## 2 Modeling of Vehicles Interactions Due to Simultaneity
### Interactions between Vehicles Simultaneously Traveling on One Lane  
The probability \(P_{i}\) that exactly \(i\) vehicles are traveling simultaneously one lane on the length \(L\) can be calculated considering the bridge or the lane as a single channel system \(\Sigma\) with a waiting queue, where the number of vehicles in the queue and the waiting time, depending on the number of vehicles in the queue, are limited. The probability \(P_{i}\) is thus the probability that \(i\ -\ 1\) vehicles are queued. Clearly, said \(d\) the minimum distance between the first axles of two consecutive vehicles, the situation corresponding to exactly \(i\) vehicles traveling simultaneously on one lane is that outlined in Figure 1, being the \((i\ +\ 1)\)-th vehicle outside the length \(L\). For the aim of the study, the evaluation of the maximum number of simultaneously traveling vehicles, \(w\) is particularly significant. Since the distance \(d\) between two consecutive lorries cannot obviously be nil, the waiting time \(t_{k}\) for the \(k\)-th vehicle in the queue depends on the maximum extent of the part of the reference length  
Figure 1: Outline of the model with exactly \(i\) vehicles traveling simultaneously on one lane. where the \(k\)-th vehicle can be located, provided that all the \(k\,-\,1\) preceding vehicles are on \(L\). With this premise, as illustrated in Figure 2, \(w\) is given by:  
\[w\,=\,\,\text{int}\!\left(\frac{L}{d}\right)\,=\,\,\text{int}\!\left(\frac{ \Theta}{t^{*}}\right)\!. \tag{3}\]  
where \(t^{*}\,=\,\,d/v\) is the time interval between the arrivals of two consecutive vehicles, and, therefore, the waiting times \(t_{k}\) result is:  
\[t_{k}\,=\,\Theta\,-\,k\,t^{*}\,1\,\leq\,k\,\leq\,w\,-\,1. \tag{4}\]  
To solve the problem, it can be hypothesized that the waiting times \(t_{k}\) are distributed again according to exponential probability density functions \(g_{k}(t)\) whose rates are \(\varrho_{k}(t)=1/t_{k}\):  
\[g_{k}(t)\,=\,\,t_{k}^{-1}\,\exp\!\left(-t_{k}^{-1}\,t\right)\,=\,\,\varrho_{k} \,\exp\!\left(-\varrho_{k}\,t\right)\!. \tag{5}\]  
The \(w\,+\,1\) realizable states of the system \(\Sigma_{i}\) are:  
* \(\Sigma_{0}\) the system is free: No vehicle on the lane;
* \(\Sigma_{1}\) the system is busy: One vehicle is traveling on the lane;
*......:
* \(\Sigma_{i}\) the system is busy with \(i\,-\,1\) requests queued: i.e., \(i\) vehicles on the lane;
*......:
* \(\Sigma_{w}\) the system is busy with \(w\,-\,1\) requests queued: i.e., \(w\) vehicles on the lane. If at a given time \(t\) the system is in the \(\Sigma_{i}\) state, in the infinitesimal time interval \((t,\,t\,+\,dt)\), it can remain in \(\Sigma_{i}\) or it can pass in one adjacent state, \(\Sigma_{i\,-\,1}\) or \(\Sigma_{i\,+\,1}\). Evidently, when \(i\,=\,0\) the unique existing adjacent state is \(\Sigma_{1}\) and when \(i\,=\,w\) the unique existing adjacent state is \(\Sigma_{w\,-\,1}\). Let \(\delta\) be the arrival rate of the process, which is the average number of vehicles in the time unit. Following the classical approach, since the probability that during \(dt\) the system jumps directly to other states not adjacent to \(\Sigma_{i}\) is infinitesimal of higher order, the variation of the probability \(dP_{i}(t)\) in \(dt\) is simply the sum of the probability \(P_{i}(t)\) that the system is in the state \(\Sigma_{i}\) at the time \(t\) and no vehicle enters or leaves the lane during \(dt\); plus, if relevant, the probability \(P_{i\,-\,1}(t)\) that the system is in the state \(\Sigma_{i\,-\,1}\) at the time \(t\) and one vehicle arrives during \(dt\); plus, if relevant, the probability \(P_{i\,+\,1}(t)\) that the system is in the state \(\Sigma_{i\,+\,1}\) at the time \(t\) and one vehicle leaves during \(dt\).