PaperlayerView exact revision on arXiv

Source-linked AI summary

Modeling Heterogeneous Network Interference Using Poisson Point Processes

Robert W. Heath Jr, Marios Kountouris, Tianyang Bai

arXiv:1207.2041v2cs.ITmath.IT

TL;DR

Heterogeneous cellular deployments complicate interference modeling, and existing analyses can make it difficult to assess performance in a particular cell. The paper models a fixed downlink cell surrounded by a Poisson interference field, then uses a Gamma approximation to simplify performance calculations. Simulations report a good fit overall, with more error at the cell edge.

  • Problem

    Heterogeneous infrastructure complicates interference modeling, while prior PPP analyses often characterize system-wide rather than particular-cell performance.

    Source: §I · p. 1

  • Method

    The hybrid model evaluates downlink performance in a fixed cell with guard-region and boundary-interferer effects, and approximates interference using a moment-matched Gamma distribution.

    Source: §I · p. 2

  • Results

    The hybrid approach matches the stochastic-geometry model's SIR distribution across locations, while the Gamma approximation fits rate and success probability well except near the cell edge.

    Source: §VIII · pp. 10–11

  • Takeaways & Limitations

    Replacing summed interference with an equivalent random variable allows average rate and success probability to be computed by numerical integration rather than Monte Carlo simulation.

    Source: §IX · p. 11

    The Gamma approximation has more significant error at the cell edge for both average rate and success probability.

    Source: §VIII · p. 11

Abstract

from arXiv · show

Cellular systems are becoming more heterogeneous with the introduction of low power nodes including femtocells, relays, and distributed antennas. Unfortunately, the resulting interference environment is also becoming more complicated, making evaluation of different communication strategies challenging in both analysis and simulation. Leveraging recent applications of stochastic geometry to analyze cellular systems, this paper proposes to analyze downlink performance in a fixed-size cell, which is inscribed within a weighted Voronoi cell in a Poisson field of interferers. A nearest out-of-cell interferer, out-of-cell interferers outside a guard region, and cross-tier interference are included in the interference calculations. Bounding the interference power as a function of distance from the cell center, the total interference is characterized through its Laplace transform. An equivalent marked process is proposed for the out-of-cell interference under additional assumptions. To facilitate simplified calculations, the interference distribution is approximated using the Gamma distribution with second order moment matching. The Gamma approximation simplifies calculation of the success probability and average rate, incorporates small-scale and large-scale fading, and works with co-tier and cross-tier interference. Simulations show that the proposed model provides a flexible way to characterize outage probability and rate as a function of the distance to the cell edge.

I. INTRODUCTION

Heterogeneous deployments make interference harder to model, while existing stochastic-geometry analyses often report system-wide performance rather than results for a particular cell. The paper introduces a fixed-cell hybrid model and uses Gamma approximations to simplify performance calculations.

I. INTRODUCTION

  • Prior stochastic-geometry approaches characterize system-wide performance, making it challenging to evaluate a particular cell or incorporate relays and distributed antennas.The hybrid model instead focuses on a fixed cell and can represent heterogeneous infrastructure.Source: §I · pp. 1–2
  • The hybrid model evaluates a fixed-size cell amid Poisson-distributed interferers, including guard-region and cross-tier effects, with interference bounded as a function of receiver location.For one tier, choosing the cell radius as a function of density makes the analysis density-invariant.Source: §I · p. 2
  • The model does not require an explicit user-association scheme, although its calculations assume users connect to the closest base station.The passage notes that shadowing can make closest-base-station association unrealistic.Source: §I · p. 2

II. MATHEMATICAL PRELIMINARIES

  • Second-order moment matching approximates distributions with Gamma variables, whose flexibility covers common fading models and facilitates interference and rate calculations.The paper uses Gamma variables for small-scale fading, composite fading, and interference approximation.Source: §I · p. 2, §II · p. 3

III. DOWNLINK NETWORK MODEL

The downlink model combines a fixed cell with a random external interference field to retain cell-specific analysis while representing heterogeneous deployments. A guard region and a dominant boundary interferer supplement the external Poisson field.

