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Poisson Hole Process: Theory and Applications to Wireless Networks
Zeinab Yazdanshenasan, Harpreet S. Dhillon, Mehrnaz Afshang, Peter Han Joo Chong
TL;DR
Homogeneous PPP models do not capture the spatial interaction created by interference-management holes, and exact typical-node interference characterization in a PHP is challenging. The paper derives tractable bounds on the interference Laplace transform using local hole geometry and explores overlap-aware refinements. The proposed bounds are reported to characterize interference and coverage accurately across the configurations considered, including cases with significant hole overlap.
Problem
Exact characterization of interference at a typical receiver in a PHP is challenging, while PPP-based approximations are typically accurate only over particular parameter ranges.
Source: A. Related Work and Applications · p. 2Method
The paper derives interference Laplace-transform bounds by modeling nearby holes explicitly, including two-hole overlap and an average correction for overlaps.
Source: §V · pp. 6–7Results
The proposed bounds closely characterize coverage across the examined configurations, including significant-overlap cases, while the first-order statistic approximation is fairly loose.
Source: §VI · pp. 9–10Takeaways & Limitations
The bounds provide accurate coverage characterization for a typical PHP receiver under the scenarios examined.
Source: §VI · p. 10The serving transmitter is fixed at distance r0, corresponding to an ad hoc network; random serving links and other hole shapes or driving processes remain future directions.
Source: §VII · p. 11
Abstract
from arXiv · showhide
Interference field in wireless networks is often modeled by a homogeneous Poisson Point Process (PPP). While it is realistic in modeling the inherent node irregularity and provides meaningful first-order results, it falls short in modeling the effect of interference management techniques, which typically introduce some form of spatial interaction among active transmitters. In some applications, such as cognitive radio and device-to-device networks, this interaction may result in the formation of holes in an otherwise homogeneous interference field. The resulting interference field can be accurately modeled as a Poisson Hole Process (PHP). Despite the importance of PHP in many applications, the exact characterization of interference experienced by a typical node in a PHP is not known. In this paper, we derive several tight upper and lower bounds on the Laplace transform of this interference. Numerical comparisons reveal that the new bounds outperform all known bounds and approximations, and are remarkably tight in all operational regimes of interest. The key in deriving these tight and yet simple bounds is to capture the local neighborhood around the typical point accurately while simplifying the far field to attain tractability. Ideas for tightening these bounds further by incorporating the effect of overlaps in the holes are also discussed. These results immediately lead to an accurate characterization of the coverage probability of the typical node in a PHP under Rayleigh fading.
I. INTRODUCTION
The paper studies Poisson Hole Processes (PHPs), which model spatial separation among transmitters, and develops tractable interference bounds that preserve the typical receiver’s local neighborhood. It applies the resulting interference characterization to coverage analysis and discusses settings where PHPs model wireless networks.
I. INTRODUCTION
- A PHP models spatial separation by carving radius-D holes from a baseline PPP, with hole locations independently distributed as a PPP.The paper considers cognitive radio, heterogeneous cellular, and device-to-device settings as applications of this model.Source: §I · p. 1
A. Related Work and Applications
- PHPs model active secondary users outside primary-user exclusion zones, small cells separated from macrocells –, and D2D transmitters outside cellular-link inhibition zones,.A related model uses PHP-distributed cluster centers for integrated D2D and cellular networks.Source: A. Related Work and Applications · pp. 1–2
- Prior analyses approximate PHPs by a density-matched PPP, ignore holes, or match first and second order statistics with a PCP; their accuracy is typically limited to particular parameter ranges.The paper identifies exact characterization of typical-node interference as challenging.Source: A. Related Work and Applications · p. 2
B. Contributions
- The approach preserves the typical point’s local neighborhood to derive tight, provable Laplace-transform bounds, while simplifying the far field for tractability.The lower bound considers the closest hole; the upper bound carves out holes individually, potentially removing baseline points more than once.Source: B. Contributions · p. 2
- Three overlap-aware approaches extend the bounds: exact overlap treatment for two closest holes, a bound on overlap area, and an average-overlap correction.These approaches aim to tighten the simpler bounds that avoid explicitly incorporating hole overlaps.Source: B. Contributions · p. 2
A. System Model
- The system model defines a PHP from independent baseline and hole PPPs, with the baseline points outside the union of radius-D holes forming the process.The hole and baseline PPP densities are λ1 and λ2, respectively, with λ2 > λ1.Source: A. System Model · pp. 2–3
- The typical receiver is placed at the origin, and its serving transmitter is assumed to be at fixed distance r0.The fixed-distance serving link is chosen to focus on interference characterization and corresponds to an ad hoc network scenario.Source: A. System Model · p. 3
B. SIR and Coverage Probability
- SIR coverage probability is the probability that the receiver’s SIR exceeds threshold γ, and under the stated fading assumption it can be obtained from the interference Laplace transform.The paper focuses on that transform as a first step toward characterizing coverage in this setup.Source: B. SIR and Coverage Probability · p. 3
III. KEY PRIOR APPROACHES
Prior methods approximate the Poisson Hole Process (PHP) with a homogeneous PPP, while the proposed approach preserves the nearest-hole geometry to derive tractable interference bounds. The section also explains how a single hole can be represented through a radially varying point density.
