## 4 Mathematical Modeling and Analysis  
### Stochastic Geometry Model for Wireless Sensor Networks  
We use a stochastic geometry model as a numerical tool to analyze interference, connectivity, and coverage in large-scale wireless sensor networks [21]. Recently, stochastic geometries have also been employed with regard to modeling D2D-enabled cellular networks [22; 23; 24]. We consider _Poisson_ point processes (PPP) to model the D2D sensor networks' underlying cellular systems. We also model SINR distributions where interference management schemes are not considered. Table 3 describes the notations used in the proposed mathematical models.  
To derive the distribution of network elements, we assume that eNBs are distributed with homogeneous PPP \(\Phi_{b}\) of intensity \(\lambda_{b}\). CUEs are located by independent stationary point process \(\Phi_{c}\) with density \(\lambda_{c}\). It is assumed that UE-Txs are distributed in a homogeneous PPP \(\Phi_{d}\) with density \(\lambda_{d}\). DUE-Rxs, the receiver of D2D communication, are uniformly distributed following DUE-Tx during the interval [\(D_{mit}\), \(D_{max}\)]. \(\mathcal{ABS}_{j}^{CUE}\) is a set of CUEs whose distance to DUE-Rx \(j\) is less than \(D_{max}^{ABS}\). The CUEs in \(\mathcal{ABS}_{j}^{CUE}\) should be silent mode on ABS subframes. Similarly, \(\mathcal{ABS}_{c}^{UE}\) is also defined as a set of DUE-Tx \(i\) in case the distance between CUE \(c\) and the DUE-Rx \(j\) is less than \(D_{max}^{ABS}\). The reference distance is the distance between CUE \(c\) and DUE-Rx \(j\). Therefore, the set of DUE-Tx \(i\) that interferes with the CUE \(c\) is determined by the distance between DUE-Rx \(j\) and CUE \(c\).  
Figure 5: Frame structure with ABS.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Definition** \\ \hline \(\Phi_{b}\) & Homogeneous _Poisson_ point processes (PPP) for eNB distribution \\ \(\Phi_{c}\) & Homogeneous _Poisson_ point processes (PPP) for CUE distribution \\ \(\Phi_{d}\) & Homogeneous _Poisson_ point processes (PPP) for DUE-Tx distribution \\ \(\lambda_{b}\) & Intensity of homogeneous _Poisson_ point processes (PPP) for eNB distribution \(\Phi_{b}\) \\ \(\lambda_{c}\) & Intensity of homogeneous _Poisson_ point processes (PPP) for CUE distribution \(\Phi_{c}\) \\ \(\lambda_{d}\) & Intensity of homogeneous _Poisson_ point processes (PPP) for DUE-Tx distribution \(\Phi_{d}\) \\ \(H_{xy}^{CUE},H_{xy}^{D2D}\) & Complex channel gain between node x to y for cellular and D2D links, respectively \\ \(\mathcal{K}_{tx}^{CUE},\mathcal{K}_{rx}^{A}\) & A set \(\{1,...,K\}\) of K DUE-Tx/DUE-Rx terminals using D2D links for frequency \(F_{n}\) \\ \(\mathcal{C}^{CUE}\) & A set \(\{1,...,C\}\) of CUE using cellular links for frequency \(F_{n}\) \\ \(D_{min},D_{max}\) & The minimum/maximum distance between DUE-Tx and DUE-Rx, respectively \\ \(D_{max}^{ABS}\) & Maximum distance between CUE and DUE-Rx for adopting ABS scheme \\ \(\mathcal{ABS}_{c}^{CUE}\) & A set of CUEs whose distance from DUE-Rx \(j\) is less than \(D_{max}^{ABS}\) \\ \(\mathcal{ABS}_{c}^{DUE}\) & A set of DUEs whose distance from CUE \(j\) is less than \(D_{max}^{ABS}\) \\ \(\gamma^{2}\) & Additive White Gaussian Noise (AWGN) noise variance \\ \(p^{CUE},p^{D2D}\) & Transmission power for CUE and DUE-Tx, respectively \\ \(\beta\) & ABS pattern ratio \\ \hline \hline \end{tabular}
\end{table}
Table 3: Notations.