## 3 Proposed Scheme
### Mathematical Modeling
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(\mathcal{K}_{CE}\) & A set of sensors using a cellular link \\ \(\mathcal{K}_{D2D}\) & A set of sensors using a D2D link \\ \(\mathcal{Z}_{ai}\) & A set of sensors in a zone which has sector \(a\) and number \(i\) \\ \(G_{j}\) & A set of sensors in a cell of eNB or sensor \(j\) \\ \(P_{t,b}\) & Tx power of a sensor for cellular link \\ \(P_{t,s}\) & Tx power of a sensor for D2D link \\ \(P_{r,m,q}\) & Rx power with mode \(m\), \(m\in\{CE,D2D\}\) and neighboring type \(q\) \\ \(N_{0}\) & Noise power \\ \(G_{kc}\) & Channel gain between sensor \(k\) and eNB \(c\) \\ \(G_{jl}\) & Channel gain between sensor \(j\) and sensor \(l\) \\ \(IF_{Xn}\) & Interference from sensors in all zone \(Xn\) \\ \(I_{1,m}\) & Experienced interference at a node with using link \(l\) on mode \(m\) \\ \(Davg(q)\) & Average distance between two zones with type \(q\) \\ \(\tau_{q}\) & Coefficient of average distance with type \(q\), where \(\tau_{q}\in\tau\) \\ \(u_{s,l}\) & the number of average users in a zone \(s\) using the link \(l\) \\ \(\mu\) & the ratio of the number of average users of using cellular case to using D2D case \\ \(r\) & distance between the transmitter and receiver \\ \(\alpha\) & path-loss coefficient \\ \(\mathcal{N}_{z}\) & A set of neighboring zones of the zone \(z\) \\ \(n\) & the number of neighboring zones \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of symbols. as a D2D link in \(Cn\). It is the same as the case of \(cn\) in the zone \(An\) symmetrically. The SINR for the frequency section \(bn\), \(SINR^{sensor_{k}(Z_{An})-sensor_{j}}_{ffr,D2D_{1}}\) is as follows:  
\[SINR^{sensor_{k}(Z_{An})-sensor_{j}}_{ffr,D2D_{1}} =\frac{P_{t_{A}}\text{G}_{ij}}{IF_{An}+IF_{Bn}+IF_{Cn}+N_{0}}\] \[=\frac{P_{t_{A}}\text{G}_{ij}}{\sum_{i\in(Z_{An})^{i}\times C_{2D2 }\cap C_{i}^{j})}P_{t_{A}}\text{G}_{ij}+\sum_{i\in(Z_{An})^{i}\times C_{2D2})}P _{t_{A}}\text{G}_{ij}+\sum_{i\in(Z_{An})^{i}\times C_{2D2})}P_{t_{A}}\text{G}_{ ij}+N_{0}}, \tag{2}\]  
where the \(IF_{An}\) case means interference from the same zone of the other cells. \(IF_{Bn}\) and \(IF_{Cn}\) mean interference from all other zones. One of them is from the D2D link and the other is from the cellular link. It is changed in the case of \(cn\), which is \(SINR^{sensor_{k}(Z_{An})-sensor_{j}}_{ffr,D2D_{2}}\). The order is not important because the two cases are symmetric. Thus, \(SINR^{sensor_{k}(Z_{An})-sensor_{j}}_{ffr,D2D_{2}}\) is given by  
\[SINR^{sensor_{k}(Z_{An})-sensor_{j}}_{ffr,D2D_{2}} =\frac{P_{t_{A}}\text{G}_{ij}}{IF_{An}+IF_{Bn}+IF_{Cn}+N_{0}}\] \[=\frac{P_{t_{A}}\text{G}_{ij}}{\sum_{i\in(Z_{An})^{i}\times C_{2D2 }\cap C_{i}^{j})}P_{t_{A}}\text{G}_{ij}+\sum_{i\in(Z_{An})^{i}\times C_{2D2})}P _{t_{A}}\text{G}_{ij}+\sum_{i\in(Z_{An})^{i}\times C_{2D2})}P_{t_{A}}\text{G}_ {ij}+N_{0}}, \tag{3}\]  
where \(n\in\{1,2\}\). In the semi-static scheme, there are inter-cell interferences from adjacent cells for both of the links, cellular and D2D links, but intra-cell interference does not exist. Therefore, the SINR of the two links, \(SINR^{sensor_{k}-eNB_{j}}_{ss,CE}\) and \(SINR^{sensor_{k}-sensor_{j}}_{ss,D2D}\) can be defined respectively as:  
\[SINR^{sensor_{k}-eNB_{j}}_{ss,CE} =\frac{P_{t,b}\text{G}_{kj}}{\sum_{i\in(\mathcal{K}_{CE}\cap C_{i }^{j})}P_{t,b}\text{G}_{ij}+N_{0}}, \tag{4}\] \[SINR^{sensor_{k}-sensor_{j}}_{ss,D2D} =\frac{P_{t,s}\text{G}_{kj}}{\sum_{i\in(\mathcal{K}_{D2D2})^{i} \cap C_{i}^{j})}P_{t,s}\text{G}_{ij}+N_{0}}. \tag{5}\]