## 3 System Model  
When proposing refinements to the Gilbert-Erasure channel model for hybrid RF/FSO communications, it is crucial to account for the inherently gradual transitions in environmental conditions, rather than assuming abrupt changes. For example, fog develops incrementally rather than appearing instantaneously, while rain often subsides progressively. In addition, when conditions that are unfavorable for using the FSO (or RF) channel occur, the decision to switch the communication technology to RF (or FSO) is usually not made immediately. The decision may depend on various factors, including the duration and severity of the adverse conditions, the current network traffic load, the specifics of the network protocols and the quality of service policies used, delays caused by the activation of the alternate channel, and the predefined thresholds or algorithms set by the system for making such transitions. Thus, in this section, we present a continuous-time Markov chain (CTMC) that comprises four states, as follows:  
* _FSO_. In this state, the FSO channel is preferred over the RF channel and is actively used for communication. This state is typical when the FSO link has favorable conditions, such as clear weather, low atmospheric turbulence, and an absence of obstacles or the ability to use repeaters to bypass them. * _FSO_\(\rightarrow\)_RF_. In this state, the RF channel is preferred due to the degradation of conditions in the FSO link, but the FSO channel is still in use. * _RF_. In this state, the RF channel is both preferred and in use. This state occurs when the FSO link experiences severe degradation, such as heavy fog or obstruction, making the RF channel the primary choice for reliable communication. * _RF_\(\rightarrow\)_FSO_. In this state, the RF channel is in use, but the FSO channel has recovered sufficiently to become preferable. This state is another transitional phase, during which the system monitors the RF channel and switches back to the FSO link if conditions are favorable. The proposed CTMC model is governed by a set of transition rates that are directly proportional to the probability of transitions between states occurring over infinitesimal time intervals; i.e., the probability of transition from one state to another over time \(\Delta t\) differs from the product of the transition rate and \(\Delta t\) by \(o(\Delta t)\). Here, \(o(\Delta t)\) represents a function that grows at a rate slower than \(\Delta t\) and thus has an insignificant impact when the value of \(\Delta t\) is small. Formally,  
\[\lim_{n\rightarrow\infty}\frac{o(\Delta t)}{\Delta t}=0 \tag{3}\]  
Let us provide notation for these transition intensities:  
* _wf:_ this transition rate increases as the quality of FSO communications deteriorates compared to RF communications;* \(\mathbf{w_{r}}\): on the contrary, this is the intensity of the transition from a situation in which it is more profitable to use the RF to a situation in which it is more advantageous to use the FSO;
* \(\mathbf{\mu_{f}}\): the rate of transition from the FSO \(\rightarrow\) RF to the RF depends on the efficiency of adopting the corresponding decision and the performance of the mechanism for switching from the FSO to the RF;
* \(\mathbf{\mu_{r}}\): the rate of transition from the RF \(\rightarrow\) FSO to the FSO is inversely proportional to the time it takes to make and implement the decision to stabilize the FSO channel, again making the FSO preferable to the RF. The resulting CTMC is shown in Figure 2. The possible transitions between states are represented by directed arrows. The transition rates at which the system transitions from one state to another are indicated next to the corresponding arrows. With probability 1, at a given moment in time, one of the states is realized. This state diagram suggests a cyclic and interconnected transition pattern, where the system can move between different states through direct or indirect paths. It is noteworthy that every state within the model is accessible from any other state.