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Poisson Cluster Process Based Analysis of HetNets with Correlated User and Base Station Locations

Mehrnaz Afshang, Harpreet S. Dhillon

arXiv:1612.07285v2cs.ITmath.IT

TL;DR

Prior HetNet spatial models often treat users and small-cell base stations (SBSs) as independent, although both may cluster around user hotspots. This paper models those coupled locations and analyzes coverage and throughput in a two-tier network. It finds that increased same-resource-block reuse within a cluster raises throughput while reducing coverage probability.

  • Problem

    Prior stochastic-geometry HetNet models generally represent users and SBSs as independent PPPs, leaving their coupled locations in user-hotspot deployments unmodeled.

    Source: §I · p. 2, A. Related Work · p. 4, §VI · p. 24

  • Method

    The paper models hotspot centers as a PPP, clusters users and SBSs around them with independent distributions, and analyzes a two-tier HetNet under two association policies.

    Source: B. Contributions and Outcomes · p. 4

  • Results

    As more SBSs in a cluster reuse the same resource blocks, throughput increases while coverage probability decreases; under Policy 2, an optimal power threshold maximizes coverage probability.

    Source: System design insights · p. 5, 2) Association Policy 2 · p. 20

  • Takeaways & Limitations

    Multiple SBSs in a cluster can reuse the same resource block when coverage probability remains acceptable, rather than relying on strictly orthogonal allocation.

    Source: System design insights · p. 5

Abstract

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This paper develops a new approach to the modeling and analysis of HetNets that accurately incorporates coupling across the locations of users and base stations, which exists due to the deployment of small cell base stations (SBSs) at the places of high user density (termed user hotspots in this paper). Modeling the locations of the geographical centers of user hotspots as a homogeneous Poisson Point Process (PPP), we assume that the users and SBSs are clustered around each user hotspot center independently with two different distributions. The macrocell base station (BS) locations are modeled by an independent PPP. This model is consistent with the user and SBS configurations considered by 3GPP. Using this model, we study the performance of a typical user in terms of coverage probability and throughput for two association policies: i) Policy 1, under which a typical user is served by the open-access BS that provides maximum averaged received power, and ii) Policy 2, under which the typical user is served by the small cell tier if the maximum averaged received power from the open-access SBSs is greater than a certain power threshold; and macro tier otherwise. A key intermediate step in our analysis is the derivation of distance distributions from a typical user to the open-access and closed-access interfering SBSs. Our analysis demonstrates that as the number of SBSs reusing the same resource block increases, coverage probability decreases whereas throughput increases. Therefore, contrary to the usual assumption of orthogonal channelization, it is reasonable to assign the same resource block to multiple SBSs in a given cluster as long as the coverage probability remains acceptable. This approach to HetNet modeling and analysis significantly generalizes the state-of-the-art approaches that are based on modeling the locations of BSs and users by independent PPPs.

I. INTRODUCTION … B. Propagation Model

The paper models user-centric HetNets by clustering users and small-cell base stations around user-hotspot centers, capturing their spatial coupling. It analyzes coverage and throughput under two association policies and derives design insights about resource reuse and power thresholds.

I. INTRODUCTION

  • The study targets user-centric deployments where SBSs are placed at user hotspots, making independent PPP models of users and base stations inadequate to capture their coupling.The paper frames a comprehensive analytical framework for these deployments as its focus.Source: §I · p. 2

A. Related Work

  • Prior coupled user–BS models include a single-tier approach and HetNet models,, but the paper identifies clustered SBS locations and exact interference analysis as unresolved needs.It proposes using Poisson cluster processes for both users and SBSs to bridge industry hotspot models, including 3GPP, and stochastic-geometry analysis.Source: A. Related Work · pp. 3–4

