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Using Poisson processes to model lattice cellular networks

Bartlomiej Blaszczyszyn, Mohamed Kadhem Karray, Holger Paul Keeler

arXiv:1207.7208v2math.PRcs.NI

TL;DR

The paper addresses whether Poisson base-station modeling can represent typical-user service characteristics beyond random deployments. It maps networks to propagation-loss processes, proves convergence under sufficiently strong log-normal shadowing, and develops Poisson-model SINR and energy-efficiency analyses.

  • Problem

    Poisson base-station modeling is widely used for tractable cellular-network quality-of-service analysis, although actual deployments may follow lattice or perturbed-lattice patterns.

  • Method

    The paper maps two-dimensional network placements to one-dimensional propagation-loss processes and analyzes their Poisson convergence under increasing log-normal shadowing.

  • Results

    Homogeneous networks, including hexagonal placements, converge to the Poisson propagation-loss process under sufficiently strong log-normal shadowing, while Poisson-model characteristics are invariant to additional fading distributions.

  • Takeaways & Limitations

    The results provide theoretical support for using Poisson models to analyze typical-user characteristics and enable SINR-distribution and mean-energy-efficiency optimization analyses.

Abstract

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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.

I. INTRODUCTION

The paper develops a tractable propagation-loss representation for typical-user service characteristics and rigorously supports Poisson modeling beyond inherently random deployments. It also derives SINR tools and applies them to energy-efficiency optimization.

  • Propagation-loss representation: Mapping base-station locations and propagation variables from R2 to propagation losses on R+ makes typical-user characteristics analytically accessible.The framework covers received-power-based quantities such as SINR, SIR, spectral efficiency, and energy efficiency.
  • Propagation-loss representation: The resulting propagation-loss process is Poisson regardless of the shadowing or fading distribution, with intensity determined by the distribution's 2/β moment.Here β is the distance-loss exponent, subject to the stated moment condition.
  • Convergence beyond Poisson deployments: Strong log-normal shadowing makes sufficiently homogeneous deterministic networks, including hexagonal and perturbed-lattice models, converge to the Poisson propagation-loss process.The convergence result extends the Poisson approximation to broad fixed network placements and sufficiently large finite patterns.
  • SINR analysis: Complementary analytic tools calculate the SINR distribution in the Poisson model for arbitrary shadowing or fading distributions.The paper positions these tools as extensions beyond results restricted to Rayleigh fading.
  • Efficiency analysis: Mean energy efficiency is evaluated as a function of base-station transmit power to support power optimization, including comparisons between Poisson and hexagonal networks.The comparison considers networks with and without shadowing.

III. CONVERGENCE RESULTS UNDER LOG-NORMAL SHADOWING

The paper studies deterministic and stationary base-station patterns under increasingly strong iid log-normal shadowing, using propagation-loss processes to establish Poisson convergence.

  • Model and assumptions: Sufficiently strong log-normal shadowing allows the infinite Poisson model to approximate typical-user characteristics for deterministic homogeneous patterns.The class includes fixed lattice placements and patterns satisfying an empirical homogeneity condition.
  • Model and assumptions: The empirical homogeneity condition is satisfied by any lattice pattern, including hexagonal networks, and by almost any ergodic point-process realization.
  • Shadowing model: Shadowing variables are iid log-normal across stations, with Gaussian logarithmic fluctuations whose standard deviation grows with the shadowing parameter.The dB-scale standard deviation is σ_dB = σ10/log 10.
  • Propagation-loss model: The distance-loss model uses constants K > 0 and β > 2, with K replaced by a σ-dependent function for the convergence analysis.
  • Propagation-loss representation: The network is mapped to a point process of propagation losses on R+, representing what the typical user experiences from the base stations.

B. Main result

The main theorem establishes weak convergence of propagation-loss processes to a Poisson process under empirical homogeneity and strong log-normal shadowing. The resulting approximation extends to large finite patterns and supports SINR, efficiency, and comparative network analyses.

  • Main theorem: As σ →∞, the propagation-loss process converges weakly to a Poisson point process with intensity measure parameter a = λπ/K^2.The analogous finite-pattern process has the same limit when additional stated conditions hold.
  • Implications: The Poisson approximation applies to homogeneous patterns including the standard hexagonal network and remains valid for sufficiently large finite patterns.
  • Scope and caveat: The distance-loss singularity at the origin can be mitigated by modifying the loss function near zero without materially affecting the Poisson approximation under the theorem’s finite-pattern condition.
  • Numerical illustration: For 9/10 shadowing realizations, a K-S test at α = 10% could not distinguish simulated SIR CDFs for a 900-station hexagonal torus from the infinite-Poisson prediction.
  • Applications: The Poisson-model analysis develops distributional tools independent of the shadowing or fading distribution and applies them to spectral and energy efficiency.
  • SINR representation: The SINR is represented using the path loss to the serving station and the corresponding SIR rather than directly using interference.

1) Path loss:

The path-loss process enables characterization of the serving link through its weakest propagation loss, yielding an explicit distribution and supporting subsequent interference and SIR analysis.

