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3GPP-inspired HetNet Model using Poisson Cluster Process: Sum-product Functionals and Downlink Coverage

Chiranjib Saha, Mehrnaz Afshang, Harpreet S. Dhillon

arXiv:1705.01699v1cs.ITcs.NImath.IT

TL;DR

The PPP-based K-tier HetNet model does not capture non-uniformity and coupling between user and small-cell locations. This paper combines PPPs and PCPs to model these configurations and shows that, as cluster size grows, the coverage result converges to the baseline PPP-based HetNet result.

  • Problem

    The PPP-based K-tier HetNet model is not rich enough to model non-uniformity and coupling across user and small-cell locations.

    Source: §VI · p. 28

  • Method

    The paper combines PPP and PCP models and analyzes typical-user downlink coverage under max-SIR association using sum-product functionals for PPPs, PCPs, and their offspring processes.

    Source: 2) Sum-product functional and coverage… · p. 7, §VI · p. 28

  • Results

    As cluster size tends to infinity, the PCP weakly converges to a PPP, and the coverage probability converges to the baseline PPP-based HetNet result.

    Source: 3) Limiting behavior · p. 7

  • Takeaways & Limitations

    The unified model generalizes the PPP-based HetNet model to represent varied user and small-cell configurations considered in practical design, including 3GPP simulation models.

    Source: §VI · p. 28

Abstract

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The growing complexity of heterogeneous cellular networks (HetNets) has necessitated a variety of user and base station (BS) configurations to be considered for realistic performance evaluation and system design. This is directly reflected in the HetNet simulation models proposed by standardization bodies, such as the third generation partnership project (3GPP). Complementary to these simulation models, stochastic geometry-based approach, modeling the locations of the users and the K tiers of BSs as independent and homogeneous Poisson point processes (PPPs), has gained prominence in the past few years. Despite its success in revealing useful insights, this PPP-based K-tier HetNet model is not rich enough to capture spatial coupling between user and BS locations that exists in real-world HetNet deployments and is included in 3GPP simulation models. In this paper, we demonstrate that modeling a fraction of users and arbitrary number of BS tiers alternatively with a Poisson cluster process (PCP) captures the aforementioned coupling, thus bridging the gap between the 3GPP simulation models and the PPP-based analytic model for HetNets. We further show that the downlink coverage probability of a typical user under maximum signal-to-interference-ratio association can be expressed in terms of the sum-product functionals over PPP, PCP, and its associated offspring point process, which are all characterized as a part of our analysis. We also show that the proposed model converges to the PPP-based HetNet model as the cluster size of the PCPs tends to infinity. Finally, we specialize our analysis based on general PCPs for Thomas and Matern cluster processes. Special instances of the proposed model closely resemble the different configurations for BS and user locations considered in 3GPP simulations.

I. INTRODUCTION

The introduction motivates a unified PPP/PCP HetNet framework to capture user–BS spatial coupling omitted by independent-PPP models, then outlines max-SIR coverage analysis and its PPP limit.

I. INTRODUCTION

  • 3GPP simulations include uniform or hotspot-clustered users and uncorrelated or correlated SBS placements, especially SBS-user hotspot coupling,,.The paper focuses on SBS deployment in user hotspots.Source: A. 3GPP Models for HetNets · pp. 2–3

B. Stochastic geometry-based approaches

1) A unified framework with PCP and PPP modeled BSs and users:

  • The unified model uses PCPs for an arbitrary number of BS tiers and a fraction of users, making the four generative models and listed 3GPP configurations special cases.User and BS clusters can share a cluster center, or a BS can occupy the user's cluster center.Source: 1) A unified framework with PCP and… · p. 7

2) Sum-product functional and coverage probability analysis:

3) Limiting behavior:

  • As cluster size tends to infinity, a PCP weakly converges to a PPP, and the framework's coverage probability converges to the independent-PPP baseline result.Source: 3) Limiting behavior · p. 7

II. SYSTEM MODEL

The model allows users and base-station tiers to follow PPPs or PCPs, including configurations that couple user and BS locations. It also introduces the point-process functionals used to analyze coverage under maximum-SIR connectivity.