B. Contributions and Outcomes

  • The framework models macro BSs as an independent PPP and clusters SBSs and users independently around PPP-distributed hotspot centers, then compares two serving-BS association policies.Policy 1 selects the open-access BS with maximum average received power; Policy 2 selects the small-cell tier above an SBS power threshold and the macro tier otherwise.Source: B. Contributions and Outcomes · p. 4
  • The analysis derives coverage and throughput using serving- and interfering-SBS distance distributions; conditioned on the serving BS and typical user, open-access interfering-SBS distances are i.i.d.That property enables the interference Laplace transform used to obtain the performance expressions.Source: B. Contributions and Outcomes · p. 4

System design insights:

  • More SBSs reusing a cluster’s resource blocks increase throughput but decrease coverage, while Policy 2 has an optimal power threshold for maximizing coverage.The analysis suggests reusing blocks across multiple SBSs when coverage remains acceptable, rather than strictly orthogonal allocation.Source: System design insights · p. 5

II. SYSTEM MODEL

  • The typical user can connect to any macro BS and to SBSs in its representative cluster; SBSs outside that cluster are closed-access interferers.The setup represents enterprise-owned SBSs restricted to authorized users and limits discussion to a two-tier network for simplicity.Source: §II · p. 5

A. Spatial Setup and Key Assumptions

  • The user and SBS processes share a parent PPP of hotspot centers but have distinct cluster distributions, reflecting hotspot users’ clustering rather than homogeneous user locations.The framework can be extended to mixed homogeneous and non-homogeneous user distributions; this analysis focuses on coupled, non-homogeneous users.Source: A. Spatial Setup and Key Assumptions · pp. 5–6

B. Propagation Model

  • All links to the typical user at the origin follow standard power-law path loss with exponent α > 2 and Rayleigh fading; for each tier, the candidate BS is the nearest open-access BS.Policy 1 serves the BS with maximum average received power among open-access BSs; Policy 2 selects the small-cell tier when its maximum RSRP exceeds a specific power threshold, and the macro tier otherwise.Source: B. Propagation Model · pp. 7–8

1) SIR at a typical user served by macrocell: · III. SERVING AND INTERFERING DISTANCES · A. Relevant distance distributions

The paper derives serving and interfering distance distributions for typical users under macrocell- and small-cell-serving conditions. Conditioning on the user’s cluster-center distance makes the clustered SBS distances tractable and supports later coverage and throughput analysis.

1) SIR at a typical user served by macrocell:

  • When a typical user is macrocell-served, interference comes from other macro BSs, active open-access SBSs in its cluster, and active closed-access SBSs outside it.The paper also defines a SIR for a user served by an SBS, partitioning interference into macro BSs, other active open-access SBSs in the cluster, and active closed-access SBSs outside it.Source: 1) SIR at a typical user served by… · p. 8

III. SERVING AND INTERFERING DISTANCES

  • The technical section derives association probabilities and characterizes distances from serving and interfering macro BSs and SBSs to a typical user.These distance distributions are used to characterize coverage probability and network throughput in the next section.Source: §III · pp. 8–9

A. Relevant distance distributions

  • The distance to the nearest macro BS is characterized using the null probability of a homogeneous PPP, while the nearest open-access SBS distance is more challenging because SBSs are clustered.The SBS distances share the cluster-center location, but conditioning on that location handles their correlation under the assumption that SBS locations are i.i.d. around the center.Source: A. Relevant distance distributions · p. 9
  • Conditioned on the typical user’s distance to its cluster center, distances to SBSs in the representative cluster are i.i.d., enabling derivation of the nearest open-access SBS distance distribution.For a Thomas cluster process, the conditional distance distribution is specialized using a Rician distribution; the nearest-SBS distribution follows from the minimum of the i.i.d. distances.Source: A. Relevant distance distributions · pp. 9–11
  • For the Thomas cluster process, distances from the user to SBSs in its representative cluster are conditionally i.i.d., with the single-distance distribution derived as Rician.The conditional distribution is given for a fixed distance to the cluster center; the cluster-center distance has a Rayleigh distribution with scale parameter σu.Source: A. Relevant distance distributions · p. 10
  • The conditional i.i.d. distances yield the CDF and PDF of the nearest open-access SBS distance by applying the minimum distribution to the sequence.The distributions are conditioned on the user’s distance to its cluster center.Source: A. Relevant distance distributions · pp. 10–11

B. Association policies

The section defines one candidate serving BS per open-access tier as the BS nearest to the typical user, then considers two policies for choosing between them.

