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Average Rate of Downlink Heterogeneous Cellular Networks over Generalized Fading Channels - A Stochastic Geometry Approach

Marco Di Renzo, Alessandro Guidotti, Giovanni E. Corazza

arXiv:1303.0529v2cs.IT

TL;DR

The paper addresses the difficulty of analytically computing downlink average rates in heterogeneous cellular networks with spatially varying interference and general fading. It introduces an MGF-based stochastic-geometry framework using aggregate-interference MGFs instead of coverage probability, and reports accurate, numerically stable evaluations across network and fading settings with reduced integral complexity.

  • Problem

    Analytical performance modeling is difficult because other-cell interference depends on base-station spatial positions and stochastic wireless-channel behavior, while heterogeneous deployments add diverse network elements and parameters.

  • Method

    The paper models multi-tier base stations as PPPs with tier-specific powers, densities, path-loss exponents, biases, and fading, then computes average rate through an MGF-based approach using aggregate-interference MGFs.

  • Results

    The framework provides accurate and numerically stable average-rate estimates across cellular-network setups and general fading models while requiring single- or two-fold rather than four-fold numerical integration.

  • Takeaways & Limitations

    The approach offers a tractable and numerically efficient way to analyze PPP-modeled heterogeneous cellular networks with varied fading and network configurations.

Abstract

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In this paper, we introduce an analytical framework to compute the average rate of downlink heterogeneous cellular networks. The framework leverages recent application of stochastic geometry to other-cell interference modeling and analysis. The heterogeneous cellular network is modeled as the superposition of many tiers of Base Stations (BSs) having different transmit power, density, path-loss exponent, fading parameters and distribution, and unequal biasing for flexible tier association. A long-term averaged maximum biased-received-power tier association is considered. The positions of the BSs in each tier are modeled as points of an independent Poisson Point Process (PPP). Under these assumptions, we introduce a new analytical methodology to evaluate the average rate, which avoids the computation of the Coverage Probability (Pcov) and needs only the Moment Generating Function (MGF) of the aggregate interference at the probe mobile terminal. The distinguishable characteristic of our analytical methodology consists in providing a tractable and numerically efficient framework that is applicable to general fading distributions, including composite fading channels with small- and mid-scale fluctuations. In addition, our method can efficiently handle correlated Log-Normal shadowing with little increase of the computational complexity. The proposed MGF-based approach needs the computation of either a single or a two-fold numerical integral, thus reducing the complexity of Pcov-based frameworks, which require, for general fading distributions, the computation of a four-fold integral.

I. INTRODUCTION

Analytical cellular-network modeling is difficult because spatially and stochastically varying other-cell interference is hard to represent, especially in increasingly heterogeneous deployments. The paper builds on PPP-based stochastic geometry and introduces an MGF-based average-rate framework intended to reduce computational complexity while supporting heterogeneous networks and general fading.

  • Accurate cellular-network performance modeling remains difficult because other-cell interference depends on random BS positions and wireless-channel variability.
  • A. Abstraction Models for Analysis and Design of Cellular Networks: Traditional abstraction models, including Wyner, single-cell-interferer, and grid models, can be oversimplified, inaccurate, or computationally demanding.
  • PPP-based stochastic geometry provides an abstraction for analyzing randomly distributed cellular networks and has been applied to interference, SIR, and related wireless problems.
  • C. Analytical Computation of the Average Rate: State–of–the–Art and Paper Contribution: The proposed MGF-based approach avoids computing coverage probability, uses the aggregate-interference MGF, and requires either one or two numerical integrals for arbitrary fading channels.
  • A. Heterogeneous Cellular Networks Model: The framework models heterogeneous deployments with tier-specific density, transmit power, path-loss exponent, biasing, and fading characteristics under PPP base-station locations.
  • The study is scoped to single-input-single-output transmission, while extensions to more advanced transmission technologies remain outside its scope.

B. Biased Long–Term Averaged Tier and BS Association

The paper models tier and BS association through long-term averaged biased received power, then computes average rate using an MGF-based framework rather than coverage probability.

