## 3 Proposed Algorithm
### Merging Collision Probability  
At any time \(t\), the probability of a merging collision of vehicle \(i\) is defined as \(P_{MC}(i,t)\) and is calculated as  
\[p_{MC}(i,t)=M_{mc}\cdot\frac{2R}{l}\cdot P_{mc}(i,j,t). \tag{7}\]  
In Equation (7), \(P_{mc}(i,j,t)\) represents the probability that vehicle \(i\) will cause a merging collision with vehicle \(j\) during time \(\Delta t\), and is calculated as  
\[P_{mc}(i,j,t)=Pr\{d_{ij}(t)>2R\}\cdot Pr\{d_{ij}(t+\Delta t)\leq 2R\}\cdot Pr \{n_{i}(t)=n_{j}(t)\}. \tag{8}\]  
The probability that the distance difference between vehicle \(i\) and vehicle \(j\) is larger than a 2-hop transmission range is calculated by  
\[Pr\{d_{ij}(t)>2R\}=1-\frac{2R}{l}. \tag{9}\]  
Suppose that the distance between vehicles \(i\) and \(j\) is given by \(d_{ij}(t)\) at time \(t\). The maximum and minimum possible distances that vehicle \(i\) can travel by during \(\Delta t\) in a direction parallel to the road are estimated as \(\Delta t\cdot v_{max}\) and \(\Delta t\cdot v_{min}\cos\theta_{i}(t)\), respectively. The distance that vehicle \(j\) travels by during \(\Delta t\) in a direction parallel to the road is \(\Delta t\cdot v_{j}(t)\cos\theta_{j}(t)\). To satisfy the condition of \(d_{ij}(t+\Delta t)\leq 2R\), \(d_{ij}(t)+\Delta t\cdot v_{j}(t)\cos\theta_{j}(t)-2R\) should be located between \(\Delta t\cdot v_{min}\cos\theta_{i}(t)\) and \(\Delta t\cdot v_{max}\). From Figure 10, the probability that the distance difference between two vehicles at time \(t+\Delta t\) is less than a 2-hops is calculated by  
\[Pr\{d_{ij}(t+\Delta t)\leq 2R\}=\frac{v_{max}-v_{j}(t)\cos\theta_{j}(t)}{v_{ max}-v_{min}\cos\theta_{i}(t)}+\frac{2R-d_{ij}(t)}{\Delta t(v_{max}-v_{min} \cos\theta_{i}(t))}. \tag{10}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Definition** \\ \hline \(M_{mc}\) & The number of vehicles in a region where a merging collision may occur \\ \(A_{ac}\) & The number of vehicles that generate a merging collision \\ \(w\) & Road width \\ \(l\) & Road length \\ \(R\) & Transmission range \\ \(d_{ij}(t)\) & The distance between vehicle \(i\) and vehicle \(j\) at time \(t\) \\ \(n_{i}(t)\) & Time slot number occupied by vehicle \(i\) \\ \(x(t)\) & \(x\)-axis coordinates of the vehicle \\ \(y(t)\) & \(y\)-axis coordinates of the vehicle \\ \(\theta_{i}(t)\) & The angle in the moving directions of the vehicles \(i\) at time \(t\) \\ \(\theta_{ij}(t)\) & The angle difference in the moving directions of the vehicles \(i\) and \(j\) at time \(t\) \\ \(N_{e}\) & The number of empty time-slots in a frame \\ \(N\) & Total number of time-slots in a frame \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters.  
The probability that two vehicles occupy the same time-slot is given by  
\[Pr\{n_{i}(t)=n_{j}(t)\}=\frac{1}{N}. \tag{11}\]  
Substituting Equations (9)-(11) into Equation (8), the probability of merging collision is calculated by  
\[P_{mc}(i,j,t)=\Big{(}1-\frac{2R}{l}\Big{)}\Big{(}\frac{v_{max}-\Delta t\cdot v _{j}(t)\cos\theta_{j}(t)}{v_{max}-v_{min}\cos\theta_{i}(t)}+\frac{2R-d_{ij}(t )}{\Delta t(v_{max}-v_{min}\cos\theta_{i}(t))}\Big{)}\frac{1}{N}. \tag{12}\]