## 5 Simulation Analysis
### Network Metrics  
Network Lifetime: The network lifetime is defined as the time elapsed until the first sensor node exhausts its energy. It is measured from the start of the simulation to the round in which the first node failed. This definition is commonly used in energy-constrained underwater networks and has been referenced in prior work [18]. The metric is computed as follows:  
\[L_{Agg}=\sum_{u=1}^{K_{u}}\frac{ST_{u}-FT_{u}}{K_{u}} \tag{12}\]  
\(ST_{u}\) represents the start time of the \(u\) simulation run, \(FT_{u}\) represents the time when the first node depletes its energy in that run, and \(K_{u}\) is the total number of simulation runs.  
Energy Consumption [30]: The total energy consumption of the network during the simulation is defined as the difference between the sum of the initial energy of all relay nodes and the sum of their residual energy at the end of the simulation. It represents the overall amount of energy expended by all nodes to support the network operation. The formula is:  
\[E_{cont}=\frac{1}{K}\sum_{u=1}^{K}\sum_{i=1}^{N}(E_{node\_initial}-E_{node\_ re}) \tag{13}\]  
In the above equation, \(E_{node\_initial}\) represents the initial energy of node \(i\), and \(E_{node\_re}\) is the remaining energy of node \(i\) at the end of the simulation. \(K\) denotes the total number of simulation runs.  
Average End-to-End Delay: This metric measures the time required for a data packet to travel from the source node to the surface receiver, including transmission and queuing  
\begin{table}
\begin{tabular}{c c} \hline \hline
**System Parameter** & **Value** \\ \hline Data Rate & 10 kbps \\ Frequency & 25 kHz \\ Number of Nodes & 100–500 \\ Communication Radius & Up to 500 m \\ Deployment Topology & Layered random deployment \\ Initial Energy (Relay Nodes) & 150 J \\ Data Payload & 50 Bytes \\ Packet Pending Interval & 40 s \\ Transmission Power & 1–10 W \\ Energy Model & Rx: 0.1 W, Idle: 1 mW \\ Mobility Model & ns3::RandomWalk2 dMobilityModel \\ \hline \hline \end{tabular}
\end{table}
Table 5: Simulation parameters.  
delays at intermediate relay nodes. This also applies to scenarios in which data flow from the receiver back to the source node. The average end-to-end delay is calculated as  
\[Delay_{Avg}=\frac{1}{P_{r}}\sum_{u=1}^{K_{u}}\sum_{m=1}^{P_{r}}\left(TP_{unt}-RP_ {unt}\right) \tag{14}\]  
\(P_{r}\) represents the total number of data packets successfully received by the surface receiver; \(RP_{unt}\) denotes the sending time of the \(u\) data packet in the \(m\) simulation run. \(TP_{unt}\) denotes the receiving time of the \(u\) data packet in the \(m\) simulation run. \(K_{u}\) is the total number of simulation runs.  
The Packet Delivery Ratio (PDR) measures the efficiency of data delivery in the network and is defined as the ratio of successfully received packets at the surface receiver to the total number of packets generated by the source nodes. The formula for calculating the PDR is as follows:  
\[PDR=\frac{1}{K}\sum_{u=1}^{K}\frac{P_{r}^{(u)}}{P_{l}^{(u)}} \tag{15}\]  
\(P_{r}^{(u)}\) is the number of packets successfully received by the surface receiver in the \(u\) simulation run. \(P_{l}^{(u)}\) represents the total number of data packets generated by the source node during the \(u\) simulation run. \(K\) denotes the total number of simulation runs. This metric effectively evaluates the robustness and reliability of the network when faced with real-world communication challenges, such as multi-hop relaying, noise interference, and packet loss.