III. DOWNLINK NETWORK MODEL

  • Unlike hexagonal layouts or random-cell PPP models, the hybrid model studies a fixed inscribing cell while retaining randomly located external base stations.The circular cell simplifies analysis but misses some poorly covered areas at the Voronoi-cell edge.Source: §III · p. 4
  • A guard region excludes other transmitters near the cell, and a dominant interferer at its outer edge represents the nearest neighboring base station.The guard region is chosen so users at the cell edge receive the strongest unfaded signal from the serving base station.Source: §III · p. 4
  • The model evaluates downlink signal power by receiver distance from the cell center, using small- and large-scale fading with a nonsingular path-loss model.The cell radius may be fixed for a particular deployment or chosen as a function of density for a typical cell.Source: §III · p. 4

IV. HOMOGENEOUS NETWORK INTERFERENCE

The homogeneous model represents interferers as a marked Poisson point process and separates interference from a dominant interferer from interference generated by the remaining field. Because the exclusion region is asymmetric for an off-center receiver, it uses a small-ball upper bound and characterizes interference through its Laplace transform.

IV. HOMOGENEOUS NETWORK INTERFERENCE

  • Interferer locations form a PPP, with channel marks combining log-normal large-scale fading and Gamma small-scale fading.The model assumes fixed transmit power and independent interferer actions.Source: §IV · p. 5
  • Total interference comprises I0 from the dominant interferer at the guard-region edge and I1 from the remaining base stations outside the guard region.The two components are modeled separately, with I1 formed by a PPP outside the guard region.Source: §IV · p. 5
  • The asymmetric exclusion distance makes exact aggregate-interference calculation difficult, so the small-ball approximation uses a reduced region and upper-bounds interference.The reduced region has radius Rc + Rg − r, leaving less area excluded from the interference calculation.Source: §IV · p. 5, Figure 2 · p. 5
  • The model uses a non-singular path-loss function and assumes Rc + Rg − r > d0 for its calculations.With a positive guard radius, the text says a singular path-loss model could instead be used without changing the results.Source: §IV · p. 5
  • The interference distribution is characterized through its Laplace transform for arbitrary identical marks, with a further evaluation available when Z follows a Gamma distribution.For Gamma-distributed Z, the expression uses the Beta Euler function and Gauss hypergeometric function.Source: §IV · pp. 5–6

V. HETEROGENEOUS NETWORK INTERFERENCE

The model extends interference analysis to heterogeneous networks with multiple infrastructure types. It distinguishes multiple kinds of out-of-cell interference from cross-tier interference contributed by low-power nodes.

V. HETEROGENEOUS NETWORK INTERFERENCE

  • The heterogeneous model considers both multiple kinds of out-of-cell interference and cross-tier interference from low-power nodes.The two scenarios are illustrated as Fig. 3(a) and Fig. 3(b), respectively.Source: §V · p. 6, Figure 3 · p. 6

A. Heterogeneous Out-of-Cell Interference

The heterogeneous out-of-cell model represents each infrastructure type as a marked PPP, with tier-specific parameters and guard regions. It combines a dominant interferer with independent shot-field contributions from the other tiers.

A. Heterogeneous Out-of-Cell Interference

  • Each of M infrastructure types is modeled as a homogeneous marked PPP with Gamma-distributed composite fading and tier-specific power and density.The model assumes common path loss across processes; non-uniformity and clustering are beyond its scope.Source: A. Heterogeneous Out-of-Cell… · p. 6
  • Different transmission powers give tiers different guard-region radii, with the fixed cell associated with tier 1.The selected configuration is stated to keep the fixed-cell edge covered by its associated base station, neglecting channel fading.Source: A. Heterogeneous Out-of-Cell… · p. 6
  • For typical-cell analysis, the fixed-cell radius is scaled to the average inscribing-ball radius of the typical Voronoi cell, making the analysis invariant to base-station density scaling.The average radius is computed from tier densities and transmit powers.Source: A. Heterogeneous Out-of-Cell… · p. 6
  • The dominant interferer may belong to any tier and lies at that tier’s guard-region edge with a tier-dependent probability.The heterogeneous case is more complex than the homogeneous one.Source: A. Heterogeneous Out-of-Cell… · p. 6
  • Total interference consists of the dominant interferer plus the out-of-guard-region contributions from the other tiers, which are independent.The heterogeneous out-of-cell interference is characterized through its Laplace transform.Source: A. Heterogeneous Out-of-Cell… · p. 7

