# Smart Bandwidth Assignation in an Underlay Cellular Network for Internet of Vehicles
## 4 Methodology
### Minimum Distance between C-UE and V-UE Transmitter  
The eNodeB can only assign the same RB to a C-UE and a V-UE if constraints Equations (3)-(6) are satisfied. Let \(\varepsilon_{n}^{c}\) and \(\varepsilon_{m}^{d}\) denote the SINR of C-UE-eNodeB \(n\) communication and V2V \(m\) communication, respectively, \(P_{n}^{c}\) and \(P_{m}^{d}\) denote the transmit power of the C-UE \(n\) and V-UE transmitter \(m\), respectively, and \(g_{nB}\), \(g_{m}\), \(h_{mB}\), \(h_{mn}\) denote channel gains of the links that participate in the system, being \(n\) the C-UE, \(m\) the V-UE transmitter and \(B\) the eNodeB. \[\varepsilon_{n}^{c}=\frac{P_{n}^{c}*g_{nB}}{\sigma^{2}+P_{m}^{d}*h_{mB}}\geq \varepsilon_{min}^{c} \tag{3}\]  
\[\varepsilon_{m}^{d}=\frac{P_{m}^{d}*g_{m}}{\sigma^{2}+P_{n}^{c}*h_{mn}}\geq \varepsilon_{min}^{d} \tag{4}\]  
\[P_{n}^{c}\leq P_{max}^{c} \tag{5}\]  
\[P_{m}^{d}\leq P_{max}^{d} \tag{6}\]  
These constraints mean that the minimum SINR requirement have to be guaranteed for both communications Equations (3) and (4) and the power transmission of the C-UE and V-UE must be lower than the maximum permitted Equations (5) and (6). The power of additive white Gaussian noise on each channel is assumed to be \(\sigma^{2}\). Moreover, there are also minimum transmission powers that the C-UE Equation (7) and the V-UE Equation (8) have to fulfill in order to guarantee the message achieves the receiver. The transmission power is minimum when there is no interference and we only guarantee the minimum SINR. \[P_{n,min}^{c}=\frac{\varepsilon_{n,min}^{c}*\sigma^{2}}{g_{nB}} \tag{7}\]  
\[P_{m,min}^{d}=\frac{\varepsilon_{m,min}^{d}*\sigma^{2}}{g_{m}} \tag{8}\]  
Therefore, the values of transmission power the system has to assign to C-UE Equation (9) and V-UE Equation (10) are limited by the minimum transmission power which depends on the propagation losses and the maximum transmission power which depends on the LTE-Advanced configuration parameters. \[P_{n,min}^{c}<P_{n}^{c}\leq P_{max}^{c} \tag{9}\]  
\[P_{m,min}^{d}<P_{m}^{d}\leq P_{max}^{d} \tag{10}\]  
The pairing process can be easily understood in Figure 3 where the rectangle delimits maximum transmission powers for C-UE and V-UE and lines \(l_{c}\) and \(l_{d}\) represents constraints Equations (3) and (4). The area on the right of \(l_{d}\) are the possible combinations of transmission powers that fulfill restriction Equation (4) and the area over \(l_{c}\) are the possible combinations of transmission powers that meet Equation (3). Therefore, the area between both lines contains all the possible power transmission combinations which fulfill all the constrains needed for sharing a RB between C-UE and a V-UE. That is, if the intersection of both lines, point A, is inside the rectangle of maximum transmission powers, C-UE and V-UE can be paired, as it is shown in Figure 3a. However, if the intersection is outside the rectangle, as in Figure 3b, those C-UE and V2V pair cannot share the same RB fulfilling all the constraints. Moreover, from Figure 3 it is deduced that point A comprises the pair of minimum transmission powers that fulfill all the pairing constraints. Since the equations of \(l_{d}\) and \(l_{c}\) are Equations (11) and (12), respectively,  
\[P_{m}^{d}=\frac{g_{m}}{\varepsilon_{min}^{d}*h_{nm}}*P_{m}^{d}-\frac{\sigma^{2} }{h_{nm}} \tag{11}\]  
\[P_{n}^{c}=\frac{\varepsilon_{min}^{c}*h_{mB}}{g_{nB}}*P_{m}^{d}+\frac{ \varepsilon_{min}^{c}*\sigma^{2}}{g_{nB}} \tag{12}\]  
calculating the intersection of both lines the coordinates of point A Equation (13) are obtained. \[\left\{ \begin{array}{l}P_{A}^{c}=\frac{\sigma^{2}(g_{i}\varepsilon_{min}^{d }+\varepsilon_{min}^{c}\varepsilon_{min}^{d}h_{mB})}{g_{j}g_{nB}-\varepsilon_ {min}^{d}h_{mB}h_{nm}}\\ \\ P_{A}^{d}=\frac{\sigma^{2}(g_{nB}\varepsilon_{min}^{d}+\varepsilon_{min}^{c} \varepsilon_{min}^{d}h_{nm})}{g_{j}g_{nB}-\varepsilon_{min}^{d}h_{mB}h_{nm}} \end{array} \right.