## 4 Performance Evaluation
### Result Evaluation
#### 4.3.1 Throughputs of the Network \(\rho_{g}\)  
The throughput of a network is the rate at which communicating data packets may send data packets/bits, and it can be described as follows:  
\[\rho_{g}=\frac{T_{L_{g}}}{T_{D_{g}}}=\frac{\sum_{i\in S_{g}}n_{\omega T_{LD_{g}H_{ lm}}}(1-b_{T_{g}})2L_{D,P}}{T_{D_{g}}}. \tag{18}\]  
where \(T_{L_{g}}\) stands for the total number of communicating data packets that the CH node properly receives during a round. This addresses all of the cluster's active communicating data packets. The features of the modem and the properties of the submerged acoustical channel have an impact on the bit error rate \(b_{T_{g}}\). The following equation may be used to obtain \(A_{H_{i}}\), which stands for the average amount of communicating data packets sent by the cluster (I) in a round:  
\[ \begin{array}{l}A_{H_{w}}=\sum_{k=1}^{T_{NW_{w}-1}}kP^{k},\\ \quad=\sum_{k=1}^{T_{NW_{w}-1}}kC_{T_{N_{w}}}^{k}P^{k}(1-P)T_{N_{w}}-kT_{N_{w}},\\ \quad=TN_{wPB}.\end{array}  \tag{19}\]  
where the total number of communicating data packets is represented by \(TN_{w}\) and the probability of generating the communicating data packets is represented by \(WB\) and \(PB\):  
\[PB=1-e^{\lambda R_{DC}}. \tag{20}\]  
The generation data packet rate of each communicating node is represented by \(\lambda\) and the acquisition delay of each round is \(R_{DC}\). This system also comprises the time transmission \(t_{w}^{d}\) for every cluster as well as the time \(P_{w}^{d}\) between the clusters. Total energy consumption takes into account energy spent to send communication data packets, energy received by the communicating nodes during communication, and energy spent while communicating nodes are in the idle state or not constantly transferring communicating data packets. Both the frequencies of transmission and the separation between the acoustical modems have a significant impact on the energy efficiency in the network. To increase energy efficiency, our chosen model modifies the energy efficiency depending on the distance and frequency during the transmission. The received energy is considered to be fixed or constant. The energy efficiency is then explained as follows:  
\[E_{t}=\sum_{i\in t}(E_{w}^{t}+E_{w}^{r})+Idle_{E}. \tag{21}\]  
The unique cluster is represented by the cluster indicator with the letter \(W\). The energy utilized for transmission by the CH communicating node and the regular communicating nodes within that specific cluster are collectively referred to as \(E_{w}^{t}\), or transmitted energy. Thus, the following formula may be used to compute the energy to transmit communicating data packets:  
\[ \begin{array}{l}E_{w}^{r}=\sum_{m_{w\in w}}p_{ww}^{t}T_{ww^{m}}^{t}\\ \quad=E_{SW}^{t}+\sum_{m_{w\in w}m_{w}}\
eq s_{w}\frac{p_{ww}^{t}p_{ww}^{t}T_{ ww^{m}}^{t}}{R},\end{array}  \tag{22}\]  
where \(S_{W}\) is the cluster's node indication and \(S_{W}\) represents the collection of communicating nodes in a cluster. Node \(n_{w}\) in cluster \(w\) has a transmission speed and time of \(p_{ww^{m}}^{t}\) and \(T_{ww^{m}}^{t}\), respectively. In cluster \(w\), the node's packet length is \(L_{ww^{m}}^{t}\), which is made up of the length of the communicating data packet. WAKE communicating data packets, ACK communicating data packets, and the energy efficiency for the transmission of inter-cluster communicating data packets are all included in the required energy for the transmission of communicating data packets \(E_{SW}^{t}\) of the cluster's central node. \(E_{SW}^{t}\) can be defined as  
\[E_{SW}^{t}=E_{\max}(L_{wake+n_{w}}L_{ack})+\frac{E_{\max}N_{www}+T_{data}}{R}. \tag{23}\]As soon as the distance exceeds its limit during the communication \(d_{c}\), the central cluster node's energy for transmission is set to \(E_{\text{max}}\). \(E_{w}^{\prime}\) is the term used to describe all of the energy that cluster w has received, including energy that has been received by both the head of the cluster node and the ordinary communicating nodes.