B. Cross-Tier Interference

Cross-tier interference is modeled from a PPP of low-power nodes with an exclusion region around the desired receiver. PPP shift invariance makes this interference independent of receiver location.

B. Cross-Tier Interference

  • Cross-tier interference comes from nodes not associated with the macro base station, modeled as a PPP outside an exclusion region around the receiver.The exclusion radius may be only a few meters and avoids a low-power node being colocated with the receiver.Source: B. Cross-Tier Interference · p. 7
  • Cross-tier interference is constant with respect to receiver location because of the PPP’s shift invariance.Source: B. Cross-Tier Interference · p. 7

VI. APPROXIMATING THE INTERFERENCE DISTRIBUTION USING THE GAMMA DISTRIBUTION

Because the interference distribution is characterized through a Laplace transform, the paper uses a Gamma approximation to obtain simpler expressions. Its parameters are based on the interference mean and variance, with extensions for heterogeneous sources and cross-tier interference.

VI. APPROXIMATING THE INTERFERENCE DISTRIBUTION USING THE GAMMA DISTRIBUTION

  • The Gamma approximation simplifies interference calculations because the Laplace-transform characterization admits further simplification only for special parameter and mark choices.The approximation is parameterized using the first two moments of the interference; for homogeneous interference, the mark is Z = GkLkPg.Source: §VI · p. 7, A. Moments of the Interference Process · p. 7

A. Moments of the Interference Process

  • For homogeneous interference, the paper derives mean and variance expressions, then approximates heterogeneous interference using moments of the sum and an equivalent mark distribution.The general-case expressions use the approximation in (14) and (15).Source: A. Moments of the Interference Process · p. 7
  • When n = 4, exact moments of heterogeneous interference can be computed without the small-ball approximation.Source: A. Moments of the Interference Process · p. 7
  • Cross-tier interference can be added by replacing Rc + Rg − r with Rcross, while different active-user counts in multiuser MIMO can use separate marked point processes.Source: A. Moments of the Interference Process · p. 8

B. Gamma Approximation of the Interference

The paper approximates interference with a Gamma distribution to make performance expressions simpler. It uses the resulting signal-to-interference ratio to characterize success probability and achievable rate.

B. Gamma Approximation of the Interference

  • The interference is approximated by a Gamma random variable matched to the mean and variance of I(r), because exact Laplace-transform characterizations are challenging for many scenarios.Source: B. Gamma Approximation of the… · p. 8

VII. CHARACTERIZATION OF THE RANDOM SIR

  • Performance is characterized using SIR, with a Gamma-distributed signal whose scale absorbs distance-dependent path loss and transmit power.Source: §VII · p. 8
  • The analysis evaluates success probability P(SIR(r) > T) and ergodic achievable rate, interpreting them as diversity performance and average rate under Gaussian interference, respectively.Source: §VII · p. 8

A. Success Probability

The section develops success-probability expressions for Gamma-distributed signals with interference represented either by its characterized distribution or by a Gamma approximation. It also discusses a more exact heterogeneous expression and its computational trade-off.

