# Depletion-of-Battery Attack: Specificity, Modelling and Analysis
## 6 Performance Analysis  
In this section, we provide simulation experiments to illustrate the theoretical results, by which we discuss the impacts of the DoB attack parameters on the network survivability metrics as well. The provided results are expected to offer insights into the enhancement of protection mechanisms. We first consider the sensor lifetime under DoB caused by both excessive packets and excessive transmission power. We set sensor battery capacity \(C=1000\ e_{0}\), and the packet transmission rate in the normal case to \(\lambda=1\). Besides the deterministic energy consumption of packet transmission, we consider the cases, when it is a random number. In some previous works the battery charge of a sensor had been described by the normal distribution (see, for example, [59]). Since the energy consumption of packet transmission cannot be negative and is limited from above, we use a truncated normal distribution. Let us introduce the following designation for doubly truncated normal distribution function [60]:  
\[\Phi_{T}\Big{(}x\Big{|}v,\sigma^{2},A,B\Big{)}=\frac{1}{\sqrt{\pi} \,\sigma}e^{-(x-v)^{2}/2\sigma^{2}}\bigg{[}\frac{1}{\sqrt{\pi}\,\sigma}\int_{ A}^{B}e^{-(t-v)^{2}/2\sigma^{2}}dt\bigg{]}^{-1},\]  
where \(A\) and \(B\) are the lower and upper truncation points, respectively. The following DoB attack scenarios have been considered. * Case 1. The attack is caused by excessive packets, so, redundant packets rate \(\lambda_{A}=7\) and it is assumed that \(p_{A}=1\). The energy consumption for transmitting each packet is not changed, and equals \(e_{0}\). * Case 2. The scenario is the same as in Case 1, except that the energy consumption of packet transmission is not deterministic. It is assumed to be a random variable having the cumulative distribution function \(\Phi_{T}(x|e_{0},1,0,2e_{0})\). * Case 3. A victim sensor is forced to increase the transmission range by three times, and in formula (6) \(\gamma=2,\ a=1\). The packet transmission rate is not changed: \(\lambda=1\). So, the attack is caused by excessive transmission power. * Case 4. This is the same as Case 3, except that the energy consumption of packet transmission is randomly distributed with the cumulative distribution function \(\Phi_{T}(x|e_{A},1,0,2e_{A})\). In all four cases, a sample of size 1000 was used to analyze the sensor lifetime degradation. The simulation results correspond to those of analytical expression: Formulas (2) and (3). The cases of deterministic energy consumption of packet transmission are consistent with the cases when this energy consumption has a truncated normal distribution. It is shown in Figure 5.  
The bottom and top of the gray boxes are the first and third quartiles. The black line inside each box shows the median. The whiskers stretch from the edge of the boxes to the furthest sample data point that is within 1.5 times the interquartile range. If some points are past the ends of the whiskers, they are displayed with dots. They can be considered to be outliers. The sample variability in Case 3 is higher than in Case 1. This is not surprising, since the variance of Erlang random variable is directly proportional to the shape parameter, and is inversely proportional  
Figure 5: Boxplot diagram for sensor lifetime. to the square of the rate parameter. The theoretically predicted MTTF in Case 1 is 225, and in Case 3 it is 200. It is consistent with the sample-based estimation of MTTF. The detailed statistics is shown in Table 1.  
We next consider the system survivability function. The behavior of \(\mathcal{H}(h,1100)\) is shown in Figure 6.  
The system survivability is relatively high up to 80% of MTTF, and then it drastically degrades. There is an inverse relationship between the cardinality of the critical sensors set and system survivability; when \(m\) decreases, SSF increases. Let us consider two DoB scenarios: an attack caused by spoofed packets, and an attack caused by excessive transmission power. In the first, case the traffic rate is increased ten times, i.e., \(\lambda^{*}=10\).