## 2 Related Works
### Markov Chain  
Evaluating the \(P_{loss}\) and \(P_{util}\) alone as common metrics for the performance analysis of communication systems is based on static and independent event assumptions, ignoring temporal and spatial correlations. The application of a Markov chain can model the dynamic channel state and can show the temporal correlation and state transfer characteristics of the communication process, so this study analyzes the data transmission process through a Markov chain. A Markov chain is a discrete stochastic process possessing the Markov property [27]. Formally, given a probability space (\(\Omega\), \(F\), \(P\)) and a sequence of random variables \(X=\{X_{n}:n>0\}\) indexed by a countable set, where each state \(X_{n}=S_{i}\) belongs to state space \(S\), the Markov property requires the conditional probability satisfie  
Figure 2: Transmission delay model. \[P(X_{t+1}\mid X_{t},\ldots,X_{1})=P(X_{t+1}\mid X_{t}) \tag{3}\]  
Then \(X\) is called a Markov chain. In a Markov chain, the change in the state of a random variable over time steps is referred to as state transition. The state transition probability, which is the most direct and crucial value reflecting the properties of the Markov chain, is defined as  
\[P_{jk}(m)=P\{X_{m+1}=k\mid X_{m}=j\} \tag{4}\]  
The term \(P_{jk}(m)\) represents the one-step transition probability of a system transitioning from state \(j\) at time step \(m\) to state \(k\) at the next time step. In stochastic processes, these probabilities are typically organized into a matrix form. Assuming the system has \(n+1\) possible states, the transition matrix \(P\) is constructed by arranging all state transition probabilities as  
\[P_{n,n+1}\equiv\left(p_{i_{n}i_{n+1}}\right)= \begin{bmatrix}P_{0,0}&\cdots&P _{0,n}\\ \vdots&\ddots&\vdots\\ P_{n,0}&\cdots&P_{n,n}\end{bmatrix}  \tag{5}\]  
For an irreducible and positive recurrent Markov chain, there exists a unique stationary distribution \(\pi=(\pi_{0},\pi_{1},\ldots,\pi_{n})\)[28], satisfying  
\[ \begin{cases}\pi P=\pi,\\ \sum_{i=1}^{n}\pi(i)=1\end{cases}  \tag{6}\]  
Here, \(\pi(i)\) represents the probability that the system is in state \(i\) under steady-state conditions. The first equation \(\pi P=\pi\) indicates that the steady-state distribution \(\pi\) is the left eigenvector of the state transition probability matrix \(P\) corresponding to the eigenvalue 1. This condition ensures the conservation of the probability distribution: once the system reaches a steady state, the probability distribution remains unchanged after any single transition step. The second equation \(\sum_{i=0}^{n}\pi(i)=1\) imposes a normalization constraint on the probability distribution, guaranteeing that the sum of probabilities over all possible states equals 1. Furthermore, if the Markov chain satisfies irreducibility (all states are mutually reachable) and positive recurrence (the expected return time to any state is finite), the steady-state distribution \(\pi\) exists uniquely. The analytical solution for \(\pi\) can then be obtained by solving the system of linear equations. In recent years, Markov chains have found extensive applications across diverse domains. [PERSON] et al. [29] developed a discrete hidden Markov mathematical model and an exponentially weighted moving average (EWMA) model to analyze and predict time delays in Wide-Area Power Systems (WAPS). These predicted delays were then fed into a predictor to mitigate the performance degradation of Load Frequency Control (LFC). [PERSON]. [30] applied the Markov chain to the driving cycles. [PERSON]. [31] investigated the reachable set synthesis for nonlinear delayed hidden semi-Markov jump systems with actuator saturation and multiple cyber-attacks. This study will employ Markov chains to analyze the transmission of Modbus data.