# An Energy Balanced and Lifetime Extended Routing Protocol for Underwater Sensor Networks
## 4 Energy Balanced and Lifetime Extended Protocol (EBLE)
### Problem Description  
We analyze the effect of energy balancing and energy efficiency in different data transmission cases firstly. Then we present some optimal principles for designing energy balanced and efficient routing protocol. We suppose the following transmission case as shown in Figure 3: one relay node is deployed between the source node \(S\) and the destination node \(D\). The distances between source and destination, source and relay, relay and destination are \(d_{SD}\), \(d_{SR}\), and \(d_{RD}\), respectively. \(\theta\) is the angle between vector \(\overrightarrow{SR}\) and vector \(\overrightarrow{SD}\). We analyze the energy consumption of one-hop direct transmission and multi-hop transmission via the relay node. **Theorem 1**. : _When the relay node is deployed on the straight line between the source and the destination (\(\theta=0\)), multi-hop transmissions are more energy efficient than one-hop direct transmission if we neglect the receiving power._  
Proof of Theorem 1.: As \(DT\), \(NL\) and \(DI\) are transmission range independent, we rewrite the transmit power as in (11). \[ \begin{array}{ll}P_{tx}&=P_{0}10^{TL/10}\\ &=P_{0}d^{k}a(f)^{d}\end{array}  \tag{11}\]  
where \(P_{0}\) is a distance independent parameter and can be expressed as in (12). \[P_{0}=4\pi\times 0.67\times 10^{-18}\times 10^{(DT+NL-DI)/10} \tag{12}\]  
Suppose that the packet length is \(L\) bits and the data rate is \(Ra\) bits\(/\)s. The energy consumption of transmitting one packet to distance \(d\) can be expressed in (13). \[ \begin{array}{ll}E_{tx}(d)&=(L/Ra)(P_{tx}+P_{rx})\\ &=\frac{LP_{0}d^{k}a(f)^{d}}{Ra}+\frac{LP_{rx}}{Ra}\end{array}  \tag{13}\]  
Figure 3: A simple transmission case in UWSNs. Here \(P_{rx}\) is the receiving power of the receiver. We assume that the relay and the sender can overhear the data packet of each other when they are located inside the transmission range. Then energy consumption for one-hop direct transmission \(E_{direct}\) can be expressed as in (14). \[E_{direct}=\frac{LP_{0}d_{SD}^{k}a(f)^{d_{SD}}}{Ra}+2\frac{LP_{rx}}{Ra} \tag{14}\]  
Similarly, the energy consumption of multi-hop transmission \(E_{multi-hop}\) can be obtained from (15). \[E_{multi-hop}=\frac{LP_{0}}{Ra}(d_{SR}^{k}a(f)^{d_{SR}}+d_{RD}^{k}a(f)^{d_{RD}} )+m\frac{LP_{rx}}{Ra} \tag{15}\]  
Here \(m\) is the number of possible receivers and \(m=2\) for \(d_{SR}>d_{RD}\), \(m=3\) for \(d_{SR}\leq d_{RD}\). As the relay node is deployed right on the straight line between the source and destination. So \(d_{RD}+d_{SR}=d_{SD}\) and we can rewrite (15) as in (16). \[E_{multi-hop}=\frac{LP_{0}}{Ra}d_{SR}^{k}a(f)^{d_{SR}}+\frac{LP_{0}}{Ra}(d_{SD} -d_{SR})^{k}a(f)^{d_{SD}-d_{SR}}+m\frac{LP_{rx}}{Ra} \tag{16}\]  
In order to minimize \(E_{multi-hop}\), we can obtain the derivation of \(E_{multi-hop}\) with respect to \(d_{SR}\) as in (17). \[ \begin{split} E^{\prime}_{multi-hop}=&\frac{LP_{0}} {Ra}\{d_{SR}^{k-1}a(f)^{d_{SR}}(k+d_{SR}\ln a(f))\\ &-k(d_{SD}-d_{SR})^{k-1}a(f)^{d_{SD}-d_{SR}}\\ &-(d_{SD}-d_{SR})^{k}a(f)^{d_{SD}-d_{SR}}\ln a(f)\}\end{split}  \tag{17}\]  
When \(E^{\prime}_{multi-hop}=0\), we can have \(d_{RD}=d_{SR}=d_{SD}/2\) and the minimal value and maximum value of \(E_{multi-hop}\) is shown in (18) and (19). \[\min(E_{multi-hop})=\frac{LP_{0}}{Ra2^{k-1}}(d_{SD}^{k}a(f)^{d_{SD}/2})+m\frac {LP_{rx}}{Ra} \tag{18}\]  
\[\max(E_{multi-hop})=E_{direct}+(m-2)\frac{LP_{rx}}{Ra} \tag{19}\]  
As the receiving power is much lower than the transmitting power [31]. So we can roughly believe \(\max(E_{multi-hop})=E_{direct}\). So it has been proved that the multi-hop transmission mode is more energy efficient in this case. Next we consider a general case in which the relay node is just deployed between the source and the destination (\(\theta\in[-\pi/2,\pi/2]\)). **Theorem 2**.