## 3 Proposed Scheme
### Coverage Probability  
Coverage probability is defined as the probability that a user can successfully transmit the signal from a transmitter to its corresponding receiver, with the SINR equal to or higher than the threshold SINR. In other words, the coverage probability of a system is complementary of the outage probability. Therefore, the coverage probability, \(P^{Cov}\), can be expressed as:  
\[P^{Cov}=1-\ P^{Out} \tag{13}\]  
where:  
\[P^{Out}=\sum_{c^{in}=1}^{N_{C}}P_{c^{in}}^{Out}+\ \sum_{g^{edge}_{t}=1}^{N_{C}}P_{ g^{edge}}^{Out} \tag{14}\]  
where \(P^{Out}\) is the outage probability of the system. Substituting \(\overline{\gamma}=\ \frac{P_{B}\cdot\ l_{c^{in},B}^{-\alpha}}{\sigma_{n}^{2}}\) into Equation (9), we have:  
\[\gamma_{c^{in}}=\ \frac{\left|H_{c^{in},B}\right|^{2}}{\frac{P_{g^{edge}}\cdot\ l_{ g^{edge}_{t},g^{edge}_{t}}^{-\alpha}}{P_{B}\cdot\ l_{c^{in},B}^{-\alpha}}\cdot \left|H_{g_{t}^{edge},B}\right|^{2}+\ \frac{L_{B}}{P_{B}\cdot\ l_{c^{in},B}^{-\alpha}}+\ \frac{1}{ \overline{\gamma}}} \tag{15}\]  
Let \(\text{Y}=\frac{\left(P_{g^{edge}}\cdot\ l_{g^{edge}_{t},g^{edge}_{t}}^{- \alpha}+l_{c,B}\right)}{P_{B}\cdot\ l_{c^{in},B}^{-\alpha}}\), then according to [32], the outage probabilities for the cellular user, \(c^{in}\) (\(P_{c^{in}}^{Out}\)), can be expressed as:  
\[P_{c^{in}}^{Out}=1-\frac{1}{1+Y\cdot\gamma_{Th}}e^{(-\frac{\gamma_{Th}}{ \overline{\gamma}})} \tag{16}\]  
Substituting all the values of \(Y\) and \(\overline{\gamma}\) in Equation (16), we have:  
\[P_{c^{in}}^{Out}=1-\ \frac{1}{1+\frac{\left(P_{g^{edge}}\cdot\ l_{g^{edge}_{t},g^{ \prime}}^{-\alpha}+l_{c,B}\right)\gamma_{Th}}{P_{B}\cdot\ l_{c^{in},B}^{-\alpha }}}e^{(-\frac{\sigma_{n}^{2}\cdot\gamma_{Th}}{P_{B}\cdot\ l_{c^{in},B}^{-\alpha }})} \tag{17}\]  
subject to:  
\[ \begin{array}{c}P_{B}^{min}\leq P_{B}\leq P_{B}^{max},P_{g^{edge}}^{min}\leq P_{g^ {edge}}\leq P_{g^{edge}}^{max}\;\forall\;g^{edge}\;\epsilon\;G,\\ \gamma_{Th}\;\leq\;\gamma_{c^{in}},\;\forall\;c^{in}\;\epsilon\;C,\\ \gamma_{Th}\;\leq\;\gamma_{g^{edge},}\;\forall g^{edge}\;\epsilon\;G,\end{array}  \tag{24}\]  
where \(P_{B}^{min}\) and \(P_{g^{edge}}^{min}\) denote the minimum transmit power of the eNB and M-D2D group, \(g^{edge}\), respectively. \(P_{B}^{max}\) and \(P_{g^{edge}}^{max}\) denote the maximum transmit power of the eNB and M-D2D group, \(g^{edge}\), respectively. In the power control mechanism, the M-D2D transmitter in a group selects its transmission power such that the signal power at the destined receivers will be under a predefined range. From the optimization problem formulated in (23), we observed that the resource allocation and power control approach are closely related with each other. During resource pairing between cellular users and D2D users, the variation of the interference scenario can impact on the transmit power. **Proposition 2**. : _The transmit power of the eNB and M-D2D group, \(g^{edge}\), in Equation (24) can be recalculated as follows:_  
\[ \begin{array}{c}P_{B}^{min}\leq\left.\frac{\sigma_{n}^{2}\cdot\gamma_{c_{in }}^{Th}\cdot\beta_{g^{edge},B}\left(\beta_{g^{edge},g^{edge}}\cdot\gamma_{c_{in }}^{Th}+\gamma_{g^{edge}}\right)+\gamma_{c_{in}}^{Th}\cdot\beta_{g^{edge},B} \cdot\gamma_{g^{edge}}\left(I_{B,\epsilon^{\prime}}\gamma_{c_{in}}^{Th}\cdot \beta_{g^{edge},g^{edge}}+I_{g,m}\right)}{\left(\beta_{c_{in},B}\cdot\beta_{g^{ edge},g^{edge}}+I_{g,m}\right)}\leq P_{B}^{max}\\ \left(\beta_{c_{in},B}\cdot\beta_{g^{edge},g^{edge}}-\gamma_{c_{in}}^{Th}\cdot \beta_{g^{edge},g^{edge}}\cdot\beta_{c_{in},g^{edge}}\right)\end{array} \right\} \leq P_{B}^{max}\\ \left.