A. Lower Bound by Ignoring Holes: Approximating Ψ by Φ2
- Ignoring holes and using the baseline PPP overestimates interference, yielding a lower bound on the interference Laplace transform.This approximation preserves the local neighborhood around the typical point.Source: A. Lower Bound by Ignoring Holes:… · p. 3, B. Approximating PHP by a PPP with the… · p. 4
B. Approximating PHP by a PPP with the Same Density
- The first-order statistic approximation independently thins the baseline PPP to match the PHP density, but this disrupts the typical point’s local neighborhood and can produce a loose approximation.A separate fitting-based approach uses a PCP matching the PHP’s first- and second-order statistics –.Source: B. Approximating PHP by a PPP with the… · pp. 3–4
III. KEY PRIOR APPROACHES
- The section also identifies a PCP fitted to match the PHP’s first- and second-order statistics as another prior approximation.The cited applications include cognitive radio and heterogeneous cellular networks –.Source: B. Approximating PHP by a PPP with the… · p. 4
IV. PROPOSED APPROACHES TO LAPLACE TRANSFORM OF INTERFERENCE IN A PHP
- The proposed approach first analyzes a PPP with one fixed hole, then uses that conditional Laplace transform to derive bounds for a typical PHP node.The fixed hole makes the intermediate interference field non-isotropic.Source: §IV · pp. 4–5
- A single hole can be dissolved into a non-homogeneous PPP with density λ2(1 − θ(r)/π), preserving the number of interferers in each thin radial strip.The removed density term λ2θ(r)/π captures the hole’s effect.Source: §IV · p. 4
A. Lower Bound on the Laplace Transform of Interference in a PHP
The lower bound keeps the hole nearest the typical point and ignores the others, thereby overestimating interference. It averages the single-hole result over the nearest-hole distance, whose minimum is D.
A. Lower Bound on the Laplace Transform of Interference in a PHP
- Ignoring every hole except the nearest overestimates interference and therefore yields a lower bound on the Laplace transform.The approach focuses on the nearest hole because path loss makes nearby holes more significant.Source: A. Lower Bound on the Laplace… · p. 5
- The nearest-hole distance V1 is at least D, and averaging the single-hole transform over its distribution gives the proposed bound.The distance distribution is derived from the PPP properties and the fact that the typical point lies outside all holes.Source: A. Lower Bound on the Laplace… · p. 5
- This construction is tighter than the known lower bound that ignores all holes because it captures the typical node’s local neighborhood more accurately.The result is the lower bound stated in Theorem 1.Source: A. Lower Bound on the Laplace… · p. 5
B. Upper Bound for the Laplace Transform of Interference in a PHP
The paper bounds the interference Laplace transform by simplifying how holes are handled, then explores overlap-aware refinements. These approaches balance tighter local modeling against the complexity or approximation introduced by accounting for overlaps.
B. Upper Bound for the Laplace Transform of Interference in a PHP
- Carving each hole separately can remove some baseline-PPP points multiple times, underestimating interference and yielding an upper bound on its Laplace transform.The authors report that this bound is remarkably tight across a wide variety of scenarios.Source: B. Upper Bound for the Laplace… · pp. 5–6
C. Ratio of the Proposed Upper and Lower Bounds
- The proposed upper-to-lower-bound ratio approximates interference removed by all holes except the closest, while ignoring overlaps.The approximation is intended to assess bound tightness; the passage says it will be shown close to one across a wide parameter range.Source: C. Ratio of the Proposed Upper and… · p. 6
V. INCORPORATING THE IMPACT OF OVERLAPS IN THE PROPOSED APPROACHES
- Theorem 3 derives a lower bound by modeling interference with the baseline process outside the union of the two nearest holes, thereby overestimating interference.The two-hole overlap is incorporated explicitly, with the distances and angle between the holes used in the derivation.Source: §V · p. 6
- Explicitly accounting for overlap between even two holes makes the expressions significantly more complex than the bounds that avoid overlap calculations.The authors identify this as a significant loss in tractability and motivate bounding overlap areas across multiple nearby holes instead.Source: §V · p. 6
- Rescaling the upper-bound expression to account for average pairwise overlap produces an approximation rather than a bound.The authors report that this approach may perform poorly when overlaps are significant because it does not capture the typical point’s local neighborhood as carefully.Source: §V · p. 7
VI. NUMERICAL RESULTS AND DISCUSSION
Across the tested network configurations, the proposed bounds and approximations closely characterize coverage, including when holes overlap substantially. The first-order statistic approximation can become loose when overlaps are significant.
