## 2 Related Works  
In UWSNs, queueing theory is widely used to analyze network performance and optimize network topology [13; 14; 15]. In [13], a data acquisition method was proposed based on the queueing model and the genetic algorithm for underwater acoustic collaborative sensor networks. This method modeled the process of each SN sending packets to the CH as a single service station hybrid M/M/1/K queueing model. The results demonstrated that this method could effectively reduce packet loss rate. In [14], a latency\(-\)sensitive underwater wireless optical network was analyzed. An M/G/1 queueing model was constructed to quantify the system performance, and the end\(-\)to\(-\)end latency and blocking rate were evaluated in different scenarios through the numerical results. In [15], a number of charging strategies were suggested for three\(-\)dimensional charging UWSNs. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbols** & **Means** \\ \hline \(K\) & Cache capacity \\ \(f\) & Access probability \\ \(N\) & Amount of packets in the system \\ \(f_{min}\) & Minimum value of \(f\) \\ \(\alpha\) & Adjustment factor for the \(f\) \\ \(H\) & Wakeup threshold \\ \(t\) & Index of time slots \\ \(p_{1}\) & Emergency packets’ arrival rate at the CH \\ \(p_{2}\) & Non\(-\)emergency packets’ arrival rate at the CH \\ \(s_{1}\) & Emergency packets’ service rate at the CH \\ \(s_{2}\) & Non\(-\)emergency packets’ service rate at the CH \\ \(R_{t}\) & Amount of total packets at time \(t^{+}\) \\ \(S_{t}\) & Amount of emergency packets at time \(t^{+}\) \\ \(T_{t}\) & CH’s state at time \(t^{+}\) \\ \(i,j,k\) & Three indexes of the system state \\ \(\Omega\) & State space \\ \(\mathbf{W}\) & One\(-\)step transition probability matrix \\ \(m\) & Amount of total packets before one\(-\)step transition \\ \(n\) & Amount of total packets after one\(-\)step transition \\ \(\mathbf{W}_{m,n}\) & Block matrix where the total number of packets changes from \(m\) to \(n\) in \\ \(x,y,z\) & Indexes of the steady\(-\)state probability distribution \\ \(\pi_{x,y,z}\) & Steady\(-\)state probability distribution \\ \(\Pi\) & Steady\(-\)state vector \\ \(\boldsymbol{e},\boldsymbol{e}^{\prime}\) & Two intermediate vectors for solving \(\pi_{x,y,z}\) \\ \(\boldsymbol{Q},\boldsymbol{Q}^{\prime}\) & Two intermediate matrixes for solving \(\pi_{x,y,z}\) \\ \(\hat{\rho}_{1}\) & Emergency packets’ blocking rate \\ \(\theta_{1}\) & Emergency packets’ throughput \\ \(\beta_{2}\) & Non\(-\)emergency packets’ blocking rate \\ \(\phi_{2}\) & Non\(-\)emergency packets’ outage and loss rate \\ \(\theta_{2}\) & Non\(-\)emergency packets’ throughput \\ \(\eta_{2}\) & Non\(-\)emergency packets’ average latency \\ \(\epsilon\) & State switching rate \\ \(E\) & Energy consumption \\ \(E_{1},\,E_{2},\,E_{3},\,E_{4}\) & Energy consumption in different states \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and corresponding meanings. An M/G/1 queueing model was used to simulate the energy mule charging process. The proposed scheme was verified through simulation experiments to be energy --saving and time --saving and to guarantee the efficient use of resources. The above literature adopted the continuous -- time queueing theory for modeling. However, we observe that the digital features of contemporary network communication are better suited to the discrete -- time queueing theory [16]. Sleep/wakeup mode is an effective energy --saving strategy to improve network lifetime. Due to the limited energy of underwater SNs, the network's lifespan can be extended significantly by reasonably arranging the wakeup and sleep states of nodes. In [17], an adaptive sleep/wakeup scheduling method was proposed. The timeline was divided into multiple time slots, and each node was allowed to autonomously decide whether to sleep, listen, or send in one -- time slot. In [18], a three -- dimensional space -- moving target tracking method based on the Kalman filter was proposed, and the sleep/wakeup strategy was used to save the system's energy consumption.