B. Association policies

  • The nearest SBS and macro BS are the respective candidate servers, at distances R_s and R_m; two association policies select between them.The candidates are drawn from all macro BSs and SBSs within the representative cluster.Source: B. Association policies · p. 11

1) Association Policy 1:

Association Policy 1 assigns a typical user to the macro or small-cell tier according to which offers the maximum received power averaged over small-scale fading, a policy that maximizes coverage probability. The section derives the associated tier and serving-distance distributions and characterizes the exclusion zones and interfering SBS distances induced by that choice.

1) Association Policy 1:

  • The typical user associates with the macrocell or small cell according to which tier maximizes averaged received power, and this policy maximizes coverage probability.The tier events are defined by the argmax over the macrocell and small-cell received powers.Source: 1) Association Policy 1 · p. 11
  • The section derives association probabilities for each tier and serving-distance distributions conditioned on association to that tier.Serving distance is the distance to the nearest BS in the associated tier; the distributions use the user’s distance from its cluster center.Source: 1) Association Policy 1 · pp. 11–12
  • Association creates a user-centered exclusion zone for open-access BSs, while the distances to open-access interfering SBSs in the representative cluster are conditionally i.i.d.The exclusion radius depends on the serving tier; conditioning the macrocell PPP is straightforward, whereas the clustered SBS distance distributions require separate characterization.Source: 1) Association Policy 1 · pp. 12–13

2) Association Policy 2:

Association Policy 2 selects the small-cell tier using a received-power threshold from the nearest open-access SBS, and otherwise selects the macrocell. Because the decision depends only on that SBS distance, tuning the threshold can balance load between tiers with greater analytical tractability than Policy 1.

2) Association Policy 2:

  • A typical user associates with a small cell when P_s R_s^-alpha ≥ P0 and with a macrocell when P_s R_s^-alpha ≤ P0, where P0 is the SBS power threshold.The supplied policy definitions state both inequalities, including equality at the threshold.Source: 2) Association Policy 2 · p. 13
  • Unlike Policy 1, Policy 2 depends only on distance to the nearest open-access SBS, and P0 can be tuned to balance load across macro and small cells.The exact impact of P0 on coverage probability is deferred to later sections.Source: 2) Association Policy 2 · p. 13
  • For a user at distance ν0 from its cluster center, the macrocell serving-distance PDF is the distance-to-nearest-macro-BS PDF, while the small-cell PDF is conditioned on small-cell association.The macrocell distance is independent of the association event; the small-cell serving distance is the nearest open-access SBS distance conditioned on that event.Source: 2) Association Policy 2 · p. 14
  • Conditioned on the serving tier, open-access interfering SBS distances are conditionally i.i.d.; the closest such interferer must be farther than the serving distance.The passage states this distance constraint for users served by either the small-cell or macrocell tier.Source: 2) Association Policy 2 · p. 14
  • Closed-access interfering SBS distance distributions do not depend on association policy; in the Thomas cluster-process case, these distances are i.i.d. with a Rician PDF.For each cluster at a given distance from the user, its closed-access interfering SBS distances are i.i.d.Source: 2) Association Policy 2 · p. 15

IV. COVERAGE PROBABILITY AND THROUGHPUT ANALYSIS

The section uses previously derived distance distributions and association probabilities to analyze typical-user coverage probability and network throughput.

  • Previously derived distance distributions and association probabilities are used to characterize typical-user coverage probability and whole-network throughput.Source: §IV · p. 15

A. Coverage probability · 1) Association Policy 1:

Coverage probability is the chance that a typical user’s SIR exceeds the target threshold. Under Association Policy 1, the analysis derives interference Laplace transforms for three sources and uses them to obtain tier-specific and total coverage probabilities.

