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Analytical Evaluation of Fractional Frequency Reuse for Heterogeneous Cellular Networks

Thomas D. Novlan, Radha Krishna Ganti, Arunabha Ghosh, Jeffrey G. Andrews

arXiv:1112.0674v1cs.ITcs.NI

TL;DR

Heterogeneous cellular networks need tractable methods to analyze coverage and rate under dense, non-uniform interference. This paper uses a spatial Poisson point process to evaluate Strict FFR and SFR in multi-tier networks. The framework captures deployment non-uniformity, yields design expressions, and characterizes tradeoffs between interference reduction, coverage, rate, and spectrum use.

  • Problem

    Heterogeneous networks require general, tractable coverage and rate models that account for non-uniform deployments and interference across tiers.

  • Method

    The paper models access-point locations with a Poisson point process and analyzes Strict FFR and SFR in multi-tier networks with open and closed access.

  • Results

    The framework produces tractable expressions for Strict FFR and SFR while capturing non-uniform heterogeneous deployments and their performance tradeoffs.

  • Takeaways & Limitations

    Strict FFR reduces edge-user interference, whereas SFR uses shared spectrum more efficiently while mitigating much of the interference.

Abstract

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Interference management techniques are critical to the performance of heterogeneous cellular networks, which will have dense and overlapping coverage areas, and experience high levels of interference. Fractional frequency reuse (FFR) is an attractive interference management technique due to its low complexity and overhead, and significant coverage improvement for low-percentile (cell-edge) users. Instead of relying on system simulations based on deterministic access point locations, this paper instead proposes an analytical model for evaluating Strict FFR and Soft Frequency Reuse (SFR) deployments based on the spatial Poisson point process. Our results both capture the non-uniformity of heterogeneous deployments and produce tractable expressions which can be used for system design with Strict FFR and SFR. We observe that the use of Strict FFR bands reserved for the users of each tier with the lowest average SINR provides the highest gains in terms of coverage and rate, while the use of SFR allows for more efficient use of shared spectrum between the tiers, while still mitigating much of the interference. Additionally, in the context of multi-tier networks with closed access in some tiers, the proposed framework shows the impact of cross-tier interference on closed access FFR, and informs the selection of key FFR parameters in open access.

I. INTRODUCTION

Heterogeneous cellular networks require tractable interference analysis because tier-specific deployments and user-association policies create non-uniform inter-cell and cross-tier interference. This paper models access points with a Poisson point process to analyze Strict FFR and SFR for coverage, rate, and system design.

  • Motivation: Heterogeneous-network analysis must account for inter-cell interference, cross-tier interference, and non-uniform access-point deployments.Different user-association policies further complicate analysis.
  • Motivation: Prior deterministic access-point models did not produce general, tractable solutions for coverage and rate analysis.The paper instead models access-point locations as a Poisson point process.
  • Fractional Frequency Reuse: Fractional frequency reuse is attractive because of its low implementation complexity and gains for low-percentile mobile users.The paper considers Strict FFR and Soft Frequency Reuse.
  • Contribution: The paper extends a single-tier PPP FFR model to multi-tier networks with open and closed access, deriving tractable SINR expressions as functions of FFR parameters.The framework addresses cross-tier interference and supports system design for these networks.
  • Fractional Frequency Reuse: Strict FFR reserves a common subband for interior users and partitions edge-user subbands across cells with reuse factor ∆.Its main benefit is reduced edge-user interference, but each cell cannot fully use all ∆+1 subbands.
  • Fractional Frequency Reuse: SFR lets each cell use all ∆ subbands, sharing them between interior and edge users while assigning higher transmit power to edge-user downlinks.The resulting increase in edge-user interference is traded for greater spectral utilization.

C. Contributions

The paper extends analytical SINR and coverage modeling for heterogeneous networks using PPP-distributed access points, covering closed and open access under Strict FFR and SFR. The framework exposes how deployment and FFR parameters shape interference and system design.

