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Price-Based Resource Allocation for Spectrum-Sharing Femtocell Networks: A Stackelberg Game Approach

Xin Kang, Rui Zhang, Mehul Motani

arXiv:1103.2240v1cs.IT

TL;DR

Cross-tier and inter-cell interference restricts spectrum-sharing femtocell network performance, motivating interference-control strategies. The paper applies interference power constraints and Stackelberg games to pricing and allocation, obtaining closed-form solutions and a rapidly convergent distributed algorithm with low complexity and minimal information exchange.

  • Problem

    Cross-tier and inter-cell interference greatly restricts femtocell network performance, creating a need for interference-control strategies.

  • Method

    The paper brings interference power constraints into uplink cross-tier interference control, prices femtocell users' resulting interference at the MBS, and adopts a Stackelberg game model.

  • Results

    The study examines optimal interference prices and power-allocation strategies, deriving closed-form solutions for sparse deployment and a rapidly convergent distributed algorithm for uniform pricing.

  • Takeaways & Limitations

    The proposed algorithm has low complexity and requires minimum information exchange for interference-control resource allocation.

Abstract

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This paper investigates the price-based resource allocation strategies for the uplink transmission of a spectrum-sharing femtocell network, in which a central macrocell is underlaid with distributed femtocells, all operating over the same frequency band as the macrocell. Assuming that the macrocell base station (MBS) protects itself by pricing the interference from the femtocell users, a Stackelberg game is formulated to study the joint utility maximization of the macrocell and the femtocells subject to a maximum tolerable interference power constraint at the MBS. Especially, two practical femtocell channel models: sparsely deployed scenario for rural areas and densely deployed scenario for urban areas, are investigated. For each scenario, two pricing schemes: uniform pricing and non-uniform pricing, are proposed. Then, the Stackelberg equilibriums for these proposed games are studied, and an effective distributed interference price bargaining algorithm with guaranteed convergence is proposed for the uniform-pricing case. Finally, numerical examples are presented to verify the proposed studies. It is shown that the proposed algorithms are effective in resource allocation and macrocell protection requiring minimal network overhead for spectrum-sharing-based two-tier femtocell networks.

I. INTRODUCTION

The paper addresses interference control in spectrum-sharing femtocell networks, where shared-spectrum operation improves practicality but cross-tier and inter-cell interference restrict performance. It proposes MBS-controlled price-based resource allocation and compares uniform and non-uniform pricing across deployment scenarios.

  • Motivation: Shared-spectrum operation is favored because spectrum is scarce, cross-tier coordination is absent, and splitting spectrum would burden mobile devices.The paper therefore considers sharing the macrocell and femtocells’ frequency band.
  • Problem: Cross-tier and inter-cell interference greatly restrict network performance in spectrum-sharing two-tier femtocell networks.This motivates interference mitigation as a central research problem.
  • Problem: Unlike cognitive-radio devices, femtocell users may lack environment-aware sensing and self-power-adaptation capabilities, making user-side interference constraints impractical.The paper relocates interference control to the MBS rather than imposing it directly on femtocell users.
  • Approach: The proposed scheme lets the MBS control femtocell transmit power by pricing interference at its receiver subject to a maximum tolerable interference margin.This applies the interference-power-constraint concept from cognitive-radio networks to femtocell uplinks.
  • Approach: The paper formulates a Stackelberg game with non-uniform or uniform pricing and develops a distributed bargaining algorithm for uniform pricing with minimal information exchange.Non-uniform pricing assigns user-specific prices, whereas uniform pricing uses one price for all femtocell users.
  • Findings: Non-uniform pricing is optimal for MBS revenue maximization, whereas uniform pricing maximizes the femtocell users’ sum-rate.The paper studies these schemes under sparse and dense femtocell channel models, obtaining closed-form sparse-scenario solutions and revenue bounds for dense deployment.

II. SYSTEM MODEL

The system model describes an uplink two-tier network with one MBS, N femtocells, single-antenna terminals, and block-fading channels. It distinguishes sparse deployments, where mutual femtocell interference is neglected, from dense deployments, where aggregate mutual interference is modeled.

