## 1 Introduction  
Road network reliability is the application of reliability theory in the field of transportation, which first appeared in system engineering. It is an index to measure the sustainable service of the road network. Reliability analysis based on road network capacity can effectively predict the growth of traffic volume, what the road network can undertake in the future, and it can help to judge whether the traffic can be sustainable. The direction of road network improvement and optimization in the future can be guided by it. The research on road network reliability can be divided into three stages. The first stage focuses on the road network connectivity reliability, which mainly studies the accessibility from one node to another [1]. In 1989, [PERSON] and [PERSON] improved it and analyzed the connectivity reliability of the whole network topology [2]. After that, [PERSON] used graph theory to analyze connectivity reliability [3]. The calculation method of network connectivity reliability is relatively simple. The topological structure of the road network is abstracted by the idea of series parallel connection, and then the calculation results are obtained [4]. In the case of series connection \(P(x)=\prod_{i}p(x_{i})\), in parallel \(P(x)=1-\prod_{i}\left(1-p(x_{i})\right)\), where \(p(x_{i})\) represents the reliability of the section \(i\), \(P(x)\) represents the overall reliability of the road network. At the beginning of the study, the connectivity reliability is limited to two states of \"pass\" and \"fail\", with the values of 0 and 1, respectively. In order to better reflect real world traffic operation, the connectivity reliability is extended to [0, 1][5]. When dealing with complex large-scale networks, it is often necessary to split them into several \"small road networks\" for analysis. Connectivity reliability is more used to evaluate the reliability after road paralysis caused by disasters [6], which is not applicable to the reliability evaluation of the conventional road network. The second stage is the development stage, which is represented by travel time reliability analysis. [PERSON] used the probability index to analyze travel time reliability [7], and then [PERSON], [PERSON], [PERSON] et al. deepened and expanded the research on this basis [8; 9; 10]. [PERSON] and [PERSON] obtained the function family based on the total travel time and the matrix of its distribution after obtaining the link flow and travel time by the stochastic user equilibrium (SUE) model, which can realize the reliability analysis based on travel time [11]. From the perspective of the network, [PERSON] analyzed the reliability of the network based on travel time rate [12]. Compared with the connectivity reliability, taking travel demand into the reliability analysis system can more directly reflect the reliability of the road network in use. Generally, it can be divided into path travel time reliability, OD (Origins and Destinations of the network) travel time reliability and system travel time reliability [13]. Taking the path travel time reliability analysis as an example, the actual calculation is the probability that the time used to select the path is less than the travel time threshold, it can be obtained by \(P_{i}=P(t_{i}\leq t_{ik})\), where \(t_{i}\) is the actual travel time of path \(i\), \(t_{ik}\) is the acceptable threshold, and is \(P_{i}\) the reliability of path \(i\). The analysis of this stage focuses on the results of road network allocation, but more consideration is given to the combing of individual paths, and the analysis results cannot directly guide the direction of road network optimization. The third stage is the improvement stage. Scholars introduced the network capacity into the travel time reliability analysis, and analyzed the reliability through the travel time ratio under congestion and unblocked conditions. Referring to the methods of [PERSON], [PERSON] and [PERSON], [PERSON] et al.