## 3 System Model and Problem Formulation  
### System Model  
In this paper, we focus on an uplink cellular network in which uplink resources are assigned to CUs by the eNB in an orthogonal manner, which avoids mutual interference among CUs. The main benefit of the uplink resource reuse phenomenon is that the eNB can exclusively coordinate the interference in a fully loaded cellular network [21]. We consider a regular multicell cellular network, where eNBs are placed at the center of the cells, as shown in Figure 1. It is also assumed that all cells are non-overlapping. The cells are divided into center zone with radius \(r\) and edge zone with radius \(R\), and different subbands are assigned to each zone. As shown in Figure 2, the available uplink spectrum \(S\) is partitioned into two bands, namely, \(S_{0}\) and \(S_{1}\). \(S_{0}\) is the frequency band assigned to users located in the center zone, and \(S_{1}\) is the frequency band assigned to users located in the edge zone. Moreover, \(S_{0}\) and \(S_{1}\) are divided into six sub-bands, namely \(S_{0,0}\), \(S_{0,1}\), \(S_{0,2}\) and \(S_{1,0}\), \(S_{1,1}\), \(S_{1,2}\), respectively. For analysis, in our network architecture, we estimate that each cell has \(M\) DPs forming a set \(D\), \(D\)={1, 2, 3,..., \(M\)}, coexisting with \(N\) CUs forming a set \(C\), \(C\)={1, 2, 3,..., \(N\)}. Thus, there are \(K\) subcarriers forming a set \(X\)  
Figure 1: The multicell cellular network with cell sectorization using three-120\({}^{\circ}\) directional antennas. X= {1, 2, 3,..., _K_}. A DP consists of a D2D transmitter and its corresponding receiver. All DPs and CUs follow a uniform distribution mechanism. Aiming at high spectrum utilization, it is assumed that DPs and CUs simultaneously share the same subchannels, and multiple DPs are allowed to reuse the same subchannel at a time. In order to mitigate uplink interference, we assume that DPs in the center cell region can reuse the resources of CUs located in the edge zone and vice versa. However, DPs in the center cell region cannot reuse the resources of CUs located in the center zone, and DPs located in the edge zone cannot reuse the resources of CUs located in the center zone. In order to indicate the resource reuse factor, we define a binary variable \(\delta^{k}_{m,n}\), where \(m\in D,\ n\in N\) and \(k\in X\). When \(\delta^{k}_{m,n}=1\), DP \(m\) can simultaneously reuse subcarrier \(k\) with CU \(n\), otherwise \(\delta^{k}_{m,n}=0\). In our proposed scheme, we have formulated the resource reuse factor (\(F\)) as follows:  
\[F=\sum_{m=1}^{N}\sum_{m=1}^{M}\delta^{k}_{m,n}+1. \tag{1}\]  
Hence, the objective function is proportionate to maximizing \(\sum_{n=1}^{N}\sum_{m=1}^{M}\delta^{k}_{m,n}\). This facilitates maximum resource reuse factor. To simplify the analysis, we have made the following assumption. We consider that the cell structure is analogous to circle. Therefore, to analyze the resource reuse scenario, the probability density function (PDF) of users in center zone in terms of polar coordinates (\(\rho_{in},\ \varphi\)) is [22]:  
\[f(\rho_{in})=\ \frac{2\big{(}\rho_{out}-\rho_{out}^{min}\big{)}}{\big{(}r-\rho_{ out}^{min}\big{)}^{2}} \tag{2}\]  
and  
\[f(\varphi)=\frac{3}{2\pi}\,\ \forall\ 0\leq\varphi\leq\frac{2\pi}{3}. \tag{3}\]  
Therefore, the corresponding cumulative distribution function (CDF) is:  
\[\left. \begin{array}{ll}P(\rho_{in})&=\int_{\rho_{in}^{min}}^{r}f(\rho_{in}) d\rho_{in}\\ &=\int_{d_{min}^{min}\frac{2(\rho_{in}-\rho_{in}^{min})}{(r-\rho_{in}^{min})}d \rho_{in}\\ &=\frac{(r-\rho_{in}^{min})}{(r-\rho_{in}^{min})}[2(\rho_{in})^{2}-2\rho_{in}.\rho_{in}^{min}\rho_{in}^{min}\rho_{in}^{min}\rho_{in}^{min}\rho_{in}^{min} \\ &=\frac{2r}{(r-\rho_{in}^{min})}\end{array}