PaperlayerView exact revision on arXiv

Source-linked AI summary

Modeling and Analysis of Cellular Networks using Stochastic Geometry: A Tutorial

Hesham ElSawy, Ahmed Sultan-Salem, Mohamed-Slim Alouini, Moe Z. Win

arXiv:1604.03689v1cs.IT

TL;DR

The paper addresses how to analytically model and evaluate cellular networks with spatially random interference. It develops a unified stochastic-geometry approach covering interference, error probability, outage probability, and ergodic rate, with extensions for cellular characteristics. The tutorial shows that, with an appropriate network model, SG can capture realistic network performance while Gaussian signaling simplifies error-rate analysis without sacrificing accuracy.

  • Problem

    Cellular-network analysis must characterize spatially random interference and performance across network realizations, while tractable point-process models may not represent all network topologies.

  • Method

    The tutorial derives interference characterizations and combines exact or approximated error analysis with abstracted outage and ergodic-rate analysis under adaptable PPP-based cellular models.

  • Results

    With an appropriate system model, SG captures realistic network performance; Gaussian signaling simplifies symbol error-rate expressions without sacrificing accuracy.

  • Takeaways & Limitations

    The framework provides average-performance expressions that support understanding cellular-network behavior and extracting design guidelines across network realizations.

  • Takeaways & Limitations

    PPP-based analysis is not sufficient for wireless networks with complex correlations among network elements and may require tractable alternative point processes.

Abstract

from arXiv · show

This paper presents a tutorial on stochastic geometry (SG) based analysis for cellular networks. This tutorial is distinguished by its depth with respect to wireless communication details and its focus on cellular networks. The paper starts by modeling and analyzing the baseband interference in a basic cellular network model. Then, it characterizes signal-to-interference-plus-noise-ratio (SINR) and its related performance metrics. In particular, a unified approach to conduct error probability, outage probability, and rate analysis is presented. Although the main focus of the paper is on cellular networks, the presented unified approach applies for other types of wireless networks that impose interference protection around receivers. The paper then extends the baseline unified approach to capture cellular network characteristics (e.g., frequency reuse, multiple antenna, power control, etc.). It also presents numerical examples associated with demonstrations and discussions. Finally, we point out future research directions.

I. INTRODUCTION

Stochastic geometry provides a unified framework for modeling spatial randomness and analyzing wireless-network behavior. The tutorial applies this framework to cellular networks, emphasizing interference, performance metrics, and communication-specific extensions.

  • SG models spatial randomness in wireless networks and can incorporate fading, shadowing, and power control.
  • Aggregate interference is generally non-Gaussian because distance-dependent path loss makes nearby interferers dominant, producing an impulsive, heavy-tailed α-stable distribution.
  • Cellular-network SG modeling addresses spatial reuse and interference exclusion regions around receivers rather than idealized hexagonal deployments.
  • The tutorial distinguishes itself through cellular-network focus, wireless-communication detail, and analytical treatment of Gaussian signaling approximation accuracy.
  • SG analysis relates cellular-network parameters and design variables to spatially averaged outage probability, error probability, and rate metrics.

B. Network Abstraction

The paper abstracts cellular networks with point processes, ordering base stations by distance from a test user and modeling the received signal and aggregate interference. It then uses Campbell’s theorem and the PGFL to characterize interference, while noting that tractability often requires a PPP approximation.

  • Network model: Base-station locations are modeled by a general two-dimensional point process, with repulsion capturing minimum-distance correlations from network planning.The process may be infinite for analytical simplicity, while finite-network modifications remain possible.
  • Network model: RSS-based association assigns the test user to the nearest base station x0, and the remaining ordered distances form the set ˜Ψ.The serving distance r0 is excluded from the interference set, and analysis initially conditions on a given r0.
  • Signal model: The received signal includes the intended transmission, aggregate interference from other base stations, and noise, with symbols and Rayleigh fading modeled as independent random variables.The test user is placed at an arbitrary origin for stationary point processes; otherwise, the analysis depends on location.
  • Interference characterization: Because interfering base-station locations and their number are uncertain, the analysis characterizes aggregate interference statistically through its PDF, characteristic function, or moments rather than instantaneously.Nearby interferers dominate because distance-dependent path loss prevents the central limit theorem from producing a Gaussian interference distribution.
  • SG tools: Campbell’s theorem converts expectations of point-process sums into intensity integrals, while the PGFL converts random products into integrals useful for the interference characteristic function.Campbell’s theorem can obtain first moments and may extend to second moments, but higher-order moments and PGFL expressions are not generally tractable for arbitrary processes.
  • Tractability: General point-process interference analysis is often analytically intractable, so the paper commonly approximates the interfering process by a PPP to retain tractability.The PPP approximation is motivated by the limited availability of closed-form higher-order moment and PGFL expressions for general point processes.

