## 3 Case Study
### Results and Discussion
The deformation joints estimated at level 1 and level 2 were characterized by a short stable working time and should be treated with strengthened inspections. Figure 10: Estimated results in the wear-out stage. Figure 9: Estimated failure rates of deformation joints in the case study. According to the estimated reliability indicator, various maintenance strategies can be tailored for joint waterproof facilities predicted with different reliability performances. Figure 11 shows the reliability changing tendencies of deformation joints numbered #12 (colored in blue) and #156 (colored in orange), respectively. The reliability and failure behaviors of these two deformation joints showed different forms. In this case study, 80% and 40% were assumed as critical reliability thresholds, based on which the service lifetime of a joint waterproof facility can be divided into three stages, i.e., stable period, unstable period and unreliable period. With reliability ranging from the initial to 80%, joint waterproof facilities were considered to work in a stable period. When reliability drops lower than 80% but higher than 40%, joint waterproof facilities will enter an unstable working period. During the unstable period, the failure rate of joint waterproof facilities will increase significantly, and inspection efforts should be strengthened to avoid unexpected waterproof failures. When reliability is estimated lower than 40%, it is required to evaluate the waterproof performance of the deformation joint and take necessary measures, such as specified testing and maintenance. As can be seen from Figure 11, the decreasing rate of reliability of joint #12 was significantly higher than that of #156. According to the estimation results in Figure 11, the #12 deformation joint reached the early warning position (80%) in about 50 days and reached the failure critical point (40%) in about 235 days. In contrast, the stable period of the #156 deformation joint was predicted to be close to 200 days, and the unstable period lasted more than two years. According to the failure characteristics of different periods, plans can be made, respectively, to optimize the inspection and maintenance strategies while ensuring waterproof quality. It should be noted, however, that it is not easy to validate the predictions of the proposed model in a real case scenario. In general, it is appropriate to compare future observed data with predicted results to verify the model accuracy. However, for critical utility tunnel facilities such as joint waterproof facilities, it is unbearable to let the facilities fail without any interventional maintenances considering that they are already predicted with high failure risks. A plausible solution is to select an experimental sample of deformation joints that have failed before but did not cause severe damages to the utility tunnel. Then we  
\[B=\frac{n}{m-1}\sum_{i=1}^{m}\left(\overline{x}_{i.}-\overline{x}_{-}\right)^{2} \tag{25}\]0 Reliability of deformation joints #12 and #156 changing over time. can use the comparison results between the observed failure time of these joints from the current time on and their predicted failure time for the validation of prediction accuracy.