Using Poisson processes to model lattice cellular networks

TL;DR

This paper rigorously justifies the Poisson distribution assumption for base station placement in cellular networks under strong log-normal shadowing, showing convergence of various network models to a Poisson process and analyzing network performance.

Contribution

It provides a mathematical proof of Poisson convergence for lattice networks with shadowing and demonstrates invariance of the distribution of service characteristics, including SINR.

Findings

Poisson convergence occurs for a broad class of networks under increasing shadowing variance.

Distribution of service characteristics is invariant to fading or shadowing distribution.

New formulas for SINR distribution enable optimization of energy efficiency.

Abstract

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.