IV. POISSON POINT PROCESS APPROXIMATION

The paper uses a PPP to approximate interfering base-station locations while retaining receiver-related exclusion through intensity and interference boundaries. Its characteristic-function analysis shows how exclusion changes aggregate-interference statistics and finiteness.

  • PPP approximation: The PPP approximation is tractable because its PGFL, stationarity, and spatial ergodicity simplify interference analysis.Studies cited in the paper report close SINR agreement between PPP analysis and simulations using actual cellular topologies.
  • PPP approximation: PPP models for repulsive cellular deployments should be parameterized by intensity λ and interference boundaries, especially the inner boundary.The outer boundary is usually taken as infinite because faraway base stations contribute negligibly.
  • Interference characterization: Using the PPP PGFL, the analysis derives the characteristic function of aggregate complex interference with an exclusion region around the receiver.The derivation uses the exclusion boundary implied by received-signal-strength association and is valid for η > 2.
  • Interference characterization: The aggregate interference is circularly symmetric but neither Gaussian nor α-stable distributed.Without an exclusion region, it has an α-stable distribution with infinite moments; receiver protection produces finite moments under suitable conditions.
  • Interference characterization: Interference power is infinite at η = 2 or r0 = 0, whereas all cumulants and moments are finite for η > 2 and r0 > 0.For η > 2, interference power decays with exclusion radius as r0^(2−η) and increases linearly with λ and P.
  • Interference characterization: For η > 2 and r0 > 0, positive finite kurtosis indicates an interference distribution with a heavier tail than the Gaussian distribution.The characteristic function, cumulants, expected power, and kurtosis expose these properties.

B. Numerical Results for iagg

Numerical results compare aggregate interference with Gaussian and α-stable distributions using PDFs and KS distances. Increasing the exclusion distance moves the interference away from α-stable behavior and toward Gaussian behavior.

  • PDF comparison: The aggregate-interference PDF has a heavier tail than the Gaussian PDF, while smaller exclusion distance makes it approach the α-stable distribution.The comparison uses Gaussian and α-stable PDFs with matching parameters.
  • KS comparison: The KS statistic compares the entire CDFs, so it does not capture deviations in tail probabilities.Fig. 6 uses relative KS distance to compare iagg with Gaussian and α-stable distributions.
  • KS comparison: The aggregate interference is neither Gaussian nor α-stable; as r0 increases, it deviates from α-stable behavior and approaches Gaussian behavior.The reported distributional relationship is summarized through the KS comparisons in Fig. 6.
  • Section summary: The section concludes that PPP networks with receiver exclusion require interference models beyond Gaussian and α-stable assumptions.The analysis derives the aggregate-interference characteristic function and moments before turning to error probability.
  • PDF comparison: Fig. 5 plots the PDF of Re{iagg} for λ = 1 BS/km2, P = 10 W, η = 4, and r0 values of 250 m and 500 m.The PDFs are obtained by numerically inverting the characteristic function in (10).

V. EXACT ERROR PROBABILITY ANALYSIS

Because cellular aggregate interference is non-Gaussian, standard Gaussian-noise SEP formulas are not directly valid. The EiD approach restores conditional Gaussian analysis by representing interference through random Gaussian components and averaging afterward.

  • Motivation: Standard SEP expressions derived for Gaussian noise are not legitimate for cellular interference because aggregate interference is non-Gaussian.The issue arises when interference is treated under the same assumptions as AWGN.
  • Conditional Gaussian representation: The EiD approach represents aggregate interference as a conditional Gaussian random variable and averages the conditional error probability to obtain the unconditional result.The representation is established by matching characteristic functions and therefore distributions.
  • Conditional Gaussian representation: Conditioning on the random Gaussian-component variances makes interference plus noise circularly symmetric complex Gaussian with total variance equal to summed component variances plus N0.This conditional representation permits AWGN-based SNR formulas to be extended to cellular SINR analysis.
  • ASEP analysis: Exact ASEP analysis conditions on the Gaussian representation, applies the AWGN expression, and then deconditions over the non-Gaussian variables.The final expression involves an additional averaging step and remains conditioned on r0.
  • ASEP analysis: Theorem 2 gives downlink ASEP for M-QAM in a PPP cellular network with Rayleigh fading, universal frequency reuse, and no intra-cell interference.The result is stated for a user at distance r0 from its serving base station.