Network configurations
- The four configurations distinguish low or high hole density from small or large hole radii; sparse holes are more benign than dense holes at either radius.The study also plots coverage as λ1 varies at D = 1 and as D varies at λ1 = 0.1.Source: §VI · pp. 7–9
Bound tightness
- The ratio of the proposed upper and lower bounds is close to one in all configurations of interest, though comparatively higher for HD-LH.This numerical result supports the tightness of both bounds.Source: Figure 6 · p. 8, §VI · p. 9
Coverage versus hole density and radius
- The proposed bounds and approximations are surprisingly tight even in HD-LH, while prior results deviate significantly when hole overlaps are significant.The first-order statistic approximation becomes very loose because independent thinning distorts the typical receiver’s local neighborhood.Source: Figure 8 · p. 8, §VI · p. 9
Coverage versus SIR threshold
- The proposed bounds remain tight across all four configurations, while the first-order statistic approximation is rather loose (Figs. 9–12).The proposed results are nearly indistinguishable except in the most extreme HD-LH case.Source: §VI · p. 9, Figure 12 · p. 9
Coverage versus hole density and radius
- The lower bounds from Theorems 1 and 4 work equally well in HD-LH, where considering even one hole characterizes interference accurately.The average-overlap approximation does not outperform the proposed bounds in this extreme configuration because it does not capture the typical point’s local neighborhood as well.Source: §VI · p. 10
Coverage versus λ2/λ1
- For coverage versus λ2/λ1, the proposed bounds remain remarkably accurate in LD-LH and HD-LH, while the first-order statistic approximation is fairly loose.These cases use D = 1.5 and λ1 = 0.05 or 0.2; the bounds preserve the local neighborhood while deriving the characterization.Source: §VI · p. 10, Figure 15 · p. 10, Figure 16 · p. 10
VII. CONCLUSIONS
The paper provides provable bounds on interference in a Poisson Hole Process by preserving the typical point’s local neighborhood while simplifying the far field. The bounds are reported as the tightest known and yield coverage characterization under independent Rayleigh fading.
Contributions
- The paper derives easy-to-use, provable upper and lower bounds on the Laplace transform of interference experienced by a typical PHP user.The authors report these as the tightest known bounds for this quantity.Source: §VII · pp. 10–11
- The approach dissolves holes into a tractable non-homogeneous PPP, preserving the typical point’s local neighborhood while simplifying the far field.The paper demonstrates bound tightness analytically and numerically against simulations and known approaches.Source: §VII · p. 11
Scope and applications
- The bounds immediately characterize coverage probability when all wireless links experience independent Rayleigh fading.The assumed fixed distance to the serving transmitter corresponds to an ad hoc network; relaxing it is identified as future work.Source: §VII · p. 11
APPENDIX
The appendix proofs reduce a baseline case to PPP interference calculations, derive a transform conditioned on hole-center distance, and use it to construct a lower bound.
A. Proof of Lemma 1
- The baseline PHP approximation uses a PPP interference-field Laplace transform, whose derivation applies the PPP PGFL, exponential fading, and a Gamma-function integral.The approximation reduces the derivation to the well-studied PPP case.Source: A. Proof of Lemma 1 · p. 11
B. Proof of Lemma 3
- Conditioning on the hole-center distance yields an interference Laplace transform, with its expression using PPP PGFL calculations and cosine-law geometry.The proof converts the integral to polar coordinates and evaluates it using the Gamma function.Source: B. Proof of Lemma 3 · p. 11
C. Proof of Theorem 1
- The lower-bound proof substitutes Lemma 3’s conditional transform and the distribution of V1, using an exclusion zone centered at the nearest hole center.The proof also uses the stated expression for g(v1).Source: C. Proof of Theorem 1 · pp. 11–12
D. Proof of Theorem 2
The proof starts from the PHP interference Laplace transform and simplifies overlap-dependent integration by ignoring hole overlaps; the typical-point exclusion sets the radial integration’s lower limit at D.
