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A Primer on Cellular Network Analysis Using Stochastic Geometry

Jeffrey G. Andrews, Abhishek K. Gupta, Harpreet S. Dhillon

arXiv:1604.03183v2cs.ITcs.NI

TL;DR

The tutorial addresses how to model and analyze cellular-network SINR distributions with stochastic geometry as a complement to simulation. It introduces Poisson base-station models and derives coverage results for downlink, uplink, and heterogeneous multi-tier downlink settings. These baseline models provide a tractable framework that has supported many extensions.

  • Problem

    Cellular-network analysis needs analytical approaches that complement simulations for benchmarking, comparison, and understanding system dependencies.

  • Method

    The tutorial uses Poisson point-process models and stochastic-geometry tools to derive SINR coverage probabilities for downlink, uplink with power control, and k-tier heterogeneous downlink networks.

  • Results

    The tutorial computes coverage probability for the three baseline network cases, including an accurately characterized uplink case and a fairly simple heterogeneous-downlink form.

  • Takeaways & Limitations

    The three baseline models and SINR derivations provide a broad framework for developing tractable analytical models of cellular networks.

Abstract

from arXiv · show

This tutorial is intended as an accessible but rigorous first reference for someone interested in learning how to model and analyze cellular network performance using stochastic geometry. In particular, we focus on computing the signal-to-interference-plus-noise ratio (SINR) distribution, which can be characterized by the coverage probability (the SINR CCDF) or the outage probability (its CDF). We model base stations (BSs) in the network as a realization of a homogeneous Poisson point process of density $λ$, and compute the SINR for three main cases: the downlink, uplink, and finally the multi-tier downlink, which is characterized by having $k$ tiers of BSs each with a unique density $λ_i$ and transmit power $p_i$. These three baseline results have been extensively extended to many different scenarios, and we conclude with a brief summary of some of those extensions.

I. INTRODUCTION

The tutorial develops stochastic-geometry tools for analyzing cellular SINR distributions under Poisson base-station models. It derives coverage results for downlink, uplink, and heterogeneous multi-tier networks as tractable complements to simulation.

  • Motivation: Analytical models complement cellular system simulations, which can be time-consuming and error-prone while still remaining indispensable.Analysis supports benchmarking, comparison, and inspection of system dependencies and trends.
  • Scope and framework: The tutorial introduces point processes, especially the Poisson point process, and computational tools for deriving cellular coverage probability.Coverage probability is the SINR CCDF, while outage probability is its complement; either characterizes the entire SINR distribution.
  • Cellular downlink: The downlink model places base stations in a PPP with i.i.d. Rayleigh fading, power-law path loss, and association with the closest base station.The closest base station is equivalently the one providing the strongest average power under the stated assumptions.
  • Cellular uplink: The uplink extends the framework to handset transmit-power control, but one active handset per cell couples the handset and base-station point processes.Under reasonable approximations, the tutorial accurately characterizes uplink coverage probability.
  • HetNet downlink: The heterogeneous downlink generalizes the model to k tiers, each with unique transmit power p_i, density λ_i, and SINR threshold τ_i.Despite the added complexity, the tutorial computes coverage probability in a fairly simple form, viewed as a generalization of the macrocell result.
  • Extensions: The three baseline SINR derivations provide a framework for tractable cellular-network models that has been extended in hundreds of ways.The tutorial concludes by discussing some of these extensions.

1) Expectation measure:

The tutorial develops expectation measures and Poisson point-process tools for converting spatial sums and products into tractable integrals, including interference calculations.

  • Expectation measure:: An expectation measure maps a set A to the mean number of process points it contains.
  • Expectation measure:: For a homogeneous PPP, the point count in A is Poisson with mean λℓ(A), and conditioned points are independent and uniform in A.
  • Expectation measure:: Campbell’s theorem converts the expected sum of point contributions into an integral, enabling mean-interference calculations.
  • Expectation measure:: For an infinite planar PPP with an exclusion radius, mean interference is finite only when α > 2; α = 2 is insufficient.
  • Expectation measure:: The PGFL converts expectations of products over PPP points into integrals and can evaluate Laplace transforms of interference.

3) Slivnyak’s theorem:

Slivnyak’s theorem preserves the statistical distribution of a Poisson point process when conditioning on or adding a point, simplifying cellular-network analysis.

  • Slivnyak’s theorem:: Slivnyak’s theorem says conditioning on a point at a location does not change the distribution of the rest of a PPP.
  • Slivnyak’s theorem:: The theorem allows a point to be added at a chosen location without changing the PPP’s statistical properties.
  • Slivnyak’s theorem:: In a cellular downlink, it permits interference to be modeled as a PPP after removing the serving base station.
  • Slivnyak’s theorem:: Independent thinning, superposition, and random displacement each produce another PPP, with thinning changing intensity to qλ.
  • Slivnyak’s theorem:: For a typical user, the serving base station is selected by highest average SNR, equivalently highest average SINR and shortest distance.

