# Multi-User AF Relay Networks with Power Allocation and Transfer: A Joint Approach
## 4 Numerical Results  
The path loss models given by \(25.17+20\times\log_{10}(d)\) dB (\(d\): distance in km) [27; 28] are adopted for the data transfer and the dedicated power transfer channels from \(S_{k}\), \(\forall k\) to \(R\), while the 3 GPP path loss model, given by \(131.1+42.8\times\log_{10}(d)\) dB, is used for the channels from \(R\) to \(D_{k}\), \(\forall k\). The distance from \(S_{k}\), \(\forall k\) to \(R\) is considered to be 1 m [28; 29]. The thermal noise density and the channel bandwidth are considered as \(-174\) dBm/Hz and 1 MHz. The value of interference suppression factor is set to 0.01. \(E_{L}=[1,15]\) mJ and \(E_{H}=[1,100]\) mJ are uniformly distributed EH profiles considered for the simulation. For simplicity, we assume that the EH profiles of all the source nodes are \(E_{s}\), whilst the EH profiles for node \(R\) is \(E_{r}\). The battery capacity at each \(S_{k}\), \(\forall k\), and \(R\) nodes is assumed to be \(E_{s,max}=E_{r,max}=10\) mJ. We consider two scenarios: (1) \(E_{s}=E_{r}\); (2) \(E_{s}\geq E_{r}\), for \(T=15\). Furthermore, we set \(\alpha=0.7\). The distances from \(S_{k}\), \(\forall k\) to \(R\) and from \(R\) to \(D_{k}\), \(\forall k\) are indicated by \(d_{sr}\) and \(d_{rd}\), respectively. In addition, we set \(L_{max}=10\) and \(\epsilon_{i}=0.01\). For better performance comparison, we also simulated the without WPT transmission.  
Figure 2 validates the convergence behavior of the algorithm for \(E_{s}=E_{H}\) and \(E_{r}=E_{L}\). Note that Figure 2 is plotted for a single channel realization. From this figure, we find that the sum rate increases monotonically when the number of iterations increases. In addition, it is noticed that the algorithm converges within four iterations which reflects the effectiveness of the algorithm.  
Figure 2: Convergence behavior of the proposed algorithm (\(N=4,d_{sr}=1\) m and \(d_{rd}=50\) m).  
Figures 3 and 4 illustrate the throughput performance of the network in terms of sum rate for each time index in scenario 1 and scenario 2. In this example, we also include the average sum rate of the network without WPT. In both scenarios, it can be observed that the sum rate performance of the algorithm with energy transfer is much better than that without energy transfer. It can also be observed in Figure 4 that for better EH conditions at the \(S_{k}\), \(\forall k\) nodes compared to the condition at the \(R\) node, the sum rate performance is remarkably enhanced at the end of the time.  
Figure 5 shows the achievable sum rate performance of the system when \(E_{s}=E_{L}<E_{r}=E_{H}\) for \(d_{sr}=1\) m and \(d_{rd}=50\) m. Since the EH rate at the source nodes is the w  
Figure 4: Scenario 2 (\(E_{s}\geq E_{r}\)): Average sum rate performance vs. time index (\(N=4\), \(d_{sr}=1\) m and \(d_{rd}=50\) m).  
Figure 3: Scenario 1 (\(E_{s}=E_{r}\)): Average sum rate performance vs. time index (\(N=4\), \(d_{sr}=1\) m and \(d_{rd}=50\) m).  
transfer from the source nodes to the relay node is almost zero and therefore the performance of the system with WPT is identical to that without WPT.  
Figure 6 illustrates the impact of the relay's position on the average sum rate performance. The parameter settings are as follows: We set \(N=1\) and \(d_{sr}+d_{rd}=2\) m. The distance from \(S_{k}\) to \(R\) and from \(R\) to \(D_{k}\) are represented b \(d_{SR}\) and \(d_{RD}\), respectively, and a distance ratio is defined as \(r_{d}=d_{SR}/(d_{SR}+d_{RD})\). From Figure 6, we can observe that when \(d_{sr}+d_{rd}=2\) m, the average sum rate performance is dominated by the energy transfer rate, \(\delta_{i}\); \(\forall i\), and the highest sum rate performance can be achieved when the relay node is placed closer to the source node.  
Figure 5: Average sum rate performance vs. time index when \(E_{s}<E_{r}\) (\(N=4\), \(d_{sr}=1\) m and \(d_{rd}=50\) m).  
Figure 6: Impact of the relay position on the average sum rate performance for different values of energy harvesting (EH) profiles with \(d_{SR}+d_{RD}=2\) m.