  • Contributions: The framework evaluates Strict FFR and SFR in downlink K-tier networks with closed access, where users associate with only one tier while other tiers contribute interference.It also proposes an open-access framework in which users may associate with access points in more than one tier.
  • Analytical framework: The resulting SINR distributions reduce, in an interference-limited special case, to expressions governed by key FFR design parameters.The expressions separate intra-tier interference before and after FFR from cross-tier interference, enabling comparisons between reuse strategies and system-design analysis.
  • System design: The models support FFR parameter selection based on tier densities, transmit powers, and resource-allocation strategies.These implications are developed for both closed- and open-access networks.
  • System model: The analytical models use independent PPPs for access-point locations, capturing non-regular heterogeneous deployments rather than deterministic grid layouts.The model includes tier densities, transmit powers, fading, pathloss, and interference sets for users served by the nearest access point.
  • FFR operation: FFR classifies users by their SINR relative to a tier-specific threshold, assigning edge users to reserved subbands and interior users to the common allocation.This classification is based on SINR rather than a fixed geographic radius, so edge and interior regions need not have the same geographic interpretation across cells.

III. COVERAGE PROBABILITY WITH CLOSED ACCESS

This section derives closed-access coverage probabilities for cell-edge users under Strict FFR and SFR in multi-tier networks. The analysis explicitly captures cross-tier interference and yields tractable expressions for comparing reuse strategies.

  • Coverage formulation: Closed access can leave a macrocell user exposed to interference from a nearby femtocell that the user is not permitted to access.Coverage probability is defined as the probability that SINR exceeds threshold T, equivalently the SINR CCDF.
  • Model relevance: The distributions from the analytical framework are reported as closer to actual base-station deployment results than simulations based on deterministic locations.This comparison motivates the PPP-based model for heterogeneous cellular analysis.
  • Analytical setup: The section derives SINR distributions for cell-edge users under Strict FFR and SFR using the PPP model for base-station locations.These results extend earlier grid-based and single-tier analyses toward heterogeneous multi-tier deployments.
  • Strict FFR: Strict FFR edge users receive a reserved subband after being selected by an initial SINR threshold, with cross-tier interference absent on the reserved macrocell subband and intra-tier interference thinned by reuse factor ∆.The resulting coverage probability is given by a closed-access edge-user theorem.
  • Interpretation: The derived expressions depend only on relevant FFR design parameters and separate intra-tier and cross-tier interference contributions.This structure provides intuitive comparisons between Strict FFR and SFR and supports system-design interpretation.

C. Multi-tier coverage with SFR

For SFR, all subbands overlap across tiers while edge users receive higher transmit power controlled by β. The analysis gives coverage expressions and compares SFR with Strict FFR for edge users.

  • SFR operation: SFR uses the entire spectrum, so its subbands overlap with those of other tiers.Users below the FFR threshold receive higher transmit power determined by β.
  • Coverage analysis: The paper provides a theorem for the coverage probability of closed-access SFR edge users whose initial SINR is below threshold T1.The expression accounts for the effective SINR and FFR thresholds shaped by β and η.
  • Special case: For α = 4 and zero noise, the coverage probability expressions reduce to simple closed forms while the general pathloss-exponent case remains valid through integrals.The special case is motivated by interference-limited urban cellular networks and enables clearer analysis of cell-edge performance.
  • Validation: The analytical model matches Monte Carlo curves exactly for a three-tier network with α = 4 and no noise.This validates the derived distributions for Strict FFR and SFR edge users in the reported comparison.
  • Comparison: FFR improves cell-edge coverage over universal frequency reuse; Strict FFR removes cross-tier interference and 1/∆ of intra-tier interference, while SFR offers lower coverage gain but uses all subbands.SFR’s coverage gain can be mitigated through higher β, and each cell fully utilizes every subband.

IV. COVERAGE PROBABILITY WITH OPEN ACCESS

The open-access analysis derives coverage probabilities for Strict FFR edge users in a two-tier network, accounting for interdependent SIRs and cross-tier interference. The resulting expressions are analytically tractable but require numerical integration.