  • Association policy: MT0 connects to the BS offering the highest long-term averaged biased received power across open-access tiers.Biasing factors modify each tier’s coverage range; values above one increase the corresponding tier’s coverage range.
  • Average-rate formulation: The average network rate is decomposed into tier-association probabilities and rates conditioned on association to each tier.The conditional rate further averages over the serving-BS distance and aggregate interference.
  • MGF-based methodology: The MGF-based approach computes the average rate directly from useful- and interference-link MGFs while avoiding coverage-probability computation.For multi-tier networks, the formulation uses aggregate-interference MGFs, including per-tier interference contributions.
  • Interpretation: The framework estimates the mean data rate achievable by a typical mobile terminal over a cell.This interpretation follows from validation of the PPP abstraction against conventional grid-based models.
  • Computational complexity: Two numerical integrals replace the four-fold integration required by general-fading coverage-probability frameworks.For many path-loss exponents, closed-form interference-MGF expressions can reduce the calculation to a single integral.
  • Density behavior: The average rate increases monotonically with BS density but has a finite upper bound.In very dense deployments, increasing transmit power does not increase the limiting average rate, and adding BSs alone is insufficient for very high rates.

A. Computation of TI (·) in (9) for General Fading Channels

The framework derives TI(·) for several standard and composite fading models, while retaining applicability to arbitrary fading distributions. Closed forms are available for many cases, but composite Rice–Log-Normal fading generally retains an infinite-series computation.

  • Standard fading models: Closed-form expressions for TI(·) are developed for Nakagami-m and Log-Normal interference fading.The Nakagami-m case models the power gain as Gamma distributed, while the Log-Normal case uses Gauss–Hermite quadrature weights and abscissas.
  • Composite fading models: Composite Nakagami-m and Log-Normal fading is handled by conditioning Gamma fading on a Log-Normal mean power.This yields the composite Gamma/LogN model used in Proposition 3.
  • Composite fading models: Composite Rice and Log-Normal fading is represented by a non-central Chi-Square distribution conditioned on a Log-Normal mean power.For K ≠ 0, TI(·) has a closed-form expression, whereas K = 0 reduces to composite Nakagami-m and Log-Normal fading with m = 1.
  • Computational caveat: For composite Rice and Log-Normal fading, computing TI(·) generally still requires an infinite series.The series can be approximated using the Nakagami-m/Rice-factor mapping with m = (1 + K)^2.
  • General fading applicability: The methodology applies to arbitrary fading distributions by evaluating integrals involving exp{−yx} f_gb(x), with closed forms possible when the transformed density remains exponential or has a Meijer G representation.For Meijer G representations, the infinite series in (9) generally cannot be avoided.

B. Efficient Computation of the Meijer G–Function in (11)

The paper improves numerical evaluation of the Meijer G-function used in average-rate calculations by switching to an asymptotic expansion for sufficiently small arguments. This produces an accurate and numerically stable computation with a tunable threshold ε.

  • Motivation: The average-rate computation may require evaluating the Meijer G-function over all positive real arguments, motivating an efficient numerical alternative.The function is available in standard mathematical software but can be difficult to compute for small arguments.
  • Piecewise evaluation: When α/2 is a ratio of positive integers, Corollary 2 computes GI(·) using an exact formula above ε and an asymptotic expansion below ε.The piecewise expression combines the exact and asymptotic terms through a threshold function.
  • Approximation: Neglecting the asymptotic term yields a computationally efficient approximation when ε is sufficiently small.The paper describes this approximation as accurate for sufficiently small ε.

C. Interference–Limited Scenario

In the interference-limited case, the average rate becomes independent of BS density and transmit power. With small nonzero noise, the rate approaches a common high-SNR upper bound with parameter-dependent convergence rates.

  • Interference-limited case: In the absence of background noise, the average rate simplifies through Corollary 3.The simplification follows from the noise-free form of the general MGF-based expression.
  • Interference-limited case: The interference-limited average rate is independent of BS density and transmit power for general fading channel models.Increasing either parameter therefore does not increase the average rate within this regime.
  • High-SNR case: With small nonzero noise, the average rate is bounded by high-SNR expressions and approaches its upper bound as SNR increases.The high-SNR approximation converges linearly, while the density-dependent approach has order α/2 convergence.
  • High-SNR case: A larger path-loss exponent α produces faster convergence to the limiting average rate as BS density increases.This is stated for the high-SNR limiting rate R(SNR∞) = R(λ∞).

E. Frequency Reuse

Random frequency reuse and correlated Log-Normal shadowing are incorporated into the MGF-based framework. Universal reuse maximizes average rate, while correlation adds only a single numerical integral under the equi-correlated model.