C. Section Summary

The EiD method provides an exact but complex route from SG interference characterization to ASEP, while Gaussian signaling offers a simpler approximation. The approximation preserves aggregate-interference distribution and power and has minor effects on SINR-dependent performance metrics.

  • Section summary: The EiD pipeline characterizes interference with SG, constructs an equivalent Gaussian representation, performs conditional AWGN analysis, and deconditions to obtain ASEP.The exact expression is computationally intensive because it includes an integral involving a sum of hypergeometric functions.
  • Gaussian signaling approximation: Gaussian signaling abstracts interfering symbols by drawing them from a unit-variance complex Gaussian constellation, simplifying error-rate analysis.It is an approximation for interfering symbols from distinct constellations, not an approximation that makes aggregate interference Gaussian.
  • Gaussian signaling approximation: Gaussian signaling directly yields the conditional Gaussian representation needed to use AWGN ASEP expressions.The paper states that this approximation avoids EiD complexity without compromising modeling accuracy.
  • Validation: The Gaussian signaling approximation preserves the aggregate-interference distribution and power of the original model.The characteristic functions and power expressions have equivalent forms, although higher-order even cumulants differ.

B. Approximate Error Probability Analysis

The paper introduces Gaussian signaling for interfering symbols to simplify ASEP analysis while preserving key aggregate-interference characteristics. It then connects the resulting interference transform to error probability, outage, and rate metrics.

  • Gaussian signaling approximation: Conditioned on network geometry and channel gains, Gaussian interfering symbols make the aggregate interference conditionally complex Gaussian.This representation permits AWGN-based ASEP expressions for the received signal.
  • ASEP analysis: The exact analysis, Gaussian approximation, and simulations are compared for ASEP across BS intensities, while Gaussian aggregate-interference modeling gives a loose estimate.The comparison confirms that the central limit theorem does not apply to aggregate interference in this setting.
  • Gaussian signaling approximation: Gaussian signaling preserves the aggregate interference distribution, odd and second cumulants, and interference power for any constellation size.Differences from exact interference occur only in even cumulants above order two.
  • ASEP analysis: The approximation reduces the ASEP exponent from a sum of M hypergeometric functions to one hypergeometric function without sacrificing ASEP accuracy.For η = 4, the ASEP further reduces to a computationally simple inverse tangent function.
  • Unified performance analysis: The Gaussian approximation replaces the exact characteristic-function route with the Laplace transform of aggregate interference power, which is easier to derive and evaluate.The same transform supports ASEP, outage probability, and ergodic-rate calculations.
  • Unified performance analysis: The SINR CDF is sufficient to characterize both outage probability and ergodic rate, providing a simpler alternative to the more involved ASEP expressions.Rate outage depends on interference and/or fading, and BER-based outage is also defined under the Gaussian signaling approximation.

B. SINR Distribution

The SINR distribution is derived through the Laplace transform of aggregate interference, using Rayleigh fading to express the SINR CDF without requiring a closed-form interference-power PDF. The resulting CDF supports ergodic-rate and outage calculations.

  • SINR CDF derivation: The SINR CDF can be derived from the Laplace transform of aggregate interference because Rayleigh fading makes the useful-channel power exponentially distributed.This avoids directly computing the generally unavailable interference-power PDF.
  • Scope and limitation: The interference-power PDF is unavailable in closed form for practical cellular-network cases with an inner interference boundary at r0.Closed-form distributions occur only in special PPP cases whose interference boundaries extend from 0 to infinity.
  • Performance metrics: Theorem 4 gives downlink ergodic rate and outage probability for a PPP cellular network with Rayleigh fading, universal reuse, and no intra-cell interference.Both metrics are stated for a user at service distance r0.
  • Performance metrics: The derivation obtains the SINR CDF from the interference Laplace transform and then uses it to compute ergodic rate and outage probability.This follows by substituting the transform expressions into the SINR result and applying the metric formulas.
  • Performance metrics: Monte Carlo validation shows that the outage and ergodic-rate expressions capture effects of interferer intensity and interference boundaries.These metrics are presented as simpler alternatives for characterizing network behavior, with abstraction potentially hiding some underlying behavior.

C. Section Summary

The unified analysis expresses ASEP, outage probability, and ergodic rate through the Laplace transform of aggregate interference, while averaging over random service distance for spatial performance. Extensions account for activity, loading, and noise conditions.

