# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 3 Single Cell User Scheduling  
The strategy proposed for the positioning of the UAVs aims at a fair distribution of the resources among all the users in the scenario subject to backhaul limitations. As it has been explained in the system overview, the total area is divided in \(P\) regions, each one served by a single UAV. According to this, a two-step strategy is proposed in this paper. First, a fair distribution of resources among all the users within a single region is performed (which is analyzed in Section 3) and, second, a fair distribution among users in different regions is addressed by taking also into account the backhaul limitations (which is analyzed in Section 4). To evaluate the distribution of resources among all the users in a given region \(A_{i}\) corresponding to the \(i\)th UAV, two TDMA schedulers have been analyzed. In both cases, the following expression provides the average rate per user \(\overline{R}\) in the \(i\)th region:  
\[\overline{R}\big{|}_{A_{i}}=\frac{\int\int_{A_{i}}t(x,y)c(x-x_{i},y-y_{i})du}{ N_{i}T_{0}}, \tag{7}\]  
where \(t(x,y)\) is the access time assigned by the TDMA scheduler to a user placed at position \((x,y)\) and \(c(x-x_{i},y-y_{i})\) is the instantaneous rate per user, which is detailed later in this section. We are assuming that the resource allocation algorithm is able to track the changes due to fast fading. Otherwise, the resource allocation would only be able to adapt to the slow fading and the rate expressions \(c(x-x_{i},y-y_{i})\) that are detailed in this section should be replaced by the corresponding ergodic rate. For the single antenna case and for fast Rayleigh fading, the ergodic rate is equal to the rate of the Gaussian channel with a signal to noise ratio (SNR) loss equal to the Euler-Mascheroni constant [24]. In other words, the link SNR budget should consider a fading margin of 2.387 dB. In the following, \(du\) denotes the users differential, \(N_{i}\) denotes the number of users to be served in region \(A_{i}\) and \(T_{0}\) denotes the total time needed to serve all the users in the region, which are expressed as follows:  
\[du = \lambda(x,y)dA=\lambda(x,y)dxdy, \tag{8}\] \[N_{i} = \int\int_{A_{i}}\lambda(x,y)dxdy,\] (9) \[T_{0} = \int\int_{A_{i}}t(x,y)du, \tag{10}\]  
where \(\lambda(x,y)\) is the spatial users' density, which may be non-uniform, and \(dA\) is a differential area. In this paper, we assume a continuous spatial users' density \(\lambda(x,y)\). This allows obtaining expressions in form of spatial integrals, as is shown below. Note that, if a discrete users' density is adopted (corresponding to the exact knowledge of all the individual users' positions and, accordingly, a density function \(\lambda(x,y)\) composed of Dirac delta functions at the users' positions), then the integrals would instead be expressed as summations. Anyway, all the proposed strategies, results and conclusions would remain the same. The boundary of the region served by each ABBS can obtained using Equation (1), where it is assumed that each user is associated to a single UAV. In other words, the boundary for region \(A_{i}\) is the boundary of the region containing the positions \((x,y)\) for which users are associated to the \(i\)th UAV.