# Modeling and Performance Analysis of mmWave and WiFi-Based Vehicle Communications
## 2 System Model
### Derivation of the PLoS  
In this subsection, we derive the PLoS expressions for 2D and 3D spaces.  
In the case of a 2D space, the PLoS represents the probability of the event, when no obstacles are presented between the \(Tx\) and the \(Rx\). Consequently, the PLoS expression, considering the presented system model, is given by [35]  
\[p_{{}_{\rm 2D}}=\exp\left(-\frac{\left[1-\exp(-\lambda_{{}_{V}}\;d_{\min}) \right]\;d}{d_{\min}}\right), \tag{12}\]  
where \(d\) is the 2D distance between \(Tx\) and \(Rx\).  
For the 3D modeling, we present in Table 3 the different conditions of LoS and NLoS scenarios.  
As shown in Table 3, and as illustrated in Figure 2, there are four cases of LoS scenarios:  
* Cases 1 and 2: In these cases, and as shown in Figure 2a,b, the obstacle's height is lower than that of the receiver and the transmitter, i.e., \(h_{{}_{V}}<h_{{}_{Rx}}\leq h_{{}_{Tx}}\) or \(h_{{}_{V}}<h_{{}_{Tx}}\leq h_{{}_{Rx}}\). Accordingly, the probability of LOS, in this case, can be defined as the probability of the event when \((h_{{}_{V}}<\min(h_{{}_{Tx}},h_{{}_{Rx}}))\). The derivation of this probability expression is detailed in Appendix A, which yields to \(p_{{}_{1,2}}=\frac{1}{3}\).
* Cases 3 and 4: The conditions of these cases can be written as \(\left\{\min(h_{{}_{Tx}},h_{{}_{Rx}})\leq h_{{}_{V}}<h_{{}_{th}}<\max(h_{{}_{Tx }},h_{{}_{Rx}})\right\}\), where \(h_{{}_{th}}\) presents the maximum vehicle's height, beyond which  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Condition** & **Scenario** \\ \hline
1 & \(h_{{}_{V}}<h_{{}_{Rx}}\leq h_{{}_{Tx}}\) & LoS \\
2 & \(h_{{}_{V}}<h_{{}_{Tx}}<h_{{}_{Rx}}\) & LoS \\
3 & \(h_{{}_{Rx}}\leq h_{{}_{V}}<h_{{}_{th}}<h_{{}_{Tx}}\) & LoS \\
4 & \(h_{{}_{Tx}}\leq h_{{}_{V}}<h_{{}_{th}}<h_  
there is no LoS between \(Tx\) and \(Rx\). As shown in Figure 2c,d, this threshold can be expressed as follows:  
\[h_{{}_{\rm fit}}=\frac{a}{r}[\max(h_{{}_{Tx}},h_{{}_{Rx}})-\min(h_{{}_{Tx}},h_{{}_ {Rx}})]+\min(h_{{}_{Tx}},h_{{}_{Rx}}), \tag{13}\]  
Based on that, Appendix B details the derivation of the corresponding PLoS expression, which is given by:  
\[p_{{}_{3A}}=\frac{11\ r}{60\ (r-d_{{}_{min}})},\ \ \forall\ r>d_{{}_{min}}, \tag{14}\]  
Now, after considering all the LoS possible cases, and taking into consideration a given number of vehicles between \(Tx\) and \(Rx\), the final expression of \(PLoS\), in a 3D model, is expressed as follows:  
\[p_{{}_{3D}}= \sum_{n=0}^{N}\frac{\left[(p_{{}_{12}}+p_{{}_{3A}})(1-\exp(- \lambda_{{}_{V}}d_{{}_{min}})d\right]^{n}}{n!\ d_{{}_{min}}^{n}}\] \[\times\exp\left(-\frac{[1-\exp(-\lambda_{{}_{V}}d_{{}_{min}})]d}{ d_{{}_{min}}}\right),\ \ \text{with}\ N\gg 1. \tag{15}\]  
Note that while our discussion focuses on LoS scenarios, our analysis inherently includes considerations for NLoS scenarios as well. This is due to the complementary relationship between LoS and NLoS conditions. Specifically, the NLoS probability (PNLoS) can be expressed as \(PNLoS=1-PLoS\). Considering the presented system model, we elaborate in the following section the corresponding performance analysis.