## 3 Case Study
### System Modelling and Availability Analysis
#### 3.3.2 Fuzzy Markov Model of the Wellhead Connector  
Before developing the Markov model of the wellhead connector, the following assumptions were made:  
1. When the wellhead connector starts running, all components are normal. 2. Both the failure rates and repair rates are constant. 3. Components failures can be detected once they fail. 4. The repaired components are considered as good as new. 5. The repair actions for the wellhead connector are independent. During oil/gas production operations, the wellhead connector is located on the wellhead. When the wellhead connector fails, the connector needs to be pulled out of the water for repair. After the repair is completed, it cannot be put into use immediately because installation testing is required. After the test is successful, it can be used normally. If the wellhead connector fails again during the installation test, it needs to be repaired and tested again until the installation test is successful. According to the failure modes and working conditions of the wellhead connector, a Markov transition diagram as shown in Figure 8 is established. The description of every state in Figure 8 is: (1) State S\({}_{0}\) indicates that the wellhead connector is working normally; (2) The states S\({}_{1}\)-S\({}_{18}\) indicate that the wellhead connector has failed, corresponding to the failure of the basic event of the fault tree X1-X18; (3) State S\({}_{19}\) indicates that the wellhead connector is installed and tested after being repaired; (4) State S\({}_{20}\) indicates that the wellhead connector has failed during installation testing or periodic testing; (5) State S\({}_{21}\) indicates periodic testing of the wellhead connector. \begin{table}
\begin{tabular}{c c c} \hline
**Basic Events Number** & **Fuzzy Possibility Score (FPS)** & **Fuzzy Failure Probability (FFR)** \\ \hline X\({}_{1}\) & 0.216236431 & \(3.44859\times 10^{-12}\) \\ X\({}_{2}\) & 0.220247538 & \(4.2376\times 10^{-12}\) \\ X\({}_{3}\) & 0.27177883 & \(4.32267\times 10^{-11}\) \\ X\({}_{4}\) & 0.227944145 & \(6.21879\times 10^{-12}\) \\ X The state transfer time functions in the Markov model obey the exponential distribution, thus, the state transfer rates are constant, and the average staying time of the system in every state is the reciprocal of the state transfer rate [41]. Assume that the system is in state \(i\) at time \(0\), the transfer rate into the next state \(j\) is \(\lambda\), then the probability that the system enters state \(j\) at time \(t\) is \(1-\mathrm{e}^{\lambda t}\). The meanings of the transfer rates between states in the model is as shown below. 1. \(\lambda_{1}\)--\(\lambda_{18}\) indicate the number of failures X1--X18 that occurred per unit time;
2. \(\lambda_{19}\) is the maintenance rate, which indicates the number of maintenance failures per unit time, and is the reciprocal of the average maintenance time. These data are the average value of the maintenance rate of the connector, derived from OREDA, thus, this paper defines it to represent the maintenance rate of failures X1--X18. 3. \(\lambda_{20}\) indicates the number of failures per unit time during the installation test;
4. \(\lambda_{21}\) is the periodic test completion rate, which indicates the number of periodic tests completed per unit time, and is the reciprocal of the average duration of the periodic tests;
5. \(\lambda_{22}\) is the periodic test rate, which indicates the number of periodic tests performed per unit time, which is the reciprocal of the test period;
6. \(\lambda_{23}\) is the installation test completion rate, which indicates the number of installation tests completed per unit time, and is the reciprocal of the average duration of the installation tests;
7. \(\lambda_{24}\) indicates the number of failures per unit time during the periodic test. Figure 8: Markov transition diagram of subsea wellhead connector. Let \(Pi\left(t\right)\) (\(i=0,\,1,\,2\,\ldots\,\,21\)) be the probability that the system is in state \(Si\) at time \(t\).