  • Frequency reuse: For FB frequency bands, random reuse modifies the interference MGF and replaces the BS density by λ(FB) = λ/FB.The tier and BS association PDF remains independent of FB.
  • Frequency reuse: The average rate is maximized at FB = 1, corresponding to universal frequency reuse.For dense deployments and interference-limited networks, the rate decreases linearly with the number of frequency bands.
  • Correlated shadowing: The MGF-based framework supports composite fading with Log-Normal shadowing, but the baseline theorem assumes i.i.d. fast fading and Log-Normal shadowing.The correlated-shadowing extension is restricted to equi-correlated fading because of the underlying stochastic-geometry interference model.
  • Correlated shadowing: The correlated-shadowing methodology extends to other fading models and multi-tier cellular networks by applying the same conditioning procedure.The illustrative construction uses composite Nakagami-m fast fading with correlated Log-Normal shadowing.
  • Correlated shadowing: Correlated Log-Normal shadowing is handled by conditioning on a common random variable, applying the independent-shadowing framework, and then averaging over the conditioning variable.The final averaging step adds one numerical integral, efficiently evaluated with Gauss–Hermite quadrature.

G. Pcov– vs. MGF–based Approach: A Comparison

The MGF-based approach reduces numerical-integration complexity relative to the Pcov-based approach while retaining applicability to general fading channels. It also provides simpler calculations and interpretable results in important operating scenarios.

  • The MGF-based approach generally requires a two-fold numerical integral, whereas the Pcov-based approach requires a four-fold integral for general fading distributions.Both approaches may involve special functions, and composite channel models may require Gauss–Hermite quadratures because of Log-Normal shadowing.
  • The MGF-based formulation reduces the number of fold integrals and avoids complex integrals compared with the Pcov-based formulation.For some path-loss exponents, Corollary 1 further reduces the MGF-based expression to a single integral.
  • In interference-limited scenarios, the MGF-based approach calculates average rate using a simple single integral, while the Pcov-based complexity is not significantly reduced.The Pcov-based simplification only sets exp=1 in one term and does not reduce the number of fold integrals.
  • The MGF-based average-rate expression supports intuitive performance analysis for dense deployments, interference-limited environments, and frequency-reuse strategies.The paper notes that the corresponding general-fading Pcov-based expression provides little insight, though it can simplify in special cases.
  • Across tested propagation and deployment scenarios, the MGF-based approach produced accurate average-rate estimates in less than five/six seconds for each SNR point.The tests used path-loss exponents covering typical cellular propagation environments and BS densities representing sparse, normal, and dense deployments.
  • When N = 0, the MGF-based computational complexity is further reduced, whereas the Pcov-based complexity is unaffected.These outcomes support the advantages of the MGF-based approach for cellular-network analysis and design.

IV. MULTI–TIER CELLULAR NETWORKS

The paper extends the MGF-based average-rate framework to generic multi-tier cellular networks with tier-dependent propagation and fading characteristics. The resulting formulation is broadly applicable but becomes simpler and more insightful when all tiers share a path-loss exponent.

  • IV. MULTI–TIER CELLULAR NETWORKS: Theorem 2 gives an average-rate expression for multi-tier networks with different path-loss exponents and fading distributions.The general expression typically requires a two-fold numerical integral and extends the framework to arbitrary tier-specific channel characteristics.
  • IV. MULTI–TIER CELLULAR NETWORKS: The aggregate-interference MGF for a tier factors across tiers because the PPPs are spatially and channel independent.Each tier contribution is computed from its density, interference MGF, and related analytical function.
  • IV. MULTI–TIER CELLULAR NETWORKS: The multi-tier formulation extends to correlated Log-Normal shadowing through the framework developed for the single-tier case.The extension follows immediately from Section III-F.
  • IV. MULTI–TIER CELLULAR NETWORKS: For equal path-loss exponents across tiers, Corollary 6 provides a simpler and more insightful analytical framework while retaining different per-tier fading distributions.The equal-exponent framework is less general than Theorem 2 but is more analytically tractable.
  • IV. MULTI–TIER CELLULAR NETWORKS: Theorem 1 reduces to Corollary 6 when T = 1, confirming consistency between the single-tier and equal-exponent multi-tier formulations.Remark 10 also identifies the same structural form between the corresponding single-tier and multi-tier expressions.
  • IV. MULTI–TIER CELLULAR NETWORKS: The equal-exponent multi-tier results generalize analyses of random frequency reuse, dense deployments, interference-limited systems, and high-SNR conditions.These extensions are given through Corollaries 7–9.

V. NUMERICAL AND SIMULATION RESULTS

The numerical results validate the MGF-based framework across single- and multi-tier networks, fading models, shadowing correlation, frequency reuse, and SNR regimes. The framework generally matches Monte Carlo simulations closely, while revealing how density, fading, path loss, correlation, reuse, and interference limitation affect average rate.