  • Network Architecture: The network contains one central MBS serving a region with N femtocells, each having a dedicated HBS and serving wireless devices.Each frequency band has at most one scheduled active user per femtocell during a signaling time-slot.
  • Transmission Model: The study focuses on uplink transmission over a single shared frequency band, while noting that the results can extend to broadband femtocell systems.Orthogonal uplink transmission is assumed within each femtocell.
  • Channel and Noise Assumptions: All terminals use a single antenna, and channels remain constant within each transmission block but may change between blocks.Channel power gains are modeled as i.i.d. continuous random variables, with independent circularly symmetric complex Gaussian receiver noises.
  • Channel and Noise Assumptions: The channel power gain from user i to HBS B_j is h_j,i, while the gain from user i to the MBS is g_i.These gains determine desired links and interference received at the MBS.
  • Sparse Deployment: In sparse deployments, mutual femtocell interference is neglected because path loss and indoor penetration loss sharply weaken cross-femtocell channels.This model is intended for rural areas where femtocells are usually far apart.
  • Dense Deployment: In dense urban deployments, femtocells are close enough that mutual interference cannot be ignored, so each receiver accounts for aggregate interference from other femtocell users.The model also assumes weak cross-femtocell gains and limited user peak transmit powers.

III. PROBLEM FORMULATION

The paper formulates price-based uplink power allocation as a Stackelberg game under an MBS interference limit. The MBS prices received interference, while femtocell users choose powers to balance transmission profit against interference cost.

  • III. PROBLEM FORMULATION: The problem formulation defines a price-based power-allocation scheme and investigates its Stackelberg equilibrium.The MBS is the leader and femtocell users are the followers.
  • A. Stackelberg Game Formulation: The MBS limits aggregate received femtocell interference to a maximum tolerable level Q.This constraint protects the MBS while allowing femtocell users to access the shared spectrum.
  • A. Stackelberg Game Formulation: The MBS assigns per-unit interference prices, and femtocell users update transmit powers to maximize their individual utilities under those prices.The MBS seeks revenue from selling interference quota within its tolerable aggregate interference margin.
  • A. Stackelberg Game Formulation: The MBS revenue depends on the interference-price vector µ and received interference powers I_i(p_i), with each user’s power selected in response to its price.The amount of interference quota purchased depends on the assigned price.
  • A. Stackelberg Game Formulation: Each femtocell utility combines transmission profit and interference cost: higher transmit power raises rate and profit but also increases MBS interference and payment.Femtocell users therefore solve power-allocation problems to maximize their own utilities.
  • A. Stackelberg Game Formulation: The Stackelberg game seeks a point where neither the MBS nor the femtocell users have incentives to deviate unilaterally.The leader’s optimization is based on the followers’ power responses.

B. Stackelberg Equilibrium

The paper characterizes Stackelberg equilibrium by solving the femtocell users’ noncooperative power-control subgame and then optimizing the MBS’s prices. It considers both user-specific and common interference prices.

  • Follower Subgame: The femtocell users form a noncooperative power-control subgame and seek a Nash equilibrium in which no user can improve utility unilaterally.This subgame supplies the followers’ responses needed by the MBS.
  • Equilibrium Solution: For a given price vector µ, the followers’ problem is solved first to obtain p*, after which the MBS problem is solved for the optimal price µ*.This backward solution procedure produces the Stackelberg equilibrium.
  • Pricing Schemes: Non-uniform pricing assigns each femtocell user a different interference price, whereas uniform pricing uses the same price for every user.Both schemes are studied under sparse and dense deployment models.

IV. SPARSELY DEPLOYED SCENARIO

The sparsely deployed scenario ignores mutual femtocell interference, simplifying the Stackelberg resource-allocation problem and enabling closed-form pricing and power-allocation solutions. The resulting equilibrium and successive-user-removal procedure determine prices according to the MBS interference margin.

