## 3 Markov Chain-Based Statistical Analysis  
Based on the mode of operation that was presented in the previous section, in the proposed selection policy, a two-state ergodic and regular Markov chain is defined, whose state 1 corresponds to the event that transmission is performed using the FSO link and whose state 2 corresponds to the event that transmission is performed with the aid of RF link (see Figure 1). This Markov chain is characterized by a unique vector of stationary probabilities given by \(\mathbf{\pi}=[\pi_{1},\pi_{2}]\). Based on the fact that the previously mentioned eventsare mutually exclusive, the CDF of the output SNR at the G using the proposed scheme, \(\gamma_{t}\), can be expressed as ([30] [Equation (3)])  
\[F_{\gamma_{t}}(\gamma)= \begin{cases}\sum\limits_{i=1}^{2}\pi_{i}\{\Pr[\gamma_{ \text{th}}\leq\gamma_{i}\leq\gamma]+\Pr[\gamma_{i}<\gamma_{\text{th}}]\\ \qquad\qquad\qquad\times\Pr[\gamma_{2}\leq\gamma]\},\gamma\geq\gamma_{\text{ th}}\\ \Pr[\max\{\gamma_{1},\gamma_{2}\}\leq\gamma],\qquad\gamma<\gamma_{\text{th}}, \end{cases}  \tag{16}\]  
where \(\gamma_{i}\), with \(i\in\{1,2\}\) denotes the instantaneous received SNR from link \(i\). Applying the definition of the CDF, i.e., \(F_{\gamma_{i}}(x)=\Pr[\gamma_{i}<x]\), on the generic expression presented in (16), the following result is obtained  
\[F_{\gamma_{t}}(\gamma)=\left\{ \begin{array}{c}\sum\limits_{i=1}^{2}\pi_{i} \{F_{\gamma_{i}}(\gamma)-F_{\gamma_{t}}(\gamma_{\text{th}})\\ \qquad+F_{\gamma_{t}}(\gamma_{\text{th}})F_{\gamma_{t}}(\gamma)\},\ \gamma\geq\gamma_{\text{th}}\\ F_{\gamma_{1}}(\gamma)F_{\gamma_{2}}(\gamma),\qquad\gamma<\gamma_{\text{th}}, \end{array} \right. \tag{17}\]  
where \(F_{\gamma_{1}}(\cdot),F_{\gamma_{2}}(\cdot)\) are given by (5) and (15), respectively, while \(\tilde{i}=3-i\). Moreover, by differentiating (17) with respect to \(\gamma\), the following expression for the PDF of \(\gamma_{t}\) can be obtained  
\[f_{\gamma_{t}}(\gamma)= \begin{cases}\sum\limits_{i=1}^{2}\pi_{i}\{f_{\gamma_{ i}}(\gamma)+F_{\gamma_{i}}(\gamma_{\text{th}})f_{\gamma_{i}}(\gamma)\},\gamma \geq\gamma_{\text{th}}\\ \qquad\qquad\sum\limits_{i=1}^{2}f_{\gamma_{i}}(\gamma)F_{\gamma_{i}}(\gamma),\quad\gamma<\gamma_{\text{th}},\end{cases}  \tag{18}\]  
where \(f_{\gamma_{1}}(\cdot)\) and \(f_{\gamma_{2}}(\cdot)\) are given by (7) and (14), respectively. Due to the ergodicity of the Markov chain of our system, the aforementioned stationary probabilities can be evaluated using \(\boldsymbol{\pi}=\boldsymbol{\pi}\cdot\mathbf{P}\) in conjunction with \(\sum_{i=1}^{2}\pi=1\), where \(\mathbf{P}\) denotes the transition matrix given by Section 7.3 in [31]  
\[\mathbf{P}= \begin{pmatrix}P_{11}&P_{12}\\ P_{21}&P_{22}\end{pmatrix} . \tag{19}\]  
Exploiting (19), the stationarity probabilities can be obtained as follows  
\[ \begin{split}\pi_{1}&=\frac{P_{21}}{P_{12}+P_{21}}\\ \pi_{2}&=\frac{P_{12}}{P_{12}+P_{21}}\end{split}  \tag{20}\]  
In (19), the transition probabilities of the corresponding Markov chain can be evaluated based on the following observations. The probability of remaining at the same State \(i\) is equal to the sum of the probability of the received SNR from State \(i\), i.e., \(\gamma_{i}\), being larger than \(\gamma_{\text{th}}\) and the probability \(\gamma_{i}\) being lower than \(\gamma_{\text{th}}\) and simultaneously being larger than the received SNR from the other state, i.e., \(\gamma_{j}\). The same approach is also followed for obtaining the probability of switching states from \(i\) to \(h\) or vice versa. From the mathematical point of view, this statement can analytically be expressed as  
\[P_{i,j}=\left\{ \begin{array}{c}\Pr[\gamma_{i}\geq\gamma_{\text{th}}]+\Pr[ \gamma_{i}<\gamma_{\text{th}},\gamma_{i}\geq\gamma_{j}],\ i=j\\ \Pr[\gamma_{i}<\gamma_{\text{th}},\gamma_{j}\geq\gamma_{i}],\ \ i\
eq j.\end{array} \right.