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Power Control in Two-Tier Femtocell Networks

Vikram Chandrasekhar, Jeffrey G. Andrews, Tarik Muharemovic, Zukang Shen, Alan Gatherer

arXiv:0810.3869v4cs.NI

TL;DR

The paper examines the largest feasible cellular SINR given femtocell SINR targets and the cross-tier interference limits imposed by shared spectrum. It develops utility-based femtocell SINR adaptation and a link-quality protection algorithm, with simulations showing limited femtocell SINR impact while protecting cellular users.

  • Problem

    The paper asks how cellular link quality is constrained by feasible femtocell SINR targets in a shared-spectrum two-tier network.

  • Method

    The paper develops utility-based distributed femtocell SINR adaptation and a link-quality protection algorithm that lowers femtocell transmission powers when cellular SINR targets are threatened.

  • Results

    16% worst-case femtocell SINR reduction accompanies the algorithm's ability to ensure a cellular user achieves its SINR target with 100 femtocells/cell-site.

  • Takeaways & Limitations

    Higher SINR targets in one tier constrain the highest SINRs obtainable in the other, while cellular link-quality protection may require femtocells to lower their SINR targets.

Abstract

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In a two tier cellular network -- comprised of a central macrocell underlaid with shorter range femtocell hotspots -- cross-tier interference limits overall capacity with universal frequency reuse. To quantify near-far effects with universal frequency reuse, this paper derives a fundamental relation providing the largest feasible cellular Signal-to-Interference-Plus-Noise Ratio (SINR), given any set of feasible femtocell SINRs. We provide a link budget analysis which enables simple and accurate performance insights in a two-tier network. A distributed utility-based SINR adaptation at femtocells is proposed in order to alleviate cross-tier interference at the macrocell from cochannel femtocells. The Foschini-Miljanic (FM) algorithm is a special case of the adaptation. Each femtocell maximizes their individual utility consisting of a SINR based reward less an incurred cost (interference to the macrocell). Numerical results show greater than 30% improvement in mean femtocell SINRs relative to FM. In the event that cross-tier interference prevents a cellular user from obtaining its SINR target, an algorithm is proposed that reduces transmission powers of the strongest femtocell interferers. The algorithm ensures that a cellular user achieves its SINR target even with 100 femtocells/cell-site, and requires a worst case SINR reduction of only 16% at femtocells. These results motivate design of power control schemes requiring minimal network overhead in two-tier networks with shared spectrum.

I. INTRODUCTION

The paper studies cross-tier interference in shared-spectrum two-tier networks, where femtocell and cellular SINR targets are coupled. It develops questions and methods for preserving cellular link quality while supporting distributed femtocell operation.

  • Motivation: Femtocells can improve indoor coverage, spatial reuse, and user data rates, but the resulting QoS improvement may reduce cellular coverage.Femtocell users are near their base stations and may seek higher SINRs than cellular users, while cellular performance can deteriorate.
  • Motivation: Cross-tier interference couples the SINR targets of macrocell and femtocell users sharing spectrum.The paper treats the macrocell as primary infrastructure whose cellular users must retain minimum SINR targets despite femtocell interference.
  • Practical constraints: The proposed setting addresses practical coordination challenges caused by limited scalability, security, and backhaul bandwidth between the macrocell and femtocell access points.The paper motivates distributed interference management because femtocells are consumer-installed and their traffic requirements are user determined.
  • System setting: Closed access limits femtocell communication to licensed home users, while interior femtocells may significantly deteriorate SINR at the macrocell base station.The work assumes closed access and focuses on protecting outdoor cellular users from cross-tier interference.
  • Research questions: The paper asks for the largest feasible cellular SINR given femtocell targets, the dependence on locations and channel gains, distributed SINR equilibria, and required femtocell reductions.These questions cover feasibility, link quality, utility-based adaptation, and cellular link-quality protection.
  • Scope: The analysis focuses exclusively on uplink operation, although portions of the analysis may apply to downlink systems with potentially different conclusions.The downlink extension is omitted for future work.

B. Prior Work

Prior work establishes feasibility and utility-based power-control foundations, while this paper extends them to closed-access femtocell SINR adaptation and cellular link-quality protection. Its contributions include Pareto SINR relations, distributed equilibria, and interference-reduction algorithms.