II. SYSTEM MODEL

  • The K-tier HetNet assigns each BS tier a PPP or PCP and allows users to be modeled as a PPP or PCP rather than requiring independence from BS locations.The analysis considers uniform users and clustered-user configurations, including clusters sharing centers with BS clusters.Source: A. K-tier HetNet Model · pp. 8–9
  • For clustered-user configurations, selecting a typical user also selects its cluster, and the model separates the associated BS point or cluster into a distinct 0th tier.The removed representative point or cluster is treated separately while the remaining BS process retains its distribution by Slivnyak’s theorem.Source: A. K-tier HetNet Model · p. 9
  • Coverage is defined for a typical user connecting to the BS with maximum SIR, assuming Rayleigh fading, path-loss exponent α > 2, and negligible thermal noise.The paper seeks an exact characterization of the resulting coverage probability.Source: A. K-tier HetNet Model · pp. 9–10

III. POINT PROCESS FUNCTIONALS

  • The paper characterizes sum-product functionals and PGFLs under original and reduced Palm distributions for PPPs, PCPs, and PCP offspring processes.The functionals are used in the next section to analyze coverage under max-SIR connectivity.Source: §III · pp. 10–11

A. Sum-product Functionals

The section derives sum-product functionals for PPPs, PCPs, and the finite offspring process associated with a PCP. The PCP expression accounts for the cluster of a selected point through a reduced-Palm PGFL.

A. Sum-product Functionals

  • For a PPP, the sum-product functional follows by applying Campbell–Mecke, with the reduced-Palm PGFL equal to the original PGFL by Slivnyak’s theorem.The PPP’s intensity measure is λdx.Source: A. Sum-product Functionals · p. 11
  • For a PCP, evaluating the sum-product functional requires tracking the selected point’s cluster center and incorporating the cluster PGFL under its reduced Palm distribution.The expression combines the PCP PGFL with the reduced-Palm PGFL of the offspring cluster.Source: A. Sum-product Functionals · pp. 11–12
  • The offspring process z + Bz is finite, so the PPP/PCP derivation does not apply directly; its functional is derived with the process conditioned to contain the selected point.Under this conditioning, the point count follows a weighted distribution.Source: A. Sum-product Functionals · pp. 12–13

B. Probability Generating Functional

The section presents PGFLs for PPPs, PCPs, and PCP offspring processes, distinguishing original and reduced Palm distributions. For PCPs, the reduced-Palm PGFL includes a contribution from the selected point’s cluster.

B. Probability Generating Functional

IV. COVERAGE PROBABILITY ANALYSIS

The section derives typical-user downlink coverage for the unified HetNet model, first for general Neyman–Scott clusters and then for Thomas and Matérn cluster processes. Under maximum-SIR association and βk > 1 for every tier, coverage is assembled from per-tier coverage probabilities.

IV. COVERAGE PROBABILITY ANALYSIS

  • The analysis first treats general Neyman–Scott clusters and then specializes coverage to Matérn and Thomas cluster processes.The typical-user coverage expression also depends on the user configuration, with separate cases for the user process.Source: §IV · p. 15, A. Neyman-Scott cluster process · p. 17

A. Neyman-Scott cluster process

  • Theorem 1 expresses typical-user coverage under maximum-SIR association for βk > 1 across all tiers.The result applies to the general K-tier setup.Source: A. Neyman-Scott cluster process · p. 15
  • Overall coverage probability Pc is the sum of K + 1 per-tier coverage probabilities; each is the joint probability that the serving BS belongs to Φk and the typical user is under coverage.Source: A. Neyman-Scott cluster process · p. 15
  • The section provides distinct per-tier coverage expressions for PCP base-station tiers and PPP base-station tiers.The PCP case uses its PGFL and reduced-Palm offspring PGFL; the PPP case uses the PPP result.Source: A. Neyman-Scott cluster process · p. 16

B. Convergence

As PCP cluster size grows, the BS process converges to a PPP, independent of its parent process. The user configurations in Cases 2 and 3 likewise converge to independent PPPs under the stated cluster-size limits.

B. Convergence

  • As cluster size tends to infinity, a PCP weakly converges to a PPP of intensity m̄_kλ_pk, provided sup(f_k) < ∞.The limiting BS tiers in K2 are PPPs.Source: B. Convergence · pp. 17–18
  • The limiting PPP and the parent PPP become independent as cluster size tends to infinity.The independence result is stated for arbitrary closed compact sets in the void-probability characterization.Source: B. Convergence · p. 18
  • In Case 2, the user PCP converges to a PPP independent of BS locations as its cluster size increases.In Case 3, the user and coupled BS processes become independent PPPs as the user and relevant BS cluster sizes increase.Source: B. Convergence · p. 19
  • When BS cluster sizes tend to infinity, coverage probability reduces to the baseline PPP-model expression.The corollary identifies the limiting coverage expression with [4, Corollary 1].Source: B. Convergence · p. 19

C. Mat´ern Cluster Process

The Matérn cluster process (MCP) models offspring uniformly within a disc around each cluster center. The section uses MCP probability-generating functionals and distance distributions to express per-tier coverage for clustered BS tiers and user configurations.

