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User Association for Load Balancing in Heterogeneous Cellular Networks

Qiaoyang Ye, Beiyu Rong, Yudong Chen, Mazin Al-Shalash, Constantine Caramanis, Jeffrey G. Andrews

arXiv:1205.2833v2cs.IT

TL;DR

Cell association in heterogeneous networks must balance macrocell congestion against the lower instantaneous SINR that may accompany small-cell selection. The paper formulates logarithmic utility-based association, develops a distributed near-optimal algorithm, and shows that carefully chosen per-tier biases nearly achieve optimal load-aware performance.

  • Problem

    Users may need to select lightly loaded small cells despite lower instantaneous SINR, while joint association and resource allocation depend on users’ SINRs and BS loads.

  • Method

    The paper formulates logarithmic utility maximization, proposes an upper bound, and develops a distributed algorithm with dynamically chosen stepsizes.

  • Results

    Carefully chosen biasing factors yield network performance that nearly achieves the optimal load-aware solution.

  • Takeaways & Limitations

    Optimal biasing depends strongly on per-tier transmit powers but is nearly independent of BS densities, while load-aware association improves resource utilization and mitigates macro-BS congestion.

Abstract

from arXiv · show

For small cell technology to significantly increase the capacity of tower-based cellular networks, mobile users will need to be actively pushed onto the more lightly loaded tiers (corresponding to, e.g., pico and femtocells), even if they offer a lower instantaneous SINR than the macrocell base station (BS). Optimizing a function of the long-term rates for each user requires (in general) a massive utility maximization problem over all the SINRs and BS loads. On the other hand, an actual implementation will likely resort to a simple biasing approach where a BS in tier j is treated as having its SINR multiplied by a factor A_j>=1, which makes it appear more attractive than the heavily-loaded macrocell. This paper bridges the gap between these approaches through several physical relaxations of the network-wide optimal association problem, whose solution is NP hard. We provide a low-complexity distributed algorithm that converges to a near-optimal solution with a theoretical performance guarantee, and we observe that simple per-tier biasing loses surprisingly little, if the bias values A_j are chosen carefully. Numerical results show a large (3.5x) throughput gain for cell-edge users and a 2x rate gain for median users relative to a max received power association.

I. INTRODUCTION

Heterogeneous networks can deploy small cells economically, but users often still attach to macrocells, creating a need for load-aware association. The paper studies optimal and near-optimal association schemes with simple coordination requirements, while addressing the trade-off between load balancing and SINR degradation.

  • Small cells enable more flexible and economical infrastructure deployment than tower-mounted macro-only systems.
  • Most users may still receive the strongest downlink signal from the macrocell even when small cells target high-traffic zones.
  • Users should be pushed toward lightly loaded small cells, which can provide higher long-term rates through greater resource availability.
  • The paper investigates optimal and near-optimal cell-association solutions, particularly schemes requiring simple coordination.
  • Heterogeneous networks make association especially sensitive to unequal cell sizes and require load balancing across tiers.
  • Cell biasing improves load balancing, but its benefit may not offset the SINR degradation experienced by some users, making bias design an open problem.

B. Contributions and Organization

The paper develops load-aware downlink association methods by relaxing the hard network-wide problem, then derives a distributed algorithm and evaluates simple biasing against the resulting benchmark.

  • Contributions and Organization: The paper jointly considers cell association and resource allocation for maximizing network utility in downlink heterogeneous cellular networks.
  • Contributions and Organization: Multi-BS association is used as an optimistic relaxation that decouples the joint utility maximization and provides an upper bound for benchmarking.
  • Contributions and Organization: Single-BS logarithmic utility maximization shows that equal resource allocation is optimal over a sufficiently large time window.
  • Contributions and Organization: Fractional association and equal resource allocation convert the previously intractable combinatorial problem into a convex optimization problem.
  • Contributions and Organization: A dual-decomposition distributed algorithm converges toward the optimal solution with a guarantee on the maximum optimality gap, providing a feasible, low-overhead implementation.
  • Contributions and Organization: Simple biasing achieves gains surprisingly close to load-aware utility maximization, with rate gains of 2-3.5x for the bottom half of users.
  • Contributions and Organization: Cell-interior users experience little to no rate gain or a small loss, while already being well-served.
  • System Model: The model focuses on downlink association, assumes fixed full-buffer BS transmit powers, and is applicable to low-mobility environments.

III. PROBLEM FORMULATION

The formulation assigns each user a strictly concave utility of its received long-term rate, making rate allocation and association central to the optimization problem.

  • Problem Formulation: User i obtains utility U_i(R_i) from receiving long-term rate R_i, where U_i is continuously differentiable, monotonically increasing, and strictly concave.

A. General Utility Maximization: Unique Association

The paper formulates unique user association and resource allocation as a joint utility-maximization problem, then uses logarithmic utility to obtain a tractable, load-balancing formulation.

