## 6 Performance Analysis
### Communication Cost
Finally, the packet will arrive at the base station through the shortest path from F  
Figure 2: The path from the phantom location to the base station. to b. Therefore, the communication cost of SLP-E is \(\text{Hop}_{\text{sp3,b}}+\text{Hop}_{\text{p3,E}}+\text{Hop}_{\text{er}}+\text{ Hop}_{\text{p,b}}\), where \(\text{Hop}_{\text{sp3}}=h\) and \(\text{Hop}_{\text{F,b}}=\sqrt{H^{2}-r^{2}}\). For \(\Delta p_{s}sb\), we have \(\text{Hop}_{\text{p3,b}}=\sqrt{h^{2}+H^{2}\cdot 2 hH\cos\gamma}\). Given \(\angle 1=\angle bp_{\text{j}}s\), \(\angle 2=\angle sbp_{\text{j}}\), \(\angle 3=\angle sbp_{\text{j}}\), \(\angle 4=\angle Esp_{\text{j}}\), we have \(\angle 1=\arcsin(H\sin\gamma/\text{Hop}_{p_{s,b}})\) and \(\angle 2=\arcsin(h\sin\gamma/\text{Hop}_{p_{s,b}})\) for \(\Delta p_{s}sb\), according to the sine theorem. Similarly, we have \(\angle 3=\arcsin(H\sin\angle 2/\text{r})\) for \(\Delta Esb\) and \(\angle 4=\angle 3-\angle 1\) for \(\Delta p_{\text{j}}Es\). We finally obtain \(\text{Hop}_{\text{p3,E}}=\sqrt{h^{2}+r^{2}-2 hr\cos\angle 4}\) according to the cosine theorem and further obtain \(\text{Hop}_{\text{p}_{E}}=(\gamma-\arccos(r/\text{H})-\angle 4)r\). According to (a) and (b), given that p4 is a point on arc \(p_{\text{j}}Cp_{\text{j}}\) and p3 is a point on the inferior arc \(p_{\text{j}}p_{\text{j}}p_{\text{2}}\), we have \(\alpha\in[0,\)\(\arcsin(\text{r/H})+\arccos(\text{r/h}))\) and \(\gamma\in[\arccos(\text{r/H})+\arccos(\text{r/h}),\)\(\pi]\). Below we will discuss the maximum, minimum and average communication cost of SLP-E.  
* When \(\gamma=\pi\), SLP-E achieves the maximum communication cost of \(2h+(\pi-1-\arccos(r/\text{H}))r+\sqrt{H^{2}-r^{2}}\). In this case, the packet first arrives at the phantom location \(p\)s and then at the base station through the line passing through \(p\)s and \(b\). * When \(\alpha=0\), SLP-E achieves the minimum communication cost of \(H\). In this case, the packet first arrives at the phantom location \(p_{6}\) and then at the base station through the line passing through \(p_{6}\) and \(b\). * The average communication cost of SLP-E is: \[h+\sqrt{H^{2}-r^{2}}\left(\arcsin(r/\text{H})+\arcsin(r/\text{ h})\right)/\pi+\int\limits_{0}^{\arccos(r/\text{H})+\arccos(r/\text{h})}(\sqrt{h^{2} +H^{2}\cdot 2 hH\cos\alpha})\ /\ \pi d\alpha\] \[+\int\limits_{\arccos(r/\text{H})+\arccos(r/\text{h})}^{\pi}( \sqrt{h^{2}+r^{2}\cdot 2 hr\cos\angle 4}+(\alpha-\arccos(r/\text{H})- \angle 4)r)\ /\ \pi d\alpha\.\]  
(3) Commucniaton Cost Comparison between SLP and SLP-E  
The communication cost of SLP and SLP-E are the same if the phantom location lies on \(p_{\text{j}}Cp_{\text{j}}\), according to the previous discussion. However, if the phantom location lies on \(p_{\text{j}}p_{\text{j}}p_{\text{2}}\), the communication cost of SLP-E becomes higher than that of SLP. Specifically, given the increased amount of communication cost \(f\), we have  
(a) \(f=0\) _if \(p_{\text{j}}\) is on \(p_{\text{j}}Cp_{\text{j}}\) ;_  
(b) \(f=\text{Hop}_{\text{er}}+\text{Hop}_{\text{F,b}}\text{-Hop}_{\text{E,b}}\) if \(p_{\text{j}}\) is on \(p_{\text{j}}p_{\text{j}}p_{\text{2}}\). Let \(\beta_{\text{j}}=\angle Esb\) and \(\beta_{\text{j}}=\angle Fsb=\arccos(r/\text{H})\), we have \(\text{Hop}_{\frac{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\where \(r\)/\(h\in(0,\,1)\) and \(r\)/\(H\in(0,\,1]\).\(f_{avg}\) grows as \(h\) decreases according to Equation (4). \(f_{avg}\) achieves the maximum value when \(h=r\). Figure 2 shows how \(f_{avg}\) varied with different \(r\)/\(H\) when \(h=r\) and \(H=100\). Since the communication radius for existing sensors such as a Mica [23] is almost 60 m, the visible area of the attacker is extremely large if \(r\) is set to 6. Therefore, we let \(r\leq 6\) in this paper. For a large sensor networks, \(r\)/\(H\) is usually no more than 1/5 [9] and therefore we have \(f_{avg}=8\). We can see from Table 2 that compared with \(\mathrm{SLP}\), the increased amount of communication cost for \(\mathrm{SLP}\)-\(\mathrm{E}\) is acceptable.