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The Intensity Matching Approach: A Tractable Stochastic Geometry Approximation to System-Level Analysis of Cellular Networks

Marco Di Renzo, Wei Lu, Peng Guan

arXiv:1604.02683v1cs.ITmath.IT

TL;DR

Cellular-network analyses that use realistic propagation and practical system features can be difficult to make tractable and insightful. The paper introduces intensity matching, approximating a PPP-based system through a mathematically convenient intensity measure to obtain tractable performance expressions. It reports empirical validation and design insights, including a local optimum BS density that depends on blockage density.

  • Problem

    Existing realistic cellular-network analyses retain important simplifications and often use multi-fold integrals that provide little direct insight into key parameter effects.

    Source: §I · pp. 2–3

  • Method

    Intensity matching approximates the system with a mathematically convenient intensity measure to produce tractable expressions for area spectral efficiency and potential throughput.

    Source: §I · pp. 3–4, B. Rationale · p. 17

  • Results

    The approach is substantiated with empirical data and various blockage models, and yields insightful expressions for cellular-network spectral efficiency and rate.

    Source: §VII · p. 33

  • Takeaways & Limitations

    The approach provides direct insight into system-parameter effects, including a local optimum BS density that depends on blockage density.

    Source: §I · pp. 3–4

    The base stations are still assumed to follow a PPP.

    Source: C. Proposed Methodology · p. 19

Abstract

from arXiv · show

The intensity matching approach for tractable performance evaluation and optimization of cellular networks is introduced. It assumes that the base stations are modeled as points of a Poisson point process and leverages stochastic geometry for system-level analysis. Its rationale relies on observing that system-level performance is determined by the intensity measure of transformations of the underlaying spatial Poisson point process. By approximating the original system model with a simplified one, whose performance is determined by a mathematically convenient intensity measure, tractable yet accurate integral expressions for computing area spectral efficiency and potential throughput are provided. The considered system model accounts for many practical aspects that, for tractability, are typically neglected, e.g., line-of-sight and non-line-of-sight propagation, antenna radiation patterns, traffic load, practical cell associations, general fading channels. The proposed approach, more importantly, is conveniently formulated for unveiling the impact of several system parameters, e.g., the density of base stations and blockages. The effectiveness of this novel and general methodology is validated with the aid of empirical data for the locations of base stations and for the footprints of buildings in dense urban environments.

I. INTRODUCTION

The introduction argues that realistic propagation, antenna patterns, and load affect cellular-network analysis, while existing tractable methods simplify important aspects and offer limited design insight. It presents intensity matching as a tractable, accurate approach that accounts for practical features and helps expose parameter effects.

I. INTRODUCTION

  • Realistic path-loss models can change predicted coverage, rate, and densification effects, while antenna radiation patterns matter for quantifying other-cell interference.Prior studies also report that densification effects depend on the path-loss and traffic-load models.Source: §I · p. 2
  • The introduction identifies a gap: prior realistic models retain important simplifications and multi-fold integrals that provide little direct insight into BS-density and blockage effects.The cited approaches omit or simplify link states, directional antennas, shadowing, fading, or traffic-load conditions.Source: §I · pp. 2–3
  • The proposed methodology aims to combine tractable, accurate cellular-network analysis with direct insight into system-parameter effects, including a local optimum BS density that depends on blockage density.It also introduces a link-state model and a general parameter-estimation approach for matching empirical propagation and blockage models.Source: §I · pp. 3–4

A. PPP-Based Abstraction Modeling

  • The system model assumes independent homogeneous PPPs for BSs and MTs, with each BS serving at most NRB MTs on orthogonal resource blocks.Power is equally divided among resource blocks, and unequal BS transmit powers are left to future research.Source: A. PPP-Based Abstraction Modeling · p. 4

B. Link State Modeling

  • Links can occupy multiple blockage-dependent states, with state probabilities varying by distance; the paper adopts the multi-ball model for tractability and flexibility.The model's accuracy is reported as experimentally validated in [7].Source: B. Link State Modeling · pp. 4–6

C. Channel Modeling

  • Path loss, log-normal shadowing, and gamma-distributed fading are modeled with parameters that depend on link state.The paper reports that bounded path loss can be accommodated, but presents formulas for the unbounded case for simplicity.Source: C. Channel Modeling · p. 6

D. Cell Association Modeling

The model associates each mobile terminal with a base station by average received power, then approximates the resulting cell areas to support tractable load modeling. It derives selection and inactivity probabilities from this approximation and the network densities.

