## 5 Performance Analysis
We evaluate the average number of samples for each tracked node under different system-configured maximum speed when a malicious node is accurately found out as cloned node. As shown in Figure 5, the average amount of samples achieves a maximum value at 8, when the maximum speed \(v_{\max}=10\) and the maximum speed error rate \(\theta=0.1\). The average number of samples reaches a minimum value at 4.25 when the predefined maximum value \(v_{\max}=100\) and the maximum speed error rate \(\theta=0.01\). There is another dimension, Figure 6 shows the average amount of samples for each tracked node under different maximum speed when a benign node is accurately detected as a normal node. As shown in Figure 6, the average amount of samples achieves a maximum value at 5.2, when the predefined maximum speed \(v_{\max}=100\) and the maximum speed error rate \(\theta=0.1\). The average amount of samples reaches a minimum value at 3 when the predefined maximum speed \(v_{\max}=10\) and the maximum speed error rate \(\theta=0.01\). As the whole, with the increase of the predefined maximum value, the average number of samples for each tracked node rises or drops slightly. The witness node would detect whether a mobile sensor node has been replicated or not with a smaller number of samples in both cases. Meanwhile, it is obvious that an increase of the maximum speed error rate \(\theta\) results in the growth of the average amount of samples for each tracked node. It can be further reasoned that the faster the movement speed, the higher the chance that the measured speed of a normal node is erroneous detected to be over the predefined maximum speed. On the contrary, the faster the movement speed, the less chance that the measured speed of a malicious node is erroneous detected to be below the system-configured maximum speed. The probability distribution of the amount of samples for each malicious node detected accurately is shown in Figure 7. When the maximum speed error rate \(\theta=0.1\) and the maximum speed \(v_{\max}=20\), about 79% of all the case falls in the range of \([4,9]\) as shown in the figure. This also indicates that the probability distribution of the amount of samples satisfies the rule reflected in Figure 5. The amount of samples for each tracked node is less than or close to the average amount for each tracked node. The probability distribution of the amount of samples for each benign node detected accurately is shown in Figure 8. When the maximum speed error rate \(\theta=0.1\) and the system-configured maximum speed \(v_{\max}=20\), about 87% of all the case falls in the range of \([3,7]\) as shown in the figure. This also indicates that the probability distribution of the amount of samples satisfies the rule reflected in Figure 1.0 The amount of samples for each benign node is below or close to the average amount for each benign node. As we can see from Figures 5 and 6, it is obvious that whether a mobile sensor node is malicious node can be decided quickly with a few samples for each tracked node. Figure 1.1 The average number of samples for each malicious node. In the process of detection, one queue for each tracked node in the tracking set is maintained, which can hold more than two corresponding detection information. Because the more samples are measured, the more accurate the detection, the length of the queue denoted as \(n_{p}\) is set to a higher value to accelerate the speed of detection. But the longer the queue, the more space costs, to avoid overload for the tracking node, therefore making the size of each queue equal to 3 is meeting demand. As is shown in Figure 9, under the same amount of nodes compromised, the scheme with \(n_{p}=3\)  
Figure 1.2 The probability distribution of the amount of samples for benign node. Figure 1.3 The average number of samples for each benign node. Figure 1.4 The probability distribution of the amount of samples for malicious node.