  We propose a model for heterogeneous cellular networks assuming a space-time
Poisson process of call arrivals, independently marked by data volumes, and
served by different types of base stations (having different transmission
powers) represented by the superposition of independent Poisson processes on
the plane. Each station applies a processor sharing policy to serve users
arriving in its vicinity, modeled by the Voronoi cell perturbed by some random
signal propagation effects (shadowing). Users' peak service rates depend on
their signal-to-interference-and-noise ratios (SINR) with respect to the
serving station. The mutual-dependence of the cells (due to the extra-cell
interference) is captured via some system of cell-load equations impacting the
spatial distribution of the SINR. We use this model to study in a semi-analytic
way (involving only static simulations, with the temporal evolution handled by
the queuing theoretic results) network performance metrics (cell loads, mean
number of users) and the quality of service perceived by the users (mean
throughput) served by different types of base stations. Our goal is to identify
macroscopic laws regarding these performance metrics, involving averaging both
over time and the network geometry. The reveled laws are validated against real
field measurement in an operational network.
