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Energy Efficient User Association and Power Allocation in Millimeter Wave Based Ultra Dense Networks with Energy Harvesting Base Stations

H. Zhang, S. Huang, C. Jiang, K. Long, V. C. M. Leung, H. Vincent Poor

arXiv:1704.07037v1cs.IT

TL;DR

The paper addresses joint user association and power allocation in mmWave ultra dense networks subject to load-balancing, QoS, energy-harvesting, energy-efficiency, and cross-tier-interference constraints. It relaxes the mixed-integer formulation, applies Lagrangian dual decomposition, and proposes an iterative gradient algorithm. Simulations report substantially higher energy efficiency than MAX-SINR and faster convergence to a near-optimal solution.

  • Problem

    User association and power allocation must jointly handle load balancing, QoS, energy harvesting, energy efficiency, and cross-tier interference in mmWave ultra dense networks.

  • Method

    The mixed-integer problem is relaxed into a convex optimization problem and solved using Lagrangian dual decomposition with iterative gradient updates.

  • Results

    10 times: user energy efficiency of the proposed algorithm versus MAX-SINR; the gradient method also reaches its optimal point in approximately 20 iterations versus approximately 35.

  • Takeaways & Limitations

    The proposed association and power-allocation scheme improves network utility and energy efficiency while supporting load balancing and minimum user-rate requirements.

Abstract

from arXiv · show

Millimeter wave (mmWave) communication technologies have recently emerged as an attractive solution to meet the exponentially increasing demand on mobile data traffic. Moreover, ultra dense networks (UDNs) combined with mmWave technology are expected to increase both energy efficiency and spectral efficiency. In this paper, user association and power allocation in mmWave based UDNs is considered with attention to load balance constraints, energy harvesting by base stations, user quality of service requirements, energy efficiency, and cross-tier interference limits. The joint user association and power optimization problem is modeled as a mixed-integer programming problem, which is then transformed into a convex optimization problem by relaxing the user association indicator and solved by Lagrangian dual decomposition. An iterative gradient user association and power allocation algorithm is proposed and shown to converge rapidly to an optimal point. The complexity of the proposed algorithm is analyzed and the effectiveness of the proposed scheme compared with existing methods is verified by simulations.

I. INTRODUCTION

The paper targets user association and power allocation in mmWave ultra dense networks under load-balancing, QoS, energy-harvesting, energy-efficiency, and interference constraints. It formulates and solves this joint optimization using relaxation, Lagrangian dual decomposition, and an iterative gradient approach.

  • Motivation: Ultra dense networks use closely deployed low-power small cells to improve network performance, reduce deployment cost, mitigate interference, and address blind spots.The paper associates these deployments with improved energy efficiency and load balancing.
  • Challenges: High deployment density makes radio resource allocation, user association, and interference mitigation challenging because cross-tier and local-tier interference cannot be ignored.These interference sources influence association rules and power allocation between users and base stations.
  • Load balancing: Load-balancing is central because SINR-based association can create serious imbalance, whereas load awareness transfers users from congested macrocells to lightly loaded small cells.Uneven power allocation and heterogeneous base-station capabilities can produce different user experiences even with uniform association.
  • Energy harvesting: RF energy harvesting enables wireless signals to transmit information and energy simultaneously through simultaneous wireless information and power transfer.The paper treats energy harvesting as a resource for devices with limited energy resources.
  • Contributions: The proposed framework jointly considers load balancing, QoS, energy efficiency, base-station energy harvesting, and cross-tier interference limits in mmWave ultra dense networks.The optimization is formulated as a mixed-integer user-association and power-allocation problem.
  • Solution approach: Relaxing the association indicators converts the nonconvex formulation into a convex optimization problem solved by Lagrangian dual decomposition and an iterative gradient algorithm.The method updates transmit power and association, with Newton-Raphson updates for Lagrange multipliers and rapid convergence toward an optimal point.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models downlink user association and transmit-power allocation in an ultra dense mmWave network where each user selects one base station. Achievable rates depend on power gain, bandwidth, interference-plus-noise, and the number of users sharing each base station.

