# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 2 System Overview  
### Scenario  
We elaborate on a scenario composed of multiple UAVs that have to serve a set of users in a given area. The positions of the users follow a generic spatial density \(\lambda(x,y)\) that may be uniform or not. We assume that the UAVs are initially located at given positions. In the simulations, we consider that the UAVs are distributed uniformly at the beginning, although this is not required. The UAVshave to be able to fly fast to quickly adapt to the static distribution of users but also have to be able to hover (stay fixed in the air) to maintain their positions when the optimum placement of the fleet of UAVs is reached. As reported in [18], there are two main types of UAVs: rotary-blade (quadcopters or hexacopters) and fixed-wing. Thus, according to our requirements, in the sequel, we shall assume the use of rotary-blades UAVs. Regarding the users-ABBS association, each user is served by a single UAV and a criterion has to be defined to select the most appropriate UAV for each user. The proposed criterion is based on the user-UAV distance \(d_{ki}\), where \(k\) and \(i\) denote the user and UAV index, respectively. The radiation pattern of the \(i\)th UAV antenna is denoted by \(G_{Ti}\) and is deeply analyzed in the following subsection. Accordingly, the UAV for the \(k\)th user is selected using the following expression:  
\[U(k)=\arg_{i}\max\left(\frac{G_{Ti}(k)G_{R}}{d_{ki}^{2}L_{s}}\right), \tag{1}\]  
where \(U(k)\) is the UAV to be associated to user \(k\), \(G_{Ti}(k)\) is the \(i\) UAV's antenna gain in the direction of the \(k\)th user, \(G_{R}\) is the gain of the receive antenna, \(d_{ki}\) is the distance between the \(k\)th user and the \(i\)th UAV, and \(L_{s}\) is the loss due to slow fading, that is, shadowing. In the previous expression, we have assumed implicitly that the transmission powers for all ABBSs are equal and constant, which is the most realistic assumption. Note, however, that, if different powers should be incorporated, this could be done by just including the transmission power for each ABBS in the numerator of Equation (1). In this association strategy, we do not include fast fading, as this fading may vary rapidly over time, whereas the association is usually required to be more stable. Note, however, that, if the association could be fast enough to follow the fading changes, then this could be formulated just by including the fading term within expression Equation (1). In the following, and for the sake of simplicity in the notation, we assume that there are no losses due to shadowing (i.e., \(L_{s}=1\)), although the general case with shadowing could be considered by just incorporating the term \(L_{s}\) in Equation (1) and the expressions of the rates that are presented in the following sections. Note that distances \(d_{ki}\) could be obtained through different ways. For example, they could be calculated using global positioning system (GPS) (in a scenario such as this one, it is expected that all the UAVs will be equipped with a GPS unit, and also all user terminals (such as smartphones) will have one, so the GPS coordinates could be used if they are sent to the central unit controlling the positions of the UAVs). Alternatively, the distances among the user devices and the ABBSs could be calculated implicitly using the time-advance mechanisms of the concrete communication standard being considered. In this case, these distances should be reported to the central unit, too. Finally, note that these distances are not strictly needed since each user terminal could estimate the term in Equation (1) for each UAV by just evaluating the equivalent channel gain that encompasses the antenna gains, the fading gain, and the loss due to distance. This association policy implies a partition of the given area in \(P\) regions, each one denoted by \(A_{i}\quad(1\leq i\leq P)\), where \(P\) is the number of deployed UAVs and each region contains all the users associated to each UAV.