# A Mobility-Aware Adaptive Duty Cycling Mechanism for Tracking Objects during Tunnel Excavation
## 4 Analysis and Evaluation
### Expected Latency  
We analyzed the expected latency for a packet transmission between two sensor nodes within the transmission range of one another. Because the latency is governed by the delay in sleep mode, we did not consider the processing, transmission, and propagation delays. We focused on the sleep latency during duty cycling mechanism. We first analyzed the latency of the periodic duty cycle mechanism with a fixed interval. Then, we evaluated our adaptive duty cycle mechanism by analyzing its ability to adapt to its mobility variations and compare it with periodic one.  
Figure 8: Comparison of energy consumption per unit time.  
Figure 7: Examples of the contour of the energy consumed per unit time for the adaptive duty cycling mechanism.  
For the periodic duty cycling mechanism, if a sensory event arrives at wakeup time [0, \(T\)], the sensor sends the data to the cluster head node without delay. However, during the sleep time [\(T\), \(T+\tau\)], the data will be sent in the next wakeup time. To be precise, let \(x\) be a random variable between [0, \(T+\tau\)] denoting the data arrival time and let \(f(x)\) be the probability density function of \(x\). As previously noted, \(\tau\) is not adjusted for the periodic duty cycling mechanism. By substituting \(1/(T+\tau)\) for \(f(x)\), we obtain the expected latency of the periodic duty cycling mechanism:  
\[\begin{array}{rcl}L_{Periodic}&=&\int_{T}^{T+\tau}(T+\tau-x)f(x)dx\\ &=&\frac{1}{T+\tau}\int_{T}^{T+\tau}(T+\tau-x)dx\\ &=&\frac{\tau}{2(T+\tau)}\end{array} \tag{7}\]  
For the adaptive duty cycling mechanism, we employed two adjusted sleep times, \(\tau/\alpha\) and \(\beta\tau\), according to change of the RSSI value. As described above, the latency for the adaptive duty cycling mechanism can be simplified to Equation (8):  
\[\begin{array}{rcl}L_{Adaptive}&=&\frac{\left(\frac{\tau}{\alpha}\right)^{2} }{2\left(T+\frac{\tau}{\alpha}\right)}.p+\frac{\left(\beta\tau\right)^{2}}{2 \left(T+\beta\tau\right)}.(1-p)\\ &=&\left(\frac{\tau}{\alpha}-\beta\tau\right).\frac{\left\{T\left(\frac{\tau} {\alpha}+\beta\tau\right)+\frac{\beta}{\alpha}\tau^{2}\right\}}{2\left(T+\frac {\tau}{\alpha}\right)\left(T+\beta\tau\right)}.p+\frac{\left(\beta\tau\right)^{2 }}{2\left(T+\beta\tau\right)}\end{array} \tag{8}\]  
Figure 9 shows the comparison between \(L_{Periodic}\) and \(L_{Adaptive}\) when \(D=1\%\) and \(T=50\) ms with various \(\alpha\) and \(\beta\). When \(\alpha\) and \(\beta\) are given, \(L_{Adaptive}\) is a linear function of \(p\) and decreases with increasing \(p\) owing to the negative coefficient of \(p\) as shown in Equation (8). In other words, \(L_{Adaptive}\) decreases with increasing movement. This is natural because the value of \(p\) should increase as the mobility of the sensor node increases, and this reduces the sleep time.  
As shown in Figure 9, by using various values of the sleep time adjusting factors, \(\alpha\) and \(\beta\), an engineer can manipulate the latency that has an advantage over the periodic mechanism. For example, when \(p\) is greater than 67%, \(L_{Adaptive}\) with \(\alpha=2\) and \(\beta=2\) has an advantage over \(L_{Periodic}\).  
Figure 9: Comparison of \(L_{Periodic}\) and \(L_{Adaptive}\) when \(D=1\%\) and \(T=50\) ms.