# A Novel Cooperative Opportunistic Routing Scheme for Underwater Sensor Networks
## 4 Energy-Reliability Analysis
If there is no limitation on the number of retransmissions to deliver a packet, the average transmissions number, \(\overline{X}\), is equal to the sum of a geometric series expressed as follows [45]:  
\[\overline{X}=\sum_{n=1}^{\infty}n.P_{s}.(1-P_{s})^{(n-1)}=\frac{1}{P_{s}} \tag{18}\]  
where \(P_{s}\) is the successful transmission probability of the data packet and \(n\) is the number of transmissions. At each hop, the expected number of transmissions is \(\overline{X}\); if the transmitter-receiver distance \(d\) is selected at each hop, the expected number of transmissions from the source to the sink node, \(\overline{TX}\), is given by:  
\[\overline{TX}=\overline{X}\frac{d_{src-sink}}{d} \tag{19}\]  
where \(d_{src-sink}\) is the distance between the source and sink. Figure 7 shows the relationship between the number of retransmissions and the delivery probability obtained from \[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]0 In this model, the increased number of retransmissions by removing \((r-j)\) nodes from the forwarding set \(F\) can be estimated as follows:  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]1  
where \(P_{r}\) and \(P_{j}\) are the packet delivery probability with \(r\) nodes and \(j\) nodes in the forwarding set, respectively. Based on the derived relations between the Figures 6 and 7, the relationship between  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]2 The relationship between the number of forwarding nodes and the delivery probability. the number of forwarding nodes and the number of retransmissions is shown in \[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]3 In an opportunistic data forwarding equipped with a retransmission mechanism, the total amount of energy consumed at cluster \(\Phi\) with \(l\) nodes to transmit one packet, \(E^{ret}\), is expressed as follows:  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]4  
The relation between the energy consumption and the number of forwarding nodes is shown in Figure 9 by assuming \(E_{tx}=2\) units, \(E_{rx}=0.75\) units. It is obvious that the number of forwarding nodes has a vital impact on the energy cost of the set. Lack of sufficient forwarding nodes may increase the  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]5 The relationship between the number of forwarding nodes and the number of retransmissions. \[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]6 The relationship between the delivery probability and the number of retransmissions. number of retransmissions and, subsequently, the energy cost. On the other hand, an excessive number of forwarding nodes can also waste energy. In this example, the energy cost reaches its minimum level, at \(l=4\), where the corresponding ordered node set is \(\langle n_{4},n_{7},n_{5},n_{8}\rangle\), which among them, \(n_{4}\) and \(n_{8}\) have the highest and lowest relay priority, respectively. Assuming that by removing \(\
u_{avg}\) nodes from each cluster, the number of retransmissions is increased by \(\theta_{avg}\), then the amount of energy loss at each hop, on average, can be estimated as follows:  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]7  
where \(r_{avg}\) indicates the maximum number of nodes, on average, in the initial forwarding set. Note that \(\theta_{avg}\) is calculated using Equation (20). If a retransmission procedure is adopted, the total energy savings in Equation (16) turns into the following equation:  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]8  
In this model, the energy savings is confined by the energy loss of the retransmissions. By removing too many nodes from each forwarding set, the packet failure probability increases, and subsequently, more energy will be required to maintain the reliability. On the other hand, if the majority of nodes are kept in the forwarding set, the reliability is enhanced, but a significant amount of energy is wasted because of involving many receiving nodes. Therefore, the number of candidate nodes within a forwarding set should be selected carefully with respect to the energy and reliability constraints. The OVAR protocol is able to satisfy these constraints by using the heuristic approach proposed in Section 3.5.4.