B. Distance to the Nearest Base Station

The nearest-base-station distance in a homogeneous PPP determines the serving link and exclusion region, supporting interference and coverage analysis through its distribution and Laplace transform.

  • B. Distance to the Nearest Base Station: The distance R from a typical user to its closest base station has CDF F_R(r) = 1 − e^(−λπr^2) and a Rayleigh distribution.
  • C. Interference Characterization: Conditioned on R = r, all interfering base stations lie outside the radius-r disc around the user.
  • C. Interference Characterization: The interference is characterized by conditioning on the nearest-base-station distance and computing its Laplace transform using PPP independence and the PGFL.
  • Downlink coverage: Combining the distance distribution, fading, and interference transform yields an integral expression for downlink coverage probability.
  • Special cases: The analysis further considers interference-limited operation and the special case α = 4, including a practically closed-form expression.

2) Interference-limited, any path loss exponent:

The interference-limited downlink coverage probability admits a simple expression, and PPP-based analysis provides a useful approximation across realistic and idealized BS layouts.

  • Interference-limited, α = 4: For α = 4, coverage probability is pc(τ, λ, 4) = 1/(1 + √τ arctan √τ).The expression depends only on the SIR threshold τ.
  • Interference-limited, α = 4: At τ = 1, the fully loaded interference-limited network has coverage probability 0.56.This threshold corresponds to 0 dB and a maximum rate of 1 bps/Hz.
  • Validation: Real LTE coverage generally lies between square-grid and PPP extremes, while the PPP curve is about 2 dB pessimistic over much of the SINR range.The gap depends on the actual BS layout and path-loss exponent.
  • Validation: PPP analysis can be calibrated with a horizontal SINR shift of 1–3 dB according to the regularity of the desired BS layout.The cited theoretical results report a 3.4 dB PPP-to-hexagonal shift and an exactly 3 dB average shift for square grids.
  • Incorporating shadowing: With sufficiently large shadowing, about 10 dB, a regular hexagonal network can effectively appear Poisson.Shadowing can be represented by perturbing BS locations, supporting PPP-based coverage analysis without explicit shadowing.
  • Incorporating shadowing: Shadowing can be absorbed into an equivalent PPP ΦD with the same coverage probability as the original shadowed PPP.The derived process preserves association, serving power, and interference, then permits reuse of the prior PPP result with λ replaced by λD.

IV. UPLINK ANALYSIS

The uplink analysis models SINR at a tagged BS while accounting for active-user association, power control, and interference-field dependencies. Because exact uplink SINR characterization is unavailable, the tutorial presents a close analytical approximation.

  • Model and objectives: The uplink studies the received SINR of a typical mobile at its tagged BS, with interference from mobile users elsewhere in the network.The tagged BS is the serving BS of the typical user.
  • Model and objectives: Exact uplink SINR characterization is unavailable, so the section provides one representative analytical approach that closely approximates the SINR distribution.The approach is intended to expose key analytical challenges while retaining tractability.
  • System assumptions: Each BS schedules one active uplink user per resource block, but selecting one user per Voronoi cell creates dependent active-user locations.This dependence causes major complications for uplink analysis.
  • System assumptions: For tractability, active users are approximated as a PPP; simulations show this approximation does not affect coverage probability too much and matches closely across power-control values.With one active user per cell, the active-user density is set equal to the BS density.
  • Power control: Fractional power control uses factor ϵ ∈ [0, 1], spanning fixed transmit power at ϵ = 0 and perfect channel inversion at ϵ = 1.As users move closer to their serving BS, the required transmit power decreases.
  • Interference modeling: The effective uplink interference field is modeled as a radially symmetric non-homogeneous PPP with intensity λIa(d) = λ(1 − exp(−πλd^2)).An interferer at distance d from the tagged BS is retained with probability 1 − exp(−πλd^2).

B. Distribution of Serving Link Distances (R and Ri)

The uplink distance model uses a Rayleigh serving-distance distribution and approximates interferer-to-serving-BS distances while accounting for their geometric dependence. These distributions support the interference and coverage derivations.