  • Conventional power control: Earlier power-control work characterized SINR feasibility using the spectral radius of a normalized channel-gain matrix.A feasible power allocation exists if and only if that spectral radius is less than unity.
  • Prior foundations: Existing cellular and tiered-network research included centralized, distributed, constrained, admission, assignment, and utility-optimization approaches.The paper positions its contribution within established power-control and game-theoretic literature.
  • Utility-based adaptation: The adaptation gives stronger femtocell interferers equilibria closer to minimum SINR targets, while less interfering femtocells obtain higher SINR margins.The resulting channel-dependent equilibrium is attained through distributed power updates.
  • Utility-based adaptation: The proposed utility models a femtocell user’s desire for higher data rates while penalizing interference caused to the macrocell.Unlike the cited sigmoidal utility, the paper gives the utility a direct interpretation tied to femtocell incentives and macrocell priority.
  • Positioning: Compared with prior approaches, the paper specifically addresses SINR adaptation and acceptable cellular performance in closed-access femtocells.The results are also presented as applicable to cognitive-radio settings involving primary and secondary users.
  • Pareto SINR contours: The paper derives the highest feasible cellular SINR for any feasible femtocell SINRs and describes Pareto SINR contours through a constant dB-sum relation.The analysis provides interpretations across path-loss exponents, femtocell counts, and user or hotspot locations.
  • Utility-based adaptation: Up to 30% higher femtocell SINRs are obtained with the utility adaptation relative to the Foschini-Miljanic update.For users seeking only their minimum SINR targets, the adaptation simplifies to the FM update.
  • Cellular link-quality protection: A distributed algorithm progressively reduces the SINR targets of strongest femtocell interferers until the cellular SINR target is met.With 100 femtocells per cell site, simulations report a worst-case femtocell SINR reduction of only 16%.

II. SYSTEM MODEL

The model characterizes two-tier SINR feasibility through the spectral radius of an interference matrix, then derives the maximum cellular SINR compatible with femtocell targets. It also expresses the macrocell–femtocell trade-off through Pareto contours and a link budget.

  • Network model: The model uses a central macrocell underlaid by N cochannel femtocells, with one scheduled active user per cell during each signaling slot.The slot interpretation applies to time or frequency resources and extends to downlink analysis.
  • Feasibility and power control: A target SINR assignment is feasible exactly when ρ(ΓG) < 1, with minimum componentwise power p∗ = (I − ΓG)^−1η.This solution satisfies all target SINRs and is Pareto efficient.
  • Feasibility properties: The spectral radius is non-decreasing in per-tier SINR targets, so increasing targets reduces the feasibility margin.Any feasible femtocell SINR tuple remains feasible in a femtocell-only network because ρ(ΓG) ≥ ρ(ΓfF).
  • Maximum cellular SINR: Theorem 1 gives the highest cellular SINR target for any feasible set of femtocell SINR targets by fixing the target spectral radius κ.The result applies when ρ(ΓfF) < κ < 1.
  • Two-tier trade-off: For one femtocell, the product of per-tier SINR targets is limited by the inverse product of the two cross-tier channel gains.The cross-tier gains are measured between the cellular user and femtocell access point in both directions.
  • Link budget: For small femtocell-to-femtocell interference, the sum of per-tier decibel SINRs equals the channel-dependent link-budget constant LdB.A cellular target of x dB leaves any feasible femtocell target no greater than LdB − x dB.

A. Design Interpretations

The link budget varies with user and hotspot locations, path-loss exponents, and the number of femtocells. Asymmetric cross-tier interference can make cell-interior SINRs worse than cell-edge SINRs.

  • Modeling choices: The link-budget analysis studies how per-tier SINRs vary with user and femtocell locations under a simplified path-loss model.Rayleigh fading and lognormal shadowing are not modeled.
  • Location effects: The approximation in (15) closely matches the exact result in (13), and configuration (b) produces the highest per-tier SINRs.Configuration (b) places the cellular user near the macrocell center and the femtocell grid farther away.
  • Location effects: Cell-interior configurations can have worse per-tier SINRs than cell-edge configurations because asymmetric interferer locations diminish link budgets.Femtocells near the macrocell BS force both tiers to lower their SINR targets through cross-tier power-control interaction.
  • Path-loss sensitivity: Higher path-loss exponents produce higher link budgets in the evaluated practical scenarios.The comparison uses α = 3.5 and α = 4.
  • Femtocell density: With 64 femtocells, regular and random placements both show diminishing link budgets in the cell interior.The result indicates significant cross-tier interference.
  • Design objective: The adaptation objectives are to maximize femtocell SINRs while limiting their cross-tier interference.These objectives motivate distributed SINR adaptation.

