## 3 System Model and Problem Formulation
### Cellular and D2D Communication Mode  
In our defined system model, communications can happen in any mentioned scenarios (eNB, SCeNB, and D2D links). According to Equations (2) and (3), the SINR in uplink for the \(j_{th}\) D2D receiver and outage probability of the \(j_{th}\) D2D link can be expressed by Equations (7) and (8) respectively [1]:  
\[\gamma_{j}^{D} =\frac{P_{j}h_{t,r}d_{t,r}^{-\alpha}}{I_{j}}\] \[=\frac{P_{j}h_{t,r}d_{t,r}^{-\alpha}}{P_{j}h_{t,r}d_{t,r}^{-\alpha }+\sum\limits_{j^{\prime}=1,j^{\prime},j^{\prime}}^{N_{D}}\zeta_{j^{\prime},f^{ \prime}}P_{j}h_{j^{\prime},r}d_{j^{\prime},r}^{-\alpha}+\sum\limits_{k=1}^{N_{ S}}\zeta_{k,f}P_{k}h_{k,r}d_{k,r}^{-\alpha}+N_{0}} \tag{7}\]  
\[P_{out,j}^{D}=1-\delta_{D}\exp(-\beta_{D}(P_{j^{\prime}}^{m}\lambda_{d}+P_{k} ^{m}\lambda_{s})) \tag{8}\]  
where for effective D2D communication, the required SINR threshold is \(\gamma_{th}\). Moreover, \(P_{out,j}^{D}\) is the outage probability of the \(j_{th}\) D2D link, \(\delta_{D}=exp\Big{(}\frac{-N_{0}\gamma_{th}d_{t,r}^{\alpha}}{P_{j}}\Big{)} \Big{(}\frac{P_{j}d_{t,r}^{\alpha}}{P_{j^{\prime},m}d_{t,r}^{-\alpha}+P_{j}d_{ t,r}^{\alpha}}\Big{)}\), \(\kappa=\pi m\Gamma(m)\Gamma(1-m)\), \(\beta_{D}=\kappa\gamma_{th}m^{2}d_{tr}^{2}/P_{j}^{m}\). \(\lambda_{d}\), and \(\lambda_{s}\) represent the density of DUEs and small cells (SCeNBs). Appendix A in [33] is referred to as a proof for the above equation. For the D2D communication mode, Equation (8) infers that small cell and DUEs density, transmit power, path loss coefficient, required SINR, distances between UEs, etc., have a significant impact on the outage probability of D2D links. It is also visible that outage probability increases with higher SINR requirements but decreases while SUEs and D2D receivers are closer to the respective transmitters. Similarly, for the macro cellular communication mode, the SINR at the receiver and outage probability of the macro cell link can be expressed as [1]:  
\[\gamma_{i}^{M} =\frac{P_{i}h_{i,r}d_{i,r}^{-\alpha}}{I_{i}}\] \[=\frac{P_{i}h_{i,r}d_{i,r}^{-\alpha}+\sum\limits_{j^{\prime}=1,j^ {\prime},j^{\prime}}^{N_{D}}\zeta_{j^{\prime},f^{\prime}}P_{j}h_{j^{\prime},r}d _{j^{\prime},r}^{-\alpha}+\sum\limits_{k=1}^{N_{S}}\zeta_{k,f}P_{k}h_{k,r}d_{k,r}^{-\alpha}+N_{0}} \tag{9}\]\[P_{out,i}^{M}=1-\delta_{M}\exp(-\beta_{M}(P_{j}^{m}\lambda_{d}+P_{k}^{m}\lambda_{s})) \tag{10}\]  
where \(\gamma_{th}\) is the required SINR for \(i_{th}\) CUE for effective cellular communication, \(\delta_{M}=\exp(-\frac{N_{0}\gamma_{th}^{m}d_{k}}{B_{1}})\), \(\kappa=\pi m\Gamma(m)\Gamma(1-m)\) and \(\beta_{M}=\frac{\kappa\gamma_{th}^{m}d_{k}^{2}}{B_{1}^{m}}\). So, from the above expressions, it is clear that the outage probability decreases with higher distances between UEs and SUEs as well as DUEs. For small cell mode communications, the link between SUE and SCeNB will suffer from the interference of other SCeNBs, D2D pairs, and eNB. Therefore, the required SINR and corresponding link outage probability of a small cell can be expressed as follows [1]:  
\[\gamma_{k}{}^{S} =\frac{(P_{k}h_{k,s}d_{k,k}^{-\alpha})}{I_{k}}\] \[=\frac{(P_{k}h_{k,s}d_{k,k}^{-\alpha})}{P_{i}h_{i,s}d_{i,s}^{- \alpha}+\sum\limits_{j=1}^{N_{D}}\zeta_{j,f}P_{j}h_{j,s}d_{j,s}^{-\alpha}+ \sum\limits_{k^{\prime}=1,K^{\prime}\
eq k^{\prime}}^{N_{S}}\zeta_{k^{\prime},f}P_{k}h_{k^{\prime},s}d_{k^{\prime},s}^{-\alpha}+N_{0}} \tag{11}\]  
\[P_{out,k}^{S}=1-\delta_{S}\exp(-\beta_{S}(P_{j}^{m}\lambda_{d}+P_{k^{\prime}}^{ m}\lambda_{s})) \tag{12}\]  
where \(\delta_{S}=\left(\frac{R_{k,s}d_{k,s}^{\alpha}}{P_{i}\gamma_{th}d_{k,s}^{- \alpha}+P_{k}d_{k,k^{\prime}}}\right)\exp(-\frac{N_{0}\gamma_{th}d_{k,s}^{ \alpha}}{P_{k}})\), \(\kappa=\pi m\Gamma(m)\Gamma(1-m)\) and \(\beta_{S}=\frac{\kappa\gamma_{th}^{m}d_{k,s}^{2}}{B_{1}^{m}}\), and \(I_{s^{\prime}}\) is the interference from all SCeNBs except for their respective small cell links.