  Stochastic geometry (SG) has been successfully used as a modelling tool for
cellular networks to characterize the coverage probability in both the downlink
(DL) and uplink (UL) systems, under the assumption that the base stations (BS)
are deployed as a Poisson point process. In the present article, we extend this
use and provide further results for interference limited and Rayleigh fading
networks, culminating in a multifaceted contribution. First, we compactly model
the two systems at once, allowing parallels to be drawn and contrast to be
created. Also, for DL we manage to obtain two closed form expressions for two
special cases. Moreover, for UL, notorious for being difficult, we develop a
clever approximation that overcomes the difficulty, yielding excellent results.
Additionally, we present two efficient Monte Carlo simulation algorithms,
designed primarily to validate the models, but can be of great use for SG
modelling of communications systems in general. Finally, we prove two theorems
at odds with popular belief in cellular communications research. Specifically,
we prove that under the SG model, the coverage probability in both DL and UL is
independent of BS density. Based on this revelation, a plethora of results in
the literature have to be re-examined to rid them of a parameter that has been
proven superfluous.
