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Analytical Evaluation of Fractional Frequency Reuse for OFDMA Cellular Networks

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

arXiv:1101.5130v1cs.ITcs.NImath.PR

TL;DR

The paper addresses limited analytical evidence for evaluating FFR in increasingly interference-limited cellular networks. It analyzes Strict FFR and SFR with a Poisson base-station model, compares them with grid and urban deployments, and derives tractable expressions. FFR improves sum rate and cell-edge coverage, while resource allocation favors Strict FFR at low traffic loads and SFR at high loads.

  • Problem

    FFR has typically been evaluated through hexagonal-grid simulations, limiting analytical characterization of coverage, rate, and interference-management tradeoffs.

  • Method

    The paper analytically evaluates Strict FFR and SFR using a Poisson point process for base-station locations and derives coverage and average-rate expressions.

  • Results

    FFR increases sum rate and improves cell-edge coverage; resource allocation yields greater Strict FFR throughput at low traffic loads and better high-load balance with SFR.

  • Takeaways & Limitations

    Strict FFR prioritizes interference reduction, whereas SFR provides greater resource efficiency and better balances the two objectives as traffic load rises.

Abstract

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Fractional frequency reuse (FFR) is an interference management technique well-suited to OFDMA-based cellular networks wherein the cells are partitioned into spatial regions with different frequency reuse factors. To date, FFR techniques have been typically been evaluated through system-level simulations using a hexagonal grid for the base station locations. This paper instead focuses on analytically evaluating the two main types of FFR deployments - Strict FFR and Soft Frequency Reuse (SFR) - using a Poisson point process to model the base station locations. The results are compared with the standard grid model and an actual urban deployment. Under reasonable special cases for modern cellular networks, our results reduce to simple closed-form expressions, which provide insight into system design guidelines and the relative merits of Strict FFR, SFR, universal reuse, and fixed frequency reuse. We observe that FFR provides an increase in the sum-rate as well as the well-known benefit of improved coverage for cell-edge users. Finally, a SINR-proportional resource allocation strategy is proposed based on the analytical expressions, showing that Strict FFR provides greater overall network throughput at low traffic loads, while SFR better balances the requirements of interference reduction and resource efficiency when the traffic load is high.

I. INTRODUCTION

The paper develops an analytical PPP-based framework for evaluating Strict FFR and SFR, addressing limitations of grid-based simulation studies. It derives tractable performance characterizations and uses them to compare interference reduction, resource efficiency, coverage, and throughput across reuse strategies.

  • Motivation: FFR partitions cellular bandwidth to reduce inter-cell interference, especially for cell-edge users, while trading off rate, coverage, throughput, and spectral efficiency.FFR uses more total spectrum than conventional frequency reuse and separates or controls spectrum access across spatial regions.
  • Motivation: Prior FFR evaluations largely relied on simulations because the hexagonal grid model does not yield tractable SINR, coverage, or rate expressions.The paper identifies the standard equally spaced grid model as a reason for numerical rather than closed-form evaluation.
  • Analytical framework: The paper models base-station locations as a Poisson point process to capture non-uniform deployments and derive more general, intuitive performance characterizations.The PPP framework is also compared with the grid model and an actual urban deployment.
  • FFR deployments: Strict FFR allocates separate interior and edge sub-bands, whereas SFR lets interior users share sub-bands with neighboring edge users at typically lower power.Strict FFR therefore reduces interference between interior and exterior users, while SFR is more bandwidth-efficient but causes more interference.
  • Analytical framework: The analysis produces coverage and average-rate expressions, including simple special-case forms that depend on key FFR design parameters.These expressions support system guidelines for comparing Strict FFR, SFR, universal reuse, and per-cell frequency reuse.
  • System guidelines: SINR-proportional resource allocation provides higher overall throughput for Strict FFR at low traffic loads, while SFR better balances interference reduction and resource efficiency at high loads.The strategy allocates frequency sub-bands according to a threshold related to network traffic load and supports increased edge coverage for given load or coverage requirements.

II. SYSTEM MODEL

The system model considers an OFDMA cellular downlink with base stations distributed as a spatial Poisson point process and users served by their closest base station. Strict FFR and SFR classify users by an SINR threshold, while SFR additionally uses power control.

