  In this paper, we analyze the signal-to-interference-plus-noise ratio (SINR)
performance at a mobile station (MS) in a random cellular network. The cellular
network is formed by base-stations (BSs) placed in a one, two or three
dimensional space according to a possibly non-homogeneous Poisson point
process, which is a generalization of the so-called shotgun cellular system. We
develop a sequence of equivalence relations for the SCSs and use them to derive
semi-analytical expressions for the coverage probability at the MS when the
transmissions from each BS may be affected by random fading with arbitrary
distributions as well as attenuation following arbitrary path-loss models. For
homogeneous Poisson point processes in the interference-limited case with
power-law path-loss model, we show that the SINR distribution is the same for
all fading distributions and is not a function of the base station density. In
addition, the influence of random transmission powers, power control, multiple
channel reuse groups on the downlink performance are also discussed. The
techniques developed for the analysis of SINR have applications beyond cellular
networks and can be used in similar studies for cognitive radio networks,
femtocell networks and other heterogeneous and multi-tier networks.