D. Proof of Theorem 2
- The proof begins with the PHP interference Laplace transform and takes expectations over channel gains and the PPP conditional on the hole region.The PPP PGFL is used to take the expectation over the interferer process.Source: D. Proof of Theorem 2 · p. 12
- Ignoring overlaps among holes makes the integral tractable for deriving the bound, after which the proof reuses the conditional-transform result from Lemma 3.The proof also solves the first integral as in Lemma 3.Source: D. Proof of Theorem 2 · p. 12
- The typical-point condition excludes points of Φ1 within distance D of the origin, so the integral’s lower limit is D.Source: D. Proof of Theorem 2 · p. 12
E. Proof of Proposition 1
The proof relates the upper and lower interference bounds through an approximation that accounts for the closest hole and the effective interference removed by other holes.
E. Proof of Proposition 1
- The ratio of the bounds uses Jensen’s inequality and an approximation because the upper- and lower-bound interference terms are not truly independent.The resulting expression is stated to provide a tight approximation numerically.Source: E. Proof of Proposition 1 · p. 12
- The difference between the upper- and lower-bound interference terms represents removal by holes other than the closest, with overlaps ignored.The upper-bound construction may virtually remove some baseline PPP points multiple times.Source: E. Proof of Proposition 1 · p. 12
- The proof completes by deconditioning over the closest-hole distance using its distribution from equation (9).The closest-hole distance is V1, the distance from the origin to the closest point of Φ1.Source: E. Proof of Proposition 1 · p. 12
F. Proof of Theorem 3
The lower-bound proof conditions on the two closest holes and their relative angle, then evaluates how their overlap affects the interference calculation.
F. Proof of Theorem 3
- The lower bound overestimates interference by retaining only the two holes closest to the typical node and conditioning on their distances and angle.The proof then deconditions over those random variables.Source: F. Proof of Theorem 3 · p. 12
- The conditional calculation uses the union of the two hole regions and treats their intersection as a separate contribution.The closest-hole, second-closest-hole, and intersection terms are identified separately.Source: F. Proof of Theorem 3 · pp. 12–13
- The holes overlap only when their centers are less than 2D apart, and for fixed distances the overlap is limited by an angle threshold.The proof finds circle-intersection coordinates from the two circle equations to evaluate the intersection integral.Source: F. Proof of Theorem 3 · p. 13
G. Proof of Theorem 4
The proof extends the lower-bound construction to the k closest holes, overestimates interference, and then averages over their distances.
G. Proof of Theorem 4
- Using the k closest holes, the proof constructs an interference region that contains the actual region, thereby overestimating interference and bounding the Laplace transform below.The construction is illustrated in Fig. 4.Source: G. Proof of Theorem 4 · p. 13
- Conditioned on the hole-center distances, the Laplace transform removes baseline PPP points in the closest hole and the remaining specified hole regions.The conditional expression uses the exponential fading distribution and the PPP probability generating functional.Source: G. Proof of Theorem 4 · p. 14
- The proof obtains the final expression by deconditioning over the k hole-center distances using their joint distribution.The joint density is derived by the same argument as equation (13).Source: G. Proof of Theorem 4 · p. 14
H. Proof of Proposition 2
The overlap correction estimates the effective removal rate by accounting for pairwise overlap area, then rescales the hole-removal term to compensate for multiple removals.
H. Proof of Proposition 2
- Theorem 2 removes overlapping circles separately, so overlapping regions are counted multiple times and the effective removal rate exceeds λ2.The overlap correction confines its estimate to pairwise overlaps for tractability.Source: H. Proof of Proposition 2 · p. 14
- To compensate for multiple removal of points, the proof rescales the second term of the Theorem 2 expression.The correction uses the estimated effective removal rate based on the pairwise overlap area.Source: H. Proof of Proposition 2 · pp. 14–15
- The overlap estimate counts circles centered inside b(y, 2D), whose number is Poisson with mean λ1 4πD².Conditioned on the number of overlapping circles, their centers are independent and uniform over that region.Source: H. Proof of Proposition 2 · p. 14
- The total pairwise overlap area is estimated as Aol = min(λ1π²D⁴, πD²).The upper cap is the area of a full hole.Source: H. Proof of Proposition 2 · p. 15