  We consider multiple unmanned aerial vehicles (UAVs) serving a density of
ground terminals (GTs) as base stations. The objective is to minimize the
outage probability of GT-to-UAV transmissions. Optimal placement of UAVs under
different UAV altitude constraints and GT densities is studied. First, using a
random deployment argument, a general upper bound on the optimal outage
probability is found for any density of GTs and any number of UAVs. A matching
lower bound is also derived to show that the optimal outage probability decays
exponentially with the number of UAVs. Next, the structure of optimal
deployments is studied when the common altitude constraint is large. For a wide
class of GT densities, it is shown that all UAVs should be placed to the same
location in an optimal deployment. A design implication is that one can use a
single multi-antenna UAV as opposed to multiple single-antenna UAVs without
loss of optimality. This result is also extended to a practical variant of the
Rician fading model recently developed by Azari et al. for UAV communications.
Numerical deployment of UAVs in the centralized and practical distributed
settings are carried out using the particle swarm optimization and modified
gradient descent algorithms, respectively.
