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Coverage and Rate Analysis for Millimeter Wave Cellular Networks

Tianyang Bai, Robert W. Heath

arXiv:1402.6430v3cs.ITcs.NI

TL;DR

MmWave cellular analysis requires models that account for blockage-sensitive LOS/NLOS propagation, which conventional UHF models do not directly provide. The paper develops a stochastic-geometry framework with distance-dependent LOS processes and a simplified equivalent LOS-ball model, finding comparable coverage and higher rates in dense networks, while optimal performance occurs at finite density.

  • Problem

    Blockage-sensitive mmWave propagation creates distinct LOS and NLOS path-loss characteristics, so conventional UHF cellular models do not directly analyze mmWave networks.

  • Method

    The paper models LOS and NLOS base stations as independent non-homogeneous PPPs under a distance-dependent LOS probability, derives SINR and rate coverage, and approximates irregular LOS regions with an equivalent fixed LOS ball.

  • Results

    Dense mmWave networks can achieve SINR coverage comparable to conventional UHF networks and significantly higher achievable rates, while optimal SINR and rate coverage require finite base-station density.

  • Takeaways & Limitations

    MmWave performance is largely determined by base-station density relative to blockage density, and further densification need not improve SINR in ultra-dense networks.

Abstract

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Millimeter wave (mmWave) holds promise as a carrier frequency for fifth generation cellular networks. Because mmWave signals are sensitive to blockage, prior models for cellular networks operated in the ultra high frequency (UHF) band do not apply to analyze mmWave cellular networks directly. Leveraging concepts from stochastic geometry, this paper proposes a general framework to evaluate the coverage and rate performance in mmWave cellular networks. Using a distance-dependent line-of-site (LOS) probability function, the locations of the LOS and non-LOS base stations are modeled as two independent non-homogeneous Poisson point processes, to which different path loss laws are applied. Based on the proposed framework, expressions for the signal-to-noise-and-interference ratio (SINR) and rate coverage probability are derived. The mmWave coverage and rate performance are examined as a function of the antenna geometry and base station density. The case of dense networks is further analyzed by applying a simplified system model, in which the LOS region of a user is approximated as a fixed LOS ball. The results show that dense mmWave networks can achieve comparable coverage and much higher data rates than conventional UHF cellular systems, despite the presence of blockages. The results suggest that the cell size to achieve the optimal SINR scales with the average size of the area that is LOS to a user.

I. INTRODUCTION

The paper develops a stochastic-geometry framework for mmWave cellular networks that models blockage-sensitive LOS and NLOS propagation, antenna directivity, coverage, and rate. Its dense-network analysis indicates that mmWave systems can provide good coverage and higher rates, while excessive densification need not improve SINR.

  • Motivation: MmWave’s large bandwidth motivates its use for fifth-generation cellular access, while antenna arrays provide array gain to offset frequency-dependent path loss.At 30 GHz, free-space path loss is 20 dB larger than at 3 GHz, but compact wavelengths permit large arrays and beamforming.
  • Motivation: Blockages create pronounced LOS and NLOS propagation differences, making lower-frequency cellular models insufficient for direct mmWave analysis.Indoor users are unlikely to be covered by outdoor mmWave base stations because building materials cause severe penetration loss.
  • Framework: The proposed framework models blockages stochastically, applies a distance-dependent LOS probability, and separates LOS and NLOS base stations into independent non-homogeneous PPPs.Different path loss laws are applied to the two processes, while directional beamforming is represented with a sectored antenna model.
  • Results: Dense mmWave networks can generally achieve good coverage and significantly higher achievable rates than conventional cellular networks, despite blockage effects.The analysis finds coverage and rate sensitivity to base-station density and blockage distribution.
  • Dense-network analysis: In dense networks, the optimal cell size scales with the LOS-region size, while increasing base-station density indefinitely need not improve SINR.The simplified model uses an equivalent fixed LOS ball and predicts a finite optimal base-station density.
  • Contributions: The paper generalizes prior analyses with general LOS functions, general small-scale fading, unrestricted LOS path loss exponents, and a more efficient coverage-probability computation.The new approach avoids numerically inverting a Fourier transform required by prior expressions.