D. Cell Association Modeling

  • Association selects the base station with the highest average received power, equivalently the link with the smallest inverse average received power.The serving link is the one with the minimum path-loss quantity across candidate links.Source: D. Cell Association Modeling · p. 7

E. Load Modeling

  • The mean area of associated coverage regions is formulated under the specified link-state, path-loss, and cell-association models.The subsequent area approximation uses the mean area 1/λBS.Source: E. Load Modeling · pp. 7–8
  • The cell-area PDF is approximated by a gamma distribution with parameters (m, Ω) = (3.5, E{area}), matching the mean area of a Voronoi tessellation.The paper reports this approximation as accurate for various setups.Source: E. Load Modeling · p. 8
  • This matching concerns cell-area distributions, not cell shapes: under the considered association, cells may be noncontiguous and differ from Voronoi cells.The paper notes the approximation would be exact in the limiting case where all points are in LOS.Source: E. Load Modeling · pp. 8–9
  • The load model defines selection and inactivity probabilities for each {λBS, λMT, NRB} triplet, reflecting blocked terminals and unused resource blocks.Associated terminals are assumed to be randomly chosen for transmission in an arbitrary resource block.Source: E. Load Modeling · pp. 9–10
  • When λBS/λMT is much greater or much less than one, the selection and inactivity expressions simplify, and active base stations are modeled by thinning the PPP.The active-base-station density is scaled by 1 − poff.Source: E. Load Modeling · p. 10

F. Antenna Radiation Pattern

The approach approximates antenna radiation patterns with a mathematically tractable multi-lobe model. It uses five lobes as a reported trade-off between complexity and accuracy.

F. Antenna Radiation Pattern

  • The generalized multi-lobe model extends the two-lobe model to approximate antenna patterns that are usually mathematically intractable.The paper reports that the approximation is sufficiently accurate based on numerical simulations.Source: F. Antenna Radiation Pattern · p. 11
  • The multi-lobe pattern provides a tractable, step-wise approximation of other antenna radiation patterns.Its parameters are obtained by solving a minimization problem.Source: F. Antenna Radiation Pattern · p. 11
  • Five lobes are reported as a good trade-off between approximation complexity and accuracy.The five-lobe approximation is applied to the patterns listed in Table III.Source: F. Antenna Radiation Pattern · p. 12, Table IV · p. 12
  • With no pointing errors, the multi-lobe approximation is needed for other-cell interference distributions, but not for the intended-link gain.Interfering links are modeled with randomly oriented radiation patterns.Source: F. Antenna Radiation Pattern · p. 12

G. Problem Formulation

The problem formulation expresses area spectral efficiency and potential throughput through scheduling probability and SINR. General expressions cover varied link, channel, and antenna models, while the intensity-matching approach supports their efficient computation.

G. Problem Formulation

  • ASE uses the Shannon rate ln(1 + SINR), while PT multiplies log2(1 + T) by the probability that SINR exceeds threshold T.Both metrics include λMT psel.Source: G. Problem Formulation · p. 12
  • The paper uses a moment-generating-function approach to formulate the rate and the Gil-Pelaez approach to formulate coverage probability.Coverage probability is C(T) = Pr{SINR ≥ T}.Source: G. Problem Formulation · pp. 12–13
  • The expressions apply to arbitrary link-state, channel, and radiation-pattern models, although their computation may not be mathematically or numerically possible for every choice.This is the stated boundary on the general formulations.Source: G. Problem Formulation · p. 13
  • The intensity-matching approach provides a methodology for efficiently computing the ASE and PT expressions under general system models.The resulting frameworks are also described as providing insight for system design.Source: G. Problem Formulation · pp. 13–14

III. THE INTENSITY MATCHING APPROACH

The approach transforms the spatial base-station PPP into path-loss PPPs whose intensity measures support system-performance calculations. It uses PPP displacement, void-probability, and Laplace-functional results to formulate the needed quantities.

