# A Mobility-Aware Adaptive Duty Cycling Mechanism for Tracking Objects during Tunnel Excavation
## 4 Analysis and Evaluation
### Energy Consumption  
We analyzed the energy consumption per unit time of both periodic and adaptive duty cycling mechanisms. The energy consumption per unit time is proportional to the ratio of wake-up time to duty cycle (\(D\)). For the simplest model, we assumed that the energy consumed for idle listening is approximately the same as the energy consumed for receiving and transmission in WSNs. Let \(P_{W}\) denote the power for idle listening, receiving, and transmission. In the periodic duty cycling mechanism, the consumed energy, \(\varepsilon_{Periodic}\), can be computed as:  
\[\varepsilon_{Periodic}\ =P_{W}\cdot D\ =\ P_{W}\cdot\frac{T}{T+\tau} \tag{4}\]  
where \(D=T/(T+\tau)\). Note that in the periodic duty cycling mechanism, a mobile sensor does not adjust its duty cycle, so the sleep time is not adjusted.  
With the adaptive duty cycling mechanism, each mobile sensor can adjust its duty cycle by decreasing or increasing the sleep time, depending on whether its change of RSSI value (\(\Delta RSSI\)) exceeds a certain threshold \(\lambda\) or not. For example, when \(\alpha=2\) and \(\beta=2\) are employed, the duty cycle is adjusted by doubling or halving the sleep time. When the mobile sensors exceed the threshold with probability, \(p\), the energy consumed per unit time for the adaptive duty cycling mechanism, \(\varepsilon_{Adaptive}\) can be derived as:  
\[\begin{array}{rcl}\varepsilon_{Adaptive}&=&P_{W}\cdot\frac{T}{T+\frac{ \tau}{\alpha}}\cdot p+P_{W}\cdot\frac{T}{T+\beta\tau}\ (1-p)\\ &=&P_{W}\cdot\frac{D}{D+\frac{1-D}{\alpha}}\cdot p+P_{W}\cdot\frac{D}{D+\beta(1 -D)}\ (1-p)\end{array} \tag{5}\]  
Figure 7 shows a few examples of the contour of the consume energy per unit time for the adaptive duty cycling mechanism with various values of \(\alpha\) and \(\beta\) when \(P_{W}=58\) mJ/s. The energy consumption parameter is selected based on the CC2420 RF module [29]. As shown in the figure, the energy consumed increases with \(p\). This is natural because the value of \(p\) should increase as the mobility of the sensor node increases, and this causes decrease of the sleep time. Accordingly, the energy consumption increases. This means that we cannot guarantee that the mobility-aware adaptive duty cycling mechanism always consumes less energy than the periodic one does.  
The mobility-aware adaptive duty cycling mechanism consumes less energy than the periodic one if and only if the following inequality \(\varepsilon_{Periodic}\ \geq\ \varepsilon_{Adaptive}\) is satisfied. This inequality can be expressed in terms of \(p\) and \(D\) as below, using Equations (4) and (5):  
\[p\ \leq\ \frac{(\alpha D+1-D)(\beta-1)}{\alpha\beta-1} \tag{6}\]  
Figure 8 shows the comparison of energy consumption per unit time with the given values of \(\alpha\) and \(\beta\). For example, when \(\alpha=2\) and \(\beta=2\) are employed and \(D\) is set smaller than 10% to save energy, approximately 36.7% probability is the upper limit that the mobility-aware duty cycle mechanism consumes less energy than the periodic one.