## 3 Throughput Performance Limits of Multimode Underwater Acoustic Networks
### The Method of Solving the Model
#### 3.2.4 Algorithms for Performance Analysis  
The throughput performance analysis algorithms of multiregime networks are shown as follows, including Algorithms 1 and 2. Algorithm 1 is used to calculate the optimal action value function, and Algorithm 2 is used to transform the results of Algorithm 1 into throughput performance limits and optimal scheduling schemes for multimode networks. ```
Input: Network node sound source level (SL), propagation loss (TL), delay matrix (\(D\)), background noise level (NL), communication rate (\(v_{i}\)) and decodable SNR threshold (\(SNR_{hit}\)) of multiple communication systems, time slot length \(\tau\) Output: Optimal policy \(\pi^{*}\) Procedure: Set the value function \(V_{\pi}(\hat{s})\) of the initial system state \(\hat{s}\) to 0, and the current policy \(\pi\) to evenly distribute all actions, While true 1. According to the current state \(\hat{s}\), combined with the parameters of the sonar equation, SL, TL, etc., the SINR of each network node at different times of action \(a\) is calculated 1.1 Combined with the reward function model corresponding to Formula (11), the profit value of each action \(a\) under the current state \(\hat{s}\) is analyzed successively. 1.2 According to the strategy evaluation Formula (13), update the value function \(V_{\pi}(\hat{s})\) of the current state \(\hat{s}\). 2. Repeat step 1 to evaluate the value function of the next state \(\hat{s}^{\prime}\) until the value function evaluation of all states is completed. 3. According to the policy promotion Formula (15), the new optimal strategy \(\pi^{\prime}\) for all states is analyzed in turn. 4. If \(\pi^{\prime}=\pi\) is true, then 4.1 The optimal policy \(\pi^{*}=\pi^{\prime}\) is obtained and output. 4.2 Break. End if End while
```  
**Algorithm 2** Throughput and scheduling result analysis algorithm  
```
Input: Optimal policy \(\pi^{*}\), temporary scheduling matrix \(C\), status number \(u\) Output: Multimode network throughput performance limit \(M^{*}\), optimal scheduling scheme \(X^{*}\) Procedure:
1.\(u\gets 0\)
2. flag \(\leftarrow\) true
3.\(\bar{s}\gets s(u),C\leftarrow\bar{s}\)
4. While flag \(4.1\ a_{now}\leftarrow\pi^{*}(\bar{s})\)
4.\(2\ \bar{s}\gets a_{now}\)
4.\(3\ C\leftarrow[C\ \bar{s}]\)
4.\(4\) if the network state saved in C has a loop, then \(X(u)\) is the part of the loop in \(C\);
```  
**Algorithm 3** Multi-rate```
Calculate the throughput of \(X(u)\) according to the formula (12), which is \(M(u)\). flag \(\leftarrow\) false. End if End while 5.\(u\gets u+1\), return 2 until all states are computed. 6.\(n\leftarrow\) argmax \(M\) 7.\(M^{*}\gets M(n),X^{*}\gets X(n)\)
```

**Algorithm 2**_Cont._

## 4 Performance Evaluation

We evaluate and compare the performance of the proposed methods in multimode networks via simulations. We compare the throughput of multimode networks under different network topologies and different channel qualities. In the simulation process, the network nodes all support three communication modes, namely, PSK, MFSK, and PRSS, with a bandwidth of 3.2 k. Other parameters are shown in Table 2.

### Performance Analysis of the Three-Node Single-Mode Network

Under the equilateral triangle network topology as shown in Figure 0(a), the proposed method is used to analyze the throughput performance of the single-mode network, and the results are compared with those of traditional methods to verify the correctness of the proposed method. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Parameter** \\ \hline Depth of water & 50 m \\ Working depth of nodes & 10 m \\ PSK rate & 1.8 kbps \\ MFSK rate & 0.8 kbps \\ PRSS rate & 0.2 kbps \\ SL & 183 dB \\ NL & 101.4 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The simulation parameter table. Figure 1: Two topology examples: (**a**) a 3-node equilateral triangle network; (**b**) a 3-node isosceles triangle network. The results obtained after strategy iteration are shown in the figure below. The diagram illustrates up to three action choices for each state.