# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 1 Introduction  
In modern telecommunications systems, the use of drones is becoming more and more relevant due to the flexibility and ease of deployment. For instance, in cellular systems, drones can assist with additional channels [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11], and also in the Internet of Things (IoT) to charge nodes or recollect data sensed by nodes in a region of interest [12]. For these cases, it is preferable to send drones to the region of interest with additional channels, in the case of cellular systems, in a temporary manner rather than setting new base stations, especially if traffic peaks also occur in this manner, i.e., temporarily. Cellular systems are usually designed to provide a blocking probability of 0.02 or 0.01 for voice services by carefully selecting the number of channels according to the peak traffic load in the coverage area of the cell. However, when certain events occur in this region, the traffic load can greatly increase, entailing a much higher blocking probability, which can affect the Quality of Service (QoS) of users. Events like emergencies (fires, earthquakes, flooding, etc. ), cultural (concerts, festivals, etc. ), sportive (games, Olympics, racing, etc. ), and many others, are expected to temporarily increase the traffic load. One alternative to tackling this problem is to provide additional channels in the Base Station. Still, the installation and operational costs are high and the extra channels would be active all the time, even when traffic returns to normal conditions. This would cause higher energy consumption in the overall system. Mobile base stations can also be implemented using vehicles. However, the advantage of using drones is that they are not constrained by terrestrial traffic nor by parking availability. Another important advantage of drones versus vehicles when introducing additional channels is that drones can be placed in the exactregion required to increase the capacity, for instance, in the case of a public event where streets are blocked to vehicle traffic. However, an important advantage of the use of vehicles is that the dwelling time is unrestricted, since it is not relying on the battery capacity. In view of this, we propose that the use of UAVs (Unmanned Aerial Vehicles) be used to provide the additional channels required, in order to maintain an adequate QoS during these events. This has the benefit of the extra channels only being provided when needed; the system operation can return to the original number of channels afterwards. However, as in the case of selecting the adequate number of fixed channels, and using the ErlangB formula [13], we develop a mathematical analysis to carefully select these additional channels and obtain the ErlangU formula. Note that if a lower number of channels are used in the UAV, the blocking probability may not be lower than 0.02. On the other hand, if a lot more channels are placed on the UAV than those required to maintain a 0.02 blocking probability, the system would increase the operational costs due to the fact that more expensive UAVs would be required. In this work, we assume that the voice calls can be switched over to the aerial base station (UAV) and the terrestrial base station following a procedure similar to a handoff procedure in mobile cellular systems. To this end, there should be an active communication between the UAV and the base station to calculate power levels and demand voice channels whenever the drone is leaving the region of interest, with the following difference: In the case of a handoff in a cellular system, the mobile user usually does not know the exact moment of departure of a specific cell. Hence, in general, the handoff procedure is performed when the power received by a neighbor base station is much higher than the received power of the attending base station.