# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 3 UAV Dwelling Times  
There are many types of drones that can be used to assist the cellular system, such as the ones proposed in [12; 15; 16]. For drones in general, the service time, i.e., the dwelling time of the drone inside the region of interest, is determined by the energy drain of the battery. Recall that for our analysis, we only need to know the average time that the drone is inside the system, since this is the time that the system will have additional channels. The energy drain depends on the type of the drone, the environmental conditions, the flying pattern, and the activities of the drone inside the service area. In our case, we assume that a single UAV will travel from the charging station, which could be placed outside the serving area, to the region of interest, where the traffic load has an unusual peak activity; it will hover over this area, providing additional channels to users and then return to the charging station. This assumes a flying pattern consisting of a vertical flight to ascend from the charging station, a horizontal forward flight, hovering in a steady state with no acceleration, and then a horizontal flight to return to the charging station, including a vertical flight to descend. The power required to perform these tasks is mainly related to the mass of the drone, vertical and horizontal speeds, and the thrust which is the required upward force. The energy consumption of the UAV can then be calculated according to the specific type of drone using the derived energy consumption models in the literature, such as the ones proposed in [2; 12; 15; 16], among others. Also in [1], the authors propose an energy model for the energy consumption of the UAV that depends on the transmitting energy, the propulsion energy, and the hovering energy required in the operation of the system. However, note that the energy consumption of the UAV is highly variable due to the environmental conditions and trajectories that the UAV must follow, which depend on the location of the traffic peak. Unlike [12], where the placement of nodes is known, in a cellular system, it is not predetermined the location where the traffic peak will occur, nor its duration. Also, these conditions may change during the UAV's flight at any point in time. As such, the use of energy consumption models, such as the ones discussed above, cannot be directly used to calculate or estimate the average dwelling time of the UAVs in the cellular system because the parameters, which are mainly constant values, have to now be random variables with an unknown distribution. In other words, in order to use the energy models reported in the literature, the times that the UAV consumed energy hovering and moving forward or backwards have to be statistically measured and known. Hence, we cannot directly use these equations, because the wind direction, trajectory, humidity, rain conditions, and many others are random variables that are unknown at this point. Indeed, in order to calculate the average time that the UAV supports the cellular system with additional communication channels, it cannot be calculated directly from constant parameters, as reported before in the aforementioned energy consumption models. One alternative to the mathematical models would be the use of simulations to calculate the average dwelling time of the drone. However, this is not straightforward; we believe that this falls outside of the scope of this work, and we leave this research area for future works. In [17], the authors present the analysis of two 3D models of cellular systems with UAVs: a truncated octahedron-based one and the binomial-Voronoi. The results show that the performance in terms of the coverage probability and average achievable rate are better in the first model, but the mathematical tractability and complexity are lower in the second.