PaperlayerView exact revision on arXiv

Source-linked AI summary

Stochastic Geometry Modeling and Analysis of Multi-Tier Millimeter Wave Cellular Networks

Marco Di Renzo

arXiv:1410.3577v1cs.IT

TL;DR

The paper addresses the limited applicability of conventional cellular models to mmWave networks, where realistic path-loss and blockage behavior must be represented. It develops a PPP-based analytical framework covering association, beamforming, and multi-tier deployments, and finds that sufficiently dense mmWave networks can outperform µWave counterparts in coverage probability and average rate.

  • Problem

    Conventional µWave cellular-network models are not directly applicable to mmWave networks because their path-loss and blockage characteristics differ.

  • Method

    The paper uses stochastic geometry with PPP-modeled base stations, realistic empirical path-loss and blockage models, alternative association criteria, beamforming alignment errors, and multi-tier deployments.

  • Results

    Sufficiently dense mmWave cellular networks can outperform µWave counterparts in coverage probability and average rate.

  • Takeaways & Limitations

    The framework provides coverage- and rate-analysis tools for realistic mmWave cellular deployments under multiple association and network-tier configurations.

  • Takeaways & Limitations

    The methodology relies on a noise-limited assumption, shown to be sufficiently accurate for typical base-station densities.

Abstract

from arXiv · show

In this paper, a new mathematical framework to the analysis of millimeter wave cellular networks is introduced. Its peculiarity lies in considering realistic path-loss and blockage models, which are derived from recently reported experimental data. The path-loss model accounts for different distributions of line-of-sight and non-line-of-sight propagation conditions and the blockage model includes an outage state that provides a better representation of the outage possibilities of millimeter wave communications. By modeling the locations of the base stations as points of a Poisson point process and by relying on a noise-limited approximation for typical millimeter wave network deployments, simple and exact integral as well as approximated and closed-form formulas for computing the coverage probability and the average rate are obtained. With the aid of Monte Carlo simulations, the noise-limited approximation is shown to be sufficiently accurate for typical network densities. The proposed mathematical framework is applicable to cell association criteria based on the smallest path-loss and on the highest received power. It accounts for beamforming alignment errors and for multi-tier cellular network deployments. Numerical results confirm that sufficiently dense millimeter wave cellular networks are capable of outperforming micro wave cellular networks, both in terms of coverage probability and average rate.

I. INTRODUCTION

The paper develops a stochastic-geometry framework tailored to mmWave cellular networks, whose propagation and deployment characteristics differ from conventional µWave models. It incorporates realistic blockage, association, beamforming, and multi-tier considerations to evaluate coverage and rate.

  • Reported implication: Numerical results indicate that sufficiently dense mmWave networks may outperform µWave networks when comparable densities and sufficient beamforming gain are available.The reported comparison concerns cellular-network coverage and rate performance.
  • Analytical approach: PPP-based stochastic geometry provides an analytically tractable abstraction for modeling base-station locations and studying heterogeneous cellular deployments.The approach models BS locations as points of a Poisson Point Process while retaining tractable system-level analysis.
  • Motivation: µWave cellular-network models are not directly applicable to mmWave systems because their path-loss and blockage characteristics differ substantially.The paper emphasizes distinct LOS/NLOS distributions, stronger spatial-blockage effects, and an additional outage state at mmWave frequencies.
  • Prior limitations: Existing mmWave stochastic-geometry approaches only partly capture realistic link states, association criteria, beamforming errors, and multi-tier interference.The paper identifies these omissions as limiting accurate system-level performance evaluation.
  • Contribution: The proposed methodology explicitly models empirical path-loss and blockage behavior, alternative cell-association criteria, beamforming pointing errors, and multi-tier deployments.This framework is designed specifically for mmWave cellular communications and performance evaluation.
  • Scope and evaluation: The framework derives coverage and rate analyses for smallest-path-loss and highest-received-power association, then extends them to beamforming errors and multi-tier networks.The paper validates the analysis numerically and compares mmWave with µWave cellular performance.

A. PPP-Based Abstraction Modeling

The system model represents base stations with a homogeneous PPP and uses directional beamforming, alignment-error models, and a three-state mmWave link model. These assumptions support tractable characterization of serving and interfering links.

