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Joint Resource Partitioning and Offloading in Heterogeneous Cellular Networks

Sarabjot Singh, Jeffrey G. Andrews

arXiv:1303.7039v2cs.IT

TL;DR

In heterogeneous cellular networks, offloading can relieve congested macrocells but degrade offloaded users’ SINR, making offloading and resource partitioning interdependent. The paper develops an analytical framework for their joint design, derives network-wide downlink rate distributions, and establishes that combining load balancing with resource partitioning is important.

  • Problem

    Offloading users to less congested small cells can reduce their SINR, creating a need to jointly determine offloading and interference-avoiding resource partitioning.

  • Method

    The paper models user association through tier-specific biases and analyzes resource sharing and partitioning in a two-tier heterogeneous cellular network.

  • Results

    The framework derives rate distributions across the network and shows that rate coverage varies oppositely across macro, small-cell, and range-expanded users as the partitioning fraction changes.

  • Takeaways & Limitations

    Load balancing alone is insufficient; combining offloading with resource partitioning is important, while SINR-only analysis can provide inconclusive insights about these techniques.

Abstract

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In heterogeneous cellular networks (HCNs), it is desirable to offload mobile users to small cells, which are typically significantly less congested than the macrocells. To achieve sufficient load balancing, the offloaded users often have much lower SINR than they would on the macrocell. This SINR degradation can be partially alleviated through interference avoidance, for example time or frequency resource partitioning, whereby the macrocell turns off in some fraction of such resources. Naturally, the optimal offloading strategy is tightly coupled with resource partitioning; the optimal amount of which in turn depends on how many users have been offloaded. In this paper, we propose a general and tractable framework for modeling and analyzing joint resource partitioning and offloading in a two-tier cellular network. With it, we are able to derive the downlink rate distribution over the entire network, and an optimal strategy for joint resource partitioning and offloading. We show that load balancing, by itself, is insufficient, and resource partitioning is required in conjunction with offloading to improve the rate of cell edge users in co-channel heterogeneous networks.

I. INTRODUCTION

Heterogeneous cellular networks use small cells to increase capacity, but effective offloading requires jointly managing association bias and interference through resource partitioning. The paper develops a tractable analytical framework to study these coupled choices and derive rate-based design insights.

  • I. INTRODUCTION: Small-cell deployment increases area spectral efficiency and complements macrocell infrastructure in heterogeneous networks.The network may include co-channel micro, pico, and femto base stations alongside macrocells.
  • I. INTRODUCTION: Limited natural small-cell coverage can leave macrocells congested, while aggressive range expansion may overload small cells with low-SINR users.Cell range expansion offloads users through a positive association bias, but excessive bias can require substantial macrocell muting.
  • I. INTRODUCTION: Load balancing and interference avoidance are tightly coupled because offloaded users experience degraded SINR in co-channel deployments.Resource partitioning mitigates interference by reserving resources on which the macrocell is inactive.
  • I. INTRODUCTION: The proposed framework models two-tier networks probabilistically and derives rate coverage as a function of offloading and resource-partitioning parameters.The model uses independent Poisson point processes for base stations and users, biased received-power association, and Rayleigh fading.
  • I. INTRODUCTION: The analysis shows that optimal association bias and offloaded-user fraction respond differently to small-cell density and backhaul capacity.With resource partitioning, optimal bias decreases as small-cell density increases, while the optimal offloaded fraction increases; reduced backhaul capacity lowers the corresponding bias.

B. Resource Partitioning

Resource partitioning mutes macro transmissions on a fraction of resources so small cells can serve range-expanded users with reduced macro interference. The model tracks how users, loads, SINR, and rates are affected under this allocation.

  • The macro cell is inactive on fraction η of resources, while small cells schedule range-expanded users on those macro-interference-free resources.The remaining 1−η fraction serves macro users and non-range-expanded small-cell users.
  • Range-expanded users share the η fraction, whereas macro and non-range-expanded users share the 1−η fraction.The effective resource fractions are represented by γ_l = 1/η for range-expanded users and γ_l = 1/(1−η) for the other groups.
  • Resource partitioning changes the SINR model through activity-dependent interference, while AP activity can alternatively be represented by thinning each tier’s PPP.The framework assumes all APs in an active tier transmit, but permits independent activity with probability p_ak.
  • For a tier-2 AP, the total load combines non-range-expanded and range-expanded users, subtracting one to account for double counting the tagged user.The model defines N_2 = N̄_B + N_B − 1, with separate resource sharing for the two user groups.
  • The baseline rate model assumes infinite backhaul bandwidth, an assumption later relaxed for limited-capacity backhaul.Rate coverage is defined as the probability that a user’s rate exceeds threshold ρ, with equivalent network-wide interpretations.

III. RATE DISTRIBUTION

The paper derives association and SINR distributions for three user groups in a two-tier network, then uses them as inputs to the rate-distribution analysis. SINR coverage is independent of the resource partitioning fraction in the model.

