# Planning-Level Optimisation of Headway Regularity
## 3 Methods
### Model Development  
According to [PERSON] [75], passenger travel time from the origin to the destination consists of \(t_{\text{od}}=t_{\text{a}}+t_{\text{w}}+t_{\text{ol}}+t_{\text{f}}+t_{\text{a}}\), where \(t_{\text{od}}\) is the origin-destination passenger travel time (min); \(t_{\text{a}}\) is the time required for reaching the PPT stop (min); \(t_{\text{w}}\) is the waiting time at the stop (min); \(t_{\text{ol}}\) is the on-line travel time (min); \(t_{\text{f}}\) is the transfer time (min); \(t_{\text{a}^{\prime}}\) is the time required for covering the distance from a PPT stop to the final destination (min). The key parameters for this model are waiting time, on-line travel time and transfer time. As noted by [PERSON] and [PERSON] [82], the expected passenger waiting time is related to both the distribution of passenger arrival times at a stop and the distribution of schedule deviations in bus arrival times at that stop. For short headways, several authors assume that passengers arrive at random times, independent of bus arrival schedules [4; 75; 82]. Under such conditions, headway regularity along the line is extremely important. When headway regularity is poor, the average passenger waiting time (\(t_{\text{w}}\)) depends on the scheduled headway (\(H_{\text{s}}\)) and headway disturbances (\(h\)), as indicated by Equation (3):  
\[t_{\text{w}}=\frac{H_{\text{s}}+h}{2} \tag{3}\]  
The time a passenger spends in a PPT vehicle--on-line time (\(t_{\text{ol}}\))--is the sum of the vehicle dwell time at the stop (\(t_{\text{d}}\)) and the vehicle running time between adjacent stops (\(t_{\text{r}}\)), i.e., \(t_{\text{ol}}=\sum t_{\text{d}}+\sum t_{\text{r}}\). According to the extant studies in this domain, dwell time is determined by the required passenger boarding time [75; 76; 83]. Hence, if there is a headway disturbance, the dwell time can be defined by Equation (4):  
\[t_{\text{d}}=\frac{\lambda}{\mu}(H_{\text{s}}+h) \tag{4}\]  
In the model developed as a part of this work, transfer time is calculated in the same way as passenger waiting time (Equation (3)) because the walking distance between two radial lines is negligible and vehicle departures are uncoordinated. In order to evaluate the passenger travel time, a deterministic model was first created for one passenger's journey and was subsequently extended to all passengers on the line. The model calculates the difference in passenger travel time between the two previously described variants. On line AB, the passenger journey is observed from the time of their arrival at stop k to their departure at stop m (Figure 2). The model does not consider the total passenger travel time, as it focuses solely on the time the passenger spends in the PPT (excluding the time spent walking to/from the stop). If the transport is realised with a single diametrical line, the time the passenger spends in the PPT (TD) is expressed by Equation (5):  
\[T_{D}=t_{\text{w}}+\sum_{j=k+1}^{m-1}t_{\text{d}}+\sum_{j=k}^{m-1}t_{\text{r}} \tag{5}\]  
Figure 2: Schematic representation of passenger travel along the route. In the case of a headway disturbance, determination of the time the passenger spends in the PPT requires consideration of several basic parameters that define the headway disturbance propagation, as shown in Equation (6). The expression in Equation (6) is obtained by incorporating Equations (3) and (4) into Equation (5).