## 4 Experimental Results and Analysis
### Modbus AUDP Gateway Analysis
This stability indicates that, under the given network conditions, the system spends the majority \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(N_{\text{max}}\) & \(P_{\text{24}}\) & \(P_{\text{45}}\) & \(\pi_{\text{0}}=\pi_{1}=\pi_{2}\) & \(\pi_{3}\) & \(\pi_{4}\) & \(\pi_{5}\) & \(\pi_{0}+\pi_{1}+\pi_{2}+\pi_{3}\) \\ \hline
1 & 24.7\% & 6.2\% & 23.55\% & 23.19\% & 5.82\% & 0.36\% & 93.84\% \\
2 & 23.5\% & 1.3\% & 23.62\% & 23.54\% & 5.55\% & 0.07\% & 94.4\% \\
3 & 23.8\% & 0.3\% & 23.61\% & 23.58\% & 5.64\% & 0.02\% & 94.39\% \\
4 & 24.9\% & 0.2\% & 23.55\% & 23.53\% & 5.82\% & 0.01\% & 94.18\% \\
5 & 24.8\% & 0.2\% & 23.55\% & 23.53\% & 5.85\% & 0.01\% & 94.18\% \\
6 & 25.6\% & 0.1\% & 23.50\% & 23.49\% & 6.02\% & 0.006\% & 93.99\% \\ \hline \end{tabular}
\end{table} 
Table 1: Statistical table of steady-state probabilities. of its operational time cycling through these essential states, with their combined probability consistently exceeding 93.8%. Such a distribution reflects the protocol's efficiency and the effectiveness of its retransmission mechanism in maintaining normal operation. The probability of being in the retransmission state (\(\pi_{4}\)) remains below 6.1% across all tested \(N_{\max}\) values, suggesting that retransmissions are infrequent and well-managed, thus preventing excessive network congestion. Most notably, the probability of entering the communication failure state (\(\pi_{5}\)) decreases sharply as \(N_{\max}\) increases, dropping from 0.36% at \(N_{\max}=1\) to just 0.006% at \(N_{\max}=6\). This trend demonstrates the significant benefit of allowing more retransmission attempts, as it greatly reduces the likelihood of unrecoverable failures. However, as derived in Equation (19), it is important to note that while increasing \(N_{\max}\) enhances reliability, it may also lead to increased average transmission delay. Therefore, a practical upper limit of \(N_{\max}=6\) is adopted to balance reliability and latency. In summary, the steady-state analysis in Table 1 quantitatively confirms that the proposed AUDP-based transmission system achieves high reliability and efficiency under weak network conditions, with the vast majority of time spent in controllable states. To further determine the optimal \(N_{MAX}\), this study employs the entropy weight method to evaluate \(N_{MAX}\in[1,6]\) through composite scoring. The parameters \(P_{24}\), \(P_{45}\), \(\pi_{4}\), and \(\pi_{5}\) are selected as negative indicators, while \(\pi_{3}\) (successful reception probability) and \(\pi_{0}+\pi_{1}+\pi_{2}+\pi_{3}\) (cumulative probability of essential operational states) serve as positive indicators. First, normalize the positive and negative indicators:  
\[ \begin{cases}X_{ij}^{\text{norm}}=\dfrac{\max(X_{j})-X_{ij}}{\max(X_{j})-\min( X_{j})}&\text{Negative Indicator}\\ X_{ij}^{\text{norm}}=\dfrac{X_{ij}-\min(X_{j})}{\max(X_{j})-\min(X_{j})}&\text{ Positive Indicator}\end{cases}  \tag{23}\]  
where \(X_{j}\) and \(X_{ij}\) represent the value of indicator \(j\) in Table 1 and the value of indicator \(j\) when \(N_{MAX}=i\), respectively. Next, calculate the weight values using the following formula:  
\[ \begin{cases}f_{ij}=\dfrac{x_{ij}^{\text{norm}}}{\sum\limits_{i=1}^{m}x_{ij}^ {\text{norm}}}\\ e_{j}=-\dfrac{1}{\ln n}\sum\limits_{i=1}^{m}f_{ij}\ln f_{ij}\\ d_{j}=1-e_{j}\\ w_{j}=\dfrac{d_{j}}{\sum\limits_{j=1}^{n}(1-e_{j})}\end{cases}  \tag{24}\]  
The optimal \(N_{MAX}\) is determined through three phases. \(e_{ij}\) and \(f_{ij}\) denote the information entropy and proportion of the \(j\)-th indicator in Table 1, \(d_{j}\) is the differentiation coefficient, \(n\) is the sample count, and \(w_{j}\) is the weight of the \(j\)-th indicator. Take the negative indicator \(P_{24}\) as an example:  
1. Normalize it: Substitute into Equation (23) the values of item \(P_{24}\) under different \(N_{\max}\) in Table 2, where \(\max(x_{j})=25.6\%\) and \(\min(x_{j})=23.5\%\), and \(x_{ij}\) is the value to be normalized.