## 5 Proposed Scheme
### Formulation of Signal-to-Interference-Plus-Noise Ratio (SINR)  
SINR is a commonly used for the analysis of characteristic of a communication system. The general expression for SINR constraint for a device can be written as  
\[\gamma\ \geq\ \gamma^{Th} \tag{5}\]  
where \(\gamma^{Th}\) is the predefined SINR threshold. A signal transmission is said to be successful when the received SINR of the user is greater than or equal to the predefined threshold SINR. To derive the SINR distribution, we assume that the resource of a cellular user \(k\) exists in inner cell region of A-1 (shown in Figure 1) is reused by two-pairs of D2D existing in edge cell region of A-1. We denote \(P_{i}\) as the transmit power of D2D-Tx \(i\), \(P_{i^{\prime}}\) as the transmit power of D2D-Tx \(i^{\prime}\), \(P_{k}\) as the transmit power of cellular user k, \(l_{i,B}\) as the distance between D2D-Tx \(i\) and eNB, \(l_{i^{\prime},B}\) as the distance between D2D-Tx \(i^{\prime}\) and eNB, \(l_{k,j}\) as the distance between cellular user \(k\) and D2D-Tx \(j\), \(l_{k,j^{\prime}}\) as the distance between cellular user \(k\) and eNB. Let \(h_{i,B}\) and \(h_{i^{\prime},B}\) represent the channel coefficients from eNB to D2D-Tx \(i\) and D2D-Tx \(i^{\prime}\), respectively, \(h_{k,j}\) and \(h_{k,j^{\prime}}\) represent the channel coefficients from cellular user \(k\) to D2D-Tx \(i\) and D2D-Tx \(i^{\prime}\), respectively, and \(h_{k,B}\) represent the channel coefficients from eNB to cellular user \(k\). Moreover, we define the interference from D2D-Tx \(i\) to D2D-Rx \(j^{\prime}\) and from D2D-Tx \(i^{\prime}\) to D2D-Rx \(j\) by \(I_{i^{\prime}}\) and \(I_{i}\), respectively, the interference from neighboring D2D-Tx to D2D-Rx \(j\) and \(j^{\prime}\) by \(I_{j,ne}\) and \(I_{j^{\prime},ne}\), respectively. Therefore, the received SINRs of D2D-Rx \(j\) and \(j^{\prime}\) are given by:  
\[\gamma_{j}=\ \frac{P_{i}\cdot l_{i,B}^{-\alpha}\cdot\left|h_{i,B}\right|^{2}}{P _{k}\cdot l_{k,j}^{-\alpha}\cdot\left|h_{k,j}\right|^{2}+I_{i^{\prime}}+\ \sum_{ne\
eq A1}I_{j,ne}+\ P_{N}} \tag{6}\]  
\[\gamma_{j^{\prime}}=\ \frac{P_{i^{\prime}}\cdot l_{i^{\prime},B}^{-\alpha}\cdot \left|h_{i^{\prime},B}\right|^{2}}{P_{k}\cdot l_{k,j^{\prime}}^{-\alpha}\cdot \left|h_{k,j^{\prime}}\right|^{2}+I_{i}+\ \sum_{ne\
eq A1}I_{j^{\prime},ne}+\ P_{N}} \tag{7}\]  
respectively, where \(\alpha\) is the path loss coefficient and \(P_{N}\) is the noise power. Similarly, the received SINR of cellular user \(k\) is given by:  
\[\gamma_{k}=\ \frac{P_{k}\cdot l_{k,B}^{-\alpha}\cdot\left|h_{k,B}\right|^{2}}{P _{i}\cdot l_{i,B}^{-\alpha}\cdot\left|h_{i,B}\right|^{2}+P_{i^{\prime}}\cdot l _{i^{\prime},B}^{-\alpha}\cdot\left|h_{i^{\prime},B}\right|^{2}+\sum_{ne\
eq A 1}I_{B,ne}+\ P_{N}} \tag{8}\]  
where \(I_{k,ne}\) is the interference from the neighboring D2D-Tx to the eNB. Here, the analytical expressions for different interference are expressed as:  
\[I_{i^{\prime}}=\ P_{i^{\prime}}\cdot l_{i^{\prime},j^{\prime}}^{-\alpha}\cdot \left|h_{i^{\prime},j}\right|^{2} \tag{9}\]  
\[I_{i}=\ P_{i}\cdot l_{i,j^{\prime}}^{-\alpha}\cdot\left|h_{i,j^{\prime}} \right|^{2}, \tag{10}\]  
\[I_{j,ne}=\ P_{ne}\cdot l_{ne,j^{\prime}}^{-\alpha}\cdot\left|h_{ne,j}\right|^{2} \tag{11}\]  
\[I_{j^{\prime},ne}=\ P_{ne}\cdot l_{ne,j^{\prime}}^{-\alpha}\cdot\left|h_{ne,j^ {\prime}}\right|^{2} \tag{12}\]  
\[I_{B,ne}=\ P_{ne}\cdot l_{ne,B}^{-\alpha}\cdot\left|h_{ne,B}\right|^{2}. \tag{13}\]In order to determine the proper set of cellular resources that can be reuse by D2D pairs, we formulate a channel coefficient relationship as listed below [23].