## 2 Related Works  
Markov chains are widely used in the design of wireless systems and performance analysis [18]. Markov chain models are versatile and, with a reasonable choice of parameters, can subtly capture the essence of time-varying processes in wireless channels. A two-state Markov chain known as the Gilbert-Elliott channel [19; 20] has been widely employed to model communication channels prone to bursty errors, where errors occur in correlated sequences rather than independently. This model is particularly useful for performance analysis and the design of error-correction mechanisms in wireless networks, optical links, and satellite communications, where environmental factors can lead to varying error rates over time. The state diagram that visually represents the dynamics of this model is provided in Figure 1.  
States are denoted as good and bad. In a good state (G), the channel is in a high-quality condition with a low error probability. In a bad state (B), the channel experiences degraded conditions, with a higher error probability. Transitions between these states are governed by fixed probabilities:  
* \(\pi_{GB}\): the probability of transitioning from the good state to the bad state;
* \(\pi_{BC}\): the probability of transitioning from the bad state to the good state. The model allows for bursts of errors (when the channel remains in state B) and long periods with a low rate of errors (when the channel remains in G). The steady-state probabilities, which represent the long-term proportion of the time spent in each state, are defined by the transitional probabilities and calculated as follows:  
\[p_{G}=\frac{\pi_{BG}}{\pi_{GB+}\pi_{BG}}\quad\quad,\quad\quad p_{B}=\frac{\pi_ {GB}}{\pi_{GB+}\pi_{BC}} \tag{1}\]  
These state probabilities allow us to calculate various performance and reliability metrics [18]. In the analysis of error rate performance (packet delivery failure rate, etc. ), it can be assumed that no errors occur in state G, making the probability of state B a key metric of interest [21]. If \(R_{1}\) and \(R_{2}\) represent the error rates in states B and G, respectively, then the overall error rate can be characterized as the following mathematical expectation:  
\[R_{1}p_{G}+R_{2}p_{G} \tag{2}\]  
Figure 1: The Gilbert–Elliott channel model [18]. In [22], the authors address the problem of minimizing the age of information for a source transmitting status updates over two heterogeneous channels. The study focuses on 5G mmWave technology, where transmissions occur via either a fast but unreliable channel (e.g., mmWave) or via a slower, more reliable channel (e.g., sub-6 GHz). The unreliable channel is modeled as a time-correlated Gilbert-Elliott channel, providing high data rates when in the \"ON\" state. Conversely, the reliable channel offers a deterministic but lower data rate. The proposed scheduling strategy optimally selects the transmission channel in each time slot, aiming to minimize the time-average age of information. The applications of the considered Markov chain include the development of a partially observable Markov-decision-process-based joint antenna selection and a user scheduling policy for a massive MIMO base station under a high workload [22]; scheduling and resource allocation for 5G platoon communications [23]; strategy of error and congestion-resilient video streaming over broadband wireless [24]; and a clustering protocol for cognitive radio sensor networks [25]. Reference [26] discusses the challenges of satellite communication due to significant path loss, particularly at higher frequencies (optical wireless and THz/mmWave bands). The focus is placed on transitions between line-of-sight (LoS) and non-line-of-sight states, as influenced by environmental factors such as buildings and trees. Unlike the discrete-time Markov chains mentioned previously, the authors present a continuous-time Markov chain (CTMC) with two states to model these transitions, using the traversed distance instead of time.