# Analytical Model for the Duty Cycle in Solar-Based EH-WSN for Environmental Monitoring
## 4 Analytical Energy Consumption Model
### Energy Consumption per Round  
The previous analysis has focused on the generation and transmission as well as the reception and forwarding of single packets. However, during a communication round, a node in a data-gathering tree is generally entailed to forward packets from multiple nodes in addition to transmitting its own packet. The scenario is described in Figure 6, where an arbitrary node \(X\) receives and forwards packets from all of its children. Every child is denoted by \(c_{i}(X)\), with \(i\) varying between \(1\) and the total number of children, namely \(CH(X)\). Let also \(\sigma(X)\) denote the total number of descendants of node \(X\) (what includes its children and, recursively, the descendants of its children). Then, according to the assumption that transmissions become sufficiently randomized, in such a way that every packet managed by a node occupies a distinct duty period (recall Section 4.1), we can set up the following equation for \(E_{round}(X)\), the total energy consumed by node \(X\) during a communication round:  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{ T}(X) \tag{15}\] \[+ \left(\left|\frac{T_{rnd}}{T_{l}+T_{slp}}\right|-(\sigma(X)+1) \right)(E_{l}+E_{sleep})\]  
In this equation, \(E_{R,i}(X)\) denotes the energy wasted by node \(X\) to receive a packet from its child node \(c_{i}(X)\), whereas the last term accounts for the total energy wasted in duty cycles that are not dedicated to transmit or receive (unused duty cycles). Note that the energy wasted in sleep periods has been included: despite power consumption is very low during such periods, the overall energy consumption may be non-negligible if \(T_{slp}\gg T_{l}\) (for low duty cycles). Also note that Equation (15) includes two types of random components: the number of tries \(k\) required by node \(X\) to upload its packets (contained in the component of energy wasted in transmission) and the fraction of duty period previous to successful reception from every child node (contained in every component of energy wasted in reception). Altogether, \(E_{round}(X)\) contains \(CH(X)+1\) random variables, which are mutually independent according to the randomization assumption. In the next subsection we compute the expected value of \(E_{round}(X)\). The stochastic nature of \(E_{round}(X)\) has ultimately its origin in the random asynchrony between the duty periods of the two endpoints of any communication pair. In particular, the randomness of the number of tries (\(k\)) results from the randomness of the asynchrony between a node and its parent node (the node to which it transmits packets), whereas the randomness of \(E_{fd}\) is due to the smaller-scale randomness (characterized by \(t^{\prime}\) in Figure 3) between a node and any node from which it receives packets (child node). Accordingly, the evaluation of the expected energy consumption per round relies on averaging \(k\) and \(E_{fd}\). In particular, from Table 1 we can easily derive the following expression for the average number of tries:  
\[\mathcal{E}[k]=\frac{\frac{\delta}{2}(\alpha+3)T_{c}+(\alpha+2)(T_{slp}-\alpha T _{c})+T_{l}}{T_{l}+T_{slp}} \tag{16}\]  
For the calculation of the expectation of \(E_{fd}\), we can use either its description in terms of the transmitter-receiver asynchrony provided in Table 2, jointly with the fact that this asynchrony is uniformly distributed, or its characterization in terms of the distribution function given in Equations (12) or (13).