## 4 Experimental Results and Analysis
### Modbus AUDP Gateway Analysis  
As illustrated in Figure 20, the transmission and reception results of Modbus AUDP data under weak network conditions demonstrate that the gateway successfully processed the data. The sender appended a CRC checksum 'd95c' to the Modbus UDP request frame, while the gateway generated a corresponding checksum 'decec' for the response frame, implementing dual-layer validation at both the transport and application layers. Furthermore, the application layer ensured data integrity through a Transaction ID matching mechanism: each request was assigned a unique Transaction ID, and responses were validated by matching the Transaction ID during parsing. Upon receiving a response, the system extracted and compared the Transaction ID; only when the ID matched was the response deemed valid, triggering data updates. Retransmission mechanisms were activated immediately upon initial request timeout, dynamically adjusting retry intervals using an exponential backoff algorithm to mitigate network congestion. As shown in the figure, the request successfully retrieved data during the second retransmission interval. For the Modbus data transmission process under AUDP, let \(S_{0}\) denote the waiting to send state, \(S_{1}\) the packet sending state, \(S_{2}\) the waiting for response state, \(S_{3}\) the successful reception state, \(S_{4}\) the retransmission state, and \(S_{5}\) the communication failure state. The state transitions are illustrated in Figure 21:  
Assume each retransmission is independent, with identical success probabilities and no differentiation between retransmission attempts. As shown in Figure 21, any state can transition to another within finite steps, and each state returns to itself in finite expected time. Thus, the system is irreducible and positive recurrent, implying the existence of steady-state probabilities analyzable via Markov chains. Based on Figure 21, the state transition matrix can be constructed as follows:  
Figure 21: State transition diagram. Figure 20: Data transmission and reception diagram. \[P= \begin{bmatrix}0&P_{01}&0&0&0&0\\ 0&0&P_{12}&0&0&0\\ 0&0&0&P_{23}&P_{24}&0\\ P_{30}&0&0&0&P_{43}&0\\ 0&0&0&P_{43}&0&P_{45}\\ P_{50}&0&0&0&0&0\end{bmatrix}  \tag{20}\]  
Substituting the state transition matrix **P** into the general balance condition Equation (6), the steady-state balance equations are derived as follows:  
\[ \begin{cases}\pi_{0}=\pi_{3}P_{30}+\pi_{5}P_{50}\\ \pi_{1}=\pi_{0}P_{01}\\ \pi_{2}=\pi_{1}P_{12}\\ \pi_{3}=\pi_{2}P_{23}+\pi_{4}P_{43}\\ \pi_{4}=\pi_{2}P_{24},\\ \pi_{5}=\pi_{4}P_{45}\\ \sum_{i=0}^{5}\pi_{i}=1\end{cases}  \tag{21}\]  
The maximum retransmission count \(N_{max}\) directly affects the system's steady-state probabilities. In a network environment with a 20% \(P_{loss}\) rate, experiments were conducted by transmitting 1000 data frames under different \(N_{max}\) values. From Figure 18, \(P_{01}=P_{12}=P_{30}=P_{50}=1\), \(P_{43}=1-P_{45}\), \(P_{23}=1-P_{24}\). Simplify the above equation to obtain Equation (22):  
\[ \begin{cases}\pi_{0}=\frac{1}{4+P_{24}}\\ \pi_{1}=\pi_{2}=\pi_{0}\\ \pi_{3}=\pi_{0}(1-P_{24}P_{45})\\ \pi_{4}=\pi_{0}P_{24}\\ \pi_{5}=\pi_{0}P_{24}P_{45}\\ \sum_{i=0}^{5}\pi_{i}=1\end{cases}  \tag{22}\]  
The steady-state probabilities are determined by the probability of entering retransmission (\(P_{24}\)) and the probability of retransmission failure (\(P_{45}\)) under a packet loss rate of 20% (the experimental values of \(P_{24}\) and \(P_{45}\) are listed in Table 1). The steady-state probabilities of each state corresponding to each \(N_{max}\) are statistically analyzed, with the results summarized in Table 1:  
As detailed in Table 1, the steady-state probabilities \(\pi_{0}\), \(\pi_{1}\), \(\pi_{2}\), and \(\pi_{3}\)--corresponding to the core communication phases of waiting to send, packet sending, waiting for response, and successful reception--remain nearly constant (23.5-23.6%) as \(N_{\text{max}}\) increases.