# Analytical Model for the Duty Cycle in Solar-Based EH-WSN for Environmental Monitoring
## 4 Analytical Energy Consumption Model
### Energy Consumption per Round
Having adopted the first alternative, we introduce the following auxiliary functions in order to simplify the analysis:  
\[E_{fd}^{(1)}(t^{\prime})=E_{x}^{pkt}\frac{T_{pkt}-t^{\prime}}{T_{ pkt}}+E_{l}\frac{W_{ack}+T_{CCA}}{T_{l}},t^{\prime}\in[0,T_{pkt}) \tag{17}\] \[I_{1}(x)=\int_{0}^{x}E_{fd}^{(1)}(t^{\prime})\,dt^{\prime},x\in[ 0,T_{pkt})\] (18) \[E_{fd}^{(2)}(t^{\prime})=E_{l}\frac{T_{c}-t^{\prime}}{T_{l}},t^{ \prime}\in[T_{pkt},T_{c}]\] (19) \[I_{2}(x)=\int_{T_{pkt}}^{x}E_{fd}^{(2)}(t^{\prime})\,dt^{\prime},x \in[T_{pkt},T_{c}] \tag{20}\]  
Accordingly, we have:  
\[\mathcal{E}[E_{fd}] = \int_{0}^{T_{l}+T_{slp}}E_{fd}(t)\frac{1}{T_{l}+T_{slp}}\,dt \tag{21}\] \[= \frac{1}{T_{l}+T_{slp}}\left[\int_{T_{CCA}}^{T_{CCA}+T_{c}}E_{fd} (t)\,dt+\ldots+\int_{T_{CCA}+(\alpha-1)T_{c}}^{T_{CCA}+\alpha T_{c}}E_{fd}(t) \,dt\right.\] \[+\int_{T_{CCA}+\alpha T_{c}}^{T_{CCA}+T_{slp}}E_{fd}(t)\,dt+\int _{0}^{T_{CCA}}E_{fd}(t)+\int_{T_{CCA}+T_{slp}}^{T_{l}+T_{slp}}E_{fd}(t)\,dt\right]\] \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]0 \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]1  
\[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]2 General scenario for the evaluation of the energy consumption per round. Note that the integral has been initially decomposed into the periodic component, which covers the region \([T_{CCA},T_{CCA}+T_{c})\cup\ldots\cup[T_{CCA}+(\alpha-1)T_{c},T_{CCA}+\alpha T_{c})\), and the rest of subdomains, namely \([0,T_{CCA})\), \([T_{CCA}+\alpha T_{c},T_{CCA}+T_{slp})\) and \([T_{CCA}+T_{slp},T_{l}+T_{slp}]\). Further development of \(\mathcal{E}[E_{fd}]\) requires considering the two cases distinguished in Table 2 (for \(k=\alpha+2\)):  
* Case 1: \(T_{slp}\leq\alpha T_{c}+T_{pkt}\) \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]3 (22)
* Case 2: \(T_{slp}>\alpha T_{c}+T_{pkt}\) \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]4 (23) \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]5  
On the other hand, the two last integrals in Equation (21) can be combined into a single integral from \(T_{CCA}+T_{slp}\) to \(T_{CCA}+T_{l}+T_{slp}\) and developed as follows:  
\[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]6  
Next, we can introduce expressions Equations (22) or (23) and (24) into the intermediate result given by Equation (21). In summary, we have:  
* Case 1: \(T_{slp}\leq\alpha T_{c}+T_{pkt}\) \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]7 (25)
* Case 2: \(T_{slp}>\alpha T_{c}+T_{pkt}\) \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]8 (26)  
Finally, the analysis can be completed by evaluating the auxiliary integrals:  
\[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]9  
\[\mathcal{E}[k]=\frac{\frac{\delta}{2}(\alpha+3)T_{c}+(\alpha+2)(T_{slp}-\alpha T _{c})+T_{l}}{T_{l}+T_{slp}} \tag{16}\]0  
\[\mathcal{E}[k]=\frac{\frac{\delta}{2}(\alpha+3)T_{c}+(\alpha+2)(T_{slp}-\alpha T _{c})+T_{l}}{T_{l}+T_{slp}} \tag{16}\]1  
\[\mathcal{E}[k]=\frac{\frac{\delta}{2}(\alpha+3)T_{c}+(\alpha+2)(T_{slp}-\alpha T _{c})+T_{l}}{T_{l}+T_{slp}} \tag{16}\]2  
Note that \(\mathcal{E}[E_{fd}]\) does not depend on the specific child node from which node \(X\) receives a packet. Now we can formulate the expectations of the energy costs in transmission and reception by recalling expressions Equations (2) and (5):  
\[\mathcal{E}[k]=\frac{\frac{\delta}{2}(\alpha+3)T_{c}+(\alpha+2)(T_{slp}-\alpha T _{c})+T_{l}}{T_{l}+T_{slp}} \tag{16}\]3  
\[\mathcal{E}[k]=\frac{\frac{\delta}{2}(\alpha+3)T_{c}+(\alpha+2)(T_{slp}-\alpha T _{c})+T_{l}}{T_{l}+T_{slp}} \tag{16}\]4In the last equation, \(d_{b,i}(X)\) stands for the backward distance between node \(X\) and its child node \(c_{i}(X)\).