# C-V2X Centralized Resource Allocation with Spectrum Re-Partitioning in Highway Scenario
## 2 Assumptions and Problem Formulation
### Problem Formulation  
In C-V2X, several packet sizes can be applied. In this work, the same packet size, the same number of resource blocks in each sub-channel, and the same broadcast power are assumed to be applied by all vehicles. However, the analysis shown below can be extended on other resource configurations as well. The sub-channel index set for the given resource sub-channels is denoted as \(Q=\{1,2,\ldots,SCH_{\text{T}}\}\), whereas \(C=\{1,2,\ldots,c\}\) is the set of indices denoting \(\mathrm{RSUs}\) in the network. Let \(i\) be the index of a vehicle located inside the coverage area of \(\mathrm{RSU}\)\(x\), and \(K_{x}\) is the number of all vehicles located inside \(\mathrm{RSU}\)\(x\) area where \(x\in C\). Furthermore, \(j\) is the index for users using the same resource \(r\) in \(\mathrm{RSU}\)\(y\) where \(y\in C\backslash x\) and \(r\in Q\). \(K_{y}\) represents all vehicles located in \(\mathrm{RSU}\)\(y\). Thus, the signal-to-interference plus noise ratio \(\gamma_{i,x}^{r}\) that is experienced by vehicle \(i\) belonging to \(\mathrm{RSU}\)\(x\) can be represented as follows:  
\[\gamma_{i,x}^{r}=\frac{\frac{\mathrm{P}_{Tx}G_{R}}{\rho+\sum_{y\
eq x,y\in C }\sum_{j\in K_{y}}\frac{\mathrm{P}_{Tx}G_{R}}{P_{loss}(D_{j,y}^{r})^{\beta}}}}{ \rho+\frac{\sum_{y\
eq x,y\in C}\sum_{j\in K_{y}}\frac{\mathrm{P}_{Tx}G_{R}}{ P_{loss}(D_{j,y}^{r})^{\beta}}}} \tag{12}\]  
where \(D_{i,x}^{r}\) and \(D_{j,y}^{r}\) are the distances between the receiver and vehicle \(i\) and vehicle \(j\), respectively. The desired signal is the one transmitted by vehicle \(i\) using resources \(r\) and the signals transmitted by vehicle \(j\) are the interference signals. This work aims to reduce the interference experienced at the receiver by increasing the frequency reuse distance. In consequence, \(\gamma_{i,x}^{r}\) will increase. For that purpose, the minimum allowed frequency reuse distance \(h_{\text{{reuse}}}\) should be determined as the minimum acceptable distance between two vehicles using the same frequency. The worst case of \(h_{\text{{reuse}}}\) can be calculated by assuming the receiver to be located on the edge of two maximum broadcasting ranges of vehicles along a horizontal line. In other words, distances \(D_{i,x}^{r}\) and \(D_{j,y}^{r}\) in Equation (12) are equal to the maximum broadcasting range \(D_{\text{max}}\). According to this assumption, \(h_{\text{{reuse}}}\) can be obtained by rewriting Equation (12) as follows:  
\[\gamma_{i,x}^{r}=\frac{\frac{\mathrm{P}_{Tx}G_{R}}{P_{loss}(D_{i,x}^{r})^{ \beta}}}{\rho+\sum_{y\
eq x,y\in C}\sum_{j\in K_{y}}\frac{\mathrm{P}_{Tx}G_{R }}{P_{loss}(h_{\text{{reuse}}}-D_{i,x}^{r})^{\beta}}}\] (13) subject to : \[D_{j,y}^{r},D_{i,x}^{r}=D_{\text{max}},\] \[D_{j,y}^{r}=h_{\text{{reuse}}}-D_{i,x}^{r},\qquad h_{\text{{ reuse}}}=2D_{\text{max}}\] \[\forall x\in C,\forall i\in K_{x},\forall r\in Q\]  
Another case of \(h_{\text{{reuse}}}\) occurs when \(D_{j,y}^{r}\
eq D_{i,x}^{r}\) and \(D_{j,y}^{r}>D_{i,x}^{r}\). If a single interfering vehicle is assumed, then in Formula (13) summation in the denominator can be omitted and then from this simplified version of (13) the minimum allowed frequency reuse distance can be derived as:  
\[h_{\text{{reuse}}}=\left[\frac{1}{\gamma_{i,x}^{r}(D_{i,x}^{r})^{ \beta}}-\frac{\rho P_{loss}}{\mathrm{P}_{Tx}G_{R}}\right]^{\frac{-1}{p}}+D_{i,x} ^{r}\] (14) subject to : \[D_{j,y}^{r}\
eq D_{i,x}^{r},\] \[D_{j,y}^{r}>D_{\text{max}},\qquad D_{j,y}^{r}=h_{\text{{reuse}} }-D_{i,x}^{r}\forall x\in C,\forall i\in K_{x},\forall r\in Q\]Let \(I_{x}^{r}\) be denoted as the interference on sub-channel \(r\) used by a vehicle located in the coverage area of RSU \(x\), and thus, the objective function can be written as follows:  
\[\min_{a\in A}\sum_{x\in C}\sum_{r\in Q}I_{x}^{r} \tag{15}\]  
where \(A\) is a set of possible strategies of band partitioning between the RSUs, with a particular strategy \(a\) referring to a split, e.g., according to FFR, PFR, or full reuse (however, other strategies can be also considered).