# An Emergency Message Routing Protocol for Improved Congestion Management in Hybrid RF/VLC VANETs
## 3 System Model
### Modeling the Channel for VLC and RF Links and the Multi-Hop Propagation  
The received signal for the VLC link can be written as, [31; 32]:  
\[y_{r}(t)=\gamma x(t)*h(t)+\sigma_{n}^{2}(t) \tag{1}\]  
where, \(y_{r}(t)\) is the received signal, \(x(t)\) is the transmitted optical signal, \(h(t)\) is the optical channel impulse response (CIR), \(*\) is the convolution. \(\sigma_{n}^{2}(t)\) is the ambient noise, and \(\gamma\) is the photo-diode responsivity. The CIR is given by [33; 34]:  
\[h(t)_{VLC}=G_{rs}\delta(t-\tau_{rs}) \tag{2}\]  
where, \(\tau_{rs}=R_{rs}/c\), denotes the time delay, and \(G_{rs}\) denotes the gain factor accounting for the losses and gains between the source \(s\) and the receiver \(r\), and \(R_{rs}\) denotes the distance between the source \(s\) and the receiver \(r\), \(\delta(. )\) denotes the impulse function, and c is the speed of light. \[G_{rs}=\frac{\cos(\phi_{rs})\cos(\theta_{rs})A_{r}T_{s}(\theta_{rs})g(\theta_{ rs})}{\pi R_{rs}^{2}} \tag{3}\]  
As shown in Figure 3, which represent transmitter, receiver, and reflector cars (Parked cars act as reflectors for the NLoS path. Reflectors have a dual property as they may act as transmitters or receivers depending on their role whether they are receiving from the source or sending to another node), where \(\phi_{rs}\) denotes the emitting angle between source \(s\) and receiver \(r\), \(\theta_{rs}\) denotes the incident angle between source \(s\) and receiver \(r\), \(A_{r}\) denotes the effective area of the receiver, \(T_{s}(\theta_{rs})\) denotes the signal transmission coefficient of an optical filter, and \(g(\theta_{rs})\) denotes the gain of the optical concentrator. For RF communication, the CIR given by the time-varying multipath channel is [35],  
\[h(t)_{RF}=\sum_{m=0}^{L(t)-1}a_{m}(t)Z_{m}(t)\exp(j\phi_{m}(t))\delta(t-\tau_{ m}(t)) \tag{4}\]  
where, \(m\) is the index of multipath components, \(L(t)\) is the number of multipath components, \(a_{m}(t)\) is the time-varying amplitude of the \(m^{th}\) multipath component, \(z_{m}(t)\) is a \"persistence\" Markov random process for the \(m^{th}\) multipath component, \(\phi_{m}(t)\) is the phase associated to the \(m^{th}\) path, \(\tau_{m}(t)=R_{m}(t)/c\), denotes the \(m^{th}\) time delay between the transmitter and the receiver, and \(R_{m}(t)\) denotes the distance between the transmitter and the receiver. Figure 3: Demonstration of angles in our model. Consider the following parameters: Received power \(P_{r}(i,j)\) at node \(j\) from transmitter \(i\), transmit power \(P_{t}\), transmitter/receiver antenna gains \(G_{t}\) and \(G_{r}\), transmitter-relay antenna gain \(G_{t,r_{n}}\), wavelength \(\lambda\), distances \(R_{ij}\) and \(R_{n}\), path loss exponent \(\eta\), maximum hop count \(K\), relay selection metric \(Q(i,r)\) with criterion \(f(\cdot)\), path loss \(PL_{i,j}^{\text{multihop}}\), and multi-hop RF CIR \(h(t)_{\text{RF,multihop}}\). For direct V2I/V2V link between nodes i and j, to model the received power \(P_{r}(i,j)\) for a direct link (It follows the standard path loss model [36]):  
\[P_{r}(i,j)=P_{t}G_{t}G_{r}\Bigg{(}\frac{\lambda}{4\pi R_{ij}}\Bigg{)}^{\eta} \tag{5}\]  
For multi-hop links with k relays, we extend this model to a multi-hop path utilizing \(k\) intermediate relay nodes. The received power is calculated as the transmit power reduced by the product of each individual hop's antenna gains \(\prod_{n=1}^{k}G_{t,r_{n}}\), and attenuated by the propagation loss accumulated over each hop distance \(R_{n}\):  
\[P_{r}(i,j)=P_{t}\prod_{n=1}^{k}G_{t,r_{n}}\Bigg{(}\frac{\lambda}{4\pi R_{n}} \Bigg{)}^{\eta} \tag{6}\]  
Equation (7) defines the relay selection metric \(Q(i,r)\) used to select the best next hop relay node \(r\) for transmitter \(i\). This considers the received power \(P_{r}(i,r)\), and may also account for interference levels and medium contention levels to choose high-quality links.