## 4 Numerical Results
### Optimal Frame Length  
In order to illustrate the probabilities that different number of packets collide in the same slot, we plot Figure 3, where the analysis results based on (31) are in line with the simulation results. In the figure, the collided probabilities of \(r=2\) are higher than that of \(r=3\) or \(4\) if the frame length \(L\) is close to \(N\). As described in [21], if the capture effect is ignored, the optimal frame length is equal to \(N\). So we analyze the collided probabilities when \(L=N\). For instance, if \(L=N=60\), the probabilities of \(r=2,3,4\) are \(0.189\), \(0.060\) and \(0.013\), respectively. In fact, from Figure 3, the collided probabilities of the same value of \(r\) are almost the same if \(L=N\). Thus, we can know that the majority of collided slots result from two packets transmitted simultaneously when the frame length is optimal. In addition, according to Figure 2 and Table 1, we can know that the capture probabilities of \(n=r=2,3,4\) decrease in sequence when the other parameters are the same. Particularly, when \(n=4\), the capture probabilities of different fading channels are all less than \(0.15\) when the value of capture ratio \(z\) is not less than \(2\), and these values gradually drop to zero as \(z\) increases. In addition, when \(n=3\) and \(z=4\), the capture probabilities are very small too. Therefore, when \(z\) is greater than \(2\), the probability that a collided slot caused by three or more packets becomes a successful slot due to the capture effect is very small.  
From the above, as long as the frame length is optimal and \(z\) is greater than two, that two packets collide in the same slot is more likely to take place, where the capture effect is more likely to occur than three or more packets collided. In this case, we can consider that the capture probability is the same in every collided slot of the frame. That is, all collided slots are caused by two packets.To achieve the maximum channel utilization, the optimal frame length can be set by (46). Let the Rician factor \(K=3\), the Nakagami-\(m\) shape parameter \(m=1.5\), we can draw Figure 4a, which gives the optimal frame lengths about different values of \(N\) based on (46) under Rayleigh, Rician and Nakagami-\(m\) fading channels when \(z=2,4,10\), respectively. In addition, we also give the optimal frame length with no capture effect (\(L=N\)). From Figure 4a, if the capture effect is taken into account, the optimal frame length is less than that of no capture effect. As \(z\) increases, the optimal frame length becomes more close to that of no capture effect. The reason is that larger \(z\) leads to smaller capture probability, which can be seen in Figure 2 and Table 1. As a matter of fact, the capture probability can be close to zero if the value of \(z\) is large enough. On this occasion, the optimal frame length is approximately equal to \(N\).  
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 cannot be ignored. Next, we take different capture probabilities of collided slots caused by different number of packets into account. Let \(K=3\), \(m=1.5\) and \(z=2,4,10\), we plot Figure 4b, where the optimal frame length can be obtained by (42) and (43). From Figure 4b, we can find the optimal frame length is more close to \(N\) as \(z\) increases like that in Figure 4a, which is also caused by the capture effect. That is to say, fewer slots are needed to transmit the same number of packets  
Figure 3: Collided probabilities when the number of packets transmitted in the same slot (\(r\))\(=2,3,4\).  
when the capture effect exists. Compare Figure 4a with Figure 4b, we find that the curves of the same value \(z\) in the former are more far from the curve of no capture effect than that in the latter. Because Figure 4a is based on (46), which factitiously magnifies the real capture probability for assuming all collided slots are caused by two packets.