# A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications
## 3 Simulation Results  
In this section, four different criteria for pairing in a D2D communication scenario are investigated. These four methods are  
* Fuzzy-based pairing (FP) method;
* Max-sum-rate pairing (MSP) method;
* Random pairing (RP) method;
* Constant pairing (CP) method. In the MSP method, the selection of the desired transmitter for D2D communication is based on the highest sum rate. According to (4) and (5), the sum rate of the co-channel cellular and potential D2D pairs are calculated as \(R_{\text{B}}+\sum_{i=1}^{N}R_{ki}\). In the RP method, the D2D transmitters are selected randomly. Finally, the nearest transmitter is considered as the paired transmitter in the CP method. The main simulation parameters are summarized in Table 2. Note that the simulation parameters are according to [40]. Now, we compare the pairing stability that is measured by (6) for all four mentioned methods. In Figure 2, the stability as a function of the number of potential D2D transmitters around a receiver, the number of co-channel D2D receivers, and the D2D search radius \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(R_{c}\) (Cellular radius) & 500 m \\ \hline \(R_{max}\) (D2D search radius) & 200 m \\ \hline N (The number of D2D receivers for each cellular users) & 1–4 \\ \hline K (The number of D2D transmitters for each receiver) & 1–4 \\ \hline \(P^{c}\) (Cellular user’s power) & 23 dBm \\ \hline \(P^{d}\) (D2D transmitters’ power) & 15 dBm \\ \hline \(\alpha\) (Path loss exponent) & 3.5 \\ \hline \(R_{0}\) (Data-rate threshold) & 0.3 \\ \hline \(E_{0}\) (Battery-level threshold) & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. are compared. In Figure 2a, the pairing stability in the FP method increases considerably when the number of potential transmitters around a receiver increases. The pairing stability increase of the MSP method is much less than the FP method, while in the CP and RP methods, this increase is negligible. In the FP and MSP methods, increasing the number of potential transmitters gives the receiver more proper choices. In Figure 2b, the stability is illustrated for all four methods. The number of the D2D transmitters is 3, and the number of receivers increase from 1 to 4. Increasing the number of D2D receivers causes us to maintain the stability of the FP method, while it decreases sharply in the other three methods. In Figure 2c, for a fixed number of D2D transmitters and receivers, the stability associated with increasing the search radius is calculated for all mentioned methods. Increasing the search radius does not have a significant effect on the stability of the FP, CP, and RP methods; however, it reduces the stability of the MSP method. The stability of the MSP method decreases with increasing the \(R_{max}\) for two reasons. First, the users' data rates decrease with increasing \(R_{max}\), and second, the battery levels are not considered for this method. In Figure 3, [PERSON]'s fairness index is computed for the number of potential D2D transmitters around a receiver, the number of co-channel D2D receivers, and the D2D search radius. We recall that [PERSON]'s fairness index can be computed as follows:  
\[\mathcal{J}(R_{1},R_{2},...,R_{N})=\frac{(\sum_{i=1}^{N}R_{i})^{2}}{N.\sum_{i= 1}^{N}R_{i}^{2}}. \tag{8}\]  
In Figure 3, it is illustrated that the FP method fairness index is maintained above all methods in a constant range, whether the number of receivers or transmitters or the D2D search radius increase. The fairness index in the MSP method behaves similarly to the FP method when \(R_{max}\) is constant. Nevertheless, increasing \(R_{max}\), the fairness index of the MSP method decreases sharply. By increasing \(R_{max}\), the range of changes in the data rate of co-channel users also increases and causes a reduction in the fairness index.