# A Practical Resource Management Scheme for Cellular Underlaid D2D Networks
## 5 Numerical Results  
In this section, we evaluate the performance of the proposed scheme in terms of average sum-rate and compare it with optimal performance. In addition, we derive the optimal threshold values to maximize the performance of the proposed scheme. Figure 3 shows the average sum-rates of the proposed scheme where all channels are assumed to be _i.i.d._ and thus \(\lambda_{ij}=1\ \forall\ i\ \text{and}\ j\). \(N=5\) and \(I_{\mathbf{th}}^{\prime}=0\) or \(5\ \text{dB}\). For the comparison, it also shows the optimal average sum-rates obtained by solving Equation (5) based on the Brute Force searching algorithm. For a higher \(I_{\mathbf{th}}^{\prime}\) denoting that a cellular network tolerates higher interference from a D2D network, the cardinality for the eligible set of D2D transmitters increases and thus average sum-rates of the D2D network also increases due to an increasing gain of user selection diversity. As the SNR of D2D transmitters increases, the average sum-rates of the D2D network also increase due to transmit power gain. However, the SNR values higher than a moderate level reduce the average sum-rate on the contrary because the cardinality for the eligible set of D2D transmitters seriously decreases and thus the user selection diversity decreases as well. The proposed scheme can achieve almost optimal average sum-rates when SNR is low or high, while the average sum-rates of the proposed scheme is lower than the optimal value for moderate SNR values. It should be also noted that the proposed scheme can dramatically reduce the complexity and feedback overhead, compared to an optimal scheme. Figure 4 also shows average sum-rates of the proposed scheme and optimal sum-rates under the same conditions as in Figure 3, except for \(N=10\). As \(N\) increases, the gain of user selection diversity increases and thus the overall performance is enhanced. The average sum-rates of the proposed and optimal schemes are tabulated and the ratios of average sum-rate obtained by the proposed scheme to optimal value are also summarized in Table 1.  
Figures 5 and 6 show the optimal threshold values to maximize the average sum-rates of the proposed scheme with the same parameter values with Figures 3 and 4, respectively. As the SNR of D2D transmitters increases to a moderate level, the optimal threshold value increases and thus \(|\mathcal{T}|\) decreases. The optimal threshold value of the proposed scheme decreases if the SNR increases above the moderate level. The excessively high SNR above the moderate level seriously decrease \(|\mathcal{E}|\) and thus a low threshold can enhance the performance of the proposed scheme by increasing \(|\mathcal{T}|\). In Figures 7 and 8, we analyze the average sum-rates of the proposed scheme when channels are non-_i.i.d._ to verify the feasibility of the proposed scheme in practical environments. Table 2 summarizes the average sum-rates of the proposed and optimal schemes and the ratios of average sum-rate obtained by the proposed scheme to optimal value. \((2\times N)\) D2D nodes are uniformly distributed in a circle with a radius of \(200\ \text{m}\) and a cellular BS is located at the center of the circle. The average channel gain between transmitter \(i\) and receiver \(j\) is determined by \(\lambda_{ij}=\min\left(-30\ \text{dB},L_{ij}^{-4}\right)\), where \(-30\ \text{dB}\) and \(L_{ij}(0\leq L_{ij}\leq 200)\) denote a minimum coupling loss and the distance between transmitter \(i\) and receiver \(j\), respectively, and \(i\in\{1,2,\cdots,N,\text{u}\}\), \(j\in\{1,2,\cdots,N,\text{b}\}\). It is assumed that the channel bandwidth is \(10\ \text{MHz}\) and thermal noise power density is \(-174\ \text{dBm}/\text{Hz}\). The transmit power of all transmitters varies from \(0\text{--}30\ \text{dBm}\) and \(I_{\mathbf{th}}=-50\ \text{or}\ -80\ \text{dBm}\). The performance gap between the proposed and optimal schemes is noticeably reduced in the entire SNR region, compared to the _i.i.d._ channel environments.