# C-V2X Centralized Resource Allocation with Spectrum Re-Partitioning in Highway Scenario
## 2 Assumptions and Problem Formulation  
### C-V2X Assumptions  
As already mentioned, SC-FDMA has been applied as the C-V2X communication technique to support high mobility in vehicular communication. The radio resources in SC-FDMA are represented in the time and frequency domains. In the time domain, a sub-frame (two time slots, 0.5 ms each) is the smallest time unit, and a resource block (12 sub-carriers, with 15 kHz distance among them) is the smallest frequency unit to be applied. The resource blocks have been grouped to form sub-channels. The number of resource blocks in each sub-channel can vary according to the applied MCS and the amount of data that are needed to be transmitted. The number of sub-channels in each sub-frame \(SCH_{sf}\) is computed as follows:  
\[SCH_{sf}=\left\lfloor\frac{RB_{T}}{RB_{scl}}\right\rfloor \tag{1}\]  
where \(RB_{T}\) and \(RB_{scl}\) are the total number of resource blocks in a given bandwidth and the number of resource blocks in each sub-channel, respectively. The number of remaining resource blocks \(RB_{free}\) in a given bandwidth that can be used later can be obtained as follows:  
\[RB_{free}=(RB_{T}-SCH_{sf}RB_{scl})N_{SF_{guide}} \tag{2}\]  
where \(N_{SF_{guide}}\) is the number of sub-frames in a CAM cycle that can be obtained by dividing the CAM duration cycle \(T_{cycle}\) by the sub-frame duration \(T_{sf}\) according to the following expression:  
\[N_{SF_{guide}}=\frac{T_{cycle}}{T_{sf}} \tag{3}\]  
The total number of sub-channels in one cycle of a CAM message \(SCH_{T}\) that can service vehicles in the awareness area of a vehicle can be computed as follows:  
\[SCH_{T}=N_{SF_{yield}}SCH_{sf} \tag{4}\]  
A fixed number of resource blocks used in each sub-channel and the same power applied by broadcasting vehicles are assumed in this paper according to [6] Furthermore, the inter-vehicle distance is kept fixed in each simulation. Similarly as in [23; 24], the Poisson distribution is applied to model the arrival rate parameter \(\lambda\), where \(\lambda\) is the vehicle's density in a fixed highway length within a given period. The number of vehicles in the given period with the mean value \(\lambda\) has been defined as a random variable for a lane with vehicles moving at a fixed speed [6]. The average inter-vehicle distance \(\Phi\) can be obtained by the following expression:  
\[\Phi=2.5\ \mathrm{s}\cdot\frac{V_{kmph}}{3600}\cdot\alpha \tag{5}\]where \(V_{kmph}\) and \(\alpha\) are the absolute vehicle speed in km per hour in each lane and a scalar coefficient added by the authors to control inter-vehicle distance in the network, respectively. Moreover, the Poisson distribution parameter \(\lambda\) for the fixed highway length \(d_{highway}\) can be found as [31]:  
\[\lambda=\frac{d_{highway}}{\alpha} \tag{6}\]  
A random number that changes according to the Poisson distribution with parameter \(\lambda\) determines the number of vehicles randomly distributed along the lane. In order to calculate the received power \(P_{Rx}\) of the broadcast signal by a vehicle with the transmitted power denoted by \(P_{Tx}\), one can use the following formula:  
\[P_{Rx}=\frac{P_{Tx}G_{R}}{P_{loss}(D)^{\beta}} \tag{7}\]  
where \(G_{R}\), \(P_{loss}\), \(D\), and \(\beta\) are the receiver antenna gain, reference path loss at the distance of 1 m, the distance between the broadcaster and the receiver, and the path-loss exponent, respectively. It is worth mentioning that this work focuses on improving the selection and re-selection of radio resources with fixed transmission power and avoiding the strong influence of interference in broadcast areas. Distance \(D\) is usually smaller than the maximum broadcast range \(D_{\max}\)[32] that can be calculated from the formula:  
\[D_{\max}=\left[\frac{P_{Tx}G_{R}}{\gamma_{\min}P_{loss}\rho}\right]^{\frac{1} {\beta}} \tag{8}\]  
where \(\gamma_{\min}\) and \(\rho\) are the minimum signal-to-interference plus noise ratio, and the noise power of a sub-channel, respectively.