  • The optimization selects binary association indicators xij and resource-allocation variables yij to maximize aggregate user utility.
  • Unique association is combinatorial, and resource allocation depends on association while association depends on achievable resources.
  • Allowing users to associate with multiple BSs relaxes the unique-association constraint and provides an upper bound on network performance, despite added implementation overhead.
  • For general utilities, the joint-association formulation provides an ultimate limit on achievable network performance, and simulations show the bound is quite tight for logarithmic utility.
  • The paper uses logarithmic utility because linear throughput maximization can make each BS serve only its strongest user, whereas logarithmic utility promotes load balancing and fairness.
  • The logarithm is concave with diminishing returns, making additional resources more valuable for users with low rates.

D. Analysis of Optimized Resource Allocation

The analysis shows that logarithmic utility makes optimized resource allocation simple and supports tractable load-aware association, while fractional association supplies a useful performance benchmark.

  • For the general joint association model, the paper proposes a tractable resource-allocation formulation and uses it as a benchmark for network performance.
  • The optimal resource allocation is equal allocation, with yij = 1/Kj for users associated with BS j.
  • Under logarithmic utility, optimal allocation is independent of SINR distribution and uniform across users served by each BS.
  • The resulting long-term rate is determined after equal resource allocation, allowing the association objective to be rewritten using per-user rates.
  • In the illustrative comparison, max-SINR overloads macro BS 1 while leaving some small BSs lightly loaded or idle.
  • The load-aware scheme moves traffic from congested macrocells to lightly loaded small cells, improving resource utilization.
  • Fractional user association reduces combinatorial complexity and upper-bounds single-BS association, but rounding is needed because multiple-BS service is harder to implement.
  • Numerical results show almost no loss after rounding, and the FUA upper bound is quite tight.

IV. PRIMAL-DUAL DISTRIBUTED ALGORITHM

The paper converts the relaxed association problem into a primal-dual decomposition, enabling separate user-side and BS-side optimization without coordination.

  • The distributed algorithm uses Lagrangian dual decomposition to split the inner maximization into user-side and BS-side subproblems.
  • The only coupling constraint links user association variables to BS loads, so a Lagrange multiplier can relax it.
  • Under feasibility-based strong duality, a dual optimum yields primal-optimal association and load solutions.
  • The decoupled subproblems can be solved separately by users and BSs without coordination.

B. The Distributed Algorithm

The distributed algorithm iteratively updates BS prices using local demand and supply information, converging to a near-optimal association with low per-iteration complexity.

  • The outer dual problem uses gradient projection, with each iteration requiring decomposed user and BS maximization subproblems.
  • BS multipliers act as load-dependent prices: overloaded BSs raise prices, while underloaded BSs lower prices to attract users.
  • The price updates motivate a rate-bias association scheme.
  • At each iteration, the distributed algorithm has complexity O(NBNU) and uses only local information for BS adjustments.
  • Information exchange per iteration is k(NB + NU), and the dynamic-stepsize gradient method generally converges in fewer than 20 simulation iterations.
  • After iterative updates, the algorithm is guaranteed to converge to a near-optimal solution.

C. Step Size and Convergence

The paper develops dynamic stepsize updates for its distributed algorithm and establishes convergence under bounded-subgradient conditions, while noting deployment sensitivity as a limitation.

  • C. Step Size and Convergence: The dynamic rule updates the target estimate D(t) using fixed positive constants β < 1 and ρ > 1.The target is adjusted relative to the best value achieved, with ε(t) increased or held constant when the target is reached and reduced otherwise.
  • C. Step Size and Convergence: A positive threshold ε keeps the stepsize δ(t) bounded away from zero when the target level is not attained.This boundedness supports the theorem's convergence argument.
  • C. Step Size and Convergence: The convergence proof uses bounded user-association variables and bounded subgradients of the dual objective.These properties satisfy the necessary conditions of the cited proposition.
  • V. Range Expansion (Biasing): The proposed approaches are sensitive to the deployment of users and base stations, while range expansion is designed to be insensitive to deployment changes.Range expansion makes uncoordinated decisions using received power and tier-specific biasing.
  • V. Range Expansion (Biasing): The optimal association motivates SINR and rate biasing; rate bias is derived from the optimal multiplier, while SINR bias requires brute-force search.Both biasing approaches achieve performance close to optimal FUA, with rate bias performing better than SINR bias.
  • V. Range Expansion (Biasing): The resulting biasing factors are insensitive to BS and user locations, making the bias schemes simple and robust to implement in practice.The paper reports this as an observed property of the biasing factors.

A. SINR Bias

SINR biasing associates each user with the base station offering the highest biased SINR, using a common factor within each tier. The paper compares this simple rule with load-aware optimal association and motivates rate bias as a tractable alternative.

