# 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  
This section presents an energy model of the proposed convergent networks. Figure 3 shows a subtree of six nodes, where parent node 3 has five child nodes. All five child nodes must be allocated in different time slots since they cannot transmit to the same parent node in the same time slot. During the five time slots, node 3 receives data from its all child nodes, and consumes reception energy for each ingress (child) node. The sum of reception energy \(E_{A}^{ingress}\) of all ingress nodes for a parent node p is expressed by Equation (1). \[E_{A}^{ingress}=\sum_{i=1}^{n}E_{rx}^{i}, \tag{1}\]  
Here, \(n\) denotes the number of child nodes, which is five, as shown in Figure 3. We assume, that the energy consumed by sensing and data processing is negligible [29] in order to focus on the problem of minimizing the data forwarding energy--a primary cause of energy consumption. In Equation (1), the reception energy \(E_{rx}^{i}\) for the received data from node \(i\) to node \(p\) is given by Equation (2) [30]. \[E_{rx}^{i}=l_{i}\cdot P_{elec} \tag{2}\]  
Here, \(P_{elec}\) is the power consumed by the transceiver and radio circuit including the channel coding and modulation circuits. \(l_{i}\) indicates the data length in seconds. The total energy consumed by each node during one active period of sensing and data forwarding is estimated by Equation (3):  
\[E_{A}^{Total}=\sum P_{rx}T_{rx}+\sum P_{p}T_{p}+P_{tx}T_{tx} \tag{3}\]  
Here, \(E_{A}^{Total}\) is the total energy consumed by a node during its active period; \(T_{rx}\), \(T_{p}\), and \(T_{tx}\) denotes the time spent on receiving, processing and transmitting respectively (sum of these time segments equal to active period), and \(P_{rx}\), \(P_{p}\), and \(P_{tx}\) indicate the amount of power consumed by receiving, processing and transmission operations, respectively. Equation (3) can also be expressed by Equation (4) assuming \(E_{p}^{Total}\) is the energy expense during the active period of the node. \[E_{A}^{Total}=\sum_{i=1}^{n}E_{rx}^{i}+\sum_{i=1}^{n}E_{p}^{i}+E_{tx} \tag{4}\]  
Here, \(E_{tx}\), \(E_{rx}\), and \(E_{p}\) denote transmission, receiving and processing energy, respectively, while \(n\) indicates the number of child nodes of the current node. In multi-hop sensor networks, the transmission power is often constrained. If the transmission power is increased beyond that constraint, it may cause interference with other nodes. The transmission distance and packet length are the main arguments for the transmission energy function. These two parameters are proportional to the energy consumption. Let's suppose that initially node-A's transmission power is set approximately to 5 dBm and it is aimed to cover a 150 m range with 95% packet delivery ratio (PDR). Node-A may create a 200 m round interference zone for other neighbor nodes that use the same slot. Hence, these nodes should not execute transmission when node-A sends its data to the parent. However, nodes within 250 m are allowed to transmit data using the same slot since their transmission is not interfered with by node-A. If node-A increases its transmission power to 10 dBm, then its interference zone obviously enlarges, and it interferes with the nodes within a 250 m range. On the other hand, now node-A consumes twice as much energy for each transmission and its battery suffers from intense drainage of energy. Thus, in strictly-scheduled TDMA MAC protocol-based WSNs, we cannot merely increase the power of transmission due to the above constraints; we can, however, reduce the transmission power when the failure recovery process replaces a failed node by a backup node in a way that the transmission distance is reduced [31]. Thus, we assume that every node is assigned a constrained transmission power, which leads to the same maximum transmission energy consumption \(E_{tx}\) for every node for its allocated time slot.