  • Model assumptions: The model assumes two access-point tiers and uses SIR, neglecting noise, to simplify the coverage expressions.The general framework can accommodate more tiers and noise, but these assumptions reduce complexity.
  • Open-access assignment: Coverage depends on thresholds T1 and T2, which determine whether users remain on common bands or move to reuse-∆ sub-bands or second-tier access points.SIR1 and SIR2 denote the SIRs from the closest first- and second-tier access points.
  • Edge-user conditioning: Users with SIR1 < T1 and SIR2 < T2 receive a new FFR sub-band, where Strict FFR and SFR determine the resulting SIR differently.The new sub-band introduces new fading and out-of-cell interference; under the described Strict FFR case, it has no cross-tier interference.
  • Strict FFR: Open-access coverage is more complex than closed-access coverage because the two SIR terms are interdependent through the distances to the nearest tier-specific access points.The Strict FFR edge-user result is given in Theorem 3.
  • Analytical tractability: The expressions generally lack closed form, with the number of tiers determining the number of integrals, but practical deployments with about three tiers remain numerically evaluable.The derivations require a double integral in the two-tier open-access case.

B. SFR

The SFR analysis derives open-access edge-user coverage and extends the framework to average edge-user rate. It shows how spectrum sharing, thresholds, and off-loading affect the resulting distributions.

  • Interference structure: Unlike Strict FFR, full ∆-reuse with SFR produces cross-tier interference for both edge and interior users.The SFR expressions also differ through the power-control factor η and effective interference parameter β.
  • SFR coverage: SFR open-access edge-user coverage is derived for users whose initial SIR is below thresholds T1 and T2.Theorem 4 gives the corresponding coverage probability in terms of the joint distance-dependent expression fn(r1, r2).
  • Model evaluation: The derived Strict FFR and SFR distributions match Monte Carlo curves exactly for the evaluated two-tier, no-noise case with α = 4.This comparison is shown in Fig. 4.
  • Off-loading: Open access shifts the coverage curves upward because some users are off-loaded onto the secondary tier.Users below T1 may be assigned an FFR band, while users with second-tier SINR above T2 are guaranteed coverage in the described case.
  • Average rate: Average edge-user rate is obtained by integrating over the SINR distribution and fading, conditioning the new-subband SINR on the previous common-subband SINR.The rate is expressed as E[ln(1 + SINR)] in nats/Hz under adaptive modulation and coding.

B. Multi-tier interference and closed access

The closed-access analysis links tier density, interference, and FFR allocation to macro-user SINR and system rate. It also uses threshold selection to control off-loading and reserve FFR resources for users most likely to benefit.

  • Multi-tier interference: Increasing the second-tier density ratio κ increases macrocell-user SINR under Strict FFR.FFR bands are reserved for macrocell users, so users moved from the common band experience reduced interference.
  • Closed access: Higher second-tier interference pushes more macro users below T1, forcing them onto FFR sub-bands because closed access prevents second-tier association.This links the density of secondary access points to FFR utilization.
  • Rate trade-off: Increasing partition size can reduce the macrocell overall sum rate because Strict FFR decreases overall spectrum usage.The rate trade-off arises even when edge-user SINR benefits from interference reduction.
  • Threshold selection: Increasing T2 restricts FFR assignment to users with the worst SIR, who can receive the greatest benefit from FFR sub-bands.The analysis reports an overall increase in edge-user SIR as T2 increases.
  • Open-access design: The framework can encode macrocell off-loading goals through T1 and T2, including a middle range TBias = T1 − T2.Raising T1 and lowering T2 can define a desired percentage of macrocell users for off-loading.
  • Framework implications: The paper presents a tractable analytical framework that captures non-uniform heterogeneous deployments and supports system-design analysis of Strict FFR and SFR.The model is also positioned as a foundation for studying dynamic FFR strategies.
  • Scope boundary: Uplink analysis remains challenging because power control, interfering-mobile mobility, and mobile-device power consumption constrain system design.These constraints are identified as an important boundary for applying the presented analysis.

