  We investigate the performance of a downlink ultra-dense network (UDN) with
directional transmissions via stochastic geometry. Considering the dual-slope
path loss model and sectored beamforming pattern, we derive the expressions and
asymptotic characteristics of the coverage probability and constrained area
spectrum efficiency (ASE). Several special scenarios, namely the physically
feasible path loss model and adjustable beam pattern, are also analyzed.
Although signal-to-interference-plus-noise ratio collapsing still exists when
the path loss exponent in the near-field is no larger than 2, using strategies
like beam pattern adaption, can avoid the decrease of the coverage probability
and constrained ASE even when the base station density approaches infinity.