  • A. SINR Bias: SINR biasing associates each user with the BS providing the highest biased SINR, with one factor shared by all BSs in a tier.Setting all factors to 1 recovers conventional max-SINR association.
  • A. SINR Bias: Biasing under-loaded small BSs expands their coverage and attracts more users, producing a fairer traffic distribution.The paper describes this as the load-balancing effect of range expansion.
  • A. SINR Bias: The empirically selected SINR bias remains stable as BS density and transmit power change, and its performance is very close to optimal FUA.The best SINR biasing factors are obtained by a high-complexity brute-force search.
  • A. SINR Bias: Rate bias multiplies the rate expression through an exponential factor, unlike SINR bias, which directly multiplies SINR.The rate-bias factor is derived from the optimal dual multiplier and reproduces the distributed algorithm's association.
  • A. SINR Bias: Using a common rate-bias factor per tier yields a solution very close to the optimal FUA solution.The factor is defined as the mean of the optimal multiplier transformation for BSs in the tier.

VI. PERFORMANCE EVALUATION

The evaluation compares load-aware association schemes in simulated three-tier heterogeneous networks under static and stochastic channels. Fractional association, rounding, distributed optimization, and biasing improve load balance and low-rate performance relative to max-SINR.

  • VI. Performance Evaluation: The simulations use a three-tier HetNet with transmit powers {P1, P2, P3} = {46, 35, 20} dBm and densities {λ2, λ3} = {5, 20} per macrocell.Macros are fixed, while small-cell locations are independently and uniformly distributed across tiers.
  • VI. Performance Evaluation: Max-SINR creates highly unbalanced loads, with overloaded macro BSs and lightly loaded or idle small BSs.Fractional association shifts users toward less congested small BSs.
  • VI. Performance Evaluation: Rounding produces loads almost identical to the fractional global optimum, while the distributed algorithm and biasing provide near-optimal load distributions with low complexity.The paper attributes rounding's effectiveness to few users being genuinely fractional.
  • VI. Performance Evaluation: Stochastic-channel rates with proportional-fair scheduling exceed static-channel rates, while macro-only networks show a distinct rate-gain pattern for uniformly distributed users.The paper explains the stochastic increase through service during favorable channel states.
  • VI. Performance Evaluation: 2–3.5x gains at low rates occur for joint association, fraction-rounding, the dual distributed algorithm, and biasing versus max-SINR in both settings.The max-SINR CDF catches up at 0.3 bits/s/Hz, while load balancing provides a more uniform user experience.
  • VI. Performance Evaluation: 3.5x rate gain versus max-SINR appears at the 10% rate point, and simple biasing remains very close to optimal association.The reported SINR bias factors are {A1, A2, A3} = {0, 6, 10.8} dB, while rate-bias factors are {B1, B2, B3} = {1.00, 1.59, 1.88}.

C. Biasing Factor

The paper develops load-aware association and biasing schemes for heterogeneous cellular networks, showing that carefully chosen bias factors can approach optimal performance. Biasing factors are nearly insensitive to small-cell density but depend strongly on transmit power, while load-aware association improves rates, especially for low-rate users.

  • Association framework: The paper formulates load-aware association through network-wide utility maximization, relaxes physical constraints, and derives a distributed algorithm via dual decomposition.The framework also extends to range expansion using biasing factors for small-cell BSs, requiring limited architectural changes.
  • Distributed algorithm: The distributed algorithm converges to a near-optimal solution with complexity linear in the number of users and BSs.The resulting association can be the same as that obtained by the distributed algorithm while retaining near-optimality.
  • Scope and extensions: The study evaluates SINR and rate biasing and identifies future extensions involving dynamic traffic, high mobility, uplink power control, analytical bias design, and additional utilities.These extensions mark boundaries of the present evaluation and formulation.
  • Performance: With optimal biasing factors, the network nearly achieves optimal load-aware performance and substantially improves resource utilization and macro-BS congestion.The numerical results report multi-fold overall-rate gains, particularly for users with previously low rates.
  • Density dependence: Optimal biasing factors are nearly independent of BS densities, so they remain almost unchanged as network infrastructure deployment evolves.The paper attributes this to more users associating with the denser BS type, leaving the needed range expansion nearly unchanged.
  • Power dependence: Biasing factors depend strongly on per-tier transmit powers: increasing a tier’s power generally reduces the bias needed for users to associate with that tier.The reported tier-specific trends differ across the second and third tiers as transmit power changes.

APPENDIX A

The appendix shows that, under the logarithmic utility formulation, the objective separates into a constant SINR term and a resource-allocation term. Equal resource allocation across users associated with a BS maximizes the latter through the arithmetic–geometric mean inequality.

  • Objective transformation: The objective is rewritten as log(c_ij) + log(y_ij), separating the channel-rate term from the resource-allocation term.The channel-rate term log(c_ij) is constant relative to SINR_ij in this step.
  • Equal allocation: Applying the arithmetic–geometric mean inequality bounds the geometric mean of users’ allocations by their arithmetic mean.Equality holds only when all associated users receive equal allocations.
  • Equal allocation: Therefore, the maximizing allocation assigns every user associated with BS j an equal share, y_ij = 1/K_j.This equal-allocation result supports the paper’s simplified load-aware association formulation.
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