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Optimizing User Association and Spectrum Allocation in HetNets: A Utility Perspective

Yicheng Lin, Wei Bao, Wei Yu, Ben Liang

arXiv:1412.5731v1cs.NIcs.IT

TL;DR

The paper addresses joint user association and spectrum allocation in multi-tier HetNets for downlink and uplink. It models network elements as spatial point processes, derives a mean proportionally fair utility, and formulates a network utility maximization framework. The analysis reveals downlink–uplink association decoupling, distance-based uplink association under a condition, and spectrum allocation matching user proportions across tiers.

  • Problem

    The paper addresses the joint load balancing, or user association, and spectrum allocation problem in multi-tier HetNets for both downlink and uplink.

  • Method

    The paper models base stations and users as spatial point processes and analytically derives a closed-form approximation of the network's mean proportionally fair utility.

  • Results

    The solution reveals downlink–uplink decoupling in user association, distance-based uplink association under a condition, and spectrum allocation matching the proportion of users associated with each tier.

  • Takeaways & Limitations

    The results illustrate the usefulness of stochastic geometry for optimizing HetNets.

Abstract

from arXiv · show

The joint user association and spectrum allocation problem is studied for multi-tier heterogeneous networks (HetNets) in both downlink and uplink in the interference-limited regime. Users are associated with base-stations (BSs) based on the biased downlink received power. Spectrum is either shared or orthogonally partitioned among the tiers. This paper models the placement of BSs in different tiers as spatial point processes and adopts stochastic geometry to derive the theoretical mean proportionally fair utility of the network based on the coverage rate. By formulating and solving the network utility maximization problem, the optimal user association bias factors and spectrum partition ratios are analytically obtained for the multi-tier network. The resulting analysis reveals that the downlink and uplink user associations do not have to be symmetric. For uplink under spectrum sharing, if all tiers have the same target signal-to-interference ratio (SIR), distance-based user association is shown to be optimal under a variety of path loss and power control settings. For both downlink and uplink, under orthogonal spectrum partition, it is shown that the optimal proportion of spectrum allocated to each tier should match the proportion of users associated with that tier. Simulations validate the analytical results. Under typical system parameters, simulation results suggest that spectrum partition performs better for downlink in terms of utility, while spectrum sharing performs better for uplink with power control.

I. INTRODUCTION

The paper optimizes joint user association and spectrum allocation in multi-tier HetNets for both downlink and uplink using a coverage-rate-based network utility. Its analysis yields asymmetric associations, distance-based uplink association under a stated condition, and spectrum partitions matched to associated-user proportions.

  • Problem and approach: The paper jointly optimizes user association and spectrum allocation for downlink and uplink multi-tier HetNets.It formulates the problem through a network utility based on coverage rate.
  • Problem and approach: Stochastic geometry models random BS and user deployments and enables a closed-form approximation of mean proportionally fair utility.The utility is averaged over BS locations and fading channels rather than a specific network realization.
  • Analytical findings: Users may choose different BSs in downlink and uplink for better performance.The resulting analysis therefore does not require symmetric user association across link directions.
  • Analytical findings: Under uplink spectrum sharing, equal target SIR across tiers makes nearest-BS association optimal under varied path loss and power-control settings.This is the paper’s distance-based association result.
  • Analytical findings: Under orthogonal spectrum partition, each tier’s optimal spectrum share equals its proportion of associated users.The result applies to both downlink and uplink and suggests a simple partition rule.
  • Simulation findings: Simulations validate the analytical optima and suggest orthogonal partition performs better for downlink utility, whereas spectrum sharing performs better for uplink with power control.The simulated association biases and spectrum allocations match numerical optima.

C. Related Work

Prior HetNet research studies stochastic-geometry performance, user association, and spectrum allocation, but this paper distinguishes itself by analytically optimizing mean user utility across multi-tier downlink and uplink settings.