C. Matérn Cluster Process

  • MCP offspring are uniformly distributed within a disc of radius r_dk around the origin relative to their cluster center.The section assumes MCPs for the BS tiers in K2 and selects this process because it closely resembles 3GPP SBS and user clusters.Source: C. Mat´ern Cluster Process · p. 20
  • For MCP coverage calculations, the offspring distance distribution is conditioned on the cluster-center location, with separate geometric cases illustrated in Fig. 5.These distance distributions support the polar-domain evaluation of per-tier coverage.Source: C. Mat´ern Cluster Process · p. 21
  • The section provides PGFL expressions for MCPs and their offspring processes as inputs to the coverage derivation.The MCP PGFL and offspring-process PGFL are given in Corollaries 2 and 3.Source: C. Mat´ern Cluster Process · pp. 20–22
  • Per-tier coverage expressions are derived for BS tiers in K2 modeled as MCPs and for tier 0 under the listed user configurations.The tier-0 calculation for Case 3 uses the sum-product functional of the representative user-coupled BS cluster.Source: C. Mat´ern Cluster Process · pp. 22–23

D. Thomas Cluster Process

The Thomas cluster process (TCP) assigns Gaussian-distributed offspring around cluster centers, extending the framework beyond the cluster geometries used in 3GPP models. Its infinite-support offspring density yields simpler coverage expressions than the Matérn cluster process.

D. Thomas Cluster Process

  • TCP offspring are Gaussian-distributed around the cluster center, and its infinite-support density yields simpler coverage expressions than MCP.The paper includes TCP to demonstrate that the framework extends beyond the cluster-based simulation models adopted by 3GPP.Source: D. Thomas Cluster Process · p. 24

CASE 2,

  • For Case 2 with TCP BS tiers, the section gives per-tier coverage expressions for both k ∈ K2 and k = 0.The expressions are presented in Corollaries 8 and 9.Source: CASE 2 · p. 25

V. RESULTS AND DISCUSSIONS

The analytical coverage results match simulation, and coverage probability varies with the HetNet model. The comparisons show that model choice matters for representing different BS and user configurations.

  • Analytical results perfectly match simulation, verifying the coverage analysis for the tested HetNet configurations.Source: §V · p. 26
  • Model 2 gives a typical user higher coverage than Model 1.Source: §V · p. 26
  • Model 1 coverage is a lower bound for Model 4 and an upper bound for Model 3, underscoring the importance of matching models to spatial coupling.Source: §V · p. 26

A. Effect of Variation of Cluster Size

Increasing PCP cluster size moves coverage toward the Model 1 result in Models 2–4. The direction of the coverage change differs between Models 3 and 4, consistent with their different user–SBS spatial relationships.

A. Effect of Variation of Cluster Size

B. Effect of Variation of Intensity of Parent PPP

Coverage probability in Models 2–4 varies with parent-PPP intensity and approaches Model 1 coverage as that intensity increases. The conclusion presents a unified PPP–PCP model, its coverage analysis, and its convergence and specialization results.

VI. CONCLUSION

  • The unified PPP–PCP HetNet model represents varied BS and user spatial configurations, including non-uniformity and coupling that the PPP-based K-tier model does not capture.Source: §VI · pp. 27–28

B. Effect of Variation of Intensity of Parent PPP

VI. CONCLUSION

  • The analysis characterizes downlink coverage under max-SIR association, evaluates sum-product functionals for PCPs and offspring processes, and proves PCP weak convergence to PPP as cluster size grows.Source: §VI · p. 28
  • The coverage results are specialized to Thomas and Matern cluster processes; future extensions include thresholds not restricted to βk > 1, other downlink metrics, uplink coverage, and more realistic channel models.Source: §VI · p. 28

APPENDIX

The proof uses the assumption βk > 1 to limit the number of BSs satisfying the coverage condition. Its subsequent steps invoke exponential fading and point-process independence to obtain the expression.

A. Proof of Theorem 1

  • Under βk > 1 for every tier, at most one BS can satisfy the coverage condition.Source: A. Proof of Theorem 1 · p. 28
  • One proof step uses exponentially distributed fading, and the final step invokes independence among the tier point processes.Source: A. Proof of Theorem 1 · p. 29
  • The interference calculation uses an i.i.d. sequence of exponential random variables and simplifies the exponential term to obtain the final expression.Source: A. Proof of Theorem 1 · p. 29
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