  • Section Summary: ASEP, outage probability, and ergodic rate under Gaussian signaling all require the Laplace transform of aggregate interference power.For η = 4, the presentation focuses on simple transform evaluations, using a modulation-dependent argument for ASEP and threshold T for outage and rate.
  • Baseline model: The baseline model equates the interference exclusion distance with the service distance, while interference intensity and exclusion region distinguish network models.These parameters determine how the aggregate-interference transform changes across models.
  • Random Link Distance r0: A random service distance requires an additional averaging step, producing a spatially averaged transform used for ASEP and the SINR CDF.The conditional transform is first obtained for r0, then averaged over its distribution.
  • Section Summary: Outage probability and ergodic rate offer simpler network-behavior characterizations than ASEP, but their abstractions may hide detailed network behavior.The comparison is made between the simpler outage/rate expressions and the more involved ASEP formulas.
  • Random Link Distance r0: Noise changes the spatial averaging step because the noise term must be averaged together with the interference transform.The simplified averaging statement therefore does not apply directly in the prominent-noise case.
  • Load-aware networks: Load-awareness is incorporated through an activity factor p that thins interfering BS intensity per channel while leaving the service-distance distribution based on λ.This reflects partially loaded networks in which some channels are idle.

C. Multi-tier Cellular Networks

The multi-tier extension models independent PPP tiers with tier-specific powers, intensities, and bias factors, then combines per-tier interference through tier-dependent exclusion regions. Under unbiased RSS association, SINR-dependent metrics reduce to the single-tier form.

  • Multi-tier model: Multi-tier cellular networks are modeled as independent PPP tiers, each with its own transmit power Pk, intensity λk, and path-loss exponent ηk.Users associate according to biased RSS controlled by tier-specific bias factors.
  • Interference analysis: Per-tier performance uses the aggregate interference from all tiers, with each tier contributing through its own intensity and interference boundary.Independence between tiers permits combining the per-tier interference transforms.
  • Association and biasing: Bias factors control tier association and can increase small-cell coverage to offload users from macro BSs to small BSs.The figure illustrates this mechanism using different bias factors with common BS locations.
  • Association and biasing: Tier-specific service-distance distributions depend on relative tier powers, bias factors, and path-loss exponents.For the common case ηk = 4, the distribution reduces to a simpler form.
  • Association and biasing: Unbiased RSS association reduces SINR-dependent performance metrics to the single-tier case, independent of tier count, transmit powers, and BS intensities.This simplification follows from the expression for η = 4.

D. Interference Coordination and Frequency Reuse

The tutorial models user-centric coordinated frequency reuse in a PPP cellular network and derives its interference transform by averaging over correlated protection and association distances. It then examines uplink power control and discusses tractability boundaries under more general fading.

  • Interference Coordination and Frequency Reuse: The available spectrum is divided into Δ sub-bands, with reuse coordinated among base stations because PPP deployments do not support traditional hexagonal-grid reuse schemes.Each base station selects a sub-band through coordination with neighboring base stations.
  • Interference Coordination and Frequency Reuse: User-centric coordination enlarges the interference-protection region to r_I = r_{Δ−1}, while association uses r_0 and their joint distribution.Averaging over the joint PDF of r_{Δ−1} and r_0 yields the spatially averaged interference Laplace transform.
  • Interference Coordination and Frequency Reuse: The reuse analysis becomes a double-integral expression, but it remains efficiently evaluable in time and complexity relative to Monte Carlo simulations.The additional integration is the main analytical cost identified for coordination and frequency reuse.
  • Interference Coordination and Frequency Reuse: Coordinated frequency reuse improves outage probability as the reuse factor Δ increases by changing both the interference boundary and interferer intensity.Figure 13 validates the corresponding interference-transform expression and shows this performance effect.
  • Uplink Case Study: In the uplink, per-UE power control maintains received power ρ at the serving base station, while each base station assigns a unique channel per user.The model assumes universal frequency reuse, no intra-cell interference, and a dense independent UE PPP.
  • Uplink Case Study: Uplink outage is higher than downlink outage because uplink transmission power is limited and association does not create geographical interference protection for uplink transmissions.Figure 15 verifies the uplink expression and the PPP approximation for interfering UEs.
  • General Fading: Changing useful-link fading from exponential can prevent outage probability and ASEP from being expressed through the aggregate-interference Laplace transform, although ergodic rate remains expressible that way.General interfering-link fading preserves the framework but may make the interference transform more involved.

