## 5 Simulation Results
### Throughput and PLR  
Figure 4 illustrates the network throughput of the proposed policy obtained by simulation and theoretical analysis. We plot the relationship between the normalized network throughput and the number of EH-STs in the network \(U\), and consider different scenarios by varying the energy harvesting rate \(\tilde{\beta}\in\{1,2,3\}\). The normalized throughput is defined as \(T(L)=(1-PLR(L))L\), where the traffic load \(L\) is normalized by \(M/N\). The capacity of the battery is assumed to be \(e_{max}=10\) and the \(PLR\) threshold is set \(10^{-3}\). The simulation results indicate that the analytical network throughput in different EH rate scenarios is well matched with that simulated in the same scenario. For the lower EH rates \(\tilde{\beta}=1\) and \(\tilde{\beta}=2\), the network throughput increases with the number of EH-STs in the network linearly, because in these scenarios, the transmission probability is constrained by the EH rate. Moreover, under a lower EH rate, although the network size increases, the average number of concurrent EH-STs is still less than what CRDSA can support, i.e., the achieved \(PLR\) does not exceed the threshold. Due to the properties of \(g(\eta)\), the network throughput increases with the network size for the lower EH rates. As the EH rate grows, each EH-ST has more transmission opportunities. Since the CRDSA protocol has a good performance on recovering collided packets under moderate traffic load, the network will gain higher throughput under a higher EH rate. When the EH rate is high enough (\(\tilde{\beta}=3\)), the throughput of the network increases with the network size first, but afterwards it tends to be flat. This is because when the network size becomes larger, more EH-STs have enough energy to transmit the packet. For the larger network size, packet collisions become the bottleneck of the performance. The policy will control the transmission probability of each EH-ST to guarantee the \(PLR\) is under the threshold. In addition, we also plot the upper bounds (UB) of the network throughput for different scenarios, which are represented by the black dashed line in Figure 4. Since \(g(\eta)\) is a concave function, \(G(\eta)\leq g(P(\eta))\) from Jensen's inequality [41]. Moreover, \(P(\eta)\leq min\left\{\beta(B\geq l),P(\eta_{PLR_{th}})\right\}\); thus, the upper bound of the network throughput can be obtained as:  
\[T_{\eta}\leq\operatorname*{arg\,max}_{x\in\left[0,min\left\{\beta(B\geq l),P( \eta_{PLR_{th}})\right\}\right]}\mathbb{E}\left(Mg(x)\binom{U}{M}x^{M-1}(1-x) ^{LI-M}\right). \tag{29}\]  
Figure 4: Normalized network throughput for different network sizes, under different EH rates \(\tilde{\beta}\in\{1,2,3\}\), \(e_{max}=10\). For random access (RA) schemes, number of time slots in each frame \(N=200\), QPSK modulation, \(E_{s}/N_{0}=10\)dB. We notice that the proposed policy calculated by Algorithm 1 closely approaches the upper bound under each scenario and the performance degradation for each case is within 5% w.r.t the upper bound. This indicates the locally optimal solution achieved by the symmetric NE is a near-global optimum. In addition to the proposed policy, we also evaluated the performance of the following policies, which are based on the baseline mentioned in Equation (12): The energy-balanced policy (EBP), where each EH-ST transmits the packets with the probability of \(\beta(B\geq l)\); the network-balanced policy (NBP), where each EH-ST transmits the packets with the probability of \(\eta_{PLR_{th}}\) so as to maximize the throughput of the network; and the greedy policy (GP), where each EH-ST transmits the packets with the probability of 1 as long as it has enough energy. The simulation results for throughput and packet loss ratio are represented in Figures 5 and 6, respectively. In addition, we also present the performance of CRDSA (without consideration of the EH process and the limitation of energy) with 2 and 3 replicas to compare with other policies.