## 3 Strategy and Model
### Parameter Optimization
It is assumed that there is a linear relationship between the two as follows:  
\[t_{d}=t_{0}-\frac{d}{l}(t_{0}-\tau) \tag{14}\] \begin{table}
\begin{tabular}{l l} \hline
**Headway Distribution Function** & **Scope of Application** \\ \hline Negative exponential distribution & Less than 250 veh/h \\ Shifted negative exponential distribution & 250–750 veh/h \\ Erlang Distribution & 250–750 veh/h \\ M3 distribution & More than 750 veh/h and the phenomenon of vehicle queuing is obvious \\ \hline \end{tabular}
\end{table} 
Table 1: Common headway distribution functions. where \(t_{d}\) is the acceptable gap when the vehicle travels to distance \(d\); \(d\) is the distance traveled by the vehicle in the merging section; \(l\) is the length of the merging section; \(t_{0}\) is the initial critical gap of lane-changing. \(P\) (\(d\)) is defined as the successful merging probability of the vehicle traveling from \(x=0\) to \(x=d\) in the merging section. Considering that the headway in adjacent lanes obeys the M3 distribution, the probability \(P\) (\(h\geq t_{d}\)) that the headway of adjacent lanes is greater than the critical gap of lane-changing at the merging section \(d\) can be calculated as follows:  
\[P(h\geq t_{d})=1-F(t_{d})=\alpha e^{-\Lambda(t_{d}-\tau)} \tag{15}\]  
Whether the headway of adjacent lanes is greater than the critical gap of lane-changing at the merging section, \(d\) is related to the waiting time of the merging vehicle there. Thus, the probability distribution function of the vehicle merging into the merging section at \(d\) can be equivalent to the geometric distribution of the waiting time of the merging vehicle at \(d\). Based on this, the differential method is applied to solve \(d\) and \(P\) (\(d\)). Assuming that \(d+\Delta d\) (\(\Delta d\)\(\rightarrow\)0) is the position where the vehicle successfully merges, the merging probability can be decomposed into the sum of the probability of merging at (0, \(d\)) and the probability of not merging at (0, \(d\)) but merging at (\(d\), \(d+\Delta d\)):  
\[P(d+\Delta d)=P(d)+(1-P(d))*\Delta t*P(h\geq t_{d}) \tag{16}\]  
where \(\Delta t\) is the travel time of \(\Delta d\) for the merging vehicle. Considering \(\Delta d\)\(\rightarrow\)0, the acceptable gap of lane-changing for the merging vehicle at \(d+\Delta d\) is the same as that at \(d\), i.e., \(t_{d}\). The driving velocity of the merging vehicle within \(\Delta d\) is also the same as at \(d\), denoted as \(v\) (\(d\)), and is described as follows:  
\[\Delta t=\frac{\Delta d}{v(d)} \tag{17}\]  
Substitute Equation (17) into Equation (16):  
\[\frac{P(d+\Delta d)-P(d)}{\Delta d}=(1-P(d))*\frac{P(h\geq t_{d})}{v(d)} \tag{18}\]  
Let \(\Delta d\)\(\rightarrow\)0 and find the limit on the left side of the above formula, as follows:  
\[P^{\prime}(d)=(1-P(d))*\frac{P(h\geq t_{d})}{v(d)} \tag{19}\]  
Solve the differential equation and get the general solution as follows:  
\[P(d)=e^{\frac{-P(h\geq t_{d})+d}{v(d)}}(e^{\frac{P(h\geq t_{d})+d}{v(d)}}+C) \tag{20}\]  
Solve the constant \(C\) in combination with physical meaning: if \(d\)\(\rightarrow\)0, \(P\) (\(d\))\(\rightarrow\)0. Take the limit \(d=0\) and \(P\) (\(d\)) = 0 on both sides of the above formula and get C = \(-\)1. Then, the equation results in the following:  
\[P(d)=1-\exp[\frac{-P(h\geq t_{d})*d}{v(d)}]=1-\exp[\frac{-\Delta d}{v(d)}\cdot e ^{-\lambda(t_{0}+\frac{d}{\tau}(\tau-t_{0})-\tau)}] \tag{21}\]  
It can be found from the above equation that the greater the distance \(d\) traveled by the merging vehicle, the greater the probability of successful merging. When \(d=l\) and \(P\) (\(d\)) is close to 1, it can be considered that the merging vehicle can change lanes smoothly in the merging section of the intersection space slice, so that the optimal length \(l\) of the merging section for the intersection space slice can be deduced. To ensure that the merging vehicle can change lanes smoothly in the merging section of the intersection space slice, the length of the merging section should be calculated in a conservative way.