# Erlang-U: Blocking Probability of UAV-Assisted Cellular Systems
## 4 Teletraffic Analysis  
In this section, we derive the analytical model to calculate the blocking probability of a UAV-assisted Cellular System. We now explain, in detail, the main assumptions and operation of the network. As shown in Figure 1, the main dynamics of the system are the following: The rate at which cellular users arrive to the system is \(\lambda\). In this regard, users do not know (and have no way of knowing) if they are going to be served by a fixed terrestrial base station or by the mobile channels provided by the UAV. As such, note that the arrival rate is the same whether the UAV is inside or outside the region of interest. Similarly, the average call duration is considered to be the same, \(1/\mu\), since, again, a user does not behave differently whether the call is served by a terrestrial base station or by a mobile base station. Then, users always leave the system at rate \(\mu\). On the other hand, the UAV arrives to the region of interest with rate \(\lambda_{0}\), which is related to two parameters: the time that it takes for the drone to reach the region of interest from the charging station and the average charging time. Indeed, we assume a single drone in the system that cannot arrive to aid the cellular system faster than the time it takes to completely charge its battery and travel to the zone where additional voice channels are needed. Also, the drone remains in the system for an exponentially distributed random time with parameter \(\mu_{0}\), which is related to the average time that the drone can stay inflight, offering additional channels to the cellular system. Note that this time is related to the drone model (weight, type of motors, etc. ), the environmental conditions, and the trajectory of the drone. From this description, we can observe that the presence/absence of the UAV device follows an ON/OFF process with rates \(\lambda_{0}\) and \(\mu_{0}\), that is, with an exponential duration of the ON (OFF) period with a mean value \(1/\mu_{0}\) (\(1/\lambda_{0}\)). Finally, the system model in Figure 1 also assumes that the cellular system is composed of s channels, while the drone has \(M\) additional mobile channels to increase the system capacity temporarily. In this regard, we consider that it is composed of \(S\) servers (channels), which are fixed and installed at the base station. We also assume that the UAVs can support \(M\) mobile channels that are available to service users in the base station's coverage area; when the UAV is flying inside the cell to provide enough channels to maintain the blocking probability in acceptable levels; and when the traffic load increases due to different events such as emergency, social, cultural, sportive, and political events, among others. In other words, we are considering scenarios where traffic arrivals increase in a temporary manner and return to normal levels in the following minutes/hours. In view of this, we develop a Continuous Time Markov Chain (CTMC) to model the main dynamics of the system, namely, arrivals of users, departures of users, arrivals of UAVs, and UAV's departures, as depicted in Figure 1. Then, the CTMC is composed of two variables: \(u\) and \(i\). The latter corresponds to the number of users with an ongoing call in the system and the former corresponds to the case when the UAV is inside the cell's coverage area (adding \(M\) channels to the cellular system). Then:  
\[u=\left\{ \begin{array}{l}0;\text{ if there is no UAV}\\ 1;\text{ if the UAV is active in the cell}\end{array} \right. \tag{1}\]  
From this description, the valid state space of the chain is described by: \(\{\Omega_{u,i}:u=1,0;0\leq i\leq S+M\}\). As such, we can also write:  
\[i=\left\{ \begin{array}{l}\epsilon[0,S];\text{ if there is no UAV ($u=0$)}\\ \epsilon[0,S+M];\text{ if the UAV is active in the cell ($u=1$)}\end{array} \right. \tag{2}\]  
Recall that the UAV can add \(M\) additional channels; then, the number of available channels in the system is given by:  
\[c=\left\{