## 3 Methodology
### A Batch-Based Queueing Model for Passenger-Vehicle Interaction in the Pick-up Zone  
There are passenger queue and vehicle queue both constrained by storage buffer existing in the pick-up zone and the connected roadway. The passenger queue is set in the passenger waiting area with \(n\) passenger waiting lanes and the vehicle queue is set across the vehicle storage park and vehicle service area with \(m\) lanes in the pick-up zone. We denote the length of connected roadway as \(l\). Vehicles of inter-city transportation arrive at the passenger transportation hub at a rate \(\lambda_{a}\), and passengers leave the hub and arrive at the passenger queue at rate \(\lambda_{p}\). Because of the management rules, we assume that vehicles move to the pick-up zone in batches (i.e., branch by branch of vehicles) from the vehicle storage park and leave in batches after carrying passengers. We set the vehicle queue has finite buffer \(N\) in batch. Based on these, we abstract the vehicle queuing process into a batch-based queue, as shown in Figure 3.  
As vehicles are served in batches in the queuing system, there is a transformation between the actual traffic parameters within the connected roadway and the traffic parameters in the vehicle queuing system in terms of quantity units; specifically, when the traffic flow of the connected roadway is \(q\), the traffic density is \(k\), and the maximum traffic density is \(k_{j}\). The relationship between the traffic flow \(q_{b}\), the traffic density \(k_{b}\) and the maximum traffic density \(k_{bj}\) of the vehicle queuing system and \(q\), \(k\) and \(k_{j}\) are as Equation (1):  
\[q_{b}=\frac{q}{(n\cdot m)}\,\ k_{b}=\frac{k}{(n\cdot m)},\ k_{bj}=\frac{k_{j}}{(n \cdot m)} \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline  & **Resolution** & **Notation** & **Type** & **Description** \\ \hline
10 & \(\lambda_{b}\) & Arrival rate of intra-city transportation vehicle by batch (batches/min) \\
11 & \(\mu_{b}\) & Service rate of vehicle queue by batches (batches/min) \\
12 & \(\mu_{1}\) & Service rate in non-congested condition (veh/min) \\
13 & \(\mu_{b1}\) & Service rate in non-congested condition by batch (batches/min) \\
14 & \(\mu_{2}\) & Service rate in congested condition (veh/min) \\
15 & \(\mu_{b2}\) & Service rate in congested condition by batch (batch/min) \\
16 & \(\mu_{b1}\) & Service load rate in non-congested situation by batch \\
17 & \(\mu_{b2}\) & Service load rate in congested situation by batch \\
18 & \(\lambda_{p}\) & Arrival rate of passenger (per/min) \\
19 & \(\lambda_{a}\) & Arrival rate of inter-city transportation vehicle (veh/min) \\
20 & \(m\) & Number of vehicle lanes \\
21 & \(l\) & Length of roadway \\
22 & \(N\) & Buffer of single vehicle lane \\
23 & \(n\) & Number of passenger lanes in queue \\
24 & \(C\) & Passing capacity of the connected roadway \\
25 & \(k_{\rm C}\) & Critical density of the connected roadway \\
26 & \(\varphi_{d}\) & Degree of capacity drop \\
27 & \(\Lambda\) & Effective arrival rate (veh/min) \\
28 & \(Ul\) & Threshold number of vehicles at which the vehicle queue becomes congested \\
29 & \(U_{b}\) & Threshold number of batches at which the vehicle queue becomes congested \\
30 & \(D\) & Threshold number of vehicles at which the vehicle queue becomes non-congested \\
31 & \(D_{b}\) & Threshold number of batches at which the vehicle queue becomes non-congested \\
32 & \(\pi(i,j)\) & The probability that vehicle queue length is \(i\) under state \(j\) \\
33 & \(S\) & The service time of a vehicle (min) \\
34 & \(S_{b}\) & Variable \\
35 & \(E(S)\) & The mean sojourn time of each vehicle in the queue \\
36 & \(E_{k}(E)\) & The mean sojourn time of each batch of vehicles in the queue \\ \hline \end{tabular}
\end{table} 
Table 3: _Cont._For the vehicle queue, vehicles are serviced in the vehicle service area and the queue is placed across both the roadway and the vehicle storage park.