  • Stochastic-geometry studies: Prior stochastic-geometry studies analyze coverage, rate, topology, access modes, and per-user-rate distributions in multi-tier networks.These works characterize performance under random network models but do not provide the same utility perspective.
  • User association: Earlier user-association methods include greedy search, utility maximization, pricing, game theory, and semi-static biased association.Dynamic methods require real-time channel and topology information, whereas biased association is simpler to implement but often empirically tuned.
  • User association: Existing biased-association studies determine optimal bias using numerical evaluation of SIR or rate coverage, while this paper derives bias factors through analytical network utility optimization.This is the stated distinction from prior bias-optimization work.
  • Spectrum allocation: Prior spectrum-allocation studies optimize area spectral efficiency, throughput, coverage-constrained objectives, or rate coverage, with some lacking an analytical optimal partition.The cited literature includes both disjoint spectrum partition and joint association-partition analyses.
  • Spectrum allocation: This paper analytically determines optimal inter-tier spectrum partition in terms of mean user utility for both downlink and uplink.It extends the spectrum-partition perspective beyond prior downlink-focused or non-utility formulations.
  • Uplink analysis: The paper analyzes and optimizes mean user utility for random multi-tier uplinks with fractional power control.This addresses an uplink setting less emphasized in the prior HetNet literature.

D. Organization

The paper models multi-tier HetNets with stochastic geometry and analyzes user association, propagation, and power-control assumptions. It then derives and optimizes utility in downlink and uplink, validating results through simulation.

  • D. Organization: The paper derives utility and optimization results for downlink and uplink, then validates them through numerical simulation before concluding.
  • A. Multi-Tier Network Topology: The paper models each BS tier as an independent homogeneous PPP, with users forming a separate homogeneous PPP.
  • B. Path Loss and Power Model: The propagation model uses a common path-loss exponent, independent unit-mean exponential fading, and ignores shadowing for tractability.
  • B. Path Loss and Power Model: Uplink transmission uses fractional power control, with ε=1 giving full power control and ε=0 giving no power control.
  • C. Biased User Association: Users are associated with BSs according to maximum biased downlink received power, where tier-specific biases encode connection preferences.

D. Spectrum Allocation among Tiers

The paper compares orthogonal spectrum partition and full spectrum reuse, then defines a coverage-rate-based proportional-fair utility for optimizing tier association and spectrum allocation.

  • Spectrum schemes: Orthogonal partition assigns non-overlapping spectrum W_k=η_kW across tiers, whereas full reuse assigns the entire band W to every tier.
  • Coverage model: The analysis assumes negligible background noise, making the network interference limited, and models coverage using tier-specific target SIRs.
  • Coverage model: Each user receives zero rate below its target SIR and a fixed spectral efficiency log(1+τ_k) otherwise.
  • Utility: The proposed utility is the logarithm of user coverage rate, decomposing into spectral-efficiency, per-user-spectrum, and coverage-probability terms.
  • Utility: Equal spectrum allocation among a BS's associated users supports proportional fairness, while the per-user spectrum term is approximated using average user counts per BS.
  • Utility: The upper bound for mean log per-user spectrum is tight when user intensity is much larger than BS deployment intensity.

B. Mean Logarithm of Coverage Probability in the Downlink

The downlink analysis derives mean log coverage probability for a typical user under both spectrum sharing and orthogonal partition. It averages over serving distance and interference using the stochastic network model.

  • Downlink coverage analysis: The downlink analysis considers a typical user served by a tier-k BS under both spectrum sharing and orthogonal spectrum partition.
  • Spectrum sharing: For spectrum sharing, interference is summed over BS point processes from every tier except the serving BS.
  • Coverage averaging: The mean log coverage probability is obtained by averaging first over interference conditional on serving distance, then over the serving-distance distribution.
  • Biased association: The serving-distance distribution is determined under biased user association, linking the downlink coverage analysis to the association biases.
  • Orthogonal partition: Under orthogonal spectrum partition, the downlink coverage expression excludes cross-tier interference.

1) Spectrum Sharing:

The uplink spectrum-sharing analysis models interference from scheduled users with fractional power control and derives mean log coverage using bounds and distance distributions. The derivation requires constraints on the power-control factor and approximates a dependency for tractability.

