## 3 System Model and Problem Formulation  
### System Model  
For our analysis, we have considered a 3-tier D2D-enabled cellular HetNet where a macro cell (eNB) is placed at the center of the coverage area, and it is surrounded by multiple small cells (SCeNBs) and DUEs. The radius of eNB is 500 m, and small cells, as well as DUEs, are distributed randomly. As small cell locations are unpredictable and random, we have modeled the small cell spatial locations of small cells by a homogeneous poison point process PPP, \(\phi_{s}\) with density, \(\lambda_{s}\). Similar to small cells, DUEs are also distributed by using another independent homogeneous PPP, \(\phi_{d}\), with density \(\lambda_{d}\). Here, \(i\in\{1,2,\ldots,N_{C}\}\), \(j\in\{1,2,\ldots,N_{D}\}\), and \(k\in\{1,2,\ldots,N_{S}\}\) represent the set of CUEs, DUEs, and SUEs respectively. Figure 1 illustrates a 3-tier heterogeneous cellular network where DUEs, SUEs, and CUEs are communicated simultaneously by sharing the same radio resources. By using the D2D communication mode, DUEs can communicate with each other directly, and CUEs and SUEs can communicate by using cellular and small-cell communication modes. In this system model, the interference scenarios can be defined as follows: (1) eNB receives signals from the transmitter of DUE and SUE; (2) DUE receiver receives signals from the transmitter of CUE and SUE; (3) DUE receiver receives signals from the transmitters of other D2D pairs; (4) SCeNB receives signals from the transmitter of DUE and CUE. We further assume that there are F orthogonal channels in the network, and all channels are used by CUEs. In other words, all the cellular resources are used by CUEs, and to enable communication, DUEs must reuse the CUE's channels. To mitigate the intra-cell interference, each cellular user uses separate RBs, and cellular resources are assigned orthogonally. In cellular communication mode, only one cellular resource will be shared with a D2D user at a time to limit the co-channel interference. For ease of analysis, here we have considered only the Rayleigh fading environment and exponentially distributed channel coefficients. In such a D2D-enabled HetNet, the received signal can be defined as follows:  
\[P_{r}=P_{t}h_{xy}D^{-\alpha} \tag{1}\]  
where \(P_{t}\), \(\alpha\), \(D\) and \(h_{xy}\) are transmission power, path loss exponent, the distance between link nodes, and channel coefficient, respectively, for that link. With the above assumptions, the SINR, \(\gamma\), at the receiver, \(y\), can be expressed as follows:  
\[\gamma_{y}=\frac{P_{t}h_{xy}d_{xy}^{-\alpha}}{I+N_{0}}=\frac{Received\ Power}{Total\ Interference} \tag{2}\]  
Here, \(I\) and \(N_{0}\) are the total interference experienced by the receiver, \(y\), and the spectral noise density, respectively. To decode a message successfully by a receiver, the SINR at the receiver has to be higher than the threshold SINR \(\gamma_{th}\), otherwise this link will experience an outage. So the outage probability of any \(x\), \(y\) link can be defined as follows:  
\[P_{out}=P_{r}\{\gamma_{y}\leq\gamma_{th}\} \tag{3}\]  
Let us assume that uplink cellular radio resources are shared among DUEs and small cells.