A. Coverage probability

  • Coverage probability is defined as the probability that a typical user’s SIR exceeds the target threshold β.The paper writes this as Pc = P(SIR > β).Source: A. Coverage probability · p. 15

1) Association Policy 1:

  • The analysis partitions interference into open-access SBSs, closed-access SBSs, and macro BSs, then combines their transforms to evaluate coverage.The proof uses independence of the three interference powers and de-conditions over the serving tier.Source: 1) Association Policy 1 · pp. 16–18
  • For open-access SBS interference, the analysis derives Laplace transforms for users served by the macrocell and small-cell tiers using the relevant distance distributions.The derivation focuses on intra-cluster interferers and conditions on the serving tier.Source: 1) Association Policy 1 · p. 16
  • The closed-access SBS interference transform uses the distance distribution of closed-access interferers and accounts for whether the serving BS is macrocell or small-cell tier.This is the inter-cluster interference component.Source: 1) Association Policy 1 · p. 17
  • The macro-BS interference transform excludes the serving BS and is conditioned on the serving tier.For numerical evaluation, the paper uses a simplified open-access SBS transform when the mean active SBS count is much smaller than the cluster SBS count; it describes this as a proxy for the exact expression across a wide range of cases.Source: 1) Association Policy 1 · pp. 16–17
  • Theorem 1 presents the coverage probability under association Policy 1 for a typical user served by the jth tier, along with the total coverage probability.Source: 1) Association Policy 1 · pp. 17–18

2) Association Policy 2:

Under association Policy 2, the paper derives coverage probabilities for typical users served by small cells and macrocells, then combines them for total coverage. The SBS power threshold trades off association between tiers, and an optimal threshold exists for maximizing total coverage.

2) Association Policy 2:

  • Theorem 2 gives coverage probabilities for typical users served by small cells and macrocells, as well as the resulting total coverage probability.The expressions are derived using the preceding interference-transform lemmas.Source: 2) Association Policy 2 · pp. 19–20
  • Decreasing the SBS power threshold reduces macrocell association and increases small-cell association, yielding an optimal threshold (or distance threshold D) for total coverage.The paper states that this optimum is demonstrated in the Numerical Results Section.Source: 2) Association Policy 2 · p. 20
  • An optimal SBS power threshold can also balance load across tiers to maximize overall rate coverage, a direction left for future investigation.Source: 2) Association Policy 2 · p. 20

B. Throughput

The section defines throughput as a measure of successfully transmitted bits per unit area and analyzes it for both association policies. It states that, in the regime of interest, throughput increases with simultaneously active SBSs even as coverage probability decreases.

B. Throughput

  • Throughput roughly characterizes the average number of bits successfully transmitted per unit area, with λ denoting active transmitters per unit area.The definition is specialized to this setup in Proposition 1.Source: B. Throughput · p. 20
  • Proposition 1 gives throughput expressions for Policy 1 using Theorem 1 and for Policy 2 using Theorem 2.Source: B. Throughput · p. 20
  • In the regime of interest, throughput increases with the number of simultaneously active SBSs, although coverage probability decreases.The resulting tradeoff suggests revisiting strictly orthogonal channelization within each cluster.Source: B. Throughput · p. 21

V. RESULTS AND DISCUSSION … VI. CONCLUDING REMARKS

In the considered setup, reusing a resource block across more SBSs per cluster increases throughput but decreases coverage probability. The paper’s framework analyzes coupled user and SBS locations under two association policies.

B. Number of simultaneously active SBSs

A. Verification of results

  • For n_s0 = 10, analytical results match simulations even at relatively large average active-SBS counts, and Policy 1 has higher coverage probability than Policy 2.Simplified intra-cluster interference Laplace-transform expressions serve as proxies for exact expressions across a wide parameter range.Source: A. Verification of results · p. 21, Figure 2 · p. 22, Figure 3 · p. 22

