Using Poisson processes to model lattice cellular networks (1207.7208v2)

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.

• The paper establishes that significant log-normal shadowing causes structured cellular networks to behave like non-homogeneous Poisson processes.
• The methodology maps 2D network propagation to a 1D point process, enabling rigorous Poisson convergence theorems and simulations.
• The results broaden analytic techniques to account for diverse fading conditions, enhancing SINR predictions in realistic urban settings.

An Analytical Study of Cellular Networks Utilizing Poisson Processes

The paper "Using Poisson processes to model lattice cellular networks" by Bartłomiej Błaszczyszyn, Mohamed Kadhem Karray, and Holger Paul Keeler provides a comprehensive analysis of using Poisson point processes to model cellular networks specifically under conditions of strong log-normal shadowing. The authors rigorously justify employing a Poisson model for cellular network analysis, particularly in settings where traditional hexagonal models (often used due to their regularity and simplicity) do not capture the real-world irregularities of base station placement.

Key Contributions and Findings

The paper's primary contribution is a formal validation of modeling propagation losses in cellular networks using Poisson processes. The authors present a convergence result demonstrating that, under substantial log-normal shadowing, a typical user's network experience converges to that of a non-homogeneous Poisson process. This convergence occurs irrespective of the actual, possibly regular, base station deployments. The result is pivotal as it extends the theoretical foundation supporting the use of the Poisson process model, which traditionally was justified through heuristic arguments about network irregularities.

The research posits that, even in structured networks such as grid-based configurations, significant shadowing variance causes the transmission environments to behave statistically like Poisson-based networks. This is demonstrated through a Poisson convergence theorem that applies to a range of network models, including hexagonal and perturbed lattice structures, when shadowing creates sufficient variance (illustratively, with a logarithmic standard deviation over 10dB).

Implications for Network Analysis

The authors' argument is anchored in robust statistical mechanics, offering a deeper understanding of how realistic environmental randomness (shadowing) impacts signal propagation. By verifying the invariance of the Poisson representation with respect to the secondary fading distribution, the paper ensures that existing technical results relying on the Poisson model hold more broadly than previously established. This encompasses extending analytic techniques originally derived under specific fading assumptions (like Rayleigh fading) to broader contexts, as indicated by the preservation of SINR distributions.

Methodological Approach

The methodologies employed involve mapping typical-user propagation characteristics from a 2D network representation to a 1D point process framework. This transformation facilitates convergence theorems, reinforcing the principal outcome of perceiving even non-Poisson, deterministic networks as Poisson in nature under strong shadowing effects. Numerical experiments corroborate these conclusions, with empirical studies and simulations aligning with the theoretical Poisson model predictions.

Future Directions

The paper sets a foundation for further exploration into effective modeling of cellular networks, especially in urban settings where shadowing is inherently variable. Its insights open avenues for optimizing network deployments, accounting for energy and spectral efficiency in scenarios where actual network topology is veiled by environmental factors.

Future work may focus on refining these models by examining broader classes of fading and shadowing distributions, potentially incorporating dynamic elements such as user mobility and adaptive power control in the base stations. Furthermore, the insights could be extrapolated to more complex heterogeneous network scenarios beyond the current multi-tier models.

Overall, the paper provides a critical reassessment of modeling assumptions in cellular networks, promoting the Poisson process as a versatile and theoretically sound candidate for capturing the stochastic nature of real-world wireless systems.