IV. UTILITY-BASED DISTRIBUTED SINR ADAPTATION

Because centralized power allocation is difficult without coordination between tiers, the paper formulates distributed SINR adaptation as a non-cooperative utility game. Users choose transmit powers as best responses to other users.

  • Motivation: Distributed utility-based SINR adaptation is introduced because centralized power allocation is likely prohibitively difficult without inter-tier coordination.The adaptation is designed for femtocell SINR control.
  • Game formulation: The network is modeled as an N + 1-player non-cooperative power-control game with individual strategy sets and utilities.Each user selects transmission power in response to the actions of other participants.
  • Equilibrium: The equilibrium is a vector of transmit powers at which each user maximizes its individual utility given the other users’ powers.At a Nash equilibrium, no user can unilaterally improve its utility.
  • Assumptions: The formulation constrains every mobile’s transmission power to the interval [0, pmax] and assumes closed-loop feedback from each serving base station.The feedback periodically reports status information for adaptation.

A. Cellular Utility Function

The cellular utility function represents a user that seeks its minimum feasible SINR while minimizing the transmission power needed to achieve it. When its SINR exceeds the target, reducing power improves utility.

  • Cellular objective: The cellular utility models the goal of achieving minimum SINR target Γ0 while using no more than the minimum required transmission power.This objective is stated under feasibility and the power constraint pmax.
  • Cellular objective: When γ0 > Γ0 under fixed interference, the cellular user can improve utility by decreasing p0 until γ0 = Γ0.The utility therefore favors meeting, rather than exceeding, the target when excess power is unnecessary.

B. Femtocell Utility Function

The femtocell utility balances SINR-based reward against a penalty for cross-tier interference at the macrocell. Under stated reward and penalty assumptions, each femtocell selects power as an individual best response.

  • Best-response optimization: Given other users’ transmit powers, each femtocell maximizes its individual utility as a best response while meeting its minimum SINR requirement.
  • Utility structure: Each femtocell utility combines a reward for its SINR with a cost for creating cross-tier interference at the macrocell.The penalty discourages transmission power that creates unacceptable interference at the primary infrastructure.
  • Utility structure: The penalty is scaled by experienced interference, so femtocells facing higher interference are penalized less.
  • Utility assumptions: The reward utility is monotonically increasing and concave in SINR, while the power-related utility is monotonically decreasing and concave in transmit power.These assumptions model declining marginal satisfaction above the SINR target and increasing penalty from additional interference.
  • Best-response optimization: Strict concavity makes the femtocell utility strictly concave with respect to its own transmit power for fixed interfering powers.The resulting optimization has at most one local optimum over the feasible power interval.

C. Existence of Nash Equilibrium

The utility-based power-control game admits Nash equilibria under standard compactness, continuity, and quasi-concavity conditions. The resulting SINR equilibrium is unique and can be reached through distributed iterative updates, with coefficients controlling the reward–interference trade-off.

  • Existence conditions: A Nash equilibrium exists when each feasible power set is nonempty, convex, and compact, and utility is continuous and quasi-concave in each user’s power.
  • Equilibrium characterization: The equilibrium power for each femtocell is the unique intersection of a decreasing SINR-reward derivative and an increasing interference-penalty derivative.
  • Convergence: The SINR equilibrium is unique, and the corresponding power-control iterates converge because the update forms a standard interference function.
  • Implementation: Distributed implementation requires each femtocell to know its target SINR and channel gains to the macrocell and its serving femtocell, while estimating the macrocell gain may require site-specific knowledge.
  • Coefficient effects: For large a_i, the equilibrium approaches Γ_i and the update becomes equivalent to the Foschini-Miljanic algorithm when the target is feasible.
  • Coefficient effects: When a_i g_i,i < b_i g_0,i, femtocells settle below Γ_i because cross-tier interference receives greater penalty.
  • Coefficient effects: The highest gains over Γ_i occur when a_i g_i,i = e b_i g_0,i, although this choice can create potentially large cross-tier interference.

D. Reducing Femtocell SINR Targets : Cellular Link Quality Protection

When cross-tier interference prevents the cellular user from reaching its SINR target, the protection procedure reduces SINR targets for selected femtocells. It prioritizes the strongest interferers and expands the selected set until the cellular SINR meets tolerance.

