## 3 Performance Analysis Methods  
### Expressions for Path Loss and Average Path Loss  
In this section, the V2I-VLC path loss expressions for LOS and NLOS links are derived based on the channel modeling in Section 2, and (5) and (9) in [48]. The weather factor and other propagation characteristics are taken into account. In addition, novel closed-form expressions for the average path loss are derived.  
Taking the LOS link as an example, the path loss of the proposed outdoor VLC consists of two main components, which are geometric loss and attenuation loss. The geometric loss is caused by the diffusion of emitted rays to areas larger than the PD receiving aperture. The geometric loss between the \(j^{th}\) Tx and PD can be expressed as follows:  
\[PL_{geo,j}=10\log_{10}\left\{\frac{D_{R}\left[\cos(\alpha_{j})^{1/\varepsilon} \right]}{\zeta D_{LOS,j}^{V_{1}-PD}}\right\}\text{ for }j=1,2, \tag{7}\]  
where \(D_{R}\) denotes the aperture diameter of the PD; \(\varepsilon\) and \(\zeta\) are weather-related correction factors [43] (pp. 6892-6894). The attenuation loss is the result of scattering and absorption. It is assumed that most of the scattered photons cannot be captured by the PD and that some of the scattered rays may be received by the PD after reflection. Based on the Beer-Lambert formula and the assumption above, the attenuation loss can be defined as follows:  
\[PL_{att,j}=10\log_{10}\left\{\exp\left[-c_{E}D_{LOS,j}^{V_{1}-PD}\left(\frac{ D_{R}}{\zeta D_{LOS,j}^{V_{1}-PD}}\right)^{\varepsilon/2}\right]\right\} \text{ for }j=1,2, \tag{8}\]  
where \(c_{E}\) denotes the extinction coefficient for a given weather type; \(D_{LOS,j}^{V_{1}-PD}\left(\frac{D_{R}}{\zeta D_{LOS,j}^{V_{1}-PD}}\right)^{ \varepsilon/2}\) denotes an additional term proportional to the geometric spread of the light source. In summary, considering the dual Txs of vehicle \(V_{1}\) and the geometric relation \(\alpha_{j}=\arccos\frac{I_{V_{1}-PD}}{D_{LOS,j}}\), the overall path loss of the LOS link can be expressed as follows:  
\[h_{LOS}^{PL}=\frac{1}{2}{\sum_{j=1}^{2}}\left[\frac{{D_{R}}^{2}\left({I_{V_{1 }-PD}}\right)^{2/\varepsilon}}{\xi^{2}\left({D_{LOS,j}^{V_{1}-PD}}\right)^{2 +2/\varepsilon}}e^{-\xi_{E}D_{LOS,j}^{V_{1}-PD}}\frac{{D_{R}}^{\varepsilon/2} }{\xi D_{LOS,j}^{V_{1}-PD}}\right]. \tag{9}\]In the proposed V2I-VLC system model, the path loss of the NLOS links is divided into two parts, \(M\) and \(RS\), based on the reflected objects. In the same way, the overall path loss of the NLOS links can be expressed as follows:  
\[h_{NLOS\_M}^{PL}=\frac{1}{2}\sum_{j=1}^{2}\left\{\left[\frac{{{A_{M}}^{2}{\left({ I_{V_{j}-M}}\right)}^{2/\epsilon}}}{{\zeta^{2}{\left({D_{NLOS,j}^{V_{i}-M}} \right)}^{2+2/\epsilon}}}\right]\left[\frac{{{D_{R}}^{2}{\left({I_{M-PD}} \right)}^{2/\epsilon}}}{{\zeta^{2}{\left({D_{NLOS,j}^{V_{i}-M}}\right)}^{2+2/ \epsilon}}}\right]e^{-\epsilon_{E}{\left({D_{NLOS,j}^{V_{i}-M}}+{D_{NLOS,j}^{M- PD}}\right)}^{2/\epsilon}}\right]^{-\epsilon_{E}{\left({D_{NLOS,j}^{V_{i}-M}}+{D_{NLOS,j}^{M- PD}}\right)}^{2/\epsilon}}\right\}, \tag{10}\]  
and  
\[h_{NLOS\_RS}^{PL}=\frac{1}{2}\sum_{j=1}^{2}\left\{\left[\frac{{{A_{RS}}^{2}{ \left({I_{V_{j}-RS}}\right)}^{2/\epsilon}}}{{\zeta^{2}{\left({D_{NLOS,j}^{V_{i }-RS}}\right)}^{2+2/\epsilon}}}\right]\left[\frac{{{D_{R}}^{2}{\left({I_{RS-PD }}\right)}^{2/\epsilon}}}{{\zeta^{2}{\left({D_{NLOS,j}^{V_{i}-RS}}\right)}^{2 +2/\epsilon}}}\right]e^{-\epsilon_{E}{\left({D_{NLOS,j}^{V_{i}-RS}}+{D_{NLOS,j}^ {RS-PD}}\right)}^{2/\epsilon}}}\right], \tag{11}\]  
where \(A_{M}\) and \(A_{RS}\) denote the effective areas (assumed to be circular) of \(M\) and \(RS\) illuminated by the rays, respectively.