# Planning-Level Optimisation of Headway Regularity
## 3 Methods
### Model Development
\[T_{\rm D}=\frac{H_{\rm s}+h_{\rm k}}{2}+\sum_{j=k+1}^{m-1}\left(H_{\rm s}+h_{\rm j }\right)\left(\frac{\lambda_{\rm j}}{\mu}\right)+\sum_{j=k}^{m-1}t_{\rm rj} \tag{6}\]  
To simplify the expressions in Equation (6), we modify \(\beta_{\rm j}=\lambda_{\rm j}/\mu\) and \(H_{\rm j}=H_{\rm s}+h_{\rm j}\), where \(\beta_{\rm j}\) is the ratio of passenger boarding intensity and passenger accumulation intensity at the \(j^{th}\) stop, and \(H_{\rm j}\) is the actual headway at the \(j^{th}\) stop, resulting in Equation (7):  
\[T_{\rm D}=\frac{H_{\rm k}}{2}+\sum_{j=k+1}^{m-1}H_{\rm j}\beta_{j}+\sum_{j=k}^ {m-1}t_{\rm rj} \tag{7}\]  
If the transport is realised with two radial lines, Equation (5) is expanded for the transfer time. As before, the time the passenger spends in the PPT (T2R) comprises several components and is defined by Equation (8):  
\[T_{\rm 2R}=t_{\rm w}+\sum_{k+1}^{l-1}t_{\rm d}+\sum_{k}^{l-1}t_{\rm r}+t_{\rm f }+\sum_{l+1}^{m-1}t_{\rm d}+\sum_{l}^{m-1}t_{\rm r} \tag{8}\]  
Once again, in the case of a headway disturbance, the time the passenger spends in the PPT (two radial lines) is described by the basic parameters that define the headway disturbance propagation (Equation (9)). Accordingly, Equation (9) is obtained by expressing Equation (8) via Equations (3) and (4). \[T_{\rm 2R}=\frac{H_{\rm k}}{2}+\sum_{j=k+1}^{l-1}H_{\rm j}\beta_{j}+\sum_{j=k}^ {l-1}t_{\rm rj}+\frac{H_{\rm l}^{\prime}}{2}+\sum_{j=l+1}^{m-1}H_{\rm j}^{ \prime}\beta_{\rm j}+\sum_{j=l}^{m-1}t_{\rm rj} \tag{9}\]  
where \(H_{\rm j}^{\prime}\) is the actual headway at the \(j^{th}\) stop on the second radial line and is given by \(H_{\rm j}^{\prime}=H_{\rm s}+h_{\rm j}^{\prime}\). The difference in passenger travel time between the two variants (Equation (10)) is obtained by subtracting Equation (9) from Equation (7). \[T_{\rm 2R-D}=\frac{H_{\rm l}}{2}-H_{\rm l}\beta_{\rm l}-\sum_{j=l+1}^{m-1} \left(h_{\rm j}-h_{\rm j}^{\prime}\right)\beta_{\rm j} \tag{10}\]  
From Equations (7), (9) and (10), it can be concluded that the influence of line splitting on each passenger's travel time depends on the position of the boarding and alighting stops on the line. By splitting the diametrical line into two radial ones, three characteristic areas have been created, which have different effects on passenger journey durations. Considering the direction of travel from terminus A to terminus B, the following sections can be distinguished (Figure 3):  
* Section from A to C (section AC)
* Line splitting point (section C)
* Section from C to B (section CB)Depending on the sections defined above and the locations of stops at which passengers enter and leave the line, three distinct passenger groups can be recognised:  
1. Passengers travelling on the AC section only and not crossing the line split point (C). As this group is not affected by the line splitting, they experience no change in their travel time. 2. Passengers travelling on section CB only. This group will experience only the positive effects of line splitting, as their waiting time at the stop and the driving time will be reduced. 3. Passengers crossing the line splitting point (C) during their journey will note both positive and negative effects on their journey duration. While their travel time will be shortened, they will be required to transfer to another vehicle at the line splitting point (C), which will lead to an additional transfer time and greater travel time uncertainty. The number of passengers transported between stops represents the passenger flow (\(q\)), while the number of passengers entering (\(U\)) at a stop \"\(j\)\" is expressed by \(U_{\text{j}}=(H_{\text{s}}+h_{\text{j}})\)\(\lambda_{\text{j}}=H_{\text{j}}\)\(\lambda_{\text{j}}\).