###### Abstract
Such emerging new methods based on the Bayesian network can to some extent provide references to prevent safety events when applied to urban utility tunnels. However, these methods rely heavily on the observed information, and risk assessment results are likely influenced by the subjectivity of experts' opinions. Moreover, it is hard for the results of risk assessment to be translated into long-term maintenance guidance for which time is a key indicator since risks are dynamic and time-varying. For instance, if a tunnel facility or a component is assessed with high risks, it is still not clear about when it will fail or how long it will take to reach the lowest required functionality. In this regard, reliability can be a better indicator than risk for the long-term safe operation of urban utility tunnels, because it is generally defined as a  
Figure 1: Leakage incidents in utility tunnels caused by failures of joint waterproof facilities. function of lifetime and can further derive other key maintenance and operation indicators such as failure rate, failure warning time and expected failure time. Reliability refers to the probability of a system operation with intended functions in the context of set time limit and specific environment [19]. It varies with operation time and system status. However, similar to the aforementioned risk assessment endeavors, conventional methods, such as set pair analysis [20] and the technique for order preference by similarity to an ideal solution (TOPSIS) [21], cannot reflect the dynamic and time-varying characteristics of utility tunnel reliability [22; 23]. In addition, these methods can only offer subjective and descriptive estimations, reducing the credibility of the results. In comparison, machine learning incorporated with Bayesian methods and Markov chain Monte Carlo (MCMC) makes it possible to achieve a dynamic and credible reliability estimation [24]. Hitherto, however, there has been an absence of research attempts made to apply these methods to the reliability estimation for the joint waterproof facilities of utility tunnels. In this sense, this study will fill such a research gap by demonstrating how these methods can be used to reveal the time-dependent reliability of critical facilities, in particular joint waterproof facilities, throughout the project life cycle of utility tunnels. The observed failure regularities of joint waterproof facilities are consistent with the classical failure mode characterized by the bathtub curve, which is widely practiced in reliability estimation and deterioration modeling for many other products. The bathtub curve is comprised of three project life stages [19], including an early failure stage with a decreasing failure rate due to improper designs and implementations, a useful life stage characterized by random failures with a relatively constant failure rate and a wear-out stage with a sharply increasing failure rate before breakdown. The three life stages of the bathtub curve can be modeled by the Weibull distribution with flexible shape parameters [19]. For this reason, Weibull distribution is chosen to model the failure and reliability of joint waterproof facilities of utility tunnels. Nevertheless, the failure rate of joint waterproof facilities is not just determined by lifetime, but also subject to the actual field conditions, such as geological and hydrological conditions and ground treatment, against which the facilities are implemented. Therefore, it is necessary to incorporate such field condition covariates into the Weibull distribution [25]. Consequently, an improved Weibull distribution model will be proposed in this study. With the aid of Bayesian statistical estimation methods and MCMC methods, the key model parameters of the improved Weibull model at any momentary positions can be derived, enabling a dynamic reliability estimation for the joint waterproof facilities of utility tunnels.