III. THE INTENSITY MATCHING APPROACH

  • For each link state, the model represents base stations as independent non-homogeneous PPPs and maps them to path-loss PPPs.The mapping invokes the displacement theorem.Source: §III · p. 14
  • The intensity measure of each path-loss PPP provides the transformed spatial quantity used in the analysis.The paper derives additional functions from PPP void probabilities.Source: §III · p. 14
  • The aggregate-interference expression is obtained from the Laplace functional of PPPs.The preceding expressions also invoke the PPP displacement theorem.Source: §III · p. 14
  • The load model accounts for active interfering base stations on a resource block by scaling their intensity by 1 − poff.This adjustment uses the inactivity probability from the load model.Source: §III · p. 15

A. Motivation

The paper replaces an exact but often numerically intractable analysis with an intensity-matching approximation, aiming to retain realistic link characteristics while making performance expressions tractable and informative for design. Its rationale is that ASE and potential throughput depend on the intensity measure for each link state.

A. Motivation

  • Simplifying the system model can make analysis tractable, but simplistic link-state and path-loss assumptions can yield inaccurate predictions of design-parameter effects.The exact expressions are generally multi-fold integrals that neither reveal performance trends nor provide design insight.Source: A. Motivation · p. 15, B. Rationale · p. 15

B. Rationale

  • The multi-ball link-state model makes antenna-pattern expectations computable without changing the resulting integral over y.For general antenna models, the expectation is usually difficult to compute, and the integral over y may remain unsolvable.Source: B. Rationale · p. 16
  • The exact integral over y is generally not closed-form because the intensity measure is complicated and includes shadowing for cell association.For multi-state path-loss models, shadowing cannot be accounted for simply by scaling base-station density.Source: B. Rationale · p. 16
  • ASE and potential throughput are determined by the intensity measure for each link state, so models with matching measures produce the same values.The approach approximates each exact measure with one more suitable for analysis, without simplifying the ASE and throughput expressions themselves.Source: B. Rationale · pp. 16–17

C. Proposed Methodology

  • The approximation selects a tractable intensity measure and a matching criterion for estimating its parameters from the exact measure, while retaining design-parameter insight.The approximated model uses a multi-ball link-state model and sets shadowing to one for mathematical tractability; shadowing’s impact remains in modified parameters.Source: C. Proposed Methodology · pp. 17–18
  • Empirical link-state intensity measures can replace estimating the link-state probability function, and may support system optimization with operator-provided data.The method still assumes base stations follow a PPP; extending it to non-PPP location models is beyond this paper’s scope.Source: C. Proposed Methodology · pp. 18–19

IV. AREA SPECTRAL EFFICIENCY AND POTENTIAL THROUGHPUT

The paper provides tractable mathematical expressions for area spectral efficiency and potential throughput using the approximated intensity measure. A two-state, single-ball blockage model gives a simplified framework for analyzing link-state effects on system design.

IV. AREA SPECTRAL EFFICIENCY AND POTENTIAL THROUGHPUT

  • The ASE expression is derived using the approximated intensity measure, with the paper presenting it as numerically easy to compute.The stated derivation inserts the approximated measure and related expressions into the general ASE framework and computes the integrals.Source: §IV · p. 19
  • The PT expression is derived using the same approximated intensity measure and is presented as numerically easy to compute.The paper gives PT as λMT p_sel log2(1 + T).Source: §IV · pp. 19–20
  • The paper attributes tractability of its ASE and PT expressions to combining the multi-ball link-state model with the intensity-matching approximation.These are identified as the approach’s main novelties.Source: §IV · p. 20
  • Under a two-state, single-ball blockage model, the general ASE and PT expressions still hold, while the intensity-measure expressions simplify.The states are taken to represent LOS and NLOS links; this corollary is used to discuss performance trends and provide system-optimization guidelines.Source: §IV · p. 20

V. PERFORMANCE TRENDS AND DESIGN INSIGHTS

The analysis uses Corollary 3 to study ASE and simplify its exact expression into approximations for understanding performance trends across cellular-network regimes.