  • A. System Model: The network contains ultra dense small cells overlaid on one macrocell, with sets of base stations and distributed users.The paper focuses on downlink user association and transmit-power allocation.
  • A. System Model: Each user associates with only one base station, represented by binary variable xij, which equals 1 for the selected base station and 0 otherwise.This single-base-station assumption avoids the added complexity of multiple-base-station association.
  • A. System Model: The channel power gain gij is modeled using the Friis transmission equation.The model includes antenna gains, wavelength, distance, a far-field reference distance, and path-loss exponent η ∈[2, 6].
  • A. System Model: The SINR expression uses transmit power pij, channel gain gij, interference from other base stations, and AWGN variance σ2.These quantities determine the received signal quality for user i from base station j.
  • A. System Model: Each user associated with base station j receives 1/Kj of its available frequency band, where Kj is the number of associated users.The allocation captures the effect of base-station load on individual user rates.

B. Problem Formulation

The problem jointly optimizes user association and power allocation in a mmWave ultra dense network while accounting for load balance, QoS, energy harvesting, energy efficiency, and cross-tier interference.

  • Each user is constrained to associate with only one base station at a time.
  • Cross-tier interference is bounded by I_j, enabling interference coordination that can adapt to traffic-load conditions.The passage links dynamic adjustment of this mechanism to spectral-efficiency and energy-efficiency improvement.
  • The objective focuses on user association and power allocation because bandwidth is assumed uniformly divided among associated users.This narrows the resource-allocation problem to power control and association for load balance and energy efficiency.
  • Each base station is equipped with energy harvesting, so received energy replenishes a rechargeable battery and contributes to net power consumption.The association matrix X and power matrix P determine the network’s consumption and harvested-energy terms.
  • The resulting optimization problem is defined with constraints C1–C7 covering power consumption, association, user counts, transmit power, QoS, variable ranges, and interference.
  • The formulation includes total power, maximum transmit power, minimum data-rate, associated-user-count, and cross-tier interference constraints.The cross-tier constraint limits interference, while the QoS constraint ensures each user meets its achievable-rate requirement.

III. LAGRANGIAN DUAL DECOMPOSITION

The original formulation is a high-complexity mixed-integer optimization because its objective is nonconvex and its association variables are binary. Relaxing those variables produces a tractable relaxed problem for dual decomposition.

  • The mixed-integer problem has high complexity due to nonconvexity and binary association variables.
  • The association variables x_ij are relaxed to nonnegative values while the user-count variables K_j remain bounded by the number of users.
  • Lagrangian dual decomposition is applied to the relaxed problem using multipliers for its coupled constraints.The multipliers are grouped as μ, λ, ν, and τ.
  • The relaxed formulation is expressed through a Lagrangian function and its corresponding dual function.

A. Dual Decomposition

Lagrangian dual decomposition separates the relaxed optimization into association/power and multiplier subproblems, with iterative criteria and subgradient updates driving convergence.

  • The original problem is divided into two independent subproblems through Lagrangian dual decomposition.One subproblem concerns X and P, while the other concerns the multipliers μ, λ, ν, and τ.
  • The association maximizer evaluates each base station by a user’s service or network-utility criterion at the inner iteration.The criterion contains the achievable-rate term log2(1 + SINR_ij).
  • K_j represents an optimal association scheme that determines how many users base station j should serve.
  • Subgradient updates adjust the Lagrange multipliers using step sizes δ1(t) through δ4(t).
  • When the multipliers converge, the dual problem reaches the global optimum.
  • The multiplier μ_j acts as a base-station price: excess demand raises the price, encouraging users to compare payoffs across stations.Overloaded base stations increase their prices to balance supply and demand.

B. Energy Efficiency and Power Allocation

Power allocation is solved with Newton–Raphson updates under the association decision and QoS constraints. Updating transmit power changes net consumption and supports near-optimal load balancing.

  • The power optimization problem is solved using the Newton–Raphson method.
  • After a user associates with base station j, x_ij is treated as the constant value 1 when deriving the power updates.
  • The Newton–Raphson treatment focuses only on the two-dimensional case because introducing the Hessian matrix increases computational complexity.
  • The transmit-power variable p_ij is iteratively updated using a step size δ4(t) and an increment Δp_ij.
  • The QoS constraint provides the minimum transmit power needed to satisfy each user’s required rate.
  • Updating transmit power changes net power consumption and, after convergence, yields a near-optimal load-balancing situation.

