# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 5 Numerical Results  
We now present the most relevant results derived by the analytical model presented above. The results presented in this section vary according to the number of mobile channels (provided by the drone), \(M\), and the drone's parameters, \(a_{D}\), which captures the relation between the times that the drone is inside and outside the region of interest. From the system description, we can observe that our mathematical analysis does not require one to know the parameters of the drone, including thrust power, landing/takeoff, weight, type of motors, trajectory, environmental conditions, etc. Indeed, for a specific condition (with the combination of these parameters), we only require knowing the average dwelling time of the drone inside the cell, which is represented in the model in the variable \(\mu_{0}\). Although the model assumes an exponentially distributed random dwelling time (which can be considered as an approximation to the real distribution), the developed model can be easily extended to consider other distributions. To the best of our knowledge, these dwelling times have not been reported before, and hence, we cannot evaluate the accuracy of this exponential assumption. In view of this, we only present analytical results. No simulations were performed, because we only evaluated the impact on the performance of the cellular system; considering specific drone or environmental conditions fall outside the scope of our work. Indeed, all of these parameters have an important impact on the average times that UAVs can remain in flight, which is the time that the cellular system will have additional channels to serve users in the region. Then, we vary \(1/\mu_{0}\) in the range of [12,22] min (approximately [769,1346] s), considering also that the UAV may have to travel additional distance to reach the charging station and that many UAVs have an approximate flying time of 20 minutes. Hence, the value of \(1/\mu\) only refers to the time that the UAV is inside the cellular system and does not consider the travel time from the charging station to the cell. We also consider that the average time between subsequent UAVs is \(1/\lambda_{0}=16\) min (960 s), again considering that the UAV charging station may be placed outside the covering cell. Then, we introduce the parameter \(a_{D}=\lambda_{0}/\mu_{0}\) that is analogous to the offered load of voice calls, i.e., the ratio of the arrival rate to the average call duration. In this case, \(a_{D}\) does not represent real traffic to the system, but rather, it represents the ratio of the durations of the ON to the OFF periods. Following this, we call \(a_{D}\) the _traffic load_ of UAVs inside the cell, that is, the arrival rate of the UAV respective to its average dwelling time in the system. As such, the range of \(a_{D}\) is [0.8, 1.4]. It is important to note that the proposed system can be used for cases where the traffic load increases temporarily but is expected to return to normal conditions. As such, we consider a traffic load, \(a\), to be higher than the traffic load for which the system was designed. In other words, the system is designed such that the number of channels, \(S\), in the base station can serve the users in the cell in peak traffic conditions with a blocking probability of 0.02 or less. Then, in our experiments, we consider traffic loads that entail higher blocking probabilities in the conventional system, i.e., in the ErlangB formula. First, we focus on the blocking probability. To this end, in Figure 2, we show the ErlangB system, which does not consider any additional channels and is the conventional formula to calculate the blocking probability in a telephone system for voice services. We also show the ErlangU system, which was derived in the previous section for different traffic loads of users, \(a=\lambda/\mu\), and the UAV traffic load, \(a_{D}=\lambda_{0}/\mu_{0}\), in the ranges commented above.