# Energy Efficient Multipath Routing Algorithm for Wireless Multimedia Sensor Network
## 3 System Model
### Problem Overview  
Sensor nodes in WMSNs are battery powered and deployed in an area where charging or replacing the battery is impossible. This situation poses a critical problem in the life of the network [12] and requires a mechanism to minimize energy usage. Therefore, given a set of \(n\) battery-poweredmultimedia sensor nodes and a base station (BS), as shown in Figure 1, the problem is to maximize the multimedia  
Figure 2: Energy model. wireless sensor network lifetimewhile at the same time minimizing energy usage between the cluster head and base station. This problem can be formulated as follows:  
\[max\ T_{ntk},\ min\sum_{i,j\in V\,,\,(i,j)\in e_{ij}}e_{ij} \tag{3}\]  
where, \(T_{ntk}\) is the network lifetime and \(e_{ij}\) is the energy consumed by the sensor node in a cluster. **Theorem 1**. : _The life span of a WMSN mainly depends on the energy of each node in the network. Furthermore, considering the multipath routing protocol, the network lifetime will be directly related to the residual energy and the number of hops along the multiple paths between the cluster head and the sink node._  
Proof. : Residual energy of a node, energy consumption, and several hops along a particular path, have a direct association with the lifetime of a network. That is to say, when energy consumption and residual energy of the nodes decreases, the network lifetime proportionally increases. Let us assume that we have \(h\) hops in a path, and each hop is supposed to have \(E_{CPU},E_{rec},\ \text{and}\ E_{tx}\) as energy expenditure by the CPU during computational processing, a packet reception from the predecessor hop, and a packet transmission to the next hop, respectively. The energy consumed in forwarding a packet along a particular path from a CH to the sink can be determined based on the radio energy modeling Equations (1) and (2), which then gives the following Equation (4):  
\[E_{path}=\ \sum_{i=1}^{h}(E_{CPU}+E_{rec}+E_{tx}) \tag{4}\]  
Suppose all the hops in a particular path have similar energy expenditure and we prove our hypothesis about the association between the lifetime of a network and the remaining energy of a hop, the energy dissipated, and several hops along the path. Similarly, we have:  
\[E_{path}=h\star(E_{CPU}+E_{rec}+E_{tx}) \tag{5}\]  
However, the energy consumed by the processor is much less than the energy consumed by the transceiver for receiving and transmitting a K-bit data packet as shown in Equation (6):  
\[E_{CPU}\ll E_{rec}+E_{tx} \tag{6}\]  
Therefore, we have Equation (7) from Equations (5) and (6):  
\[E_{path}=h\star(E_{rec}+E_{tx}) \tag{7}\]  
We can deduce the following from Equation (7):  
\[E_{rec}+E_{tx}=\ \frac{E_{path}}{h} \tag{8}\]  
A particular path between a CH and the sink serves only until the energy level of a particular hop along the path has the least required energy. Otherwise, the failure of one or more nodes along the path will result in breaking up the path. The minimum energy of a node in the path can be used to partially determine the minimum working time of the path. Similarly, the minimum working time for a hop in a path \(p\), \(T_{p}\) can be determined as shown in Equation (9):  
\[T_{p}=\ \frac{E_{min}(p)}{E_{rec}+E_{tx}} \tag{9}\]From Equations (8) and (9):  
\[T_{ntk}=\ \frac{E_{min}(p)}{E_{path}}*h \tag{10}\]  
In most literature, the lifetime of a particular WMSN is defined based on \"first node die\" (FND). Hence, the minimum network lifetime will also be associated with the minimum hop energy, as shown in Equation (11). \[T_{ntk}=\underset{i\in h}{min}\ T_{p} \tag{11}\]  
\[max\ T_{ntk},\ min\sum_{i,j\in V\,,\,(i,j)\in e_{ij}}e_{ij} \tag{3}\]0  
Thus, \(T_{ntk}\) is directly associated with the energy consumption in a path, the residual energy of the minimum energy node, and the hop distance to the base station. When combining Equations (3) and (12), maximizing the minimum time utilized in a path will maximize the overall network lifetime as shown below in Equation (13).