  • Model and solution: Negligible mutual interference between femtocells simplifies the price-based resource-allocation problem.The model sets cross-femtocell interference to zero, enabling closed-form solutions for the Stackelberg game.
  • Pricing schemes: The analysis considers both non-uniform and uniform interference pricing schemes.The schemes are compared for implementation advantages, disadvantages, and suitable application conditions.
  • Non-uniform pricing: Closed-form expressions are obtained for the optimal interference price and power allocation in the sparse scenario.The equilibrium is represented by (µ∗, p∗), with the price from the optimal-price theorem and power from the user-allocation expression.
  • System design: An infinite interference price prevents a femtocell user from transmitting, while admitting all users requires Q to exceed T_N.The interference tolerance margin determines which users remain active under the price-based allocation.
  • Implementation: The successive-user-removal algorithm computes the optimal non-uniform interference-price vector from network-state information.The MBS collects channel and weighting information, computes thresholds, selects prices using Q, and returns them through backhaul links.
  • Implementation: The non-uniform implementation incurs high complexity and feedback overhead because the MBS must collect information for each femtocell user.This burden motivates the uniform-pricing approach, which requires less information.

B. Uniform Pricing

Uniform pricing assigns one interference price to all femtocell users and uses a distributed bargaining algorithm to meet the MBS interference constraint. The algorithm converges because total received interference decreases monotonically with price, while requiring substantially less exchanged information than centralized implementation.

  • Uniform pricing: Uniform pricing assigns the same interference price to every femtocell user.Each user computes its optimal transmit power from the received common price.
  • Equilibrium: The uniform-pricing Stackelberg equilibrium is unique and consists of the optimal price and power allocation.The equilibrium is denoted by (µ∗, p∗).
  • Distributed algorithm: The distributed bargaining algorithm updates the common price using total received interference relative to Q.The MBS increases or decreases the price by Δµ and broadcasts the new value until the interference constraint is met within ε.
  • Convergence: Algorithm 4.2 has guaranteed convergence because the target constraint is met with equality and its left-hand side decreases monotonically with µ.These properties support iterative price adjustment toward the unique optimal price.
  • Implementation: The distributed implementation requires the MBS to measure total received interference, while each femtocell user needs its own-HBS channel gain.The exchanged network information is greatly reduced compared with the centralized approach.

C. Non-Uniform Pricing vs. Uniform Pricing

The two pricing schemes optimize different objectives and impose different implementation requirements. Uniform pricing maximizes femtocell sum-rate and supports decentralized implementation, whereas non-uniform pricing maximizes MBS revenue but requires centralized implementation.

  • Implementation comparison: Non-uniform pricing requires centralized implementation, whereas uniform pricing can be implemented in a decentralized way.Uniform pricing is more favorable when network-state information is unavailable.
  • Objective trade-off: Non-uniform pricing maximizes MBS revenue, while uniform pricing maximizes femtocell sum-rate.The schemes therefore favor different system objectives.
  • Performance comparison: For a fixed interference constraint Q, uniform pricing maximizes the femtocell users’ sum-rate.

V. DENSELY DEPLOYED SCENARIO

The densely deployed scenario accounts for non-negligible mutual femtocell interference while retaining non-uniform and uniform pricing schemes. Worst-case and ideal-case analyses bound the MBS revenue, with the bounds converging as ε decreases to zero.

  • Dense deployment requires accounting for mutual interference between femtocells, while assuming each receiver’s aggregate interference remains bounded.
  • The MBS considers non-uniform pricing with user-specific interference prices and uniform pricing with one common price.
  • For a fixed price vector, each femtocell user’s power-selection problem is convex and the resulting noncooperative game has at least one Nash equilibrium.
  • In general, multiple Nash equilibria make obtaining the optimal power-allocation vector NP-hard.
  • The worst-case and ideal-case analyses provide lower and upper bounds, respectively, on the MBS’s maximum achievable revenue.
  • The two revenue bounds approach each other as ε decreases and coincide when ε = 0.

