## 3 System Model
### Model Building  
In UWSNs, random events (such as temperature change, humidity change, object movement, etc.) perceived by SNs occur randomly. The occurrence of these events can be regarded as a random process, and its probability distribution can be used to describe the frequency of occurrence of events. When these events are perceived by the SNs and packets are generated, the arrival of packets is also random. Therefore, we assume that the arrival process of packets is a Bernoulli process. Specifically, the Bernoulli process is a simple stochastic process in which the occurrence or absence of events within each time interval is independent, and the probability of events occurring within each time interval is constant. In this way, we can more accurately analyze and optimize the performance of UWSNs, ensuring that the network can efficiently process and transmit the perceived event information. We divide the time axis into infinite time slots with the same interval, expressed as \(t=1,2,3,\cdots\). We assume that arrival intervals of emergency and non\(-\)emergency packets follow the geometric distribution, and the arrival rates are \(p_{1}(0<p_{1}<1,\bar{p}_{1}=1-p_{1})\) and \(p_{2}(0<p_{2}<1,\bar{p}_{2}=1-p_{2})\), respectively. The forwarding time of the CH for emergency and non\(-\)emergency packets follows geometric distribution, and the service rates are \(s_{1}(0<s_{1}<1,\bar{s}_{1}=1-s_{1})\) and \(s_{2}(0<s_{2}<1,\bar{s}_{2}=1-s_{2})\). Based on the regulation of the Early Arrival System, the packet arrival event and departure occur at the start \((t,t^{+})\) and end \((t^{-},t)\) of each time slot, respectively [27]. Let \(R_{t}=i(i=0,1,2,\ldots,K+1)\), \(S_{t}=j(j=0,1)\) and \(T_{t}=k(k=0,1)\), where \(i\), \(j\), and \(k\) are three state\(-\)variables, \(R_{t}\) represents the amount of total packets in the system at time \(t^{+}\), \(S_{t}\) represents the amount of emergency packets in the system at time \(t^{+}\), and \(T_{t}\) represents the state of the CH at time \(t^{+}\). When \(T_{t}=0\),\[W=\left( \begin{array}{ccccccccc}W_{0,0}&W_{0,1}&W_{0,2}&&&&0\\ W_{1,0}&W_{1,1}&W_{1,2}&W_{1,3}&&\\ &W_{2,1}&W_{2,2}&W_{2,3}&W_{2,4}&&\\ &&\ddots&\ddots&\ddots&\ddots&\\ &&&W_{K-1,K-2}&W_{K-1,K-1}&W_{K-1,K}&W_{K-1,K+1}\\ &0&&&&W_{K,K-1}&W_{K,K}&W_{K,K+1}\\ &&&W_{K+1,K}&W_{K+1,K+1}\end{array} \right)_{(2K+H+3)\times(2K+H+3)} \tag{4}\]  
We will give each block matrix \(W_{m,n}\) in detail, where \(m\) and \(n\) represent the amount of total packets before and after the one\(-\)step transition. 1. When \(m=0\), it indicates that the amount of packets is zero before the one\(-\)step transition, and the transition submatrices are given as follows. \[W_{0,0}=(\hat{p}_{1}\hat{p}_{2})_{1\times 1}\] (5) \[W_{0,1}=\left\{ \begin{array}{ccc}\left(\begin{array}{cc}\hat{p}_{1}p_{2} &p_{1}\hat{p}_{2}\end{array}\right)_{1\times 2}&H=0\\ \\ \left(\begin{array}{cc}\hat{p}_{1}p_{2}&0&p_{1}\hat{p}_{2}\end{array}\right) _{1\times 3}&1\leq H\leq K\end{array} \right.\] (6) \[W_{0,2}=\left\{ \begin{array}{ccc}\left(\begin{array}{cc}0&p_{1}p_{2} \end{array}\right)_{1\times 2}&H=0\\ \\ \left(\begin{array}{cc}0&0&p_{1}p_{2}\end{array}\right)_{1\times 3}&1\leq H \leq K\end{array} \right.\] (7)
2. When \(m=1\) and \(n=0\), due to the special dimension of the submatrix, its form is given as follows. \[W_{1,0}=\left\{ \begin{array}{ccc}\left(\begin{array}{cc}s_{2}\hat{p}_{1}( \hat{p}_{2}+p_{2}\frac{1}{a})&s_{1}\hat{p}_{1}(\hat{p}_{2}+p_{2}\frac{1}{a}) \end{array}\right)_{2\times 1}^{\rm T}&H=0\\ \\ \left(\begin{array}{cc}0&s_{2}\hat{p}_{1}(\hat{p}_{2}+p_{2}\frac{1}{a})&s_{1 }\hat{p}_{1}(\hat{p}_{2}+p_{2}\frac{1}{a})\end{array}\right)_{3\times 1}^{\rm T}&1 \leq H\leq K\end{array} \right.\] (8)
3. When \(1\leq m\leq K\) and \(n=m-1(n\
eq 0)\), \(W_{m,n}\) is expressed as follows. \[W_{m,n}=\left\{