# On Performance Analysis of Protective Jamming Schemes in Wireless Sensor Networks
## 4 Analysis on Eavesdropping Probability
### Analysis of Fri-Jam Schemes
#### 4.2.3 Case III: FJ-PC Scheme
Then, from the definition of \(P(C)\), we have  
\[P(C) =\int\limits_{r>0}P\left[\left.\frac{P_{t}hr^{-\alpha}}{\sigma^{2} +I_{t}+I_{c}}>\beta|r\right]e^{-\lambda\pi r^{2}}2\pi\lambda rdr\right.\] \[=\int\limits_{r>0}P[h>\beta_{p}r^{\alpha}(\sigma^{2}+I_{t}+I_{c})| r]e^{-\lambda\pi R^{2}}2\pi\lambda rdr.\]According to the channel model (given in Section 3.2), we have  
\[P[h >\beta_{p}r^{\alpha}(\sigma^{2}+l_{t}+l_{c})|r]\] \[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]0 \[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]1 \[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]2  
where \(L(\mu\beta_{p}r^{\alpha})=\exp(-\pi r^{2}\lambda\rho(\beta_{p},\alpha))\), \(\rho(\beta_{p},\alpha)=\beta_{p}{}^{-2/\alpha}\int_{\beta_{p}}^{\infty}\frac{1} {1+(\mu)^{\alpha/2}}d\mu\) and \(E[I_{j}]\) is given by Equation (8). We then have the eavesdropping probability \(P_{e}\) in FJ-PC scheme as the following theorem. **Theorem 5**: _In FJ-PC scheme, the eavesdropping probability \(P_{e}\) that one transmission has been eavesdropped is_  
\[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]3  
_where \(E[I_{j}^{\prime}]=E\left[4\sum\limits_{k=1}^{n}P_{j}(k)\Bigg{(}2\bigg{(}d\cdot \sqrt{\left(2k-3\right)^{2}+\left(2k-1\right)^{2}}\bigg{)}^{-\alpha}+\bigg{(} d\cdot\sqrt{2\left(2k-1\right)^{2}}\bigg{)}^{-\alpha}\Bigg{)}\right]\)._  
The derivation of eavesdropping probability \(P_{e}\) in FJ-PC scheme is similar to the derivation of Equation (15) in Theorem 4 and the main difference is the cumulative interference from jammers. In particular, the calculation of interference from \(n\)th layer jammers in FJ-PC scheme \(I_{j}^{\prime}\) is similar to Equation (A3) in Appendix A, which is shown in the following equation:  
\[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]4 \[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]5  
Then we have the averaged cumulative interference from all the jammers as follows,  
\[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]6  
According to the definition of the probability of eavesdropping attack \(P(E)\), we have  
\[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]7  
where \(P_{e}\) can be replaced by the different values as specified in Non-Jam scheme, FJ-Reg scheme, FJ-Ran scheme and FJ-PC Scheme, which can be obtained by Theorem 1, Theorem 2, Theorem 3 and Theorem 5, respectively. In the next section, we will present numerical results of \(P(E)\) based on the above schemes.