  • Interference model: Uplink interference at a typical BS comes from users scheduled by other BSs on the same spectrum resource block.
  • Approximation: For analytical tractability, the derivation ignores the dependency between serving-link distance and uplink interference.
  • Interference model: The scheduled-user intensity matches the corresponding BS intensity because each BS schedules at most one user per resource block.
  • Interference geometry: Unlike downlink interference, uplink interfering users can be arbitrarily close to the typical BS, so the interference integral starts at zero.
  • Power-control constraint: The uplink mean-interference integral converges only when the fractional power-control factor satisfies the stated constraints; otherwise mean interference is unbounded.
  • Coverage averaging: Mean log coverage is averaged over user, BS, and interferer locations and interference-channel randomness using the serving-distance distribution.

2) Orthogonal Spectrum Partition:

The paper formulates spectrum-partition optimization for downlink HetNets and shows that the optimal spectrum share for each tier matches its associated-user proportion.

  • Downlink spectrum partition: The optimization is generally nonconvex and lacks a closed-form solution, so numerical methods obtain a local optimum.The resulting bias factors are recovered after optimizing association probabilities.
  • Downlink spectrum partition: The analysis considers orthogonal spectrum partition for downlink and optimizes the mean user utility over tier-specific spectrum shares.The optimization imposes positive shares whose sum is one.
  • Downlink spectrum partition: The optimal proportion of spectrum allocated to a tier equals the proportion of users associated with that tier.The user association probability is equivalent to the mean proportion of users associated with the tier.
  • Downlink spectrum partition: The spectrum-partition problem is solved using a Lagrangian and KKT conditions under the unit-sum spectrum constraint.The derivation introduces a dual variable for the constraint that all tier shares sum to one.

2) Optimal User Association:

The paper optimizes user association through tier association probabilities and derives downlink and uplink rules, including distance-based uplink association under specific conditions.

  • Downlink association: For downlink under orthogonal partition, the optimal bias solution follows from a concave association-probability objective and KKT optimization.The resulting association is positive for tiers whose λk log(1 + τk) exceeds a threshold.
  • Uplink association: For uplink under spectrum sharing, equal target SIRs and ϵα = 2 or ϵα ≥ 4 make distance-based association optimal.Each user communicates with its closest base station under these conditions.
  • Uplink association: Distance-based uplink association limits users’ serving distances, reducing high transmit powers and interference to nearby small-cell base stations.The paper gives this as the intuition behind connecting users to their closest base stations.
  • Uplink association: The uplink theorem is analytically established only for ϵα = 2 and ϵα ≥ 4, while simulations find distance-based association across feasible regimes when target SIRs are equal.The objective is neither convex nor concave in the intermediate regimes.
  • Downlink–uplink asymmetry: Optimal downlink and uplink bias factors may differ, allowing users to associate with different base stations for downlink and uplink.The paper identifies this as downlink–uplink association asymmetry.

D. Orthogonal Spectrum Partition in the Uplink

For uplink orthogonal spectrum partition, the optimal spectrum share equals each tier’s user-association proportion, and full power control yields symmetry with the downlink solution.

  • Uplink spectrum partition: The uplink orthogonal-partition problem maximizes mean utility over positive spectrum shares constrained to sum to one.The optimization substitutes the uplink utility into the spectrum-partition constraints.
  • Uplink spectrum partition: The optimal proportion of spectrum allocated to a tier equals the proportion of users associated with that tier.This uplink result parallels the corresponding downlink theorem.
  • Uplink–downlink symmetry: Under orthogonal partition, full-power-control uplink shares the downlink system’s optimal spectrum partition ratio and association bias for identical network parameters.The matched parameters are density, power, target SIR, and path-loss exponent.
  • Simulation setting: The simulations use two tiers, with macro base stations at lower intensity and higher power than femto base stations.The study uses PPP tier intensities and a common target SIR for both links.
  • Simulation setting: Coverage rate is represented through the number of subcarriers whose SIR exceeds the threshold under round-robin scheduling.The simulations use Monte Carlo spatial snapshots, time slots, Rayleigh fading, and a 2048-subcarrier bandwidth.

A. Validation of the Optimization of User Association Bias and Spectrum Partition Ratio

Simulations validate the analytical association and spectrum-partition solutions, showing distinct sharing-versus-partition preferences for downlink and uplink.