  • The analysis derives load and SINR distributions before using them to obtain network-wide rate coverage.Association probabilities determine the fractions of users in macro, non-range-expanded, and range-expanded groups.
  • Association probabilities quantify membership in the three disjoint user sets and simplify when all tiers share the same path-loss exponent.Increasing association bias decreases the macro-user population and increases the range-expanded-user population.
  • The network-wide SINR coverage is a weighted sum of conditional coverages for macro, non-range-expanded, and range-expanded users.The weights are the corresponding association probabilities.
  • Under ignored noise and equal path-loss exponents, the SINR coverage expression admits a simplified corollary.The general result requires a numerical integration and a lookup table for Q.
  • SINR coverage is independent of η because the model treats SINR as independent of the amount of resources allocated to a user.The SINR distribution of non-range-expanded small-cell users is also independent of association bias.

B. Main Result

The main result expresses network-wide rate coverage by combining user-group association probabilities with their conditional rate coverages. The framework also provides load modeling and a mean-load simplification.

  • The rate distribution is obtained by combining association, load, and conditional SINR information for a typical user.Rate coverage is the probability that the user rate exceeds threshold ρ.
  • Theorem 1 gives network-wide rate coverage as a weighted sum across macro, non-range-expanded, and range-expanded users.The weights are A_1, Ā_B, and A_B, respectively.
  • The analysis ignores load–SINR correlation for tractability, although larger association regions imply both higher load and lower SINR.The resulting rate expression sums over possible loads using their PMF and conditional SINR coverage.
  • The load PMF is characterized through the distribution of association areas in the Poisson-Voronoi model.The analysis uses a linear-scaling association-area approximation and a Poisson-Voronoi area distribution.
  • Increasing η increases range-expanded-user rate coverage because those users receive a larger fraction of macro-interference-free resources.The mean-load approximation removes the summation over load values and further simplifies numerical evaluation.

C. Rate Coverage with Limited Backhaul Capacities

The limited-backhaul extension models backhaul as an additional rate bottleneck shared among users at each access point. It shows that reduced backhaul bandwidth lowers rate coverage and changes the optimal offloading bias.

  • Limited backhaul bandwidth caps each user’s peak rate at the access point’s backhaul bandwidth divided by its load.The allocation assumes equal sharing of O_k among N_k associated users.
  • A necessary condition for positive rate coverage is that the access-point load not exceed the backhaul-imposed threshold.This follows because a user’s maximum rate is O_J(l)/N_J(l).
  • Exact analysis of wired backhaul allocation among competing TCP flows remains future work.
  • Rate coverage decreases as backhaul bandwidth decreases.The limited-backhaul rate expression modifies the infinite-backhaul user rate through the backhaul constraint.
  • Decreasing backhaul capacity lowers the optimal association bias for the corresponding tier.Unlike the infinite-backhaul case, the optimal bias may increase with increasing small-cell density.

D. Extension to Multi-tier Downlink

The two-tier framework extends to K-tier networks by modeling each base-station tier as a Poisson point process with biased association and resource partitioning. It derives SINR and rate coverage for this generalized setting.

  • Multi-tier model: Users associated with tier j are divided into disjoint sets according to whether they belong to the range-expanded user group.The generalized association classification follows the two-tier construction.
  • Resource partitioning: With resource partitioning, tier-j access points schedule offloaded users on an η fraction of resources protected from macro-tier interference.Non-range-expanded users use the remaining 1 − η fraction.
  • Coverage analysis: The generalized model specifies the SINR of a tier-j user under resource partitioning and derives its SINR coverage using the two-tier analysis techniques.The resulting expressions are given in the cited equations.
  • Coverage analysis: The K-tier rate is W log(1 + SINR), and rate coverage follows from the generalized SINR-coverage expressions and a generalization of Lemma 3.This extends the rate-coverage derivation beyond the two-tier case.

E. Validation of Analysis

The analysis is validated against simulations and used to examine how bias, resource partitioning, and small-cell density affect SINR and rate coverage. The results show that these controls interact and that optimal settings depend on their joint configuration.

  • Validation of analysis: Theorem 1, Corollary 2, and Lemma 4 are validated by comparing analytical rate distributions with simulations across rate thresholds.Figure 2 uses λ_1 = 1 BS/km^2, λ_2 = 5 BS/km^2, and λ_u = 100 users/km^2, with α_1 = 3.5 and α_2 = 4.
  • SINR coverage trends: SINR coverage behaves differently with and without resource partitioning when offloading is varied.The paper uses SINR coverage to explain trends in the primary rate-coverage metric.
  • SINR coverage trends: Without resource partitioning, any positive association bias is suboptimal for SINR coverage under the stated setting.The general claim is stated for b > 1 and τ ≥ 1, while Figure 3 uses τ = 0.5.
  • SINR coverage trends: With resource partitioning, the SINR-maximizing bias can be positive, but its upper bound and the optimal bias decrease as small-cell density increases.Increasing small-cell density also increases interference on protected resources for offloaded users.
  • SINR coverage trends: When all users are offloaded to small cells, SINR coverage is below the unbiased case; at very large bias, shutting off the macro tier would be preferable.The paper attributes this to protecting only offloaded users while most macrocell users have been removed.
  • Rate coverage trends: Rate coverage has an optimal association bias because increasing bias first relieves macrocell load and increases small-cell participation, then increases sharing and reduces contributions.With partitioning, changing η improves macrocell and ordinary small-cell coverage while reducing range-expanded-user coverage because fewer resources are available.