1) Nakagami-m:

The tutorial extends its SG framework to Nakagami-m fading, slow fading, and a receive-diversity MIMO example. Integer Nakagami-m fading preserves Laplace-transform-based expressions, while MIMO introduces additional analytical complexity and is treated through a focused case study.

  • Nakagami-m: For integer Nakagami-m fading, the SINR CDF can be expressed using the Laplace transform of aggregate interference.The derivation uses the gamma CDF with integer shape parameter and the Laplace-transform identity.
  • Slow Fading: With slow fading, tractability remains when association selects the base station offering the highest received signal strength.The displacement theorem captures shadowing by scaling the PPP intensity with a shadowing fractional moment.
  • MIMO Receive Diversity: The tutorial presents receive diversity as a simple MIMO case study because a unified analytical framework for all MIMO configurations is difficult to provide.The treatment intentionally avoids the broader MIMO-system details available elsewhere.
  • MIMO Receive Diversity: With Nr receive antennas, the intended effective channel gain is gamma distributed with shape parameter Nr, while each interfering-link gain remains unit-mean exponential.This follows from summing Nr independent unit-mean exponential desired-link gains and independence between desired and interfering channels.
  • MIMO Receive Diversity: The receive-diversity SINR CDF uses the aggregate-interference Laplace transform with r_I = r_0, and Figure 17 validates the resulting expression.The example connects receive diversity to the earlier Nakagami-m analysis because the intended gain is gamma distributed.
  • MIMO Receive Diversity: MIMO analysis is more involved because precoding alters interfering signals and correlations may arise across antenna-branch interference.Even under Rayleigh fading, MIMO fading is no longer generally exponential.

H. Network MIMO

Network MIMO cooperation excludes the nearest cooperating base stations from interference, enlarging the interference-protection region to the distance of the farthest cooperating base station. The discussion also shows that practical network-management choices can make PPP-based SG predictions realistic, while identifying broader modeling and evaluation directions.

  • Network MIMO: CoMP transmission serves each user from its nearest n base stations, whose signals therefore do not contribute to interference.The model considers a downlink single-tier cellular network with single-antenna base stations and user-centric CSI-agnostic cooperation.
  • Network MIMO: Cooperation increases the geographical interference-protection region to rI = rₙ₋₁.The nearest n cooperating base stations are excluded from the interference field.
  • Discussion: With two receive antennas and reuse factor 3, outage probability at T = 0 dB drops from almost 50% to below 5%.The combined use of receive diversity and frequency reuse produces the reported reduction.
  • Discussion: With an appropriate system model, PPP-based SG analysis can capture realistic network performance and provide acceptable performance characterization.The discussion attributes pessimistic results to assumptions such as universal frequency reuse, saturation, and peak base-station transmit power rather than to PPP itself.
  • Discussion: Simple system models can reveal parameter trends, but their results are illustrative and do not provide true numerical performance values.This scope boundary applies when trends are prioritized over absolute values.
  • Future directions: PPP has limited ability to represent complex wireless topologies, motivating new tractable point processes for networks with D2D, V2V, and M2M communication.PPP is characterized by intensity and an interference boundary, which provide limited degrees of freedom for more complex correlations.

APPENDIX I THE POISSON POINT PROCESS

Appendix I collects probability distributions, the probability generating functional, Voronoi-cell relations, association probabilities, and averaging expressions used in PPP-based analysis. It also lists expressions involving nearest-point distances, Gamma variables, Laplace transforms, and PPP moments.

  • PPP distributions: The appendix gives the distance distribution from a generic location in R² to the nearest point of a PPP with intensity λ.
  • PPP distributions: The joint distance distribution to the nearest and nth points of a PPP is provided.
  • PPP identities: The probability generating functional is stated for a measurable function f and a PPP Φ.
  • Voronoi geometry: The appendix gives the area distribution of a generic PPP-Voronoi cell, using c = 3.575 for the R² Voronoi tessellation.
  • Voronoi geometry: For independent PPPs of base stations and users, the appendix provides the probability mass function for the number of users in a generic base-station Voronoi cell.The point-process intensities are λb and λu.
  • Auxiliary expressions: The appendix includes the probability that a user associates with tier k and an averaging technique for a unit-mean Gamma variable, a random variable specified by its Laplace transform, and a constant.It also lists a PPP expression involving intensity, distance, path-loss exponent, and moments of the interfering symbol.
Loading 1604.03689v1…