  • PPP network model: Base stations form a homogeneous PPP of density λ, and the probe mobile terminal is served by the smallest-path-loss or highest-received-power base station.Full-frequency reuse is assumed, with the serving BS separated from the interfering set.
  • Directional beamforming: Directional antenna arrays are modeled with sectored patterns to represent main- and side-lobe gains and interference isolation.The model specifies beamwidth and angle-dependent gains for both base stations and mobile terminals.
  • Directional beamforming: The intended link can exploit maximum directivity without alignment errors, whereas interfering-link orientations are modeled as random.Interfering directivity gains therefore follow a distribution induced by random beam orientations.
  • Alignment errors: Beamsteering errors are modeled as additive, symmetric, independent random variables for the BS and MT, with intended-link gains obtained from their gain distributions.The framework also considers practical degradation caused by imperfect alignment, especially for narrow beams.
  • Three-state propagation model: Each link follows a three-state model comprising LOS, NLOS, or outage, where outage represents an unestablishable link with effectively infinite path-loss.State probabilities depend on link distance and propagation scenario or carrier frequency.
  • Three-state propagation model: Independent distance-dependent thinning partitions the original PPP into independent non-homogeneous LOS, NLOS, and outage PPPs.Their radial densities are λ_LOS(r)=λp_LOS(r), λ_NLOS(r)=λp_NLOS(r), and λ_OUT(r)=λp_OUT(r).

E. Path-Loss Modeling

The paper models mmWave propagation with distance-dependent LOS and NLOS path loss, an infinite-loss outage state, and Log-Normal large-scale shadowing. It analyzes association by smallest path loss or highest received power, while simplifying some channel effects for tractability.

  • Path-loss model: The close-in path-loss model represents LOS and NLOS links using state-specific intercepts and power path-loss exponents.The intercepts describe path loss at 1 meter, while the exponents determine distance dependence.
  • Link states: Outage links are assigned infinite path loss, l_OUT(r) = ∞, preventing communication through that state.The three-state formulation includes LOS, NLOS, and OUT links.
  • Fading model: Each link includes a state-dependent Log-Normal power gain that captures large-scale shadowing, with different parameters for LOS and NLOS links.The gain parameters have state-specific means and standard deviations in dB.
  • Modeling assumptions: Shadowing correlations between links are ignored, and fast fading is neglected for mathematical tractability.The paper states that fast fading could be incorporated later and that independent fading causes only a minor loss of SINR accuracy in cited simulations.
  • Cell association: Two association criteria select either the smallest path loss or the highest received power, with the latter incorporating transmit power and directivity gains across tiers.The highest-received-power criterion can provide the best achievable performance but requires estimating large-scale shadowing.
  • Cell association: Under the smallest-path-loss formulation, the serving base station is selected from LOS, NLOS, and OUT candidates according to their path-loss values.The association framework explicitly accommodates the three link states.

H. Problem Formulation

This section formulates coverage probability and average rate for a downlink mmWave network, then adopts a noise-limited approximation and develops path-loss transformation results supporting both association criteria.

  • H. Problem Formulation: The received-power variable U(0) is the intended power from the serving base station and becomes zero in communication blockage.When U(0) = 0, the mobile terminal has no usable serving link.
  • H. Problem Formulation: Coverage probability and average rate are formulated from the SINR using a reliability threshold T > 0 and expectation operator E{·}.The average rate is subsequently obtained from the coverage probability.
  • H. Problem Formulation: Gauss-Chebyshev Quadrature provides an approximation using closed-form weights and abscissas, especially when coverage lacks a closed-form expression.The quadrature approximation is useful for evaluating otherwise non-closed-form coverage formulas.
  • H. Problem Formulation: The subsequent coverage analysis uses a noise-limited approximation supported by numerical simulations and field measurements for mmWave communications.Monte Carlo simulations also include other-cell interference to assess approximation accuracy.
  • H. Problem Formulation: For communication blockage, SNR equals zero, so coverage probability and average rate are zero for every threshold T > 0.This follows because no base station is available to serve the mobile terminal.
  • III. ANALYSIS AND APPROXIMATIONS OF TRANSFORMATIONS OF THE PATH-LOSS: The transformed path-loss framework covers LOS, NLOS, and outage base stations and supports both smallest-path-loss and highest-received-power association.The association cases correspond to different transformation parameters, including (A_LOS, A_NLOS) = (1, 1) for smallest path loss.
  • III. ANALYSIS AND APPROXIMATIONS OF TRANSFORMATIONS OF THE PATH-LOSS: The transformed path-loss process is analyzed as a Poisson point process, with its intensity and minimum-element distribution derived using PPP results.The intensity is expressed through LOS and NLOS components and the minimum distribution follows from the void probability theorem.
  • III. ANALYSIS AND APPROXIMATIONS OF TRANSFORMATIONS OF THE PATH-LOSS: The framework generalizes prior single-state results by incorporating distinct LOS and NLOS distributions together with an outage state.It reduces to the earlier result when those additional mmWave-specific features are removed.

A. Two-Ball Approximation

The paper introduces a two-ball approximation for three-state mmWave links and fits its parameters by matching path-loss intensities. The approximation improves tractability while preserving key connectivity behavior across distance regimes.

