  An almost ubiquitous assumption made in the stochastic-analytic study of the
quality of service in cellular networks is Poisson distribution of base
stations. It is usually justified by various irregularities in the real
placement of base stations, which ideally should form the hexagonal pattern. We
provide a different and rigorous argument justifying the Poisson assumption
under sufficiently strong log-normal shadowing observed in the network, in the
evaluation of a natural class of the typical-user service-characteristics
including its SINR. Namely, we present a Poisson-convergence result for a broad
range of stationary (including lattice) networks subject to log-normal
shadowing of increasing variance. We show also for the Poisson model that the
distribution of all these characteristics does not depend on the particular
form of the additional fading distribution. Our approach involves a mapping of
2D network model to 1D image of it "perceived" by the typical user. For this
image we prove our convergence result and the invariance of the Poisson limit
with respect to the distribution of the additional shadowing or fading.
Moreover, we present some new results for Poisson model allowing one to
calculate the distribution function of the SINR in its whole domain. We use
them to study and optimize the mean energy efficiency in cellular networks.
