## 2 Modeling of Vehicles Interactions Due to Simultaneity
### Interactions between Vehicles Simultaneously Traveling on One Lane
Explicitly written, the expressions of \(dP_{i}(t)\) lead, after some trivial operations, to the following system of ordinary differential equations:  
\[\left\{\begin{array}{r}\frac{dP_{0}}{dt}\,=\,\,-\delta\,P_{0}(t)\,+\,\, \alpha\,P_{1}(t)\\ \frac{dP_{1}}{dt}\,=\,\delta\,  
The probabilities \(P_{i}(t)\) are then the solutions of the system of Equation (5) under the constraint  
\[\sum_{i\,=\,0}^{w}P_{i}(t)\ =\ 1\qquad t\,\geq\,0, \tag{7}\]  
for the initial conditions  
\[\left\{ \begin{array}{c}P_{0}(0)\ =\ 1\\ P_{i}(0)\ =\ 0\qquad 1\ \leq\ i\ \leq\ w\end{array} \right.. \tag{8}\]  
Generally, parameters \(\alpha\), \(\delta\), and \(\varphi_{i}\) depend on \(t\), but in stationary conditions they assume constant values; moreover, since for \(t\to\infty\) derivatives \(dP_{i}/dt\to 0\), the probabilities \(P_{i}(t)\) are independent on \(t\), so that the system (6) plus constraint (7) reduces to the system of \(w\ +\ 1\) linear equations in the \(w\ +\ 1\) unknowns \(P_{i}\):  
\[\left\{ \begin{array}{c}-\delta\ P_{0}\ +\ \alpha\ P_{1}\ =\ 0\\ \delta\ P_{0}\ -\ (\delta\ +\ \alpha)\ P_{1}\ +\ (\alpha\ +\ \varphi_{1})\ P_{2}\ =\ 0\\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\\ \delta\ P_{i-1}\ -\ \Big{(}\delta\ +\ \alpha\ +\ \sum_{j\,=\,1}^{i\ -\ 1}\varphi_{j}\Big{)}\ P_{i}\ +\ \Big{(}\alpha\ +\ \sum_{j\,=\,1}^{i\ }\varphi_{j}\Big{)}\ P_{i}\ +\ 1\ =\ 0\\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\\ \delta\ P_{w\ -\ 1}\ -\ \Big{(}\alpha\ +\ \sum_{j\,=\,1}^{w\ -\ 1}\varphi_{j}\Big{)}\ P_{w}\ =\ 0\\ \sum_{i\,=\,0}^{w}P_{i}\ =\ 1\end{array} \right.. \tag{9}\]  
The solution of (9) can be obtained by repeated substitutions, with some cumbersome but elementary calculations. Starting from the first equation and going forward in increasing order of \(i\), it is possible to express each \(P_{i}\) (\(i>0\)) as a function of \(P_{0}\); thus \(P_{0}\). is deduced by the last equation of system (9) and from it the expression of each unknown \(P_{i}\):  
\[\left\{\begin{array}{c}P_{0}\ =\ \Big{\{}1\ +\ \frac{\delta}{\alpha}\ +\ \sum_{j\,  
Further specializing Equation (12), the annual flow \(\underline{N}_{(v_{1},v_{2},\ldots,v_{l})_{l}s_{l}}\) of a specific \(i\)-vehicles convoy on the lane, composed by \(i\)-vehicles belonging to the \(v_{1}\), \(v_{2}\),..., \(v_{i}\) categories of the \(q\) standardized types of vehicles in the given order, results  
\[\underline{N}_{(v_{1},v_{2},\ldots,v_{l})_{l}s_{l}}\ =\ \underline{N}_{s_{l}}\prod_{j \ =\ 1}^{i}\frac{N_{v_{j}s}}{N_{s}}\ =\ \frac{P_{i}}{(1\ -\ P_{0})}\frac{N_{s}}{i}\prod_{j\ =\ 1}^{i}\frac{N_{v_{j}s}}{N_{s}}\qquad 1 \leq\ v_{j}\ \leq\ q. \tag{14}\]  
In this way, finally, a modified vehicle flow on each bridge lane can be obtained, composed by vehicles and vehicle convoys, under the specific constraint that only one vehicle or one vehicle convoy can travel the lane. Consequently, for these modified flows only simultaneity on different lanes is relevant. For practical applications, Equation (14) can be markedly simplified; in effect, only the case \(w\ =\ 2\) is relevant, since the effects caused by the simultaneous presence of more than two vehicles are insignificant. In this situation, Equation (10) becomes  
\[\left\{ \begin{array}{c}P_{0}\ =\ \Big{[}1\ +\ \frac{\delta}{\alpha}\Big{(}1\ +\ \frac{\delta}{\alpha\ +\ \varphi_{1}}\Big{)}\Big{]}^{-1}\\ P_{1}\ =\ \frac{\delta}{\alpha}P_{0}\\ P_{2}\ =\ \frac{\delta^{2}}{\alpha(\alpha\ +\ \varphi_{1})}P_{0}\end{array}