## 4 Numerical Results
### Channel Utilization  
In order to show the capture effect impacts on the channel utilization, we plot Figure 5a based on (47) first, where the simulation results are also showed. Here, we assume that the number of packets transmitted by the users in the current frame keeps unchanged, i.e., \(N=100\). The other parameters are set as Figure 4a in Section 4.2, which implies that the collided slots are only caused by two packets. As seen in Figure 5a, which is one special case of literature [11], the maximum channel utilization with considering the capture effect is larger than 0.37 with no capture effect. In fact, if the durations of idle slot, successful slot and collided slot are not the same, the maximum channel utilization can also be larger than 0.37 even the capture effect is ignored [34]. Besides, the channel utilization decreases as capture ratio \(z\) increases. The reason is that \(z\) affects the capture probability. In addition, according to (38) and (47), we know that the capture probability affects the channel utilization. As a result, \(z\) affects the channel utilization indirectly. The above results are based on the contrived assumption that the collided slots are all caused by two packets. However, if \(z\) is not large, such as \(z=2\), the capture probabilities of \(n=3\) are all greater than 0.12 in Table 1, which can not be ignored. In fact, the majority of collided slots are caused by two packets while the others are caused by three or more packets even the frame length is optimal. Next, we take different capture probabilities of slots caused by different number of packets into account. Then, the analysis results of channel utilization can be obtained by (38) or (41). For the sake of simplicity, we just give the analysis results and simulation results of channel utilization for \(z=2,4,10\) when \(N=100\), and the maximum values of channel utilization are specifically given in Table 2. From Figure 5b, the curves of different fading become more and more close to that of no capture effect as \(z\) increases. The reason is as described in the previous paragraph. In a word, the capture effect can improve the channel utilization, and if we set the optimal frame length by (42) and (43), the maximum channel utilization can be obtained. Figure 4: Optimal frame length when the capture ratio \((z)=2,4,10\). (**a**) Assume all the collided slots are only caused by two packets; (**b**) Collided slots are caused by two or more packets in practice. Compare Figure 5a with Figure 5b, we find that the maximum channel utilization of the same value \(z\) in the former is greater than that in the latter. Because compared with the capture probability in the latter, the larger capture probability in the former leads to more collided slots turning into successful ones, which results in shortening the optimal frame length. In order to verify the frame lengths of Figure 4b are optimal, we plot the actual channel utilization versus different values of \(N\) in Figure 6. For persuasive experimental results, Monte Carlo simulation method is adopted. From Figure 6, we can see that the actual channel utilization can be maximized in different fading channels by setting the optimal frame length. Actually, the values of actual channel utilization are all greater than 0.37 of no capture effect and approximately equal to the theoretical values in Table 2, respectively. In addition, the channel utilization decreases as \(z\) increases. Because smaller \(z\) leads to larger capture probability, which results in more collided slots turning into successful slots and eventually improves the channel utilization. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fading** & \(z=2\) & \(z=4\) & \(z=10\) \\ \hline Rayleigh & 0.56 & 0.46 & 0.41 \\ Rician & 0.49 & 0.41 & 0.38 \\ Nakagami-\(m\) & 0.52 & 0.43 & 0.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Theoretical maximum channel utilization. Figure 5: Channel Utilization when \(z=2,4,10\).