  • Network model: Users are served by their closest base station, with SINR determined by fading, path loss, interference, and noise.Small-scale fading is i.i.d. exponential, corresponding to Rayleigh fading.
  • Network model: Base station locations follow a spatial Poisson point process, producing Voronoi cells with random areas that more closely reflect irregular deployments.The PPP model can lower-bound performance metrics because base stations may be arbitrarily close together.
  • User classification: FFR separates users into edge and interior classes using the threshold TFR: below it is edge, above it is interior.Under the PPP model, these labels do not have the same concentric geographic interpretation as in grid models.
  • SFR model: SFR uses all sub-bands while assigning edge transmissions power βP and interior transmissions power P, with β ≥ 1.Interferers are likewise divided into interior and edge classes according to the users they serve.
  • Performance metric: Coverage probability is the probability that instantaneous SINR exceeds threshold T and is equivalently the SINR CCDF.The analysis develops coverage expressions for Strict FFR and SFR under the PPP model.

A. Strict FFR

Strict FFR assigns edge users below TFR to new sub-bands reserved for edge users, while interior users share a common sub-band. The PPP analysis derives coverage distributions for both classes.

  • Edge users: The edge-user analysis accounts for dependence between SINR before and after sub-band reassignment because the user’s geometry is unchanged.The dominant path loss therefore remains the same even though interference and fading change.
  • Edge users: Strict FFR edge users with initial SINR below TFR receive one of Δ sub-bands reserved for edge users.The edge-user coverage probability is given by Theorem 1.
  • Interior users: Strict FFR interior-user coverage does not depend on Δ because all interior users share the same sub-band.Their SINR CCDF is closely related to the no-frequency-reuse case.
  • Interior users: Interior users are selected as users whose SINR is at least TFR, yielding the max{T, TFR} threshold in the coverage expression.This threshold reflects the classification rule imposed by Strict FFR.

B. SFR

SFR reuses all sub-bands and differentiates edge and interior transmissions through power control rather than disjoint edge frequency reuse. Its PPP coverage expressions depend on the effective interference factor η and β.

  • SFR deployment: SFR lets base stations reuse all sub-bands while applying β to one of the δ sub-bands for edge users.This replaces frequency reuse for edge users with power control governed by β.
  • SFR deployment: The effective interference factor is η = (∆−1 + β)/∆, consolidating interference from higher- and lower-power downlinks.SFR interference is represented as ηPIr.
  • Coverage expressions: SFR edge-user coverage is derived for users whose initial SINR is below TFR.Theorem 3 gives the corresponding coverage probability expression.
  • Coverage expressions: SFR interior-user coverage has a structure similar to Strict FFR but contains the effective interference factor η.Interior transmissions do not receive the extra β power control, so only η remains in the expression.
  • Special case: Under interference-limited urban-network assumptions, Strict FFR edge-user SINR distribution depends only on T and TFR.The same special case yields similarly simple expressions for interior users, while SFR depends additionally on β through η.

B. SFR: No-noise and α = 4

The no-noise, α = 4 special case reduces the analytical distributions to simple forms that clarify SFR’s relation to other reuse strategies. Increasing β moves SFR from reuse-1 behavior toward reuse-∆ behavior and can make it outperform Strict FFR in coverage.

  • Analytical simplification: Under the special case, SINR distributions depend on T for SFR and on T and the design parameters for comparison across reuse strategies.The simplified structure enables performance comparisons as design parameters vary.
  • Strict FFR comparison: When ∆ = 3, only 33% of universal-reuse interfering base stations are active on the same resources under Strict FFR.The number of interfering sources drives outage because interference degrades coverage multiplicatively.
  • Coverage comparison: No-reuse and reuse-∆ edge users have sharp coverage cutoffs because they do not receive a new sub-band and their SINR remains below TFR.This behavior differs from FFR, which reallocates edge users to a new sub-band.
  • Coverage comparison: As β increases, SFR coverage approaches and surpasses Strict FFR when β ≥15, equivalent to a 12 dB edge-downlink power increase.The bounds in Fig. 3 are reported as quite tight for β = 1 and β approaching ∞.
  • SFR bounds: SFR is bounded by reuse-1 as β →1 and reuse-∆ as β →∞.At β →1, SFR matches a no-frequency-reuse strategy with a new sub-band assigned below TFR.

E. Comparison with Grid Model

The paper compares Poisson, grid, and actual urban deployments for FFR coverage and develops analytical rate expressions. The comparisons expose geometric differences between models and tradeoffs between coverage, rate, and resource allocation.