C. Iterative Gradient Algorithm

The paper proposes an iterative gradient algorithm for user association and power allocation, alternating association updates with power optimization under the problem constraints. The method is designed for rapid convergence and practical feasibility.

  • The iterative gradient algorithm finds an optimal user association solution under the constraints in (13).
  • User association: Association updates calculate power consumption, select j∗, update x_ij, and adjust the Lagrange multipliers.
  • Power allocation: Power allocation updates the gradient step and transmit powers using an optimized step size until convergence or Imax iterations.
  • The algorithm alternates between fixing p_ij while updating multipliers and updating p_ij, yielding relatively fast convergence and practical feasibility.

D. Complexity Analysis

Each iteration performs user association and power allocation over users and base stations, with both stages requiring O(B × U) operations. If convergence takes I iterations, total complexity scales with those repeated updates.

  • User association and power allocation each require O(B × U) operations per iteration.
  • The algorithm’s total computational cost depends on the number I of iterations required for convergence.

IV. SIMULATION RESULTS AND DISCUSSION

Simulations compare the proposed gradient association with MAX-SINR in an ultra dense network. The proposed method balances users across macrocell and small cells while reducing macrocell load pressure and improving network energy efficiency.

  • Simulation setup: The simulation uses an ultra dense network with 1500 small cells and 6000 users per macrocell, a 100 m macrocell radius, and −134 dBm noise power.
  • Simulation setup: The energy harvester efficiency is set to ψ = 0.8.
  • Association comparison: MAX-SINR associates many users with the higher-power macrocell, producing a seriously unbalanced load.
  • Association comparison: The proposed gradient algorithm assigns equal user proportions to the macrocell and each small cell, promoting load balancing and energy efficiency.
  • Performance comparison: Figures 3 and 4 compare performance from the user and small-cell perspectives, respectively.

SINR ·

The proposed gradient-based association and power-allocation scheme improves energy efficiency, balances cell loads, and converges rapidly while maintaining user-rate requirements, including under mmWave blockage.

  • Energy efficiency: 10 times: User energy efficiency under the proposed algorithm is 10 times that of MAX-SINR.User energy efficiency is mostly distributed between 0.5×10^15 and 3×10^15 bits/Joule, versus 0.3×10^14 to 1.8×10^14 bits/Joule for MAX-SINR.
  • Energy efficiency: Almost 10 times: The gradient algorithm achieves nearly tenfold higher accumulated energy efficiency than MAX-SINR.The comparison is attributed to coordinating system load with users’ minimum service rates, whereas MAX-SINR prioritizes user rate without load balancing.
  • Convergence: Approximately 20 versus 35 iterations: The proposed gradient method reaches its optimal point faster than MAX-SINR.The gradient method also has higher energy efficiency at each iteration, while MAX-SINR’s energy efficiency gradually decreases as iterations increase.
  • User rates: 6.44×10^9 to 6.48×10^9 bits/s/Hz: Most user rates under the proposed algorithm fall in this range and satisfy the minimum user-rate requirement.MAX-SINR user rates range from 2.9×10^10 to 3.4×10^10 bits/s/Hz, while the proposed method provides relatively constant rates.
  • Blockage effects: Approximately 20 iterations: With blockage included, the proposed algorithm still converges rapidly, whereas MAX-SINR appears not to converge.At 1500 base stations and 13 dBi antenna gain, the proposed algorithm’s energy efficiency is about 9 times that of MAX-SINR.

V. CONCLUSION

The paper formulates joint user association and power allocation for energy-harvesting mmWave ultra dense networks under interference, power, and QoS constraints. Its decomposed gradient-based solution improves network utility, energy efficiency, and user experience in simulations.

  • The proposed mechanism coordinates interference between base stations and users while optimizing network utility under power and QoS constraints.
  • Lagrangian dual decomposition separates the dual optimization into two subproblems that can be solved independently.
  • The algorithm outperforms MAX-SINR, while load-balancing and price-control association improve network utility and energy efficiency.
  • Pricing rules provide satisfactory user experience and user rates under the minimum user-rate constraint.
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