B. Number of simultaneously active SBSs

C. Impact of SBS standard deviation

D. Optimal distance threshold

  • An optimal SBS power threshold maximizes total coverage probability and decreases as the average number of simultaneously active SBSs per cluster increases.The threshold is unchanged across the considered SBS scattering standard deviations; its optimum reflects the conflicting effects on macrocell and small-cell association.Source: D. Optimal distance threshold · pp. 23–24, Figure 8 · p. 24, Figure 9 · p. 24

VI. CONCLUDING REMARKS

  • The framework models hotspot centers as a PPP, clusters users and SBSs around them, and derives coverage and throughput for two power-based association policies.It supports general user and SBS distributions relative to cluster centers; new distance distributions enable the analysis of user-centric small-cell deployments.Source: §VI · p. 24

APPENDIX … D. Proof of Lemma 5

The appendix proves lemmas by deriving distance distributions, conditional association probabilities, and the conditional distribution of interfering SBS distances. For macro-served users, the proof restricts open-access interferers to lie beyond the association threshold.

A. Proof of Lemma 1

  • A. Proof of Lemma 1: The proof locates a uniformly selected SBS relative to its representative cluster center and derives the typical user’s cluster-center distance density using the same argument as for the SBS distance.It models the SBS location as z0 = x0 + ys, with x0 = (ν0, 0); the typical user is at the origin and the cluster-center offset has the distribution of Yu.Source: A. Proof of Lemma 1 · p. 25

B. Proof of Lemma 3

  • B. Proof of Lemma 3: The proof obtains the conditional macro-tier association probability for a given ν0 and uses it to obtain the small-cell-tier association probability.Source: B. Proof of Lemma 3 · pp. 25–26

C. Proof of Lemma 4

  • C. Proof of Lemma 4: For a typical user at distance ν0 from its cluster center, the macro-serving event is expressed as Rm > xm, yielding the conditional CCDF and PDF of XP1_m.Source: C. Proof of Lemma 4 · p. 26
  • C. Proof of Lemma 4: The conditional PDF for XP1_s is derived by the same method as for XP1_m, so its derivation is omitted.Source: C. Proof of Lemma 4 · p. 26

D. Proof of Lemma 5

  • D. Proof of Lemma 5: When the typical user is served by a macro BS, Policy 1 implies that no open-access BS lies within ξs_m x_m, so interfering open-access SBS distances are conditioned to exceed that threshold.Source: D. Proof of Lemma 5 · p. 26
  • D. Proof of Lemma 5: The conditional joint CDF factorizes, making the interfering SBS distances conditionally i.i.d.; their PDF follows by differentiating the CDF, and the small-cell case is derived analogously.The conditional density has zero support below ξs_m x_m.Source: D. Proof of Lemma 5 · p. 26

E. Proof of Lemma 9 · F. Proof of Lemma 10

The proofs derive intra-cluster interference transforms for macro- and small-cell-served users, then show that the inter-cluster interference transform is independent of the serving tier.

E. Proof of Lemma 9

  • The intra-cluster interference Laplace transform is derived for a typical user served by a macrocell, conditioned on V0 and XP1.Source: E. Proof of Lemma 9 · p. 27
  • The derivation averages over fading and the active SBS count, using a change of variables, polar coordinates, and conditionally i.i.d. distances with the PDF from Lemma 5.Source: E. Proof of Lemma 9 · p. 27
  • The active SBS count is Poisson-distributed conditioned on its total being less than ns0.Source: E. Proof of Lemma 9 · p. 27
  • For a small-cell-served user, the intra-cluster derivation follows the macrocell case but removes the serving SBS from potential interferers.Source: E. Proof of Lemma 9 · p. 27
  • Conditioned on at least one active SBS serving the user, the cluster’s active SBS count follows a truncated weighted Poisson distribution.Source: E. Proof of Lemma 9 · p. 27

F. Proof of Lemma 10

  • The inter-cluster interference Laplace transform does not depend on which base-station tier serves the user.Source: F. Proof of Lemma 10 · p. 27
  • Its derivation uses independent fading, exponential fading averages, a coordinate change and polar integration, Poisson cluster counts, and the PPP probability generating functional.Source: F. Proof of Lemma 10 · p. 28
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