# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 6 Evaluation and Results
### Positioning Strategy Evaluation
To fit it in the multicell scenario, Equation (51) has been derived taking into account that at each region \(A_{i}\) there are \(N_{i}\) users with the same rate:  
\[f=\frac{(\sum_{i=1}^{P}N_{i}\cdot\overline{R}\big{|}_{A_{i}})^{2}}{\sum_{i=1}^ {P}N_{i}\cdot\sum_{i=1}^{P}N_{i}\cdot\left(\overline{R}\big{|}_{A_{i}}\right)^ {2}}. \tag{51}\]  
In our simulation, the fairness index has been increased from 0.77006 at the initial step up to 0.99999 in iteration 3000, which was the purpose of the proposed strategy. The rates are 9.29, 9.29, 9.28, 9.28, 9.37, 9.27, 9.27, 9.27, and 9.27 kbps and the average user rate in the entire scenario is 9.29 kbps, which is computed as follows:  
\[\overline{R_{T}}=\frac{\sum_{i=1}^{P}\overline{R}\big{|}_{A_{i}} \cdot N_{i}}{\sum_{i=1}^{P}N_{i}}. \tag{52}\]  
These rates are not very high, but we have to consider that the number of users per cell after 3000 iterations is high, around 10,000. Moreover, in real life, not all the users are transmitting simultaneously. Thus, assuming that only 10% of users are transmitting simultaneously, the obtained 10,000 active users per region would correspond to a total number of 100,000 persons in the region, which is a huge population. This amount of users can easily be found in a large sporting facility (there exists some stadiums with a capacity of 100,000 people) placed in a city (the population density of a city such as Barcelona is 15,000 people per km\({}^{2}\)). In this way, the regions placed outside the stadium still contain a huge concentration of users due to the population density of the city. However, as this density is lower than inside the stadium, these regions are larger. Note that the proposed equal rate scheduler assures equal rate among the set of active users at a given time. That means that, if either the composition of the group of active users or the traffic  
Figure 4: Evolution of the rate per user, for all coverage areas. Each color represents one coverage area using the same colors as the ones assigned to letters in Figure 3.  
profiles change, then the achieved rate may change. A possible solution entails elaborating a more sophisticated resource allocation strategy that forces fairness among all possible sets of active users and traffic profiles. Note, however, that it is very complicated since the scheduler would have to work based on predictions and fairness could only be assured with a certain probability. In fact, fairness should be defined in a different way for each kind of traffic profile. Anyway, the development of this sophisticated scheduler is out of the scope of this paper, whose main objective is to show a framework in which it is proved that performance can be improved by considering jointly resource scheduling and navigation. Regarding the constraints imposition, although in some iterations there can be unfulfilled constraints, the most important thing is that all of them are fulfilled on average to ensure a proper distribution of the resources. After 3000 iterations the links achieve the following average backhaul capacities \(\overline{R_{i}^{BH}}\): 100.01, 111.87, 124.36, 111.77, 143.38, 155.72, 124.57, 155.99, and 173.29 Mbps, which are only slightly higher than their average aggregated rate \(\overline{\overline{R}|_{A_{i}}\cdot N_{i}}\): 99.31, 106.43, 107.31, 106.28, 139.58, 150.30, 106.99, 151.51, and 135.75 Mbps. The average total shared backhaul bandwidth \(\overline{TSBB}\) (179.94 MHz) is lower than total backhaul bandwidth \(B_{BH}^{total}\) (180 MHz). Thus, analyzing the results after 3000 iterations, it can be concluded that all the constraints are fulfilled in average, even the constrains in Equation (24), which were not fulfilled in the initialization in the regions far away from the backhaul ground antenna (6100 m, 6100 m).