  • The comparison uses SINR CCDFs from a Poisson model, a uniformly spaced grid, and an actual urban base-station deployment for Strict FFR and SFR.
  • The grid model is more optimistic in terms of coverage probability than the actual deployment.
  • Strict FFR provides better coverage than SFR to edge users because first-tier interfering base stations contribute no interference under Strict FFR.
  • SFR distributions follow a similar sloping shape across models, whereas Strict FFR distributions differ because the Poisson model lacks a fixed reuse plan.
  • Coverage and rate expressions derived analytically can be evaluated numerically and reduce to simple expressions in the same special cases.

2) SFR:

SFR improves resource efficiency by allowing all sub-bands to be reallocated between edge and interior users, while its coverage and rate depend on power allocation and traffic conditions. Compared with Strict FFR and reuse baselines, SFR can balance interference reduction with broader resource use.

  • Average rate: Strict FFR provides the highest average edge-user rates in the reported threshold comparison because it provides the highest edge-user coverage.
  • Average rate: As β increases, SFR edge users can achieve a higher rate than Strict FFR, while increasing β also increases η.
  • Resource allocation: SFR can achieve 100% allocation unlike Strict FFR because resources are shared between interior and edge users.
  • Power allocation: Increasing β beyond the reported operating point produces diminishing performance gains compared with substantially increased required transmit powers.
  • Spectral efficiency: Under high traffic loads, reserving bandwidth for reuse partitions can reduce resource efficiency and peak cell throughput.
  • Sum rate: SFR balances the SINR gains of Strict FFR while using more available sub-bands, and SFR with β = 2 can exceed standard-system sum-rate.
  • Sum rate: Strict FFR sum-rate decreases as more edge-user sub-bands are allocated, and it remains below reuse-∆ when both allocate the same number of sub-bands.

D. SINR-Proportional Resource Allocation

The paper proposes allocating FFR sub-bands from analytical SINR distributions rather than geometric cell regions. Simulations show that the SINR-proportional strategy improves performance and that the preferred strategy changes with the threshold and traffic load.

  • Allocation principle: In the Poisson model, geometric intuition for sub-band allocation does not apply, so allocation should use SINR distributions.
  • Allocation principle: Given a threshold TFR, Nedge is chosen proportional to the CCDF evaluated at TFR.
  • Simulation results: Both SFR and Strict FFR outperform standard reuse strategies under the SINR-proportional algorithm.
  • Simulation results: SFR outperforms Strict FFR for smaller TFR, while Strict FFR provides greater sum-rate as TFR increases.
  • Conclusion: The framework yields tractable coverage and average-rate expressions and highlights Strict FFR’s interference reduction against SFR’s greater resource efficiency.
  • Scope: The analysis is scoped to downlink cellular networks; uplink evaluation would need to include fine-granularity power control and total power consumption.

APPENDIX A PROOF OF THEOREM 1

The appendix derives the Strict FFR edge-user coverage expression by conditioning on serving distance, fading, and interference, then applying the PPP probability generating functional. The resulting expression is de-conditioned over distance.

  • A user with SINR below TFR is assigned an FFR sub-band δy, while interference indicators identify other base stations transmitting edge users on the same sub-band.
  • The edge-user SINR CCDF is conditioned on the previous SINR and distance, then evaluated through a joint Laplace transform of new and prior interference.
  • The derivation uses exponential fading assumptions and the PPP probability generating functional to obtain a Laplace transform of interference.

APPENDIX B PROOF OF THEOREM 3

The proof derives the SFR edge-user SINR distribution by conditioning on the user’s previous SINR and distance to the nearest base station, then obtains the average rate by integrating that distribution.

  • SFR assigns a new sub-band to a user whose SINR is below TFR, with the sub-band index δ selected from {1, ..., ∆}.
  • The derivation conditions the edge-user CCDF on the previous SINR and applies Bayes’ rule to relate the prior and new SINR states.
  • Conditioning on the distance to the nearest base station enables evaluation of the interference-related expectation and Laplace transform.
  • The proof then de-conditions on distance and carries the resulting expression through the corresponding theorem-based substitutions.
  • The average edge-user rate is obtained by integrating over the derived SINR distribution, using the tail-probability representation of the expectation.

APPENDIX D PROOF OF THEOREM 6

The proof of Theorem 6 obtains the SFR average-rate expression by starting from the edge-user SINR distribution and following the preceding theorem’s integration method.

  • The SFR average rate is derived by integrating the edge-user SINR distribution using the same method as Theorem 3.
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