III. COVERAGE AND RATE ANALYSIS IN GENERAL NETWORKS

This section analyzes coverage and rate in the general mmWave model by deriving SINR and rate-coverage expressions, then simplifying arbitrary LOS probabilities with a moment-matched step function.

  • General-network analysis: The analysis first establishes SINR ordering for antenna-pattern parameters and derives coverage and rate-coverage expressions for a general LOS probability function.It then introduces a systematic approximation of the LOS function by a moment-matched equivalent step function.

A. Stochastic Ordering of SINR With Different Antenna Geometries

The section characterizes how antenna geometry affects SINR and develops the probabilistic machinery for coverage analysis under LOS/NLOS association. It culminates in a computable SINR-coverage expression whose approximation agrees favorably with simulations and is more efficient than prior results.

  • Antenna geometry: Directional-array performance depends on beamwidth, directivity gain, and back-lobe gain, motivating stochastic SINR-ordering results for antenna geometries.The ordering analysis focuses on transmitter geometry but states that the same results apply to the receiver.
  • Directivity gains: Larger main-lobe directivity gain or front-to-back ratio yields a better SINR distribution when the other antenna parameters are fixed.This result is stated as stochastic ordering with respect to directivity gains.
  • Beamwidth: A smaller beamwidth provides a better SINR distribution when main-lobe gain and front-to-back ratio are fixed.Under the model, narrower beams reduce the number of interfering base stations transmitting through their main lobes while leaving the desired signal unchanged.
  • Beamwidth: The beamwidth ordering applies when beamwidths are larger than channel angle spread; narrower beams can otherwise lose signal energy outside the main lobe.The paper gives beamwidths above 55° as an example condition based on measurements.
  • Coverage analysis: The coverage analysis splits outdoor base stations into independent LOS and NLOS PPP tiers and associates the user with the base station having the smallest path loss.Conditional serving-distance distributions and LOS/NLOS association probabilities support the subsequent coverage calculation.
  • Coverage analysis: Theorem 1 computes SINR coverage as a weighted combination of conditional LOS and NLOS coverage probabilities.The conditional terms are evaluated using the serving-distance distributions and Nakagami small-scale-fading parameters.
  • Coverage analysis: Theorem 1’s coverage expressions compare favorably with simulations and compute more efficiently than prior results requiring numerical Fourier-transform inversion.The expressions may still require numerical evaluation of multiple integrals, especially for complicated LOS probability functions.

C. Rate Analysis

The paper defines achievable rate from SINR with a bandwidth and distortion threshold, then derives average-rate and rate-coverage expressions from SINR coverage.

  • Rate definition: Achievable rate is defined as Γ = W log2(1 + min{SINR, Tmax}).W is the bandwidth assigned to the typical user, while Tmax limits exploitable SINR because of constellation and RF-front-end distortions.
  • Rate definition: The distortion threshold Tmax accounts for very high mmWave SINRs that may be limited by RF-front-end linearity and related factors.
  • Average rate: Average achievable rate E[Γ] is computed from the SINR coverage probability Pc(T).The resulting lemma provides a first-order characterization of the rate distribution.
  • Rate coverage: Rate coverage probability PR(γ) is the probability that the achievable rate exceeds threshold γ, and it is obtained from Pc(T) by a change of variables.
  • Rate coverage: The rate-coverage expression enables comparisons between mmWave and conventional systems using different bandwidths.

D. Simplification of LOS Probability Function

The paper replaces a general distance-dependent LOS probability with a fixed equivalent LOS ball to make SINR analysis and dense-network design more tractable.