B. Uniform Pricing

Uniform pricing is evaluated through equilibrium analysis and numerical comparisons with non-uniform pricing. Non-uniform pricing generally favors MBS revenue, whereas uniform pricing generally favors femtocell-user sum-rate and requires less information exchange.

  • B. Uniform Pricing: Under uniform pricing, all femtocell users face the same interference price, and their optimal power allocation is obtained from the corresponding noncooperative-game formulation.
  • B. Uniform Pricing: The dense-scenario analysis reuses worst-case and ideal-case methods from the sparse scenario, but optimal power allocation remains NP-hard in general.
  • B. Uniform Pricing: For the same maximum tolerable interference margin Q, non-uniform pricing generally yields higher MBS revenue, while uniform pricing generally yields higher femtocell-user sum-rate.
  • B. Uniform Pricing: When Q is sufficiently small, the two schemes produce equal MBS revenue and femtocell-user sum-rate because only one femtocell is active.
  • B. Uniform Pricing: When Q is very large, the MBS revenues under the two pricing schemes converge to the same value.

C. Example 3: Convergence Performance of Distributed Interference Price Bargaining Algorithm

The numerical section evaluates distributed interference-price bargaining and dense-deployment revenue bounds. Bisection accelerates convergence, while the ideal case provides the largest dense-scenario MBS revenue.

  • C. Example 3: Convergence Performance of Distributed Interference Price Bargaining Algorithm: The distributed bargaining algorithm converges for all tested Q values, with convergence speed increasing as Q increases.
  • C. Example 3: Convergence Performance of Distributed Interference Price Bargaining Algorithm: Bisection converges much faster than the simple subgradient-based method.
  • C. Example 3: Convergence Performance of Distributed Interference Price Bargaining Algorithm: The bargaining procedure has the MBS broadcast a trial price, receive users’ resulting powers, and update price bounds using measured aggregate interference.
  • C. Example 3: Convergence Performance of Distributed Interference Price Bargaining Algorithm: In dense deployment, the ideal case ε = 0 gives the largest MBS revenue among ε = 0, 0.5, and 2, confirming it as a revenue upper bound.
  • C. Example 3: Convergence Performance of Distributed Interference Price Bargaining Algorithm: For ε = 0, 0.5, and 2, MBS revenue increases with Q; at fixed Q, revenue also increases as ε decreases.
  • VII. CONCLUSION: The proposed resource-allocation framework uses Stackelberg games with an MBS interference-power constraint to jointly study MBS and femtocell-user utility maximization.

APPENDIX

The appendix establishes optimization properties and conditions underlying the pricing solutions. It uses convex duality, KKT conditions, and threshold-based arguments to characterize optimal interference prices.

  • APPENDIX: Problem 4.4 is convex, so its zero dual gap permits solution through the dual optimization problem.
  • APPENDIX: The appendix constructs the Lagrangian, dual function, dual problem, and KKT conditions for Problem 4.4.
  • APPENDIX: The proof derives constraints on interference prices using nonnegative dual variables and contradiction arguments.
  • APPENDIX: The optimal interference-price solution is characterized by substituting derived expressions into the relevant inequalities and conditions.
  • APPENDIX: For threshold intervals, the appendix shows that the candidate price vector is optimal only when Q exceeds the specified threshold T_N.

C. Proof of Proposition 4.3

The proof reformulates the sum-rate problem through convex duality and dual decomposition, showing that the dual variable plays the same role as the uniform interference price. It then establishes convergence of the optimal dual price to the Stackelberg equilibrium price.

  • The femtocell sum-rate maximization is formulated as a convex optimization problem with a Lagrangian for the interference constraint.
  • The dual function decomposes into independent subproblems, one for each femtocell user.This decomposition enables distributed power-allocation analysis.
  • The nonnegative dual variable ν has the same role as the uniform interference price µ.
  • When the interference constraint is met with equality, the optimal dual variable converges to the equilibrium uniform price µ*.
  • The section concludes the proof of Proposition 4.3.
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