# A Novel Cooperative Opportunistic Routing Scheme for Underwater Sensor Networks
## 4 Energy-Reliability Analysis  
In this section, we evaluate the optimal trade-off between two objectives: energy consumption and reliability, with regard to the opportunistic data forwarding in OVAR. The analysis carried out indicates that adjusting the number of nodes involved in the data forwarding has a significant impact on the energy consumption and reliability. Without loss of generality, we aim to analyse an opportunistic data forwarding similar to the scenario presented in Figure 4.  
As can be seen, a packet is relayed with candidate node collaboration in each cluster. It is assumed that initially, the number of nodes in each cluster is \(r\). Let \(E_{tx}\) and \(E_{rx}\) denote the energy consumed by a node to transmit and receive a packet, respectively. The total amount of energy consumed at the forwarding set \(F\) with \(r\) nodes to transmit one packet is given by [44]:  
\[E(F,r)=E_{tx}+r.E_{rx} \tag{13}\]  
Figure 4: Opportunistic routing in opportunistic void avoidance routing (OVAR). By removing \((r-j)\) nodes from the cluster, the total amount of energy consumed by the new forwarding set is calculated as follows:  
\[E(F,j)=E_{tx}+j.E_{rx}+(r-j)E_{re} \tag{14}\]  
where \(E_{re}\) is the energy used to read the header of the packet for early rejection. By subtracting the \(E(F,j)\) from the \(E(F,r)\), the amount of energy savings by removing \((r-j)\) nodes is expressed as:  
\[ES(F,(r-j))=(r-j)(E_{rx}-E_{re}) \tag{15}\]  
Consider a network (e.g., Figure 4) with a single-direction progressive routing protocol when the distance between the source and sink is \(N\) hops. By removing \(\
u_{avg}\) nodes from each cluster at each hop, the total savings of energy, \(ES_{total}\), for \(K\) packets can be expressed as:  
\[ES_{total}=K.N.\
u_{avg}(E_{rx}-E_{re}) \tag{16}\]  
The above equation shows the total energy savings when no retransmission mechanism is adopted in the forwarding nodes. If a retransmission procedure is applied, the amount of energy savings depends on two factors: the amount of energy savings by decreasing the number of receiving nodes and the amount of energy loss due to retransmissions. Note that by decreasing the number of receiving nodes at a cluster, the number of retransmissions may be increased due to the increasing of the packet failure probability [45]. Thus, the transmission errors play an important role in reliability and energy savings. A transmission is successful if at least one node in each cluster can receive the packet without any error. Instead of using an acknowledgement mechanism, which is costly in UWSNs, underwater nodes can easily notice a successful delivery just by listening to the neighbouring nodes' transmissions. For packet transmission suppression, candidate nodes only read the header of the packet. If a packet is relayed with the collaboration of \(l\) nodes in the forwarding set, the successful transmission probability can be calculated as follows [45]:  
\[P_{s}=\sum_{i=1}^{l}P_{i}\prod_{j=1}^{i-1}(1-P_{i}) \tag{17}\]  
Figure 5: Example of opportunistic data forwarding in one hop. where \(P_{i}\) is the packet delivery probability between the forwarding node and receiving node. Note that the forwarding priority of candidate nodes affects the value of \(P_{s}\). To clarify the matter, consider the opportunistic data forwarding in Figure 5, as an example. The forwarding set of \(R_{i}\) includes a number of candidate nodes for which each of them is associated with a pair, \((\beta_{ij},P_{ij})\), where \(\beta_{ij}\) is the packet advancement and \(P_{ij}\) is the packet delivery probability between \(R_{i}\) and \(n_{j}\). If candidate nodes are sorted based on the value of \(\beta_{ij}.P_{ij}\) and added to the forwarding set, the successful transmission probability is changed in accordance with the diagram shown in Figure 6, which shows the effect of adding every individual node into the forwarding set on the probability of successful packet delivery.