  • Simulation metrics: The simulations evaluate mean utility, mean rate, cell-edge 5th-percentile rate, and zero-rate-user proportion across optimization settings.The analysis accounts for finite-simulation outages when interpreting log utility.
  • Spectrum sharing: For downlink spectrum sharing, mean finite utility and mean rate vary little across the simulated bias range, while many users receive zero rate.The zero-rate-user proportion dominates the resulting utility.
  • Spectrum sharing: For uplink spectrum sharing with full power control, the analytically optimal distance-based bias nearly maximizes utility, rate, and cell-edge rate.The simulation reports no zero-rate users under full power control and complete path-loss compensation.
  • Spectrum partition validation: When both tiers have equal deployment intensity, the optimal spectrum partition allocates equal spectrum to each tier.The simulations also show analytically optimal partition values achieving approximately the highest utility.
  • Spectrum comparison: Orthogonal spectrum allocation significantly improves downlink utility and cell-edge rate, whereas spectrum sharing gives better uplink utility and cell-edge rate.The comparison uses λ2 = 9λ1 in both spectrum schemes.
  • Spectrum comparison: The paper attributes the opposing downlink and uplink preferences to power control: downlink is more interference-sensitive, while uplink is less sensitive.The uplink therefore benefits from retaining high bandwidth through spectrum sharing.

B. Optimal Bias and Spectrum Partition Ratio under Different System Parameters

The paper examines how deployment density, power, power control, and spectrum regime affect optimal association biases and spectrum proportions in multi-tier HetNets. Results show distinct downlink and uplink behaviors, including distance-based uplink association under stated conditions and tier-dependent spectrum allocation.

  • Spectrum sharing: Under uplink spectrum sharing with α = 4, ϵ = 1, and equal target SIRs, distance-based association is optimal.The associated bias condition is reported as B1 = P2 under the stated setting.
  • Spectrum sharing: When target SIRs are equal, the optimal uplink bias remains equivalent to distance-based association across feasible ϵα regimes, including nonconvex cases.For unequal target SIRs, the association becomes close to distance-based at about ϵ > 0.6.
  • Spectrum partition: More tier-2 BSs reduce the optimal tier-1 spectrum proportion, while the tier-1 bias first increases and then decreases under downlink orthogonal partition.The corresponding uplink behavior under full power control is similar with respect to BS deployment intensity.
  • Power control: As the uplink power control factor increases, tier-1 bias and spectrum proportion first decrease and then increase under orthogonal partition.The initial decrease reflects stronger interference from farther tier-1 users transmitting at higher power; improved tier-1 signal quality eventually compensates for that interference.
  • Utility comparison: Simulation results suggest spectrum partition provides higher downlink utility, whereas spectrum sharing provides higher uplink utility with power control.The analytical framework derives utility-based association and allocation results using spatial point-process models and stochastic geometry.

APPENDIX A

The appendix develops approximations and optimization steps for the multi-tier utility problem. It models cell areas and associated-user counts, then analyzes subproblem curvature and derives optimal association solutions.

  • User association modeling: Conditioned on cell size, the number of users associated with a BS follows a Poisson distribution.This distribution supports the subsequent derivation of the typical user's mean spectrum proportion.
  • User association modeling: The multi-tier cell topology is treated as a multiplicatively weighted Voronoi tessellation to derive the user-count distribution for each tier-k BS.The derivation uses an area approximation together with the distribution of normalized Voronoi-cell sizes.
  • Optimization: For ϵα ≥ 4, both relevant subproblem objectives are convex, and their solutions yield the optimal solution to the original problem.The appendix obtains these solutions using the same optimization procedure applied to the earlier subproblem.
  • Optimal association: When all tiers have equal target SIRs, the derived bias factors satisfy PkBk = PjBj for k ≠ j, which is equivalent to distance-based user association.The result follows from the association-probability transformation used in the appendix.
  • Optimization: The optimization objective is neither convex nor concave when ϵα < 2 or 2 < ϵα < 4.The two component subproblems have opposing curvature in these regimes, complicating direct optimization of the original problem.
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