# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 4 Teletraffic Analysis
Building on this, user arrivals only occur until state \((0,S)\) when the UAV is not in the system, and arriving calls are blocked, while in the case where the UAV is in the system, arrivals continue to arrive until the \(S+M\) channels are occupied. Therefore, a user is blocked when in state \((S+M)\) if the UAV is serving the cell. Also note that if there are no more UAV arrivals once a UAV is already active, then they are only transitioning from state \((0,j)\) to state \((1,j)\) for \(0\leq j\leq S\); there are no more transitions from state \((0,S)\) to states \((1,j)\) for \(j>S\) since the UAV is already in the system. Finally, note that when the UAV leaves the system, if there are channels of the UAV occupied (i.e., in states \((1,j)\) for \(j>S\)), the system goes to state \((0,S)\), dropping all those calls. We now calculate the steady state probabilities, with \(\pi_{u,i}\) using the rate equalization method [13], i.e., solving the following linear equation system. First, for the upper row, state (0,0), we have:  
\[[\lambda+\lambda_{0}]\pi_{0,0}=\mu\pi_{0,1}+\mu_{0}\pi_{1,0} \tag{4}\]  
For state (0,1), we have:  
\[[\lambda+\lambda_{0}+\mu]\pi_{0,1}=\lambda\pi_{0,0}+2\mu\pi_{0,2}+\mu_{0}\pi_ {1,1} \tag{5}\]  
We follow a similar procedure until the  
For state (1,1), we have:  
\[[\mu_{0}+\lambda+\mu]\pi_{1,1}=\lambda\pi_{1,0}+\lambda_{0}\pi_{0,1}+2\mu\pi_{1,2} \tag{8}\]  
We follow a similar procedure until the last state of the second row, i.e., state (1,S+M), with the following equation:  
\[[\mu_{0}+(S+M)\mu]\pi_{1,S+M}=\lambda\pi_{1,S+M-1} \tag{9}\]  
By numerically solving this linear system, we obtain the steady state probabilities \(\pi_{u,i}\) for \(u=0,1\) and \(0\leq i\leq S+M\). Hence, the blocking probability when the UAV is not serving the cell is \(\pi_{0,S}\), while the blocking probability when the UAV is in the system is given by \(\pi_{1,S+M}\). Since we consider exponential times for both the voice calls and UAV activities, we find the blocking probability based on the PASTA property. Specifically, the portion of the time that the system has all the channels occupied is the same proportion that the users have to view the system with no available channels whenever they arrive. This occurs at states \((0,S)\) and \((1,S+M)\) of the proposed Markov chain. Then, the total blocking probability is given by:  
\[P_{BU}=\pi_{0,S}+\pi_{1,S+M} \tag{10}\]  
The probability that there is a UAV serving the system, \(T_{D}\), is given by the average dwelling time of the UAV divided by the total time of the system operation, i.e., the average time that the UAV is not active (or in other words, the average time required for the UAV to arrive to the system), plus the average time that the UAV is active in the system. Then:  
\[P[\text{There is a UAV}]=T_{D}=\frac{\frac{1}{\mu_{0}}}{\frac{1}{\mu_{0}}+ \frac{1}{\lambda_{0}}} \tag{11}\]  
and  
\[P[\text{No UAV}]=1-P[\text{There is a UAV}]=\frac{\frac{1}{\lambda_{0}}}{ \frac{1}{\mu_{0}}+\frac{1}{\lambda_{0}}} \tag{12}\]  
On the other hand, the average number of calls supported by the UAV device, and consequently, the average number of calls that have to forcedly terminate their service when the UAV leaves the region of interest, can be calculated as:  
\[N_{D}=\sum_{j=S}^{S+M}\left[(j-S)\pi_{1,j}\right] \tag{13}\]  
Note that, in state \(j\), for \(j=S,S+1,\ldots,S+M\), the UAV is always serving \((j-S)\) users that are forced to terminate their service when the UAV leaves the system. Also, in state \(j(j<S)\) when the UAV leaves the system, we assume that all calls served by the drone can be accommodated to the terrestrial BS and no calls are dropped.