# Depletion-of-Battery Attack: Specificity, Modelling and Analysis
## 4 DoB Analysis
### DoB based on Excessive Transmission Power  
In this case, an intruder does not use additional packets. However, the radio transceivers of critical sensors are forced to increase transmission power. Assume that in the normal state, a sensor passes packets to a sink through a set of intermediate sensors, \(I_{S}\). Under a DoB attack, this sensor transmits packets directly to the sink at a longer distance. In this case, packet transmission rate \(\lambda\) is not changed. However, the energy consumption for transmitting one packet is increased. Let \(e_{A}\) be the energy amount consumed for one packet transmissioin under a DoB attack, \(e_{A}\gg e_{0}\). Now, the total number of packets transmitted by a sensor, \(n_{A}\), can be estimated as follows:  
\[n_{A}=C/e_{A}.\]  
The sensor battery behavior shown in Figure 2 is the same, except for the number of states, which is drastically reduced: \(n_{A}\ll N_{0}\). Therefore, in this scenario:  
\[\text{MTTF}=\frac{n_{A}}{\lambda}, \tag{4}\]  
and  
\[\text{SSF}=\mathcal{H}(h,\lambda,n_{A}). \tag{5}\]  
It is easy to see that the MTTF of the attacked sensor degrades \(N_{0}/n_{A}\) times. At the same time the system QoS can be improved. To illustrate this, we consider part of a wireless sensor network, shown in Figure 4.  
The sensor \(s_{A}\) monitors an event of interest, generates a report, and transmits the report to the sink node through sensors set \(I_{S}\). Here, set \(I_{S}\) is a chain of \(n_{C}\) sensors.  
The delivery latency in packets generated by sensor \(s_{A}\) is random variable \(t_{L}\) and:  
\[t_{L}=\sum_{j=1}^{n_{C}}t_{j}+t_{0},\]  
where \(t_{j}\) is the packet service time at node \(j\). It includes time expenses for packet receiving, data treatment, waiting, and packet transmitting. The variable \(t_{0}\) is the interaction time between the sink and adjacent nodes. The variables \(t_{j}\), where \(j\in I_{S}\), are assumed to be identically distributed. Therefore, the average delivery latency is:  
\[Et_{L}=n_{C}\overline{t}+Et_{0},\]  
where \(\overline{t}\) is the average service time at a node.  
When under DoB attack, sensor \(s_{A}\) sends data directly to the sink. Therefore, the average delivery latency becomes \(Et_{0}\).  
Generally, \(\overline{t}\gg Et_{0}\). However, even if \(\overline{t}\approx Et_{0}\), the packet delivery latency, which is an important QoS parameter, is reduced significantly: \(n_{c}+1\) times.  
If \(I_{S}\) also receives packets from sensors other than \(s_{A}\), and packets losses are possible, then DoB decreases the offered load for \(I_{S}\). Therefore, the loss rate is decreased as well. Thus, the system QoS can be improved under DoB. At the same time, system lifetime is essentially degraded.  
The harm from DoB attacks based on excessive transmission power can be stronger, compared to DoB attacks caused by excessive packets, even if not interrupted (\(p_{A}=1\)). To confirm this proposition, consider MTTF:  
\[\frac{n_{A}}{\lambda}<\frac{N_{0}}{\lambda_{A}+\lambda}.\]  
Taking into account equality (1), we get:  
\[e_{A}>e_{0}\bigg{(}1+\frac{\lambda_{A}}{\lambda}\bigg{)}.\]  
If the above inequality is satisfied, then DoB accompanied by QoS improvement is more destructive than powerful DoB attacks accompanied by QoS degradation.