambda_{i}\) and \(\gamma_{i}\) are scale and shape parameters of components taking full loads, respectively, and \(\lambda_{i}^{{}^{\prime}}\) and \(\gamma_{i}^{{}^{\prime}}\) represent scale and shape parameters of components taking sharing loads, respectively (\(i=1,2\)).  
For standby redundancy subsystems, alike events \(E_{11},E_{12}\), the corresponding reliability function can be obtained from Equations (22) and (24)  
\[R_{E_{j}}^{sb}(t)=e^{-(\lambda_{1}t)^{\gamma_{1}}}+\int_{0}^{t}\lambda_{1} \gamma_{1}(\lambda_{1}u)^{\gamma_{1}-1}\cdot e^{-(\lambda_{1}u)^{\gamma_{1}}} \cdot e^{-(\lambda_{2}Qu)^{\gamma_{2Q}}}\cdot e^{-[\lambda_{2E}(t_{1e}+t-u)]^{ \gamma_{2E}}-(\lambda_{2E}t_{1e})^{\gamma_{2E}}}du \tag{27}\]  
where \(\lambda_{i}\) and \(\gamma_{i}\) represent scale parameters and shape parameters of component \(i\) (\(i=1\), SE, 2Q and 2E). Considering \(2/3\) redundancy subsystems, the system reliability is equal to the probability that the number of working components is greater than or equal to 2, which can be expressed as follows  
\[R_{E_{k}}^{2/3}(t)=\sum_{r=2}^{n=3}\binom{n}{r}\left[R_{k}(t)\right]^{r}\left[ 1-R_{k}(t)\right]^{n-r} \tag{28}\]  
where \(R_{k}(t)\) is the reliability of components of subsystem \(E_{k}\) at time \(t\) (\(k=15,16,17\)).