  • Step-function approximation: Approximating p(t) by a step function makes LOS probability one within radius RB and zero outside it.This replaces the irregular LOS region with a fixed equivalent LOS ball.
  • Benefits and interpretation: The fixed-ball approximation enables faster numerical computation and simpler dense-network analysis than integrating over an irregular LOS region.Simulations are reported to show that its approximation error is generally small in dense mmWave networks.
  • Radius criteria: The equivalent LOS-ball radius RB can be chosen by matching the average number of LOS base stations when the relevant first moment is finite.For a step function, the average LOS base-station number is ρ = πλR2.
  • Benefits and interpretation: The general LOS model produces an irregular, random user-observed region because building locations are random.
  • Radius criteria: When the first-moment condition is not satisfied, RB is instead chosen to preserve the LOS association probability.For the step function, that probability is 1 − e^−λπR2_B.

IV. ANALYSIS OF DENSE MMWAVE NETWORKS

The dense-network analysis is motivated by numerical evidence that substantial mmWave coverage requires dense deployments, and it derives simplified SINR insights for this regime.

  • The paper specializes its framework to dense networks because numerical results indicate that significant coverage requires dense mmWave deployments.
  • The dense-network analysis derives simplified SINR expressions and examines system-performance insights in this asymptotic regime.

A. Dense Network Model

Dense networks are modeled through relative LOS-base-station density and simplifying assumptions that remove NLOS links, thermal noise, and small-scale fading from the analysis.

  • Dense network model: A network is dense when the average observed LOS base-station number ρ exceeds K or LOS association probability AL exceeds 1 − ϵ.For illustration, the paper sets K = 1 and ϵ = 5%.
  • Dense network model: The equivalent LOS-ball model confines LOS base stations to B(0, RB), yielding ρ = λπR2_B.
  • Dense network model: Relative density ρ equals the average number of LOS base stations, the LOS-area-to-cell-size ratio, and normalized base-station density.The paper states that dense-network SINR coverage is largely determined by ρ.
  • Model assumptions: The dense-network analysis ignores NLOS base stations and thermal noise because performance is assumed to be limited by other LOS interferers.Simulations report negligible evaluation error from these omissions.
  • Model assumptions: Small-scale fading is ignored because measurements found nearby mmWave LOS-transmitter signal power to be almost deterministic.
  • SINR analysis: When αL = 2, the dense-network SINR-coverage expression admits a further simplification.

C. Asymptotic Analysis in Ultra-Dense Networks

The paper analyzes asymptotic SIR behavior in ultra-dense mmWave networks and finds that densification can eventually reduce performance when all base stations remain active.

  • The asymptotic SIR depends on the LOS path loss exponent: it vanishes for αL ≤ 2 but converges to a nonzero random variable for αL > 2.
  • Increasing base station density beyond a threshold hurts performance, so the SINR-optimal base station density is finite.
  • At very high density, users observe several LOS base stations and experience significant interference.
  • The asymptotic trends assume that all base stations are active; turning off some base stations is proposed as simple interference management.

A. General Network Simulations

Simulations examine how antenna geometry, cell size, LOS association, approximation accuracy, and density affect mmWave coverage and rate. They show that denser deployments improve LOS association and rates up to an operating point, while excessive densification can reduce SINR coverage.

  • A. General Network Simulations: Increasing main-lobe gain and decreasing main-lobe beamwidth improve SINR when side-lobe gain is fixed.
  • A. General Network Simulations: LOS association probability increases as average cell radius decreases, with received power mostly determined by LOS base stations below 100 meters average cell size.
  • A. General Network Simulations: Theorem 1 matches simulations with negligible errors, while mmWave SINR coverage remains sensitive to base station density and generally requires small cells.
  • A. General Network Simulations: The step-function LOS approximation provides a lower bound on SINR coverage, with smaller errors at higher density and faster coverage evaluation.
  • B. Dense Network Simulations: Coverage curves peak near ρ = 5, corresponding to an average cell radius about half the LOS range, then decline as density becomes very large.
  • B. Dense Network Simulations: The optimal base station density is generally insensitive to LOS path loss exponent changes from 1.5 to 2.5.
  • A. General Network Simulations: At relative density ρ ≈ 1, mmWave provides comparable spectrum efficiency to conventional UHF systems; at high density, achievable rates are an order of magnitude better.

