###### Abstract  
Safety issues are a major concern for the long-term maintenance and operation of utility tunnels, of which the focal point lies in the reliability of critical facilities. Conventional evaluation methods have failed to reflect the time-dependency and objectivity of the reliability of critical facilities, hence reducing the credibility of the analysis results and posing serious risks to the safety of utility tunnels. Taking joint waterproof facilities as an example, this paper focuses on the scientific problem of how to achieve a dynamic estimation of the reliability of critical facilities throughout the project life cycle of utility tunnels. To this end, an improved Weibull distribution model is proposed to incorporate the actual field conditions that affect the reliability of joint waterproof facilities of utility tunnels. Bayesian methods and Hamiltonian Monte Carlo methods are used to realize the posterior estimation of the model parameters via the observed failure data. The case study shows that the posterior prediction results fit well with the actual observation data. The proposed model can be used to estimate in real time such key reliability indicators as failure rate, failure warning time and expected failure time, which facilitate the safe operation and targeted maintenance of utility tunnels. reliability estimation; joint waterproof facilities; utility tunnel; improved Weibull model +
Footnote †: journal: Applied  
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phenomenon for the deformation joints of utility tunnels. As can be seen from the photos in Figure 1, water leakages further induce various safety risks to the operation of utility tunnels, including structural and metal corrosion, electrical short circuit and invasion of toxic and harmful substances. In order to effectively reduce water leakage incidents and control joint failure risks, it is necessary to undertake a safety assessment or reliability estimation for the joint waterproof facilities of utility tunnels. In recent years, more attention has been paid to the safety issues of urban utility tunnels. Conventional approaches have resorted to various fuzzy set theories for the safety evaluation of utility tunnels. For instance, [PERSON] et al. (2019) [10] combined such methods as weighted fuzzy Petri nets, even trees and a fire dynamics simulator to assess the cable fire risks in utility tunnels. [PERSON] et al. (2022) [11] adopted the grey clustering method for the safety evaluation of urban utility tunnels. However, these approaches are static and fail to consider the uncertainty characteristics of safety events related with utility tunnel facilities. To better address the safety uncertainties in utility tunnels, the Bayesian network model combined with other models, such as the Dempster-Shafer (D-S) evidence theory [12], bow-tie model composed of a fault tree and event tree [13; 14; 15] and the Work Breakdown Structure and Risk Breakdown Structure method [16], is widely used to predict the comprehensive risks of all pipelines [5] or of specific pipelines and safety incidents in utility tunnels. The predicted risks derived from the Bayesian network can also be periodically updated using the cumulative observed information recorded in a specific period (e.g., a year). The Bayesian network can also be integrated with numerical simulation methods to quantitatively evaluate the safety event consequences inside and outside utility tunnels. For example, [PERSON] et al., (2022) [17] used the computational fluid dynamics method to quantify the potential gas dispersion and explosion consequences for utility tunnels housing natural gas pipelines. In addition, considering that there are complex types of risks in utility tunnels, a system dynamics simulation was used to achieve a dynamic coupling risk assessment for utility tunnels [18].