## 5 Performance Analysis  
The detection probability is the probability that the replica nodes are accurately detected. The effect of different amount of epochs on the success rate has been an important performance index. Here an epoch is a random time interval, in this time interval a node keeps moving in the identical direction and at a constant velocity. The movement pattern adopted in the proposed protocol is Random Waypoint Mobility Model. Some random characteristics of RWP are adopted in the security analysis. In the following, the lower bound of detection probability would be analyzed. **Theorem 1**. : _Assume node \(e\) is a compromised node, and \(e_{1}\) is a cloned node of \(e\), \(e\) runs into one of its witness node with the time-location claim \(\{ID_{e},t_{e},l_{e},\text{SIG}_{\text{SK}_{\text{K}}}(H(ID_{e}\parallel t_{e }\parallel l_{e}))\}\), \(e_{1}\) encounters the same witness with the time-location claim \(\{ID_{e1},t_{e1},l_{e1},\text{SIG}_{\text{SK}_{\text{K}_{1}}}(H(ID_{e1} \parallel t_{e1}\parallel l_{e1}))\}\), then the chance of the witness node detecting the node replication attacks by utilizing the two time-location claims is:_  
\[P_{d}(t)=1-\frac{\pi}{D}\times(v_{\max}\times t)^{2} \tag{28}\]  
_in which \(t=\mid t_{e}-t_{e1}\mid\)._  
Proof. : Assume \(f(x,y)\) is the probability density function that a node appears in a position \((x,y)\) in the networks, and nodes in the network are uniformly distributed, therefore \(f(x,y)=1/D\). Node \(e\) is in the position \((xe,ye)\), the conditional probability that node \(e\) and node \(e_{1}\) are contradictory in their locations is:  
\[P_{d}(t|l_{e}=(x_{e},y_{e}))=\iint\limits_{U}f(x,y)dxdy \tag{29}\]  
where \(U=\bigg{\{}(x,y)\in S\bigg{|}\sqrt{\big{(}x-x_{e}\big{)}^{2}+(y-y_{e})^{2}}>v_ {\max}\times t\bigg{\}}\), and:  
\[ \begin{array}{c}P_{d}(t)\hskip 56.905512 pt=P_{d}(t|l_{e})P(l_{e})\\ \hskip 56.905512 pt=P(|l_{e}-l_{e_{1}}|>v_{\max}\times t)\\ \hskip 56.905512 pt=\iint\limits_{U}f(x_{e},y_{e})\times P\times d(t|l_{e}=(x_{e },y_{e}))dx_{e}dy_{e}\end{array}  \tag{30}\]  
Then:  
\[P_{d}(t)=1-\frac{\pi}{D}\times(v_{\max}\times t)^{2} \tag{31}\]  
\(\square\)Assume there are only two nodes with the same identity, they are the compromised node \(e\) and its cloned node \(e_{1}\). Only one witness node is pre-distributed to detect the node replicas attacks. Let \(N_{d}\) denote the amount of epochs until the witness node detects the replica sensor. \(N_{m}\) indicates the number of epochs that is the only time the witness node encounters malicious node before the witness node detects the replica node, then:  
\[P(N_{m}=i)=( \begin{array}{c}2\\ 1\end{array} )(1-p)^{i-1}p(1-p)^{k-i}=( \begin{array}{c}2\\ 1\end{array} )(1-p)^{k-1}p \tag{32}\]  
where \(p\) is the probability that any two nodes encounter in one epoch. Let \(P(N_{d}=k)\) indicate the chance that the node replication attacks are found out until the \(k\)th epoch, only if the witness node receives two time-location claims coming from the different nodes with the same identity, the malicious nodes can be detected with a specified probability, and half time-location claims are useful. The probability:  
\[P(N_{d}=k)=\frac{1}{2}\sum_{i=1}^{k}\sum_{j=i}^{k}P(N_{m}=i)P(N_{m}=j)P_{d}(i,j) \tag{33}\]  
where \(P_{d}(i,j)\) denotes the probability that the witness node detects node replication attacks when the witness node encounters node \(e\) at \(i\)th epoch and encounters node \(e_{1}\) at \(j\)th epoch. According to Theorem 1, \(P_{d}(t)\) represents the probability that the witnesses detect node replication attacks by utilizing the two time-location claims received from \(e\) and \(e_{1}\) respectively. When the two time-location claims are retrieved in different epoch, \(P_{d}(t)\) is 0.