# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 6 Evaluation and Results
### Positioning Strategy Evaluation  
The positioning strategy derived in Section 4 is evaluated in this subsection assuming that the intra-cell resource allocation is based on the equal rate scheduler described in Section 3.2. A squared scenario with a side of 6100 m and a spatial distribution of users \(\lambda(x,y)\) with a Gaussian spatial shape centered at the point (4500 m, 4500 m) has been considered, which can simulate, for instance, a sporting event scenario. Moreover, nine UAVs have been uniformly distributed in the scenario as starting point, the backhaul ground antenna has been placed at the corner (5900 m, 5900 m) and the total backhaul bandwidth \(B_{BH}^{total}\) has been uniformly allocated among all the UAVs as initial distribution (\(B_{i}^{BH}=B_{BH}^{total}/P\)). Figure 3 shows the scenario in the first iteration and after 500, 1000, 1500, 2000, and 3000 iterations. The peak of the users distribution is plotted as a red cross, the backhaul ground antenna is plotted as an orange circle, the value of the rate for each user at each region \(\overline{R}\) is plotted from blue to yellow depending on its value (although it is constant within each region as the equal rate scheduler achieves a fair distribution of the resources) and, finally, UAVs \((x,y)\) coordinates (in meters) are plotted as letters of different colors to be able to track their movements over the iterations. Figure 4 shows the evolution along the iterations of the rate per user (\(\overline{R}\)) of each region. The colors are the ones defined in the letters in Figure 3.  
By observing both figures, it can be seen that, after 3000 iterations, which is equivalent to 15 min (\(T_{it}=0.3\) s), the positioning strategy is able to reduce the difference among the average rate per user of each region by moving almost all the UAVs towards the area with the highest concentration of users (4500 m, 4500 m), which leads to a more uniform distribution of the resources. Figure 3 shows qualitatively how, as the iterations go on, the UAVs move so that the rates for users at different regions tend to be equal. In fact, Figures 3 and 4 show that convergence is almost achieved at iteration 2000. Moreover, as concluded in the previous experiment, the altitudes of the UAVs serving large regions have increased, whereas the altitudes of the UAVs serving small regions, which are the ones closer to the high density of users area, have been reduced. Regarding the numerical results of the experiment, we provide two types of results: (1) results evaluating the fairness of the resources distribution among users of different regions; and (2) results evaluating the constraints imposition. In the evaluation of the fairness, we use the Jain's Fairness index [37], which has the following definition for a vector \(\mathbf{I}\) of length \(L\):  
\[f=\frac{\left(\sum_{i=1}^{L}l[i]\right)^{2}}{L\cdot\sum_{i=1}^{L}\left(l[i] \right)^{2}}. \tag{50}\]  
Figure 3: Initial UAV positions and service areas (**top left**); after 500 iterations (**top right**); after 1000 iterations (**center left**); after 1500 iterations (**center right**); after 2000 iterations (**bottom left**); and after 3000 iterations (**bottom right**). This index takes values between zero and one. Zero denotes a totally unfair distribution, whereas one denotes a totally fair distribution corresponding to all the elements in vector **I** being equal. This index is independent of the scale, the metric and the population size. The index is continuous, which implies that any slight change modifies the fairness index also slightly.