## 3 Materials and Methods
### The Gilbert-Elliott Channel Model  
We implemented the Gilbert-Elliott channel model in our work to model the noisy communication channel [37]. This model is computationally efficient and produces an error burst to demonstrate the accurate effect of errors on an application without involving the physical processes [38; 39; 40]. These errors may result in loss or alteration in packets, frames, or bits from the transmitted bitstreams [41; 42]. The Gilbert-Elliot (GE) channel model is a discrete time hidden Markov chain model that consists of two states, i.e., the good state and the bad state. Suppose that \(S=\{G,\,B\}\) is the state space of the wireless communication channel, where \(G\) and \(B\) are the good and the bad states, respectively. The probability of error occurrence in good state \(G\), denoted by \(P(G)\), is relatively lower than the probability of error occurrence in bad state \(B\), denoted by \(P(B)\). When the error occurrence in the good state does not happen, then \(P(G)\) will be 0, indicating the error-free transmission of bits through the channel, and \(P(B)\) will become 1, indicating the occurrence of error in the bad state. In order to generate fewer erroneous bits, it is assumed that \(P(G)>P(B)\), i.e., the good state is more likely to experience error bits than the bad state. The probability of shifting from one state to the other state is known as _transition probability_. Let \(P_{GG}\) be the probability that the next state is again a good state and \(P_{BB}\) be the probability that the next state is again a bad state. \(P_{GG}\) and \(P_{BB}\) are known as self-transition probabilities. Similarly, \(P_{GB}\) and \(P_{BG}\) denote cross-transition probabilities, i.e., \(P_{GB}\) is the probability that the next state is bad state \(B\) from the current state \(G\), whereas \(P_{BG}\) is the probability that the next state is a good state considering the present state \(B\). A state transition diagram of the GE channel model is given in Figure 4.  
Figure 3: Error detection process in block coding. The Gilbert-Elliot Model uses a two-state transition matrix of order two-by-two to determine the state transition probabilities. The two-state transition matrix determines the state transition probabilities and is represented by M in (5), as follows:  
\[M= \begin{bmatrix}P_{GG}&P_{GB}\\ P_{BG}&P_{BB}\end{bmatrix} , \tag{5}\]  
The sum of all probabilities from a particular state is always 1. Thus, from the state-transition diagram shown in Figure 4, it is concluded that:  
\[P_{BB}+P_{BG}=1\text{ and }P_{GG}+P_{GB}=1 \tag{6}\]  
From (6), the self-transition probabilities (7) and cross-transition probabilities (8) can be calculated as follows:  
\[P_{BB}=1-P_{BG}\text{ and }P_{GG}=1-P_{GB} \tag{7}\]  
\[P_{BG}=1-P_{BB}\text{ and }P_{GB}=1-P_{GG} \tag{8}\]  
The expected amount of time for which the channel remains in one state before moving to the other state (either in \(G\) or \(B\)) is known as the mean sojourn time of being in that state. There were only two states in our GE channel model; therefore, the mean sojourn times of good state \(G\) and bad state \(B\) are denoted by \(T_{G}\) and \(T_{B}\), respectively. The mean state sojourn times \(T_{G}\) of state \(G\) and \(T_{B}\) of state \(B\) can be estimated by Equation (9), as follows:  
\[T_{G}=\frac{1}{1-P_{GG}}\text{ and }T_{B}=\frac{1}{1-P_{BB}} \tag{9}\]  
The probability that the channel errors occur in steady state is known as the steady state probabilities, denoted by \(P_{GG}\) and \(P_{BB}\) depending on the steady state being \(G\) or \(B\), respectively. Steady state means that the state of the channel remains unchanged.