A. Success Probability

  • For a Gamma-distributed signal and the characterized out-of-cell interference, the success probability is derived using results from and parameters supplied by Theorem 4.Source: A. Success Probability · pp. 8–9
  • For ks = 1, the success-probability expression has a simplified form.The regularized hypergeometric function can be computed with numerical packages, and Monte Carlo evaluation may be faster depending on the package used.Source: A. Success Probability · pp. 8–9
  • The Gamma approximation yields a simpler success-probability expression for Gamma-distributed signal and interference, while the exact expression is described as involved and uninformative.The Gamma-interference parameters are obtained from the moment expressions, with heterogeneous interference handled by replacing I(r) by J(r).Source: A. Success Probability · p. 8
  • The heterogeneous expression that models each interferer separately is more exact but requires truncating an infinite series; related work reports the approximation often outperforms that truncation unless many terms are retained.Source: A. Success Probability · pp. 8–9

B. Average Rate

Average rate is expressed as the expected logarithm of one plus SIR. The section derives expressions for homogeneous and heterogeneous interference and offers a moment-matched alternative when sums become costly to compute.

B. Average Rate

  • Average rate is computed as E ln(1 + SIR(r)); for homogeneous interference, the derivation evaluates expected logarithms of sums of Gamma random variables.Source: B. Average Rate · p. 9
  • The homogeneous average-rate expression uses θmin, ρ, C, and recursively found coefficients, and the same approach extends to heterogeneous interference.Source: B. Average Rate · p. 9
  • When the number of terms is too large for practical computation, the sum of signal and interference can instead be approximated by a Gamma variable matched on mean and variance.The heterogeneous extension modifies the mean and variance calculations.Source: B. Average Rate · pp. 9–10

VIII. SIMULATIONS

Simulations compare the hybrid model with a PPP layout and test Gamma-approximated interference in homogeneous and heterogeneous settings. The approximation performs well overall, with larger errors near the cell edge.

VIII. SIMULATIONS

  • The hybrid approach matches the PPP-based stochastic-geometry model across locations, while omitting the dominant interferer makes the cell-edge result too optimistic.Fig. 4 plots coverage probability against distance from the cell center.Source: §VIII · p. 10, Figure 4 · p. 10
  • The Gamma interference approximation and the bound using interferers outside radius Rc + Rg − r provide relatively tight coverage-probability and SIR-distribution fits.Fig. 5 evaluates coverage probability for a receiver at 0.7Rc; its caption notes larger error at the cell edge for β = 1 at high SIR.Source: §VIII · p. 10, Figure 5 · p. 10
  • The heterogeneous simulation includes macro-base-station, distributed-antenna, and femtocell interference, with femtocells allowed inside the cell and a 10dB penetration loss assumed.The setup uses a 5m femtocell guard region; performance is evaluated as a function of distance from the fixed-cell center.Source: §VIII · pp. 10–11
  • The effective fixed-cell coverage area shrinks in the heterogeneous case as the overall interferer density increases.The heterogeneous cell radius and guard-region size are calculated for each tier.Source: §VIII · p. 11
  • The Gamma approximation fits average rate and success probability well under homogeneous and heterogeneous interference, with more significant error at the cell edge.In the heterogeneous case, cross-tier interference shifts the rate curve by a relatively small, consistent offset.Source: §VIII · p. 11, Figure 6 · p. 11

IX. CONCLUSIONS

The paper models composite interference outside a fixed cell and uses a moment-matched Gamma distribution to simplify rate and success-probability calculations. Simulations show a reasonable fit, with small cell-edge error when the Gamma approximation is used.

IX. CONCLUSIONS

  • Moment matching approximates the interference distribution with a Gamma random variable, yielding relatively simple expressions for success probability and ergodic rate.The paper says the moments are finite under its nonsingular path-loss model.Source: §IX · p. 11
  • With a guard region and dominant interferer, the model reasonably fits the stochastic-geometry model; the Gamma approximation has small error at the cell edge.Fig. 7 compares Gamma-approximation and Monte Carlo success probabilities at a 5dB target SIR, with slight cell-edge error.Source: §IX · p. 11, Figure 7 · p. 11
  • The sum-interference term can be replaced by an equivalent interference random variable for numerical integration, although those integrals may still require substantial computational power.The authors identify bounding approximation errors and incorporating more complex propagation models as future work.Source: §IX · p. 11
Loading 1207.2041v2…