## 3 Throughput Performance Limits of Multimode Underwater Acoustic Networks
### The Method of Solving the Model
#### 3.2.4 Algorithms for Performance Analysis
The optimal scheduling scheme is determined through the analysis of Algorithm 1, as shown in Figure 2. When the initial state sequence number is three, adopting the optimal action (serial number 8) leads to the formation of a new state (serial number 8). Subsequently, the optimal action for this new state is identified, and this process continues until a closed loop exists. The scheduling result in Figure 2 can be represented by Formula (17): a cycle comprises four scheduling time slots with status serial numbers 3, 8, 9, and 19, along with their corresponding specific status values [0;1;0], [0;3;2], [2;0;0], and [3;0;1]. \[s(3)\overset{a(8)}{\rightarrow}s(8)\overset{a(9)}{\rightarrow}s(9)\overset{a( 19)}{\rightarrow}s(19)\overset{a(3)}{\rightarrow}s_{3\times 3}\Rightarrow \left[ \begin{array}{cc}0&0&2&3\\ 1&3&0&0\\ 0&2&0&1\end{array} \right] \tag{18}\]

There are three optimal scheduling schemes obtained through Algorithm 2, respectively. \[\left[ \begin{array}{cc}2&3&0&0\\ 0&0&1&3\\ 0&1&0&2\end{array} \right];\ \left[ \begin{array}{cc}3&2&0&0\\ 0&1&0&3\\ 0&0&1&2\end{array} \right];\ \left[ \begin{array}{cc}0&3&0&2\\ 0&0&3&1\\ 2&1&0&0\end{array} \right] \tag{19}\]

Only one scheme is given in the paper [14], which is the same as the first scheme in this paper, and all three possible schemes are given in this paper, which verifies the validity of the method in this paper. ### Performance Analysis of Multimode Networks with Different SLs in Arbitrary Topologies

In the following, the throughput performance of the multimode network is analyzed using the method proposed in this paper under an irregular topology. The topology of an irregular triangular network is shown in Figure 3.

The maximum time slot length of the irregular triangle network is G = 2. This network includes three transmission links \(l_{1}\), \(l_{2}\), and \(l_{3}\), with corresponding link lengths rounded to \(l_{3}=2\), \(l_{2}=2\), and \(l_{1}=1\) respectively. The sound source level is measured at 186 dB. The optimal scheduling results are illustrated in Figure 4 where different lines represent different nodes and different columns represent various time slots. Figure 2: Result of the policy iteration of the network in Figure 1a. In the scheduling results:

* \"Send_\" indicates that the node is transmitting signals during this time slot; the number corresponding to
* represents the destination node of the transmitted signals. * \"Rcv_\" indicates that the node receives signals during this time slot; the number corresponding to
* represents the source node of the received signals. The corresponding scheduling result consists of eight slots with an upper limit output of 2400 bps per slot. To achieve maximum throughput, two communication modes PSK and MFSK are utilized within the network. Figure 5 shows a comparison of the throughput analysis results between the proposed method and the traditional method at different emission source levels. The traditional method refers to the method presented in [16]. The throughput performance of the proposed method surpasses that of the method in [26], with a continued improvement observed as the number of iterations increases. After the fourth iteration of the algorithm, convergence is achieved in reaching the upper limit of throughput across all sound source levels. When comparing the throughput performance limits under different sound source levels, the results of the two methods are similar for small and large sound sources. The largest disparity between the two methods appears when the sound source level is 179 dB, at which point the method proposed in this paper achieves a 1.5 kbps per slot higher throughput compared to the method in [16]. On average, the performance is improved by 68%. Table 3 presents the results of the algorithm throughput analysis at various sound source levels. It is apparent that there is a direct correlation between the sound source level and the performance limit of network throughput and the communication rate of the associated communication link.