  • A. Two-Ball Approximation: The approximation remains tractable when Log-Normal gains prevent closed-form evaluation of the relevant expectation and Laplace transform.The paper replaces the original link-state probabilities with their two-ball counterparts and derives corresponding intensity expressions.
  • A. Two-Ball Approximation: The two-ball model approximates LOS, NLOS, and outage probabilities with piecewise-constant values over distance intervals separated by D1 and D2.Each link is constrained to occupy exactly one of the three states.
  • A. Two-Ball Approximation: The distance regions correspond to mostly LOS/NLOS links below D1, any state between D1 and D2, and mostly outage links beyond D2.The construction follows empirical connectivity patterns with two breaking distances.
  • A. Two-Ball Approximation: Compared with earlier single-ball approximations, the model explicitly includes the mmWave outage state and introduces path-loss intensity matching.The parameter-estimation technique is presented as new in the paper.
  • A. Two-Ball Approximation: The fitting procedure first solves an unconstrained 15-parameter problem, then refines it with the three-state approximation constraint.The unconstrained solution initializes the constrained search.
  • A. Two-Ball Approximation: The resulting approximation is reported for the empirical three-state model and is described as more mathematically tractable without losing accuracy.Its accuracy is evaluated later in the paper.
  • A. Two-Ball Approximation: For distances below about 50 meters, no outage occurs, LOS probability exceeds 80%, and links are either LOS or NLOS.This regime corresponds to r < D1.
  • A. Two-Ball Approximation: Between about 50 and 200 meters, all three states are possible and the mobile terminal is most likely served by an NLOS base station.Beyond about 200 meters, outage becomes most likely, while approximately 200 meters marks a critical mmWave operating regime.

B. Communication Blockage Probability

The three-state link model assigns LOS, NLOS, and blockage probabilities, with blockage occurring when neither LOS nor NLOS base stations can serve the mobile terminal. A closed-form blockage probability is provided, and complete blockage can force coverage and rate to zero.

  • Communication blockage is the event that no LOS or NLOS base station is available to serve the mobile terminal.
  • Pblockage = exp(−Λblockage) gives a closed-form expression for the probability of communication blockage.
  • The blockage probability is independent of the cell-association criterion and equals zero when δOUT = 0.
  • The three-state model satisfies PLOS + PNLOS + Pblockage = 1, while PLOS + PNLOS ≤ 1.
  • Coverage probability and average rate can both be zero when PLOS = PNLOS = 0 and Pblockage = 1; this regime corresponds to δOUT → +∞.For the considered mmWave channel model, the critical distance where this operating regime emerges is 200 meters.

IV. MODELING COVERAGE PROBABILITY AND AVERAGE RATE

Under smallest-path-loss association, the paper derives integral formulations for coverage probability and average rate using the SNR and distinct LOS/NLOS link distributions. Special assumptions yield simpler formulations, while performance improves with transmit power, directivity gain, and base-station density.

  • Smallest Path-Loss Association: The formulation distinguishes LOS and NLOS links through their different distributions and corresponding state-dependent quantities.
  • Smallest Path-Loss Association: Proposition 1 provides an exact single-integral expression for coverage probability under smallest-path-loss association and no beamsteering errors.
  • Average Rate: The average rate is computed from the coverage formulation, generally requiring a two-fold integral in the exact setup.
  • Approximations: The general formulation cannot be further simplified using the two-ball approximation, although special cases produce simpler expressions.
  • Special Cases: Assuming equal LOS and NLOS fading parameters yields a simpler integral formulation that is easier to compute numerically and remains distribution-general.
  • Average Rate: The average-rate approximation becomes a single-integral expression, with accuracy expected to increase as directivity gain and base-station density increase.
  • Performance Trends: Coverage probability and average rate increase with transmit power, intended-link directivity gain, and base-station density, and decrease with noise power.

B. Highest Received Power Cell Association

Highest-received-power association provides coverage at least as large as smallest-path-loss association, while requiring knowledge of instantaneous shadowing power gains. Exact and approximate coverage formulations lead to corresponding rate integrals.

  • Exact Formulation: The exact coverage expression is a single integral, whereas the resulting exact average-rate formulation is a two-fold integral.
  • Approximation: The two-ball approximation provides an approximated closed-form coverage expression and reduces the average-rate computation to a single integral.
  • Coverage: The coverage probability under highest-received-power association is at least that under smallest-path-loss association: P(cov, path−loss)(T) ≤ P(cov, power)(T).
  • Trade-off: The higher-performance association requires knowledge of the instantaneous shadowing power gains.
  • Performance Trends: A similar performance trend is expected when LOS and NLOS links have different distributions, although its proof is not straightforward from the general formulation.

