# A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications
## 2 System Model
### Communication Models  
The system model channels are supposed to be independent and identically distributed (i.i.d.) Rayleigh channels with perfect channel state information (CSI) only available for the D2D receivers mentioned as follows:  
1. \(h_{c,B}\) and \(d_{c,B}\) are the direct channel and distance between the cellular user and evolved node base station (eNB), respectively.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(r_{i}\) & \(i^{th}\) D2D receiver \\ \hline \(t_{ki}\) & \(k^{th}\) Potential D2D transmitter for \(r_{i}\) D2D receiver \\ \hline \(N\) & The maximum number of co-channel D2D receivers \\ \hline \(K\) & The number of D2D transmitters around each receiver \\ \hline \(R_{max}\) & The maximum search radius of D2D receivers \\ \hline \(h_{c,B}\) & The direct channel between the cellular user and eNB \\ \hline \(d_{c,B}\) & The distance between the cellular user and eNB \\ \hline \(h_{t_{ki},r_{j}}\) & The channel between the D2D transmitter \(t_{ki}\) and the D2D receiver \(r_{j}\) \\ \hline \(d_{t_{ki},r_{j}}\) & The distance between the D2D transmitter \(t_{ki}\) and the D2D receiver \(r_{j}\) \\ \hline \(h_{t_{ki},B}\) & The interference channel between the D2D transmitter \(t_{ki}\) and eNB \\ \hline \(d_{t_{ki},B}\) & The distance between the D2D transmitter \(t_{ki}\) and eNB \\ \hline \(h_{c,r_{i}}\) & The interference channel between the cellular user and D2D receiver \(r_{i}\) \\ \hline \(d_{c,r_{i}}\) & The distance between the cellular user and D2D receiver \(r_{i}\) \\ \hline \(\mathrm{SINR}_{B}\) & The cellular link SINR \\ \hline \(R_{B}\) & The cellular link data rate \\ \hline \(\mathrm{SINR}_{ki}\) & The D2D link (consists of transmitter \(t_{ki}\) and receiver \(r_{i}\)) SINR \\ \hline \(R_{ki}\) & The D2D link (consists of transmitter \(t_{ki}\) and receiver \(r_{i}\)) data rate \\ \hline \(p^{c}\) & Cellular user’s power \\ \hline \(p^{d}\) & D2D transmitters’ power \\ \hline \(\alpha\) & Path loss exponent \\ \hline \(N_{0}\) & The received noise power \\ \hline \(E_{ki}\) & The battery level of transmitter \(t_{ki}\) \\ \hline \(E_{0}\) & The battery level threshold \\ \hline \(R_{0}\) & The date-rate threshold \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notation summary.  
2. \(h_{t_{ki};r_{j}}\) and \(d_{t_{ki};r_{j}}\) are the channel and distance between the D2D transmitter \(t_{ki}\) and the D2D receiver \(r_{j}\), respectively where \(t_{ki}\in\mathcal{D}_{t_{i}}\), \(r_{j}\in\mathcal{D}_{r}\), \(k\in\{1,\ldots,K\}\) and \(i,j\in\{1,\ldots,N\}\).
3. \(h_{t_{ki};B}\) and \(d_{t_{ki};B}\) are the interference channel and distance between the D2D transmitter \(t_{ki}\) and eNB, respectively.
4. \(h_{c,r_{i}}\) and \(d_{c,r_{i}}\) are the interference channel and distance between the cellular user and D2D receiver \(r_{i}\), respectively. The cellular and D2D links SINR can be noted as follows: \[\text{SINR}_{B}=\frac{S_{B}}{I_{B}+N_{0}},\] (2) \[\text{SINR}_{ki}=\frac{S_{ki}}{I_{i}+N_{0}},\] (3) where \[S_{B}=P^{c}|h_{c,B}|^{2}|d_{c,B}|^{-\alpha},\] \[S_{ki}=P^{d}\beta_{ki}|h_{t_{ki};r_{i}}|^{2}|d_{t_{ki};r_{i}}|^{-\alpha},\] \[I_{B}=\sum_{j=1}^{N}\sum_{k=1}^{K}\beta_{kj}P^{d}|h_{t_{kj};B}|^{2}|d_{t_{kj };B}|^{-\alpha},\] \[I_{i}=\sum_{j=1}^{N}\sum_{k^{\prime}=1}^{K}\beta_{kj^{\prime}}P^{d}|h_{t_{kj ^{\prime}};r_{i}}|^{2}|d_{t_{kj^{\prime}};r_{i}}|^{-\alpha}+P^{c}|h_{c,r_{i}}|^ {2}|d_{c,r_{i}}|^{-\alpha}.\]  
Note that \(N_{0}\) is the received noise power and \(\beta_{kj}\) is the binary pairing variable denoted by  
\[\beta_{kj}=\left\{\begin{array}{ll}1,&\text{if $t_{kj}$ is paired with the receiver $r_{j}$;}\\ 0,&\text{otherwise.}\end{array}\right.\]  
According to the above SINR denotations, the cellular and D2D users data rates are written as  
\[R_{B}=\log(1+\text{SINR}_{B}); \tag{4}\]  
\[R_{ki}=\log(1+\text{SINR}_{ki}). \tag{5}\]