# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 4 Teletraffic Analysis
\begin{array}{l}S;\text{ if there is no UAV ($u=0$)}\\ S+M;\text{ if the UAV is active in the cell ($u=1$)}\end{array} \right. \tag{3}\]  
which can also be written as \(c=(1-u)S+u(S+M)\). The transitions of the chain are as follows:  
* From state \((u,i)\) to state \((u,(i+1))\): This occurs when there is a user arrival with rate \(\lambda\) and \(i<c\). Note that when the UAV is (not) serving the system, \(i\) can take values up to \(S+M\) (\(S\)). * From state \((u,i)\) to state \((u,(i-1))\): This occurs when there is a user departure with rate \(i\times\mu\). * From state \((0,i)\) to state \((1,i)\): This occurs when there is a UAV arrival with rate \(\lambda_{0}\times(1-u)\). Note that when the UAV is serving the system, there are no further arrivals of UAVs. Indeed, we assume that a single UAV is serving the cellular system at a particular time. Otherwise, there should be active communication among UAVs or even swarms of UAVs which would greatly increase the complexity of the system; we believe this falls outside the scope of this work. * From state \((1,i)\) to state \((0,i)\), when \(i\leq S\): This occurs when there is a UAV departure with rate \(\mu_{0}\). Note that, as in the case of UAV arrivals, there are no departures unless a UAV is inside the cell. In this case (\(i\leq S\)), the only channels used in the system are  
Figure 1: Markov Chain of the UAV-assisted cellular system. provided by the serving base station. As such, when the UAV leaves, there are no dropped calls. * From state \((1,i)\) to state \((0,i-S)\), when \(i>S\): This occurs when there is a UAV departure with rate \(\mu_{0}\), and there are users using channels of the UAV. Hence, when it leaves the system, all the calls being coursed by the UAV are dropped. From this description, it is important to note that we assume UAVs remain in the system for a random exponential distributed time with mean \(1/\mu_{0}\), and UAV arrivals occur in random inter-arrival times that are exponentially distributed with mean \(1/\lambda_{0}\). Hence, we assume that the presence/absence of the UAV device follows an ON/OFF process, with an exponential distribution for both periods of activity and inactivity. As we mentioned before, the rationale behind this is twofold. First, this assumption allows us to simplify the analysis by considering a birth/death process for both the UAV and user's arrivals and departures. Second, to the best of our knowledge, there is no public data related to the dwelling and inter-arrival times of UAVs to increase the capacity of a cellular system. Hence, many measurements have to be conducted in practical scenarios in order to have an accurate statistical modeling of the UAV behavior, which most likely depend on the cellular coverage area, weather conditions, trajectory of the UAV, and obstacles in the serving cell, among others. In this regard, the exponential distribution can be considered as the base model, where the Coefficient of Variation (CoV) is equal to 1 (given the exponential distribution properties). For the case where practical measurements prove that the \(CoV\
eq 1\), an extension of this base model can be developed considering phase-type distributions (such as the Erlang, Hyper-Exponential or Coxian distributions) which are based on the exponential distribution and would only require minor modifications to the model presented in this work. The aforementioned Markov chain is numerically solved to find the blocking probability as we describe it now. First note that, the first row of the chain corresponds to the case where the UAV is not active in the system. Hence, for all states in this row, there are only \(S\) available channels. While on the second row, the UAV is active in the cell, and there are \(S+M\) channels. The system goes from the first row to the second one when a UAV arrives in the system.