## 3 System Model
Since a Markov process is considered, the system remains in each state for a duration governed by an exponentially distributed random variable. Let us analyze the probability of the system being in the FSO state at time \(t+\Delta t\) by enumerating all possible scenarios. At the preceding time \(t\), the system is in state FSO and remains in this state at the current time \(t+\Delta t\), with a probability of \(1-w_{f}\Delta t+o(\Delta t)\). Additionally, at time \(t\), if the system is in state FSO \(\rightarrow\) RF, it transitions to state FSO within the interval \(\Delta t\) with a probability of \(w_{r}\Delta t+o(\Delta t)\). Similarly, if the system is in state RF \(\rightarrow\) FSO at time \(t\), it transitions to state FSO with a probability of \(\mu_{r}\Delta t+o(\Delta t)\). The probabilities of all remaining scenarios are in the order \(o(\Delta t)\). We denote the probabilities of the states FSO, FSO \(\rightarrow\) RF, RF, and RF \(\rightarrow\) FSO as \(p_{0}\), \(p_{1}\), \(p_{2}\), and \(p_{3}\), respectively. According to the law of total probability:  
\[p_{0}(t+\Delta t)=p_{0}(t)\Big{(}1-w_{f}\Delta t\Big{)}+p_{1}(t)w_{r}\Delta t+ p_{3}(t)\mu_{r}\Delta t+o(\Delta t). \tag{4}\]  
After expanding the brackets, collecting like terms, and dividing the resulting expression by \(\Delta t\), we obtain  
\[\frac{p_{0}(t+\Delta t)-p_{0}(t)}{\Delta t}=-p_{0}(t)w_{f}+p_{1}(t)w_{r}+p_{3 }(t)\mu_{r}+\frac{o(\Delta t)}{\Delta t} \tag{5}\]  
Figure 2: Continuous-time Markov chain for hybrid FSO/RF communications. Taking the limit as \(\Delta t\rightarrow\) and noting that  
\[\lim_{\Delta t\to 0}\frac{p_{0}(t+\Delta t)-p_{0}(t)}{\Delta t}=\frac{dp_{0}}{dt} \tag{6}\]  
we derive the following:  
\[\frac{dp_{0}(t)}{dt}=-w_{f}p_{0}(t)+w_{r}p_{1}(t)+\mu_{f}p_{3}(t) \tag{7}\]  
Similarly, by considering the remaining states of this CTMC, we obtain the differential equations, as follows:  
\[\frac{dp_{1}(t)}{dt}=w_{f}p_{0}(t)-(w_{r}+\mu_{r})p_{1}(t) \tag{8}\]  
\[\frac{dp_{2}(t)}{dt}=\mu_{r}p_{1}(t)+w_{f}p_{3}(t)-w_{r}p_{2}(t) \tag{9}\]  
\[\frac{dp_{3}(t)}{dt}=w_{r}p_{2}(t)-(w_{f}+\mu_{f})p_{3}(t) \tag{10}\]  
The resulting system of differential equations is a variation on [PERSON]'s equations. The CTMC in Figure 2 is irreducible, i.e., every state is reachable from every other [36]. Hence, the system always reaches a stationary regime. In this phase, the state probabilities are time-independent and invariant with respect to the initial state of the Markov chain. Therefore,  
Figure 2.0  
The system of Equations (1)-(4) in the steady state is not independent. To obtain a solution, it is necessary to replace one of the equations with a normalization equation:  
Figure 2.1  
We chose Equation (3) for replacement. Thus, the stationary probabilities for the stochastic process shown in Figure 2 are determined from the system of linear algebraic equations:  
Figure 2.2  
where \(\mathbf{A}\) is the transition matrix  
\[A= \begin{pmatrix}-w_{f}&w_{r}&0&\mu_{r}\\ w_{f}&-\begin{pmatrix}w_{r}+\mu_{f}\end{pmatrix}&0&0\\ 0&0&w_{r}&-(w_{f}+\mu_{f})\\ 1&1&1&1\end{pmatrix} , \tag{14}\]  
\(\mathbf{p}\) is a column vector of the state probabilities we are looking for, while \(\mathbf{b}\) is a column vector of the following form: \((0,0,0,1)^{T}\). This system has the following unique solution:  
\[p_{1}=\left(1+\frac{w_{r}+\mu_{f}}{w_{f}}+\frac{\mu_{f}}{\mu_{r}}\biggl{(} \frac{w_{f}+\mu_{r}}{w_{r}}+1\biggr{)}\right)^{-1}, \tag{15}\]  
\[p_{0}=\frac{w_{r}+\mu_{f}}{w_{f}}p_{1}, \tag{16}\]\[p_{2}=\frac{\Big{(}w_{f}+\mu_{r}\Big{)}\mu_{f}}{w_{r}\,\mu_{r}}p_{1}, \tag{17}\]  
\[p_{3}=\frac{\mu_{f}}{\mu_{r}}p_{1}, \tag{18}\]  
The obtained state probabilities allow us to analyze various system efficiency metrics.