APPENDIX A

The appendix derives Strict FFR coverage using conditioning, Laplace transforms, and the Poisson point process representation of interference. It separates newly allocated subband interference from the original common-band terms.

  • Subband assignment: A macrocell user with SINR1 < T1 is assigned a FFR sub-band δy, selected uniformly from the available reuse sub-bands.The new sub-band uses new fading and out-of-cell interference terms.
  • Two-stage conditioning: The derivation conditions the edge-user CCDF on the previous SINR and uses Bayes’ rule to combine the two stages.The previous common-band SINR and the new-subband SINR are treated as distinct stages.
  • Interference averaging: Conditioning on the nearest-base-station distance enables evaluation using exponentially distributed fading variables and a joint Laplace transform of interference.The interference terms include the new and original first-tier interference together with other tier contributions.
  • PPP evaluation: The probability generating functional of the Poisson point process is used to evaluate the interference terms.Independence between first-tier interference components and other-tier interference is used during the factorization.
  • Final expression: Substituting the evaluated terms and changing variables yields the stated Strict FFR coverage expression.The appendix explicitly identifies this substitution as the final step to obtain the result.

APPENDIX B

This appendix derives the Strict FFR edge-user distribution by conditioning on low initial SINR, assigning a random FFR sub-band, and evaluating the resulting interference terms.

  • Strict FFR derivation: A macrocell user with SINR below T1 is assigned one of Δ FFR sub-bands uniformly at random.The reassigned user experiences new fading, transmit power βP1, and out-of-cell interference.
  • Conditional CCDF: The edge-user CCDF is conditioned on the assigned sub-band, new fading power, transmit power, and out-of-cell interference.The derivation focuses on the resulting numerator and denominator after applying Bayes’ rule.
  • Interference analysis: The interference terms are handled by conditioning on the serving distance and evaluating their joint Laplace transform.The transform includes the interference components from the tiers and the noise power.
  • Final expression: Substituting the derived numerator and denominator terms yields the Strict FFR edge-user expression.The appendix completes the result by applying the intermediate expressions and substituting r12 = v.

APPENDIX C

This appendix derives the open-access Strict FFR edge-user distribution by jointly conditioning on users’ tier-specific SIR events, serving distances, and interference transforms.

  • Open-access Strict FFR: A user failing both tier-specific SIR thresholds is assigned an FFR sub-band uniformly across Δ sub-bands.The reassigned user experiences new fading, transmit power βP1, and out-of-cell interference.
  • Conditioning: The conditional CCDF is formed from the previous SIR values and the distances to the nearest tier 1 and tier 2 access points.Bayes’ rule is applied while evaluating the conditional denominator and numerator.
  • Interference analysis: The derivation evaluates joint Laplace transforms for the tier-specific interference terms conditioned on the two serving distances.The transforms are obtained using exponentially distributed fading and subsequent de-conditioning.
  • Final expression: Substituting the derived transform and numerator terms produces the open-access Strict FFR distribution.The final step plugs the intermediate expressions into the CCDF and substitutes r12 = v.

APPENDIX D

This appendix derives the open-access SFR edge-user distribution using joint tier interference transforms, while the accompanying figures describe the evaluated Strict FFR and SFR scenarios.

  • Open-access SFR derivation: A user below both tier-specific SIR thresholds receives an FFR sub-band uniformly across Δ sub-bands.The reassigned user experiences new fading, transmit power βP1, and combined interference ηP1I1 + P2I2.
  • Conditional CCDF: The SFR edge-user CCDF incorporates the combined interference from both tiers and the two tier-specific desired signals.The relevant threshold conditions are expressed through the two SIR inequalities.
  • Final expression: Substituting the numerator and auxiliary terms yields the open-access SFR edge-user expression.The final expression uses the auxiliary function fn and the preceding intermediate result.
  • Evaluated scenarios: The figures cover edge-user SINR or SIR distributions for Strict FFR and SFR under closed or open access across different network settings.The scenarios include three-tier and two-tier networks, tier-density ratio κ, and SFR threshold T2.
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