[Paper Review] Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures

Bartłomiej Błaszczyszyn, Holger Paul Keeler|ArXiv.org|Jan 16, 2014

Advanced MIMO Systems Optimization62 references55 citations

TL;DR

This paper introduces a novel framework for analyzing the SINR process in Poisson wireless networks by deriving explicit, numerically tractable factorial moment measures for all orders. The key contribution is a complete characterization of the SINR process, enabling exact computation of k-coverage probabilities and joint distributions of order statistics under general signal combining and interference cancellation, with finite-series expansions replacing Laplace transform inversions.

ABSTRACT

Based on a stationary Poisson point process, a wireless network model with random propagation effects (shadowing and/or fading) is considered in order to examine the process formed by the signal-to-interference-plus-noise ratio (SINR) values experienced by a typical user with respect to all base stations in the down-link channel. This SINR process is completely characterized by deriving its factorial moment measures, which involve numerically tractable, explicit integral expressions. This novel framework naturally leads to expressions for the k-coverage probability, including the case of random SINR threshold values considered in multi-tier network models. While the k-coverage probabilities correspond to the marginal distributions of the order statistics of the SINR process, a more general relation is presented connecting the factorial moment measures of the SINR process to the joint densities of these order statistics. This gives a way for calculating exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations. The presented framework consisting of mathematical representations of SINR characteristics with respect to the factorial moment measures holds for the whole domain of SINR and is amenable to considerable model extension.

Motivation & Objective

• To develop a general mathematical framework for analyzing the SINR process experienced by a typical user in Poisson-based wireless networks.
• To characterize the full distribution of the SINR process via its factorial moment measures, which are expressed as explicit, numerically tractable integrals.
• To enable exact computation of k-coverage probabilities and joint distributions of the strongest SINR values without relying on Laplace transform inversions.
• To extend the framework to model signal combination and interference cancellation in multi-tier and heterogeneous networks.
• To leverage propagation invariance properties to unify analysis across different fading and shadowing distributions.

Proposed method

• Derives factorial moment measures of the SINR process using stochastic geometry and shot noise theory on a stationary Poisson point process.
• Introduces the STINR (signal-to-total-interference-and-noise ratio) process as a more tractable alternative to SINR, with a simple mapping to recover SINR results.
• Employs determinant identities and matrix calculus (e.g., Sherman–Morrison formula) to derive closed-form expressions for the factorial moment measures.
• Applies the Schuette-Nesbitt formula to express coverage probabilities as finite expansions in terms of factorial moment measures.
• Uses the algebraic structure of the SINR process to show that only finitely many base stations can simultaneously provide positive SINR above a threshold, enabling finite-series expansions.
• Leverages propagation invariance—where the distribution of the SINR process is invariant to the distribution of fading and shadowing under power-law path loss—enabling model generalization.

Experimental results

Research questions

• RQ1 How can the full distribution of the SINR process in Poisson wireless networks be characterized in a mathematically tractable way?
• RQ2 Can k-coverage probabilities be computed exactly using factorial moment measures, rather than approximations via Laplace transforms?
• RQ3 What is the relationship between the factorial moment measures of the SINR process and the joint densities of its order statistics?
• RQ4 How can signal combining and interference cancellation be modeled within this framework to compute exact coverage probabilities?
• RQ5 To what extent do propagation invariance properties simplify the analysis of SINR processes under general fading and shadowing?

Key findings

• The factorial moment measures of the SINR process are derived as explicit, numerically tractable integrals, enabling exact analysis without approximation.
• The k-coverage probability—defined as the probability that at least k base stations provide SINR above a threshold—can be computed as a finite series using the factorial moment measures.
• The joint distribution of the k strongest SINR values is fully characterized through the factorial moment measures, allowing exact analysis of signal combining and interference cancellation schemes.
• The framework reveals that only finitely many base stations can simultaneously provide SINR bounded away from zero, which justifies finite-series expansions and avoids infinite sums.
• The STINR process is shown to be equivalent to the SINR process under a simple transformation, and its determinant-based representation enables efficient computation of partial derivatives.
• Propagation invariance ensures that the SINR process distribution is invariant to the distribution of fading and shadowing under power-law path loss, simplifying model generalization.