  • Serving distance R: The typical user’s distance R to its serving BS is Rayleigh distributed under the PPP model.The resulting density is obtained by differentiating the PPP null probability.
  • Dependence structure: The interfering users’ serving distances are identically distributed but generally dependent because of the Poisson-Voronoi tessellation and one-active-user-per-cell constraint.This dependence complicates exact uplink analysis.
  • Interferer distance Ri: For an interfering user at distance Di from the tagged BS, its serving distance Ri is bounded above by Di.Otherwise the tagged BS would provide smaller average path loss and become the serving BS.
  • Interferer distance Ri: The conditional distribution of Ri given Di is approximated by a truncated Rayleigh distribution.The approximation reflects the geometric restriction imposed by association.
  • Coverage validation: The analytical uplink coverage expression achieves an almost perfect match with Monte Carlo simulations across power-control factors under the stated interference-limited setup.The comparison uses λ = 4 × 10^-6 BS/m^2, α = 4, p = 1, and negligible thermal noise.

E. Special Case

The full-power-control uplink case yields a simplified interference-limited coverage expression, while the tutorial places this result among related uplink approaches before introducing heterogeneous downlink networks.

  • E. Special Case: For full power control, ϵ = 1, and no noise, σ^2 = 0, the uplink coverage probability has a specialized closed-form derivation.The derivation substitutes the full-power-control interference transform into the coverage expression.
  • E. Special Case: In this special case, the Laplace-transform argument is independent of the serving distance r and can be taken outside the integral.This enables the subsequent simplification of the coverage calculation.
  • E. Special Case: The tutorial compares full-power-control uplink and interference-limited downlink coverage, finding faster uplink decay with SINR threshold.Downlink coverage decreases as 1/(1 + ρ(τ, α)), whereas uplink coverage decreases as exp(−ρ(τ, α)).
  • Related approaches: The uplink literature includes variants addressing channel inversion, curve-fitted thinning, TDD two-tier networks, MIMO combining, and interference-aware power control.These approaches differ in mathematical formulation and assumptions while often retaining distance-dependent thinning.
  • Heterogeneous cellular networks: The heterogeneous downlink model considers k overlaid BS tiers differing in transmit power, deployment density, and target SINR.The tiers represent macro, micro, pico, femto, and distributed-antenna deployments.
  • Heterogeneous cellular networks: The tutorial emphasizes analytical tools rather than covering all possible heterogeneous-network generalizations.This is an explicit scope boundary of the presentation.

A. HCN Model

The HCN model represents each BS tier by an independent PPP with its own density and transmit power, while users connect under specified cell-selection rules. Coverage analysis proceeds through tier-specific serving distances, association probabilities, SINR, and interference transforms.

  • HCN assumptions: Each tier is modeled as an independent homogeneous PPP with density λ_i, and all tier BSs transmit at common power p_i.A tier is characterized by {λ_i, p_i, τ_i}.
  • HCN assumptions: The model uses open access, so a mobile may connect to any BS, providing a best-case SINR coverage setting.Users follow a stationary point process independent of BS locations, and all BSs are active.
  • HCN assumptions: Received power follows power-law path loss with i.i.d. Rayleigh fading, with an ith-tier BS contributing p_iHr^-α.Shadowing can be incorporated through the same framework discussed earlier.
  • Cell selection: Maximum average received-power association produces a multiplicatively weighted Voronoi tessellation rather than an ordinary Poisson-Voronoi diagram.Lower-power femtocells therefore have smaller coverage footprints than macrocells.
  • Cell selection: The analysis considers average-power and instantaneous-power cell selection, with the latter connecting users to the BS offering the highest instantaneous received SINR.Coverage is derived for a typical user at the origin using tier-specific distance and exclusion-region calculations.
  • Coverage analysis: For average-power association, coverage derivation combines the serving-tier distance distribution, association probability, conditional SINR, and interference Laplace transform.Interference is decomposed across tiers, whose transforms combine using tier independence.

1) Special Cases:

The tutorial develops special cases for multi-tier downlink coverage and instantaneous-power cell selection. Interference-limited assumptions and SINR thresholds above 1 yield particularly tractable expressions and derivations.

  • Average-power association: In interference-limited networks, setting σ2 = 0 simplifies the average-power HCN coverage expression and exposes its relation to the single-tier result.The multi-tier expression differs from the single-tier form through terms associated with multiple tiers and their parameters.
  • Average-power association: With equal per-tier thresholds τ_i = τ, the average-power coverage probability becomes independent of tier transmit powers and BS densities.Scaling BS density changes serving and interfering distances together, leaving SIR invariant.
  • Instantaneous-power association: The closest BS in each tier need not be the serving BS under instantaneous-power selection because fading can change which BS provides the strongest SINR.Open access defines coverage as receiving SINR above threshold from at least one BS.
  • Instantaneous-power association: For instantaneous-power selection with τ_i > 1, at most one BS can satisfy the coverage condition, making the union-bound step exact.More generally, at most m BSs can meet the condition when the target SINR exceeds 1/m.
  • Instantaneous-power association: Instantaneous-power analysis uses indicator sums and the Campbell-Mecke theorem because each candidate BS’s SINR depends on interference from the remaining PPP.Standard Campbell’s theorem cannot be applied directly to this point-process-dependent function.

