# On Performance Analysis of Protective Jamming Schemes in Wireless Sensor Networks
## 5 Numerical Results
### Impacts of Friendly Jammers on Legitimate Transmissions  
Another concern is to investigate whether friendly jammers will significantly affect the legitimate transmissions. In order to differentiate the effect with jammers and the effect without jammers in terms of the eavesdropping probability and the transmission probability, we define the _eavesdropping deviation_\(\mathcal{D}_{E}\) and the _transmission deviation_\(\mathcal{D}_{C}\) as follows. **Definition 4**: _Eavesdropping deviation \(\mathcal{D}_{E}\) is equal to the difference between the eavesdropping probability \(P(E)\) without jammers and the eavesdropping probability \(P(E)\) with jammers._  
**Definition 5**: _Transmission deviation \(\mathcal{D}_{C}\) is equal to the difference between the transmission probability \(P(C)\) without jammers and the transmission probability \(P(C)\) with jammers._  
We then derive the eavesdropping deviation \(\mathcal{D}_{E}\) and the transmission deviation \(\mathcal{D}_{C}\) in the first case of comparing FJ-Ran scheme with Non-Jam scheme. In particular, we have \(\mathcal{D}_{E}(\mathrm{Ran})=P_{\mathrm{Non-Jam}}(E)-P_{\mathrm{FJ-Ran}}(E)\), where \(P_{\mathrm{Non-Jam}}(E)\) denotes the eavesdropping probability of Non-Jam scheme and \(P_{\mathrm{FJ-Ran}}(E)\) denotes the eavesdropping probability of FJ-Ran scheme. Besides, we have \(\mathcal{D}_{C}(\mathrm{Ran})=P_{\mathrm{Non-Jam}}(C)-P_{\mathrm{FJ-Ran}}(C)\), where \(P_{\mathrm{Non-Jam}}(C)\) denotes the transmission probability  
Figure 4: Probability of eavesdropping attacks \(P(E)\) with FJ-Reg scheme (Grid) versus Non-Jam scheme when \(\alpha=3,4,5\) with SINR threshold \(T\) ranging from 0 dB to 20 dB. Figure 5: Probability of eavesdropping attacks \(P(E)\) with FJ-PC scheme versus Non-Jam scheme when \(\zeta=2,10,20\) with SINR threshold \(T\) ranging from 0 dB to 20 dB. of Non-Jam scheme and \(P_{\text{FJ-Ran}}(C)\) denotes the transmission probability of FJ-Ran scheme. Note that \(P(C)\) can be calculated by [31] and we omit the detailed derivations in this paper. Table 2 shows the comparison results. As shown in Table 2, the eavesdropping deviation is much larger than the transmission deviation at the same network settings, implying that _using jammers in WSNs will not significantly affect the legitimate communications_ compared with the reductions on the eavesdropping probability. For example, when \(\lambda_{2}=2.0\), \(\mathcal{D}_{E}=0.5178\) while there is less than \(0.1\) reduction on the transmission probability (i.e., \(\mathcal{D}_{C}=0.0963\)). Similarly, we derive the eavesdropping deviation \(\mathcal{D}_{E}\) and the transmission deviation \(\mathcal{D}_{C}\) in the second case of comparing FJ-Reg scheme with Non-Jam scheme. Table 3 shows the comparison results. It is shown in Table 3 that FJ-Reg scheme can also significantly reduce the eavesdropping probability with only minor influence on the legitimate transmissions (e.g., the reduction of \(P(E)\) is 0.6650 while the reduction of \(P(C)\) is only 0.1143 when \(d=0.2\)). We next derive the eavesdropping deviation \(\mathcal{D}_{E}\) and the transmission deviation \(\mathcal{D}_{C}\) in the third case of comparing the FJ-PC scheme with the Non-Jam scheme. Table 4 shows the comparison results. It is shown in Table 4 that the FJ-PC scheme can significantly reduce the eavesdropping probability with only minor influence on the legitimate transmissions. For example, the \(P(E)\) is 0.6128 and \(P(c)\) is 0.1729 in the Non-Jam scheme and they become 0.1770 and 0.1367, respectively when FJ-PC scheme with \(d=0.6\) is applied. At this time, the reduction of \(P(E)\) is 61.8% while the reduction of \(P(C)\) is only 12.5% when \(d=0.6\) implying that the FJ-PC scheme can significantly reduce the eavesdropping probability while maintaining the minor impairment to the legitimate communications.