V. PERFORMANCE TRENDS AND DESIGN INSIGHTS

  • The analysis focuses on ASE, using the framework in Corollary 3 as its starting point.The same methodology could be used to study Potential Throughput, but space limits the discussion to ASE.Source: §V · p. 20
  • The exact ASE expression separates contributions by serving-link state (LOS or NLOS) and whether the serving base station lies inside or outside the LOS/NLOS ball.Interfering base stations are not restricted to the same inside-or-outside region as the serving base station.Source: §V · p. 21
  • Negligible LOS contribution from outside the ball permits a tight ASE approximation, but the expression remains too complex for engineering insight.The paper therefore develops four asymptotic approximations for distinct operating regimes; accurate and weaker approximations are provided for each case study.Source: §V · pp. 21–22

+ FLOS

  • Table VII summarizes rate and ASE trends across the very dense, dense, sparse, and very sparse regimes.The section uses these regime-specific approximations to discuss performance trends and optimization guidelines.Source: §V · p. 21, + FLOS · p. 24, Table VII · p. 24

A. Impact of the Density of Base Stations

The effect of increasing base-station density depends on the operating regime: it improves rate and ASE in very dense networks, while the dense-regime impact can vary.

A. Impact of the Density of Base Stations

  • Across operating regimes, increasing base-station density does not have a uniform impact on ASE.The paper notes that comments on density assume antennas are not very directive.Source: A. Impact of the Density of Base… · pp. 24–25
  • In very dense networks, increasing λBS raises both rate and ASE because base stations move closer without increasing other-cell interference.The interference term in the approximation does not depend on λBS.Source: A. Impact of the Density of Base… · p. 25
  • In the dense regime, the ASE integrand's response to λBS is not always positive or always negative across y and z.The resulting impact on ASE therefore depends on the parameter pair discussed in the section.Source: A. Impact of the Density of Base… · p. 25

B. Impact of the Density of Mobile Terminals

Mobile-terminal density affects rate and ASE differently across regimes, with its most noticeable system-level impact occurring in very dense networks.

B. Impact of the Density of Mobile Terminals

  • Mobile-terminal density noticeably affects rate and ASE only in very dense networks, where many base stations may be inactive.In other regimes, base stations are likely active and contribute to other-cell interference.Source: B. Impact of the Density of Mobile… · p. 25
  • In the very dense regime, increasing λMT decreases rate because it determines the other-cell interference.The paper attributes this to many base stations being inactive when they have no mobile terminals to serve.Source: B. Impact of the Density of Mobile… · p. 25
  • The passage concludes that ASE increases as λMT increases, noting that a quantity is positive for z ≥ 0.Source: B. Impact of the Density of Mobile… · p. 25

C. Impact of the Number of Resource Blocks

The section examines resource-block count, antenna directivity, and blockage density, finding regime-dependent effects for resource blocks and blockages but consistent gains from greater antenna directivity.

C. Impact of the Number of Resource Blocks

  • In sparse networks, increasing NRB raises the LOS-inside contribution and the overall ASE, despite opposing effects of more served terminals and lower transmit power per resource block.The paper also states that ASE increases with NRB in the sparse and very sparse regimes.Source: C. Impact of the Number of Resource… · pp. 25–26

D. Impact of the Antenna Radiation Pattern

  • Increasing antenna directivity raises rate and ASE in all four regimes, reducing interference in very dense and dense networks and enhancing the intended link in sparse and very sparse networks.For highly directive antennas, increasing λBS also raises ASE in the dense regime.Source: D. Impact of the Antenna Radiation… · p. 26

E. Impact of the Density of Blockages

  • Greater blockage density shortens the LOS/NLOS ball radius; blockages help rate and ASE in very dense and dense regimes, have minor impact in very sparse networks, and lower rate and ASE in sparse networks.The blockage effect is expected to be limited in the very dense regime; the sparse-regime result follows because rate and ASE decrease as bD1 decreases.Source: E. Impact of the Density of Blockages · p. 26

F. Impact of the Shadowing Severity

The analysis traces how shadowing, blockage density, and BS density shape rate and ASE, then uses those trends to derive optimization guidance. Numerical results examine the approximations and their validation across blockage models and empirical data.