R2 LI

The instantaneous-power HCN result is derived by factorizing interference across independent tiers and applying PPP probability-generating functionals under Rayleigh fading.

  • Interference transform: The derivation computes the interference Laplace transform by exploiting independent PPP tiers and independent fading variables.Rayleigh fading permits the relevant expectations to be evaluated through exponential channel-power gains.
  • Interference transform: The PPP probability-generating functional converts products over interfering BSs into integrals, which are then simplified using polar coordinates and a variable change.This produces the closed-form coverage expression stated after Theorem 4.
  • Theorem 4: Theorem 4 gives downlink coverage for an open-access k-tier HCN under maximum instantaneous power-based selection when τ_i > 1.The expression includes ζ(α) = 2π^2/α csc(2π/α).

1) Special Cases:

The tutorial surveys extensions beyond its baseline HCN analysis, including generalized fading, path loss, access rules, and cell-selection strategies. These extensions preserve some tractability while introducing scope constraints or more complex expressions.

  • Special cases: Interference-limited instantaneous-power HCN coverage is independent of BS density, tier count, and tier transmit powers, so the SIR distribution is invariant to these quantities.This matches the density-and-power invariance identified for average-power association.
  • Fading and shadowing: General fading can be incorporated through higher-order derivatives of the interference Laplace transform, while Rayleigh fading on the serving link remains especially useful for tractability.Interfering-link fading can be generalized without loss of tractability when the serving link remains Rayleigh.
  • Fading and shadowing: Shadowing and general fading can be represented as random spatial perturbations that transform the BS PPP into an equivalent PPP with modified density.The transformed density depends on a fractional moment of the shadowing distribution.
  • Path loss: More general path-loss models can be handled through equivalent PPP transformations or multislope power laws whose exponent increases with link distance.The multislope model is described as more flexible and accurate, with implications for dense networks.
  • Cell selection: Cell-selection strategies include average power, instantaneous power, and selection bias; biasing can push users toward small cells and achieve near-optimum results when chosen correctly.The motivation is to reduce macrocell congestion and small-cell underutilization caused by many users selecting high-power macrocells.
  • Access assumptions: The baseline HCN discussion assumes open access, whereas closed access can reduce coverage probability and violate SIR invariance.This boundary is particularly relevant for femtocells and multi-RAT networks.
  • Special cases: Extending the instantaneous-power result to τ_i < 1 requires different approaches and produces less simple expressions than the τ_i > 1 case.The τ_i > 1 assumption makes the union-bound coverage analysis exact and tractable.

C. More general spatial setups

The tutorial’s stochastic-geometry framework extends beyond independent homogeneous Poisson deployments to dependent, correlated, multi-antenna, and mmWave network models. These extensions preserve tractability in some cases but introduce additional analytical complexity through spatial dependence, serving/interferer differences, blocking, and generalized fading.

  • Dependent and correlated deployments: Repulsive point processes, including determinantal and Matérn processes, generalize the framework beyond Poisson assumptions; a fixed 2–3 dB SINR shift can approximate some alternatives and hexagonal grids.The reported SINR shift is presented as an accurate approximation for more general point processes and even hexagonal layouts.
  • Dependent and correlated deployments: BS deployments with tier dependence can be modeled by carving holes around macro BSs, preventing certain small cells from being placed nearby.This Poisson Hole Process models repulsion between macrocell and picocell locations.
  • Dependent and correlated deployments: User locations should be correlated with BS locations in capacity-centric deployments, where BSs concentrate in areas of high user density and traffic.Poisson cluster processes model clustered users around small-cell cluster centers, including general user-cluster distributions.
  • Multi-antenna extensions: Multiple antennas complicate analysis because serving and interfering nodes have different received-power distributions due to beamforming or precoding gains.For zero-forcing precoding with m antennas serving one user per resource block, the serving-link gain is Gamma distributed while similar interfering-link gains are exponential.
  • Multi-antenna extensions: MIMO effects can be captured by assigning different effective fading distributions to serving and interfering links, while Gamma serving gains require higher-order derivatives of the interference Laplace transform.These derivatives can be evaluated using Faà di Bruno’s rule, Bell polynomials, or lower triangular Toeplitz matrices.
  • mmWave extensions: mmWave analysis separately models LOS and NLOS links, blocking-dependent non-homogeneous BS processes, directional antenna arrays, and large-degree-of-freedom Nakagami-m fading.The framework can represent LOS and NLOS BSs as a superposition of two non-homogeneous PPPs, but blocking and Nakagami-m fading add significant complexity.
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