# An Energy-Efficient Fail Recovery Routing in TDMA MAC Protocol-Based Wireless Sensor Network
## 2 Network Topology and Energy Model
### Energy Model of Convergent Network
Although the receiving and processing energy may increase during the fault recovery (backup node selection) procedure, we assume the processing power is negligible compared with the receive and transmit power [32]. On the other hand, an increase in selected backup parent node's receiving energy is inevitable since its number of child nodes grows due to the failure recovery process. When a sensor node is selected as a backup node that has a larger number of child nodes, it consumes higher energy to receive the additional sensed data from the additional child nodes. For the calculation of a network lifetime, we calculate each sensor node's battery lifetime \(L\). The lifetime \(L\) of a node is defined as the time length from the power-up time until the battery outage time of the node, which is expressed by Equation (5). \[L=\frac{E_{initial}}{E_{consumed}}\times t_{cycle} \tag{5}\]  
Here, \(E_{initial}\) is the initial energy of node \(n_{i}\) at its power-up time. \(t_{cycle}\) is one full cycle time including data sensing, receiving all child data, and transmitting the aggregated data to \(n_{i}\)'s parent node. \(E_{consumed}\) denotes the energy consumed by the node \(n_{i}\) during \(t_{cycle}\). \(E_{consumed}\) is expressed by Equation (6) ignoring the energy consumption for sensing and data processing. \[E_{consumed}=E_{tx}(l,d)+nE_{rx}(l) \tag{6}\]  
In Equation (6), \(n\) denotes the number of child nodes and the transmission power \(E_{tx}(l,d)\) for data length \(l\), and distance \(d\) is defined as follows. Using [PERSON]'s path loss model, we assume that the transmission power of node \(n_{i}\) is selected in a way that the path loss is compensated [33]. According to  
Figure 3: Power consumption of parent node in duty cycle. the radio model used in Reference [34], data transmission usually depends on the distance and packet length, as expressed in Equation (7). \[E_{Tx}(l,d)=\left\{ \begin{array}{ll}l\cdot E_{dec}+l\cdot E_{Fs}d^{2},\;if\;d<d _{0}\\ l\cdot E_{dec}+l\cdot E_{amp}d^{4},\;if\;d>d_{0}\end{array} \right. \tag{7}\]  
Here, \(E_{Fs}\) and \(E_{amp}\) are the amplifier energy consumption for the distances in free space (\(d^{2}\) power loss) and the channel with multi-path fading (\(d^{4}\) power loss) respectively. It is mentioned above that a variable \(l\) is a length of data in Equation (7) (or it denotes a time that is required to send a sensed data). Since we are concerned with finding backup nodes with a short distance, the free space fading (\(E_{Fs}\times d^{2}\)) is a more appropriate model [35], and thus, it is used in the remainder of the paper. Using Equations (6) and (7), Equation (5) can be expressed by Equation (8). \[L=\frac{E_{initial}}{E_{tx}+nE_{rx}}\times t_{cycle}=\frac{E_{initial}\;t_{ cycle}}{lE_{elec}+lE_{Fs}d^{2}+nlE_{elec}}=\frac{E_{initial}\;t_{cycle}}{lE_{Fs}d^{2}+(n+1)lE_{ elec}} \tag{8}\]  
Equation (8) indicates that the distance and the number of child nodes are the main components of energy consumption. Using Equation (8), the proposed algorithm selects a set of optimal backup nodes for all parent nodes in the network as described in the next section.