1) Impact of resource partitioning:

Jointly tuning association bias and resource partitioning improves network rate performance, with the optimal operating point depending on small-cell density and backhaul constraints.

  • Impact of resource partitioning: B = 15 dB and η = 0.47 significantly increase rate coverage over B = 0 dB and η = 0.Increasing η raises the optimal association bias because more macro-interference-free resources become available to offloaded users.
  • Impact of resource partitioning: The same optimal (B,η) pair applies to fifth-percentile and median rate, providing one network-wide operating region across these metrics.
  • Impact of infrastructure density: At fixed association bias, increasing small-cell density raises fifth-percentile rate by reducing the load per access point.
  • Impact of infrastructure density: With resource partitioning, the optimal association bias and partitioning fraction both decrease as small-cell density increases.Without partitioning, the optimal bias remains 5 dB across densities; with partitioning, the optimal bias decreases with density.
  • Impact of infrastructure density: Increasing small-cell density raises the optimal traffic offload fraction, while limited backhaul bandwidth lowers the optimum bias and offload fraction.The comparison considers infinite backhaul and limited small-cell backhaul bandwidth of O2 = 5 Mbps.
  • Conclusion: The framework establishes that combining load balancing with resource partitioning is necessary, and that rate-based analysis can differ from SINR-based conclusions.

APPENDIX A

Appendix A derives association probabilities and distance distributions for the relevant network tiers using Poisson point-process void probabilities.

  • APPENDIX A: The proof obtains association probabilities from the definitions of three disjoint user sets.
  • APPENDIX A: For each tier, the distance survival function follows from the probability that the corresponding PPP contains no point inside a radius-z ball.The resulting expression is P(Zk > z) = P(Φk ∩ b(0,z) = ∅) = exp….
  • APPENDIX A: Differentiating the distance survival function yields the PDF fZk(z).
  • APPENDIX A: Substituting the survival function and PDF into the preceding association expression completes Lemma 1.

APPENDIX B

Appendix B characterizes tagged-cell association areas and user loads, relying on an assumed palm-inversion relation for multiplicatively weighted Poisson–Voronoi cells.

  • APPENDIX B: The proof assumes that palm inversion for ordinary Poisson–Voronoi cells also holds for multiplicatively weighted cells.
  • APPENDIX B: Using the assumed relation, the distribution of the association area of a tagged tier-k access point is derived.
  • APPENDIX B: The formal proof of this weighted-cell palm-inversion relation is outside the paper’s scope and deferred to future work.
  • APPENDIX B: The load PMF for a typical user is then obtained from the tagged access point’s association area and the associated-user counts.
  • APPENDIX B: The approach is related to prior single-tier and multi-tier analyses.

APPENDIX C

Appendix C derives distance, interference, SINR-coverage, and overall coverage expressions for a typical user conditioned on its serving tier.

  • APPENDIX C: The proof first derives the distribution of the distance between a typical user and its tagged serving access point when conditioned on association.
  • APPENDIX C: The PDFs of the relevant distances are obtained using the corresponding association and nearest-access-point distributions.
  • APPENDIX C: Conditioned on the serving tier, interference is represented through the Laplace functional of the interfering PPP.
  • APPENDIX C: The closest-interferer distance bounds zkl(y) depend on the serving tier and association bias, with explicit cases for l = 1 and l = B̄.
  • APPENDIX C: A change of variables simplifies the interference integral and yields its Laplace transform.
  • APPENDIX C: Substituting the distance distribution and interference expressions into the SINR formulas produces conditional and overall SINR coverage via total probability.

APPENDIX D

The appendix analyzes SINR coverage and the offloading bias through derivative conditions and polynomial-root bounds. It concludes that coverage decreases for τ ≥ 1 when b ≥ 1, while the upper bound on optimal bias decreases with small-cell density.

  • For τ ≥1, SINR coverage decreases for all offloading biases b ≥1.
  • The derivative condition ∇xS = 0 is characterized by the zeros of a polynomial after approximating tan−1(a) ≈a and substituting x for √b.
  • The polynomial P(x) has one positive root and up to three negative roots by Descartes' sign rule.
  • The appendix relates negative roots of P(x) to positive roots of P(−x), with bounds determined using the quantities U and L.
  • The upper bound on optimal offloading bias is inversely proportional to the density of small cells a.
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