V. GENERALIZATIONS

The framework is generalized beyond single-tier, error-free settings to include beamsteering errors and multi-tier deployments while retaining tractable coverage and rate calculations. Closed-form coverage relies on noise-limited operation and a two-ball approximation, whose accuracy is evaluated by Monte Carlo simulation.

  • Generalizations: Removing the single-tier and no-beamsteering-error assumptions does not increase the complexity of the coverage and rate frameworks.
  • Multi-Tier Networks: Each tier is modeled as a homogeneous PPP with its own density, transmit power, and maximum and minimum directivity gains.
  • Cell Association: The multi-tier mobile terminal is served by the base station providing the highest received power, accounting for transmit power and directivity gain.
  • Multi-Tier Networks: For general multi-tier networks, the two-ball approximation provides an approximated closed-form coverage expression.
  • Beamsteering Errors: Beamsteering errors can be incorporated while retaining a closed-form coverage expression, and the rate generally requires a single integral.
  • Approximation Assumptions: The closed-form formulation assumes noise-limited mmWave systems and approximates empirically derived link-state models with a two-ball model.
  • Validation: Monte Carlo simulations are used to investigate the accuracy of the noise-limited and two-ball approximations.

VI. NUMERICAL AND SIMULATION RESULTS

Numerical results validate the proposed frameworks and show that dense mmWave deployments can outperform µWave systems, while performance depends on density, blockage, beamforming errors, and tier structure.

  • Simulation setup: Monte Carlo simulations validate the proposed mathematical frameworks and compare mmWave with µWave cellular networks.The simulations do not enforce the analytical tractability assumptions used by the system simulator.
  • Noise-limited approximation: For Rc ≥100 meters, the noise-limited approximation is accurate for the considered setup; at higher BS densities, it may no longer hold.The gap from Monte Carlo simulations remains tolerable, indicating that the networks are likely not interference-limited.
  • Coverage and density: The outage state generally reduces coverage probability, especially at small reliability thresholds, whereas denser deployments improve performance.Coverage and rate are compared under the noise-limited assumption across varying average cell radii.
  • Cell association: Smallest-path-loss and highest-received-power cell association provide generally very close performance.The two association criteria are evaluated using the corresponding analytical propositions.
  • mmWave versus µWave: Sufficiently dense mmWave systems can outperform µWave systems, while µWave remains preferable at lower densities, especially for small T.At 28 GHz, mmWave slightly outperforms 73 GHz because of smaller path-loss.
  • Beamforming and multi-tier networks: Beamsteering errors degrade performance, with noticeable degradation when pointing-error standard deviation exceeds 6 degrees; multi-tier networks improve performance in specified regimes.Multi-tier gains are strongest for small T and large higher-tier cell radii, while the noise-limited approximation still holds in the evaluated setup.

VII. CONCLUSION

The paper introduces an analytical framework for mmWave coverage and rate using empirical propagation models and a systematic two-ball approximation. It shows that the noise-limited assumption is accurate for typical BS densities and that sufficiently dense mmWave networks can outperform µWave counterparts.

  • Contribution: The paper proposes a mathematical framework for computing mmWave coverage probability and average rate.Its novelty is the use of realistic channel and blockage models based on empirical literature data.
  • Method: A systematic two-ball approximation models mmWave link states by matching the intensities of empirical three-state and approximated two-ball PPP models.The framework represents the propagation structure through LOS, NLOS, and outage-related modeling.
  • Scope and assumption: The methodology relies on a noise-limited assumption shown sufficiently accurate for typical BS densities and envisioned transmission bandwidths.The approach also supports different cell association criteria, multi-tier deployments, and beamforming pointing errors.
  • Results: Numerical examples confirm that sufficiently dense mmWave cellular networks can outperform their µWave counterpart.

APPENDIX I – PROOFS OF THE RESULTS IN SECTION III

The appendix derives the paper’s blockage, propagation-loss, coverage, and average-rate results using PPP transformations, independent link states, and expectation-based calculations.

  • Propagation-loss process: The scaled propagation-loss process remains a PPP, enabling intensity-based analysis of LOS, NLOS, and outage components.The outage component has zero intensity because outage links have infinite path-loss.
  • Blockage probability: The blockage probability follows from independence of LOS and NLOS processes and the PPP void probability theorem.The resulting integral can be evaluated in closed form under the stated parameter choice.
  • Coverage and rate: Coverage is obtained by inserting the link-state probabilities into the relevant expectation, while average rate follows from the coverage expression and a high-SNR approximation.The rate derivation swaps integration order and uses a closed-form integral.
Loading 1410.3577v1…