# Modeling and Performance Analysis of mmWave and WiFi-Based Vehicle Communications
## 3 Performance Analysis
### Successful Transmission Probability (\(Stp\))
#### 3.1.2 mmWaves Communication Scenario  
This subsection details the derivation of the STP expressions in LoS and NLoS mmWave communications scenarios.  
1. LoS Scenario: In this scenario, the STP expression can be written as follows: \[STP_{{}_{mmWaves}}^{{}^{LoS}}=Pr\Bigg{\{}\Bigg{(}\frac{\mathcal{P}_{T}A_{{}_{M }}\mathcal{S}_{1}\mathcal{F}_{1}r^{-\alpha_{1}}}{\mathcal{P}_{N}}\Bigg{)}{>} =\gamma_{th}\Bigg{\}}.\] (24) By employing the PDF expressions of \(\mathcal{S}_{1}\) and \(\mathcal{F}_{1}\), \(STP_{{}_{mmWaves}}^{{}^{LoS}}\) is rewritten as follows: \[STP_{{}_{mmWaves}}^{{}^{LoS}}=\int_{0}^{+\infty}p_{\mathcal{S}_{1}}(y)\int_{v_ {1}}^{+\infty}p_{\mathcal{F}_{1}}(x)\;dx\;dy,\] (25) where \(v_{1}=\frac{\gamma_{th}}{\mathcal{P}_{T}A_{{}_{M}}\mathcal{P}_{N}}\).  
By using the expression of \(p_{\mathcal{F}_{1}}(x)\) and the change in variables: \(x=\frac{m_{1}}{\Omega_{1}}v_{1}\), and based on (Equation 3.381.3 [34]), the integration in (14), with respect to \(x\), is derived as follows:  
\[STP_{{}_{mmWaves}}^{{}^{LoS}}=\Gamma(m_{1})\int_{0}^{+\infty}p_{\mathcal{S}_{1 }}(y)\Gamma(m_{1},\frac{m_{1}}{\Omega_{1}}v_{1})\;dy, \tag{26}\]  
where \(\Gamma(.,.)\) is the incomplete gamma function (Eq. 8.350.2 [34]).  
Now, by substituting the PDF expression of \(p_{\mathcal{S}_{1}}(y)\), the \(STP_{{}_{mmWaves}}\) expression becomes  
\[STP_{{}_{mmWaves}}^{{}^{LoS}}=\frac{1}{\Gamma(m_{1})\;\sqrt{2\pi\
u_{\text{ln} (\mathcal{S}_{1})}^{2}}}\int_{0}^{+\infty}\frac{1}{y}\exp\Bigg{(}\frac{-\left[ \ln(y)-\mu_{\text{ln}(\mathcal{S}_{1})}\right]^{2}}{2\sigma_{\text{ln}( \mathcal{S}_{1})}^{2}}\Bigg{)}\Gamma\bigg{(}m_{1},\frac{m_{1}}{\Omega_{1}}v_{1 }\bigg{)}dy. \tag{27}\]  
Finally, by applying Laguerre theorem, the final expression of \(STP_{{}_{mmWaves}}\) is given by  
\[STP_{{}_{mmWaves}}^{{}^{LoS}}{=}\frac{1}{\Gamma(m_{1})\;\sqrt{2\pi\
u_{\text{ln }(\mathcal{S}_{1})}^{2}}}\sum_{n=1}^{N}\Bigg{\{}\frac{w_{{}_{x}}}{x_{{}_{n}}} \exp\Bigg{(}x_{{}_{n}}-\frac{\left[\ln(x_{{}_{n}})-\mu_{\text{ln}(\mathcal{S}_ {1})}\right]^{2}}{2\sigma_{\text{ln}(\mathcal{S}_{1})}^{2}}\Bigg{)}\Gamma\bigg{(} m_{1},\frac{m_{1}}{\Omega_{1}}v_{1,n}\bigg{)}\Bigg{\}}, \tag{28}\]  
where \(v_{1,n}=\frac{\gamma_{th}\mathcal{P}_{N}P_{N}^{\alpha_{1}}}{\mathcal{P}_{T}A_{ {}_{M}}x_{{}_{n}}}\), \(x_{n}\) is the roots of the Laguerre polynomial of order \(N\) and \(w_{n}\) are the corresponding weights.  
1. NLoS Scenario:By analogy with the derivation of the STP expression in an LoS scenario, the corresponding expression in an NLoS mmWaves communication scenario is given by:  
\[STP_{{}_{mml_{\text{reson}}}}^{{}^{\text{NLoS}}} =\!\!\frac{1}{\Gamma(m_{2})}\sum_{\sqrt{2\pi\sigma_{\text{ln}(S_{2} )}^{2}}}\sum_{n=1}^{N}\left\{\frac{w_{n}}{x_{{}_{n}}}\exp\left(x_{{}_{n}}-\frac {\left[\text{ln}(x_{{}_{n}})-\mu_{\text{ln}(S_{2})}\right]^{2}}{2\sigma_{\text{ ln}(S_{2})}^{2}}\right)\Gamma\left(m_{2},\frac{m_{2}}{\Omega_{2}}v_{2,n} \right)\right\}, \tag{29}\]  
where \(v_{2,n}=\frac{\gamma_{n}\mathcal{P}_{N}v_{0}^{\text{\tiny{22}}}}{\mathcal{P}_{ T}A_{M}{}^{\text{\tiny{22}}}}\).