## 9 Performance Evaluation
### Energy Cost  
In our work, we will assume a simple energy dissipation model [27, 42, 48, 49]. The radio dissipates \(\xi\) nJ/bit for both transmission and reception by the sensors circuitry. Moreover, it consumes \(\varepsilon\) nJ/bit/m\({}^{2}\) for  
Figure 23: The accumulated delay is a function of the hop count (hc) and the size of \(\omega\). Figure 22: Average accumulated delay per one node. ECAT shows the good performance of a minimum average delay since it sends the message only one subinterval after the message arrival. the transmitter amplifier to achieve an acceptable signal to noise ratio. Therefore, to transmit \(k\) bits for \(r\) distance, the total transmission energy dissipation will be:  
\[\mathrm{E_{t}}=\mathrm{k}\;\times\;\mathrm{\varepsilon}+\mathrm{k}\times\;r^{2} \times\mathrm{\varepsilon} \tag{48}\]  
In addition, the receiver would consume for reception of k-bit message:  
\[\mathrm{E_{r}}=\mathrm{k}\times\;\mathrm{\varepsilon} \tag{49}\]  
SGAT assumes that every node would send one message at the end of each interval where the message could be either real or fake. If we have \(N\) nodes in the network, then we expect \(N\) messages during each interval. The energy spent for transmission or reception is similar per one message because we have unified-size messages to prevent size correlation attacks by the adversary. If we have \(p\) percent of the nodes issue or forward real data at each interval, then _1 - p_ percent of the energy and the bandwidth is wasted on fake messages. We can adjust the amount of energy consumed by increasing the interval time (\(\mathrm{\omega}\)). However, increasing (\(\mathrm{\omega}\)), would increase the delay. The consumption of transmitting fake messages is a double fold since the transmitter will consume \(\mathrm{E_{t}}\) for every message and all the neighbors (_N\({}_{i}\)_) will consume (_N\({}_{i}\)_\(\times\)_E\({}_{r}\)_). When the transmission range increases, _N\({}_{i}\)_ increases. The total energy consumed in the network to send real and fake messages in one interval [48]:  
\[\mathrm{E_{R}}=(\mathrm{N})(\mathrm{k}\;\times\;\mathrm{\varepsilon}+\mathrm{ k}\;\times\;r^{2}\;\times\;\mathrm{\varepsilon})+(\mathrm{N}\;\times\;N_{i})( \mathrm{k}\;\times\;\mathrm{\varepsilon}) \tag{50}\]  
ECAT has improved the energy dissipation by reducing the amount of fake messages transmitted while maintaining the required temporal security. The number of total messages transmitted per interval has reduced from 100% to a certain percentage (_p_). We have simulated the WSN in Figure 15 as presented in Section 9.1 using SCAT to calculate the energy dissipation. Figure 24 exhibits the total energy dissipation per one message considering the transmitter, the recipients and the range of transmission. The size of the messages is 8000 bits, \(\mathrm{\varepsilon}\) is 50 nJ/bit and \(\mathrm{\varepsilon}\) is 100 pJ/bit/m\({}^{2}\). The simulation shows that the energy dissipation due to the increase of sensor range is marginal compared to the increase in energy dissipation due to the increase of neighbors (_N\({}_{i}\)_). However, increasing the range could increase N\({}_{i}\) if the WSN has uniform sensor distribution. We have also simulated the network to see how the transmission of fake messages has improved using ECAT. Figure 25 exhibits the simulation of 40 subintervals (\(\mathrm{\omega}\)). The graph shows the maximum possible fake message at each subinterval (\(\mathrm{\omega}\)). For instance, the total fake messages during \(\mathrm{\omega}=10\) is 16 messages while during \(\mathrm{\omega}=32\) is 20 messages. The mean of fake transmissions is 19.5 (compared to 48 messages in SGAT). The average fake messages for the completely simulated period is 19.5 messages which shows about 59% reduction of possible fake messages compared to SGAT. The number of fake message will be reduced further as the network gets busy transmitting real messages since a sensor node do not send a fake message at a subinterval where it has a real message to convey.