  • Serving path loss: The serving base station is associated with the minimum propagation loss L among the points of the loss process.The weakest propagation loss is almost surely attained because the process has finitely many points in every bounded interval.
  • Serving path loss: The CDF of L is P{L ≤ t} = 1 − exp[−a t^(2/β)], with a determined by the propagation-loss intensity measure.
  • Serving path loss: The distribution of L corresponds to a Fréchet distribution with shape parameter 2/β.
  • Interference factor: The conditional Laplace transform of the interference factor, together with the distribution of L, characterizes their joint distribution.
  • Interference factor: Conditioned on L = s, removing the smallest loss leaves a Poisson process on (s, ∞), enabling analysis of the interference factor.
  • SIR: For t ≥ 1, the complementary SIR CDF has an explicit expression, and the result remains valid for arbitrary shadowing distributions satisfying E[S^(2/β)] < ∞.

1) SINR:

The SINR analysis introduces noise explicitly and provides a route to evaluate its CDF, enabling study of SINR-based network functionals.

  • SINR: The SINR section evaluates the CDF of the signal-to-interference-and-noise ratio, where N denotes the noise power.

2) Spectral efficiency:

The paper defines spectral efficiency for a typical cellular-network user and develops numerical tools for evaluating SINR distributions and related efficiency measures. It also introduces energy efficiency as a power-dependent quantity with a non-trivial optimization.

  • Spectral efficiency: Spectral efficiency is defined as S := log(1 + SINR), measuring bits per second per Hertz delivered to the typical user.The definition is given for the AWGN setting with optimal theoretical link performance.
  • Energy efficiency: Energy efficiency is E = W log(1 + SINR(P))/P′, with consumed power P′ = cP + d.Because E(0) = E(∞) = 0, energy efficiency has a non-trivial optimization in transmitted power P.
  • SINR distribution: The CDF of Y := NL + f is computed because it suffices for evaluating the SINR distribution.The method combines the distribution of L with the conditional CDF of f and Laplace-transform inversion.
  • SINR distribution: The conditional CDF can be numerically evaluated with the trapezoidal rule, while the parameter γ controls approximation error.Other inversion techniques can also retrieve the conditional CDF.
  • SINR distribution: For t ≥ 1, the complementary CDF of the SINR admits an explicit expression, whereas the cited expression is not valid for t < 1.The restriction is noted in connection with Figure 2.

5) Numerical examples:

The numerical examples compare finite hexagonal, finite Poisson, and infinite Poisson models for SINR and energy efficiency. They show that sufficiently strong shadowing makes the infinite Poisson model a good approximation to the finite hexagonal network, including the power maximizing expected energy efficiency.

  • Simulation setup: σdB = 12dB is the default logarithmic standard deviation for log-normal shadowing in the numerical examples.The simulations use a COST-Hata urban distance-loss setting with β = 3.52 and a finite hexagonal network of 900 base stations.
  • SINR comparison: Figure 2 compares SINR CDFs for finite hexagonal networks with and without shadowing, finite Poisson networks, and the infinite Poisson model.The infinite Poisson curve is obtained by Laplace-transform inversion, while the explicit expression (20) is valid only for SINR > 1.
  • Energy-efficiency comparison: Figure 3 compares mean energy efficiency against transmitted power for finite hexagonal networks with and without shadowing and the infinite Poisson model.The consumed-to-emitted power relation uses c = 21,45 and d = 354.44W.
  • Numerical findings: High shadowing makes the infinite Poisson model a reasonable approximation of the finite hexagonal network in both figures.The approximation is especially accurate for the transmitted power at which the shadowed hexagonal network reaches maximal expected energy efficiency.
  • Numerical findings: Logarithmic shadowing standard deviation greater than approximately 10dB yields good approximations in realistic urban scenarios.The conclusion connects this numerical threshold with the paper’s Poisson-convergence result for actual, including regular hexagonal, networks.

A. Proof of Theorem 3

The proof establishes convergence of the logarithmically transformed propagation-loss process by verifying the conditions of a classical convergence theorem. The argument controls contributions from bounded and annular regions as the shadowing variance increases.

  • Proof of Theorem 3: The proof studies the propagation-loss point process on the logarithmic scale after setting n := σ^2.This transformation is used to analyze the image measure under increasing log-normal shadowing variance.
  • Proof of Theorem 3: A change of variables and integration by parts reduce the required integral estimates to Gaussian-tail expressions.The proof uses t = s − β log(Kr) − n/β over √n and the relation G(−t) = 1 − G(t).
  • Proof of Theorem 3: Conditions (10) and (11) ensure that boundary terms vanish and the relevant integral converges as n → ∞.The proof identifies equivalent limiting requirements involving the auxiliary sequences a_n and b_n.
  • Proof of Theorem 3: The annular decomposition A_k = B_0(r_{k+1}) \ B_0(r_k), with r_k = e^{ϵk}, bounds contributions from distant network points.Local finiteness supplies a maximizer in the inner region, while the remaining sums are controlled using bounds on ν_n.
  • Proof of Theorem 3: Letting ϵ → 0 and δ → 0 completes the bounds needed for the integral estimates.The same strategy, with a straightforward modification, proves the companion result.
  • Proof of Theorem 3: A classical convergence theorem then yields convergence for all bounded Borel sets after the two required conditions are verified.The first condition follows from (26), and the second follows from Lemmas 13 and 14.

B. Representation of the function ϕβ(z) given by (14)

This appendix rewrites the function ϕβ(z) using modified incomplete gamma functions and provides an expansion used to establish positivity.

  • Function representation: The function ϕβ(z) is represented through recurrence relations and the modified incomplete gamma function γ∗.The modified incomplete gamma function is holomorphic on C × C in the stated representation.
  • Function representation: The expansion implies ϕβ(z) > 0 for β > 2 and z ∈ R+.Positivity follows because γ∗(α,z) > 0 when α > −1 and z ∈ R+.
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