F. Impact of the Shadowing Severity

  • As shadowing severity rises, bθ[0, bD1]LOS and bD1 decrease while bϕ[bD1,∞]NLOS increases; in very sparse networks, rate and ASE increase.In dense and very dense regimes, shadowing’s impact on rate and ASE is determined by bD1; in sparse networks it depends on the blockage model.Source: F. Impact of the Shadowing Severity · p. 27

G. Existence of a Local Minimum and Maximum of the Rate

  • Rate has opposite density trends in dense versus very dense regimes and in dense versus sparse regimes, implying a local minimum near λBS = λMT/NRB and a blockage-dependent local maximum.The local maximum is expected when the average cell radius is proportional to the LOS/NLOS-ball radius; higher blockage density reduces that radius.Source: G. Existence of a Local Minimum and… · p. 27

H. Guidelines for System-Level Optimization

  • A useful trade-off is an average cell radius on the order of the LOS/NLOS-ball radius, which depends on blockage density and environment.Setups with λBS > λMT/NRB may be considered only when economically convenient and cost-effective, since rate decreases as λBS increases.Source: H. Guidelines for System-Level… · pp. 27–28

VI. NUMERICAL AND SIMULATION RESULTS

  • The numerical study evaluates the approach across blockage models and against rate and ASE estimates for actual BS locations and building footprints in dense urban London.The simulation setup uses a two-state LOS/NLOS blockage model for practical relevance.Source: §VI · p. 28

A. IM-based Approach: Approximations and Impact of Shadowing and Density of Blockages

  • Three- and four-ball models approximate blockage models more accurately than the one-ball model at higher computational complexity, while the one-ball model gives similar performance trends.The reported ball radii decrease as blockage density or shadowing standard deviation increases.Source: A. IM-based Approach: Approximations… · p. 29
  • Figures 2 and 3 compare rate and ASE for one- and two-state blockage models, with Monte Carlo simulations and three-ball IM-based approximations.The setup assumes PPP-distributed BSs and the RS blockage model.Source: Figure 2 · p. 30, Figure 3 · p. 30

B. IM-based Approach: On the Importance of Modeling Assumptions

The numerical results test the effects of blockage-model choices and validate the IM-based approach against simulations and an empirical urban deployment. They support density-dependent design conclusions, while showing that ASE’s response to densification can depend on setup.

B. IM-based Approach: On the Importance of Modeling Assumptions

  • With LOS and NLOS links modeled, an optimal BS density emerges for rate, while ASE may grow sub-linearly or super-linearly with BS density depending on operating regime.The comparison includes one-state LOS-only and NLOS-only cases and a two-state model.Source: B. IM-based Approach: On the… · p. 30

E. Validation of the IM-based Approach Against the Density of MTs

C. Validation of the IM-based Approach Against the Empirical Dataset in [7]

  • The empirical-data comparison shows good accuracy for rate and ASE, with the network operating near its local optimum where Rcell ≈ bD1.The study reports Rcell ≈ 83.4 m, RMT ≈ 7.6 m, and bD1 ≈ 87.6 m; rate is almost independent of NRB but decreases slightly, while ASE increases.Source: C. Validation of the IM-based Approach… · pp. 30–31, Figure 4 · p. 32

E. Validation of the IM-based Approach Against the Density of MTs

  • ASE can decrease with BS density in a special omni-antenna setup despite practical load, although the other analyzed load-model cases show monotonic ASE growth.Directional antennas provide a monotonic ASE increase in the reported special case.Source: G. ASE in the Dense Regime: On the… · p. 33

VII. CONCLUSION

  • The conclusion is that the approach provides accurate system-level approximations for PPP-based cellular networks and supports insight into densification, blockage, and antenna directivity.Generalization to non-PPP BS locations and spatial blockage correlations is ongoing work.Source: §VII · p. 33
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