# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 5 Numerical Results
We can observe that the blocking probability for the conventional cellular system (i.e., with S fixed channels) remains constant for different values of \(a_{D}\), since the dynamics of the UAV only provides additional channels for the UAV-assisted system. As such, the blocking probability only increases as the traffic load, \(a\), increases. However, for the UAV-assisted system, as the UAV continues to have longer times in the region of interest (as the UAV continues to have longer times, \(1/\mu\) increases, and then, \(a_{D}\) increases), the blocking probability decreases considerably, since there is more time with additional channels to serve users. To further study the impact of the number of mobile channels supported by the UAV, we present, in Figure 3, the blocking probability for different numbers of channels added by the UAV, \(M\). From these results, it is easy to observe the number of channels required to serve the cell by the UAV to increase the system capacity in cases of an increased traffic load, given certain events like earthquakes, sporting events, and others. Form these results, we can clearly observe that as the traffic load increases, we require more than 10 mobile channels provided by the UAV to maintain an acceptable blocking probability. In this regard, both Figures 2 and 3 offer a design tool for the deployment of such UAV-assisted cellular systems, in the sense that it is possible to determine the type of drone required in terms of the weight and battery capacity to provide specific flight times for the region of interest, as well as the number of mobile channels required to offer an adequate service to users in increased traffic load conditions. In other words, for a service provider that launches UAVs with mobile channels without considering these results may not significantly improve the system performance if the UAV does not leave  
Figure 3: Blocking probability for the ErlangB system and ErlangU system for different users’ traffic load and additional channels provided by the UAV, \(M\). Figure 2: Blocking probability for the ErlangB system and ErlangU system for different users’ traffic load, \(a\) and UAV traffic load, \(a_{D}\). sufficient time in the region or if it does not carry the required mobile channels to reduce the blocking probability. Now, we focus on the drop call probability. In this case, there are no drop calls in the conventional cellular system. In Figures 4 and 5, we show the fraction of the calls that are forced to terminate when the UAV leaves the cellular system, calls that are admitted, and the total number of calls that arrive, respectively, for different UAV _traffic loads_, as well as the number of channels provided by the UAVs. Clearly, the ratio of calls forced to finish to the total calls is higher than the ratio of calls forced to terminate to the calls that are admitted (\(P_{DO}>P_{DA}\)), but both can be used to determine an adequate Quality of Service to users under this proposed system, as well as to adequately select the type of UAV to be used in such applications. Considering that a dropped call has a very annoying effect on users with an ongoing call, these results allow a careful system design to keep this number as low as possible. Furthermore, this performance metric should be given preference over the blocking probability. Recall that a call is dropped during an ongoing call when the UAV leaves the system to return to the charging station, affecting the user's calls served by the UAV if the base station has no capacity to absorb such calls. Then, we can observe that as the number of mobile channels, \(M\), increases, the number of dropped calls increases accordingly. The reason for this is that when all the channels at the base station are occupied and the drone is serving a high number of users inside the region of interest (high values of \(M\)), and when the drone must leave the area, all these users will have to terminate their call, increasing the drop call probability.