## 3 Proposed Scheme
### Network Model  
We consider a multi-cell network that has seven hexagonal-shaped cells in which eNB are placed at the center of the cell as shown in Figure 1a. Figure 1b shows the deployment of the available spectrum using the FFR method. In the FFR scheme, the cell coverage region is divided into two regions: Inner \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(R\) & Radius of outer region of cell \\ \(r\) & Radius of inner region of cell \\ \(P_{B}\) & Transmission power of eNB \\ \(P_{g^{edge}}\) & Transmission power of M-D2D group \(g^{edge}\) \\ \(l_{e^{in},B}\) & Distance between cellular user \(c^{in}\) and eNB \\ \(l_{g^{edge},B}\) & Distance between M-D2D transmitter \(g^{edge}_{t}\) and eNB \\ \(l_{e^{in},g^{edge}}\) & Distance between M-D2D transmitter \(g^{edge}_{t}\) and receiver \(g^{edge}_{r}\) \\ \(l_{e^{in},g^{edge}}\) & Distance between cellular user \(c^{in}\) and M-D2D receiver \(g^{edge}_{r}\) \\ \(l_{e^{in},B}^{min}\) & Minimum possible distance of cellular user from the eNB \\ \(H_{e^{in},B}\) & Channel coefficient between cellular user \(c^{in}\) and eNB \\ \(H_{g^{edge},B}\) & Channel coefficient between M-D2D transmitter \(g^{edge}_{t}\) and eNB \\ \(H_{g^{edge},g^{edge}}\) & Channel coefficient between M-D2D transmitter \(g^{edge}_{t}\) and receiver \(g^{edge}_{r}\) \\ \(H_{c^{in},g^{edge}}\) & Channel coefficient between cellular user \(c^{in}\) and M-D2D receiver \(g^{edge}_{r}\) \\ \(a\) & Path loss coefficient \\ \(\sigma_{n}^{2}\) & Noise power \\ \(\mathcal{T}_{Th}\) & Minimum SINR requirement of a user defined by the system \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations used in the problem formulation. and outer, which is covered by three 120\({}^{\circ}\) directional antennas. We consider two different radiuses for the inner and outer region of each cell to minimize the interference between cellular users and D2D users. Moreover, the network spectrum is partitioned into two bands, namely \(S_{0}\) and \(S_{1}\). The spectrum band, \(S_{0}\), is used to serve users in the inner region and the spectrum band, \(S_{1}\), is used to serve users in the outer region. Additionally, these bands are divided into six equal sub-bands, namely \(S_{0,0}\), \(S_{0,1}\), \(S_{0,2}\), \(S_{1,0}\), \(S_{1,1}\), and \(S_{1,2}\). Figure 2 shows the M-D2D transmission model and uplink interference scenarios generated due to the integration of D2D communications into the conventional cellular networks. We consider whether the network allows both cellular communications and D2D communications at the same time. Figure 1: (**a**) The structure of the multi-cell D2D communication network; (**b**) spectrum partitioning. Figure 2: (**a**) M-D2D transmission by reusing uplink cellular resource; (**b**) uplink interference scenarios. For easy analysis, we assume cellular users as primary users that can access the available resources assigned by the eNB and D2D users as the secondary users that can reuse the cellular resources to generate a transmission. Suppose in each cell there are totally \(N_{C}\) cellular users constituting a set, \(C\), where \(C\) = {1, 2,..., \(N_{C}\)}, the total number of D2D users constituting a set, \(D\), where \(D\) = {1, 2,..., \(N_{D}\)}, and the total number of M-D2D groups constituting a set, \(G\), where \(G\) = {1, 2,..., \(N_{C}\)}. We assumed that all cellular users and D2D users are uniformly distributed in each cell. In each M-D2D group, we selected a D2D user as the multicast transmitter, which transmits the same signal to multiple receivers in the group. Thus, the total sum throughput increases with the number of successful communications of the group. We assumed that one M-D2D group can reuse only one cellular link at a time. In the network model, we assume that the available resources are allotted to cellular users by the eNB in an orthogonal manner. Therefore, there is no intra-cell interference between the cellular users.