  • Framework Validation for Single-Tier Cellular Networks: The MGF-based approach provides very accurate average-rate estimates for single-tier networks across Rayleigh, Nakagami-m, Log-Normal, and composite fading models.Monte Carlo markers and analytical curves are compared for the considered fading distributions.
  • Framework Validation for Single-Tier Cellular Networks: Average rate increases with BS density, depends on the fading distribution, and improves with SNR before approaching a high-SNR asymptote.The high-SNR plateau corresponds to the interference-limited operating regime.
  • Impact of Fading Model and Fading Parameters: Average rate is slightly sensitive to Nakagami-m and Rice K parameters but decreases significantly as Log-Normal shadowing deviation σ increases.Less severe fading improves rate, while stronger shadowing substantially reduces it.
  • Correlated Log-Normal Shadowing: Correlated Log-Normal shadowing is accurately handled; at high SNR, rate is density-independent and increases with correlation, whereas at low SNR it slightly decreases.The high-SNR increase is attributed to the closest-BS association policy used in the paper.
  • Impact of Frequency Reuse: Frequency reuse is detrimental to average rate, despite significantly improving coverage probability.The resulting rate–coverage trade-off is outside the paper’s scope.
  • Framework Validation for Multi-Tier Cellular Networks: In two-tier networks, increasing the lower-tier BS density significantly increases average rate, and the framework remains very accurate for the tested composite fading models.The two fading models show negligible differences for the selected parameters.
  • High-SNR Scenario: The high-SNR approximation is accurate for SNR ≥ SNR*, with SNR* depending on BS density, path-loss exponent, and fading model; for α = 4 in Fig. 1, SNR*≈20 dB.Typical operating SNRs satisfy this condition in many analyzed deployments, though not universally for sparse networks with larger path-loss exponents.

VI. CONCLUSION

The paper introduces a comprehensive PPP-based framework for average-rate analysis in multi-tier cellular networks with general fading and numerically efficient computation.

  • The framework analyzes average rates in multi-tier cellular networks whose BSs follow a PPP spatial distribution.It supports general fading models with arbitrary fading parameters.
  • The methodology includes numerically efficient and stable algorithms for computing transcendental functions such as the Meijer G-function.
  • The framework is applicable to general fading channel models with arbitrary fading parameters.

APPENDIX I PROOF OF Theorem 1

The proof rewrites the average-rate expression using MGFs and independence, eliminating the need to compute coverage probability and reducing the problem to aggregate-interference analysis.

  • Conditioning on the serving distance makes the desired signal and interference independent, allowing the expectation to be rewritten through their MGFs.
  • The identity in (33) avoids computing coverage probability and differentiates the analytical development from prior approaches.
  • The aggregate-interference MGF accounts for interferers outside an association-induced exclusion region around the probe mobile terminal.The exclusion region contains no interfering BSs allowed to transmit.
  • The MGF of the aggregate interference is derived for general fading channels, addressing a gap where prior work provided only Rayleigh-fading bounds.
  • Closed-form expressions for the interference term are available for several fading models, including Nakagami-m, Log-Normal, and composite channels.

APPENDIX II PROOF OF Remark 1

The proof transforms the average-rate integral and derives bounds and limiting behavior by exploiting the rate integrand's bounds and asymptotic decay.

  • The average-rate expression is transformed using the substitution SNRy = z and a change of variables in the integral.
  • The transformed average rate is bounded using the fact that the relevant exponential term lies between 0 and 1.
  • For every α > 2, the rate contribution vanishes as λ approaches infinity, yielding the stated limiting result.
  • The interference term can be evaluated from derivatives of the interference MGF or through closed-form expressions for specific fading models.

APPENDIX IV PROOF OF Proposition 2

The proof obtains tractable interference-MGF expressions for Log-Normal and composite fading by approximating Log-Normal distributions with Gauss-Hermite quadrature and simplifying the resulting series.

  • The Log-Normal interference term is computed using an approximate Log-Normal PDF and special-function identities.
  • Composite Nakagami-m and Log-Normal fading is made analytically tractable through a Gauss-Hermite quadrature approximation of the Log-Normal distribution.
  • Composite Rice and Log-Normal fading is treated with the same Gauss-Hermite-based approximation and additional special-function identities.
  • The composite Rice and Log-Normal expression is further rearranged because its initial infinite series is not fast converging.
  • The asymptotic limit of the resulting expression is obtained by taking z toward zero and applying generalized hypergeometric-function limits.
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