## 3 Materials and Methods
### Estimation of Travel Time Reliability  
\"Buffer time\" is used to reflect reliability which is composed by expected travel time and the potential variation. Travel time reliability will decrease with the increasing of the buffer time [37]. Passenger travel time includes reliability-based in-vehicle time (IVT) and reliability-based waiting time (WT). As Figure 2 shows, IVT and WT are estimated respectively.  
\[t_{ij}=t_{j}-t_{i} \tag{1}\]  
\[t_{ij}^{in-vehicle}=\mathrm{E}\left[t_{ij}\right]+\rho\sigma(t_{ij}) \tag{2}\]  
\[\mathrm{E}[t_{ij}]=\frac{\sum_{n=1}^{N}t_{ij}(n)}{N} \tag{3}\]  
\[\sigma(t_{ij})=\sqrt{\frac{1}{N-1}{\sum_{n=1}^{N}\left(t_{ij}(n)-E(t_{ij}) \right)}^{2}} \tag{4}\]  
\[w=t_{i}-t_{0} \tag{5}\]  
\[t^{waiting} =\mathrm{E}[w]+\rho\sigma_{w} \tag{6}\]  
\[\mathrm{E}(w)=\mathrm{E}(t_{i}-t_{0})=\mathrm{E}(t_{i})-\mathrm{E}(t_{0}) \tag{7}\]  
\[\sigma_{w}=\sqrt{D(w)} \tag{8}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Descriptions** \\ \hline \(t_{j_{i}}\), \(t_{i}\) & Bus arrival time samples of stop \(i\) and \(j\). \\ \(t_{ij}\) & Sample time from \(i\) to \(j\). \\ \(t_{ij}^{in-vehicle}\) & Reliability-based in-vehicle time. \\ \(E(t_{ij})\) & Expectation of IVT. \\ \(\sigma(t_{ij})\) & Standard deviation (SD) of \(ij\). \\ \(\rho\) & Reliability preference parameter. \\ \(N\) & Number of samples. \\ \(w\) & Sample waiting time. \\ \(t^{waiting}\) & Reliability-based waiting time. \\ \(E(w)\) & Expectation of WT for all the samples. \\ \(D(w)\) & Deviation of sample waiting time \\ \(\sigma_{w}\) & Standard deviation of \(w\). \\ \(z_{w}\) & Expectation of passenger arrival time/bus arrival interval. \\ \(s\) & Travel arc. \\ \(Cs\) & General cost of Travel arc s. \\ \(u\) & Unit travel time cost (yuan/h). \\ \(S_{in-vehicle}\) & Set of in-vehicle travel arc. \\ \(S_{waiting}\) & Set of waiting arc. \\ \(d_{s}\) & Extra delay time. \\ \(y_{s}\) & Number of waiting passengers. \\ \(\delta_{od}^{ps}\) & Parameter that determines whether travel arc \(s\) passes on path \(p\) from \\  & node \(o\) to node \(d\) \\ \(y_{od}^{p}\) & Number of passengers on available path \(p\) from nodes \(ij\) \\ \(K\) & Vehicle capacity. \\ \(a,b,\ \beta,\ \beta_{1},n\) & Parameters to be estimated. \\ \(S_{transfer}\) & Set of transfer arc. \\ \(u^{\prime}\) & Unit construction cost. \\ \(l_{s}\) & the length of EBL. \\ \(t_{s}^{0}\) & Unit travel time of car under free flow velocity \\ \(y^{\prime}\)ar & Volume of car. \\ \(H\) & Passenger arrival headway \\ \(\varphi_{s}\) & whether the EBL is set on the arc \(s\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters and variables.  
\[P(T=t)=\frac{\left(z_{w}H\right)^{t}e^{-z_{w}H}}{t!} \tag{9}\]  
\[E(t)=\int_{0}^{H}tP(T)dT=z_{w}H \tag{10}\]  
\[\sigma(t)=\sqrt{z_{w}} \tag{11}\]  
Equations (1)-(4) reflect the estimation of IVT, and Equations (5)-(8) reflect the estimation of WT. It is mentioned that Equation (2) reflects that the reliability-based IVT of stop \(i\) and \(j\) is composed of expected time and \"buffer time\". Equation (3) reflects the expectation of IVT. Equation (4) reflects standard deviation. Equation (6) reflects the reliability-based WT. The variation of WT is caused by bus arrival time and passenger arrival time. The bus interval \(h\) is assumed to follow the certain probability distribution. E(\(h\)) is estimated by field GPS data. In public transportation systems, transit passengers' arrival at bus stops can be regarded as a stochastic process that follows Poisson distribution [0, H][38].