C. Comparison with Real-scenario Simulations

The proposed analytical models are compared with simulations using the University of Texas at Austin campus building distribution. They generally characterize the real scenario closely, while the study reports density- and blockage-dependent coverage and rate conclusions.

  • Simulation setup: β = 0.0063 m−1 and R_B = 225 m are selected to match the area’s building statistics.
  • Simulation setup: The real-scenario simulations use the University of Texas at Austin campus building distribution and a modified directional antenna pattern.The analytical models fit LOS probability parameters to the campus building statistics and use a sectored beamforming model.
  • Comparison with simulations: The analytical models generally closely characterize the real-scenario results, despite some deviations in the high-SINR regime.The comparison uses the campus building distribution and fitted LOS parameters.
  • Conclusions: SINR coverage can be comparable to conventional UHF networks when the base station density is sufficiently high.
  • Conclusions: Achievable rates can be significantly higher than in conventional networks because of the larger available bandwidth.
  • Conclusions: Performance is largely determined by relative base station density, while increasing density can transition the network from power-limited to interference-limited operation.Relative base station density is the ratio of base station density to blockage density.
  • Conclusions: Optimal SINR and rate coverage can be achieved at a finite base station density in ultra-dense mmWave networks.Further density increases need not improve SINR.

APPENDIX A

Appendix A derives coverage expressions by conditioning on LOS or NLOS association and applying stochastic-geometry calculations to the independent base-station processes.

  • Coverage derivation: The user associates with a LOS base station when one exists and its nearest LOS station has lower path loss than the nearest NLOS station.
  • Coverage derivation: The LOS association analysis uses the probability of observing at least one LOS base station and the distance density to the nearest LOS station.
  • Conditional coverage: The LOS and NLOS conditional coverage probabilities are derived separately using the same general approach.
  • Interference analysis: The LOS interference term is evaluated using the Laplace functional of the LOS PPP and the moment-generating function of gamma fading.
  • Overall coverage: Overall coverage combines LOS- and NLOS-associated conditional coverage through the law of total probability.The combination is P_c(T) = A_LP_c,L(T) + A_NP_c,N(T).

APPENDIX D

Appendix D develops coverage approximations for general LOS path-loss exponents and simplifies the α_L = 2 case using gamma-distribution and PPP interference calculations.

  • General exponent: For general α_L, the coverage probability is conditioned on the serving-base-station distance and interference power.
  • Fading approximation: The desired-link fading is approximated using convergence of a normalized Gamma distribution toward an identity variable as its parameter grows.
  • Interference calculation: The LOS interference contribution is obtained from the Laplace functional of the LOS PPP after a change of variables.
  • Special case α_L = 2: When α_L = 2, integration by parts and Lemma 7 further simplify the coverage expression.

APPENDIX E

Appendix E analyzes dense-network SIR through an equivalent-network scaling that preserves relative density, then bounds the asymptotic SIR using interference directivity assumptions.

  • Equivalent-network scaling: Equivalent networks scale base-station density and LOS range while preserving the relative density ρ = λπR_B^2.
  • Equivalent-network scaling: All networks in the equivalent class have the same SIR distribution because the scaling factor cancels in the SIR expression.
  • Asymptotic regime: The asymptotic analysis fixes base-station density and examines the equivalent network as the LOS-ball radius grows without bound when ρ tends to infinity.
  • SIR bounds: Interfering links are bounded by assigning either maximum or minimum directivity gain to all interferers.
  • Asymptotic regime: For α_L > 2, the appendix computes a lower bound on asymptotic SIR coverage for T > 1.
  • Asymptotic regime: For α_L ≤ 2, an upper bound on SIR converges to zero in probability in the asymptotic equivalent network.
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