  • Protection procedure: The protection algorithm selects a femtocell subset whose SINR equilibria are reduced by a factor t > 1 to protect the cellular SINR target.
  • Set expansion: If reducing one femtocell set does not achieve Γ0, the algorithm chooses a larger set containing it.
  • Implementation: Centralized selection of reduction factors and femtocell sets can be difficult in OFDMA systems because the macrocell must communicate them for each frequency subband.Periodic macrocell feedback enables a simpler distributed adaptation.
  • Stopping rule: The SINR-reduction procedure repeats after every M updates until the cellular user’s SINR exceeds (1−ǫ)Γ0.
  • Interferer selection: Femtocells are ranked by interference at the macrocell, and the selected set contains the strongest interferers above threshold y.Lowering y after every M iterations expands the set of femtocells whose targets are reduced.

V. NUMERICAL RESULTS

Numerical experiments evaluate utility-based SINR adaptation and cellular link-quality protection under dense femtocell deployments. The adaptation improves femtocell SINRs, while protection preserves cellular performance with bounded femtocell degradation.

  • Utility adaptation: Nearly 30% higher mean femtocell SINRs are obtained relative to the average minimum SINR target with a = 0.1, b = 1, and N = 64.
  • Utility adaptation: Nearly 2 dB improvement in mean femtocell SINRs above the mean SINR target occurs with a higher interference penalty at femtocells.
  • Utility adaptation: When a >> 1 and N ≥64, mean equilibrium femtocell SINRs fall below the mean target because users reduce transmit power to improve cellular link quality.
  • Cell-edge protection: Nearly 8% SINR improvement is obtained for a cell-edge cellular user and femtocells when N = 64.
  • Dense deployment protection: With N = 100 femtocells and a cell-edge location, the worst-case femtocell SINR reduction is only 16%, although nearly 45% operate below their minimum target.
  • Dense deployment protection: In all other evaluated cases, mean femtocell SINR reduction is less than 6%.The experiments use 5000 different SINR trials in each experiment.
  • Overall outcome: The cellular link-quality protection algorithm is reported to guarantee reliable cellular coverage without significantly affecting femtocell transmit power.

VI. CONCLUSION

The paper characterizes cross-tier SINR trade-offs, develops utility-based femtocell adaptation and cellular link-quality protection, and reports limited femtocell impact under protection.

  • VI. CONCLUSION: The work asks how cellular link quality depends on femtocell SINR targets and user locations, and how utility-based adaptation determines femtocell equilibria.It also considers whether these equilibria can be achieved distributively.
  • VI. CONCLUSION: Higher SINR targets in one tier fundamentally constrict the highest SINRs obtainable in the other tier because of near-far effects.The effect arises from asymmetric positions of interfering users relative to nearby base stations.
  • VI. CONCLUSION: Utility-based femtocell adaptation can require femtocells to deliberately lower their SINR targets to protect an active cellular user.The paper provides an algorithm for progressively reducing targets at strong femtocell interferers when the cellular user cannot meet its target.
  • VI. CONCLUSION: 16% was the worst-case femtocell SINR reduction reported for the cellular link-quality protection algorithm.Simulation results confirm the algorithm’s efficacy and minimal impact on femtocell SINRs.
  • VI. CONCLUSION: The distributed algorithm ensures minimal network overhead in a practical two-tier deployment.The conclusion connects distributed operation with reduced coordination requirements.

SYSTEM PARAMETERS

This section presents SINR contours, link-budget plots, utility-based adaptation results, and cellular link-quality protection procedures for two-tier femtocell networks.

  • SYSTEM PARAMETERS: Per-tier SINR contours are evaluated for different cellular-user and femtocell locations.The contours use a common femtocell SINR target in the reported analysis.
  • SYSTEM PARAMETERS: Mean femtocell SINR targets are compared for different reward and cost coefficients.The reported setup places the grid center at the cell edge and assumes SINR targets uniformly distributed in dB.
  • SYSTEM PARAMETERS: The distributed power-control procedure maintains cellular link quality by adapting cellular and femtocell transmit powers.The algorithm uses a pre-specified tolerance ε ∈ [0, 1] and identifies femtocells causing excessive cross-tier interference.
  • SYSTEM PARAMETERS: Fig. 7 reports mean femtocell SINR reduction and the percentage of femtocells below their SINR target.The cellular SINR target is uniformly distributed in [3, 10] dB.
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