# Communication Infrastructure Design for Reliable UAV Operations in Air Mobility Corridors
## 2 Materials and Methods
### Markov Modeling of Communication Reliability and Estimation of Availability Metrics
#### 2.2.2 Reliability of Communication Service Estimation  
Based on the state-transition diagrams presented in Figures 6-9, the system of Chapman-Kolmogorov equations can be constructed in accordance with standard modeling principles [30]. These equations describe the dynamics of the Markov process governing the system behavior under each strategy.  
For the continuous-time Markov chains used in this study, state transitions are governed by an infinitesimal generator matrix \(Q=\left[q_{(ij),(s,l)}\right]\), where each element represents the instantaneous transition rate from state \(H_{ij}\) to state \(H_{sl}\).  
Assuming each point in the corridor is covered by nnn radio stations and can serve up to mmm UAVs simultaneously, the transition rates are defined as follows:  
* UAV arrival (increasing load): \(q_{(ij),(i+1)}=\phi(n-i),\text{ for }j<m,\) where \(\phi\) is the arrival rate and \((n-i)\) reflects the number of operational stations available to accept new UAV connections.
* UAV departure (releasing load): \(q_{(ij),(i,j-1)}=j\psi,\text{ for }j>m,\) where \(\psi\) is the UAV service rate, and \(j\) is the number of active connections.
* Station failure (loss of communication capacity): \(q_{(ij),(i+1,j)}=(n-i)\lambda,\text{ for }i<n,\) where \(\lambda\) is the per-unit failure rate.
* Station repair (recovery of capacity): \(q_{(ij),(i-1,j)}=i\mu,\text{ for }i>0,\) where \(\mu\) is the per-unit repair rate.  
Let \(P_{ij}\) denote the steady-state probability of being in state \(H_{ij}\). These probabilities satisfy the global balance equations:  
\[\sum_{(s,l)}P_{sl}\cdot q_{(s,l),(i,j)}=0\quad\forall(i,j) \tag{7}\]  
Figure 9: Markov chain representation of UAV communication for Strategy 4.  
By solving the resulting system of steady-state equations, we derive an expression for the stationary probabilities, \(P_{ij}\), of being in state \(H_{ij}\), where the following applies:  
\(i\) is the number of failed radio stations (\(0\leq i\leq n\)).  
\(j\) is the number of UAVs currently in communication (\(0\leq j\leq m\)).  
\(n\) is the number of radio stations covering each point of the corridor:  
\(n=1\) for Strategy 1 (single coverage).  
\(n=2\) for Strategy 2 (fully overlapping coverage).  
\(n=3\) for Strategy 3 (alternating redundancy).  
\(n=4\) for Strategy 4 (complete duplication with full overlap).  
The stationary probabilities \(P_{ij}\) are given as  
\[P_{ij}=\begin{pmatrix}i\\ K\end{pmatrix}\begin{pmatrix}j\\ m\end{pmatrix}\gamma^{i}\rho^{j}P_{00} \tag{8}\]  
where \(\gamma=\lambda/\mu\), \(\rho=\phi/\psi\), \(K=n+m\), \(P_{00}\) is the stationary probability of the system being in state \(H_{00}\) (all stations operational, no active UAV sessions).  
To determine \(P_{00}\), the normalization condition is used:  
\[\sum_{i=0}^{n}\sum_{j=0}^{m}P_{ij}=1 \tag{9}\]  
Substituting the expressions for \(P_{ij}\) and solving the resulting equation yields  
\[P_{00}=\left[\sum_{j=1}^{m}\begin{pmatrix}i\\ n\end{pmatrix}\begin{pmatrix}j\\ m\end{pmatrix}\gamma^{i}\rho^{j}\right]^{-1} \tag{10}\]  
The availability, \(A\), of the communication channel is defined as the probability that at least one station is operational and that the system has sufficient capacity to serve the incoming UAVs. This is calculated as  
\[A=1-\sum_{i=b}^{n}\sum_{j=n+1-i}^{m}P_{ij} \tag{11}\]  
where  
\[b=\begin{cases}0,&n<m\\ n-m+1,&n\geq m\end{cases} \tag{12}\]  
To complement this measure, the unavailability, \(U\), is introduced as a direct indicator of the risk of communication interruption or service denial at a given point in the corridor:  
\[U=1-A \tag{13}\]  
This parameter represents the total probability of the system being in a failed or blocked state, making it a key indicator in evaluating the reliability and continuity of communication service for UAVs across different strategies.