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Downlink Energy Efficiency of Power Allocation and Wireless Backhaul Bandwidth Allocation in Heterogeneous Small Cell Networks
Haijun Zhang, Hao Liu, Julian Cheng, Victor C. M. Leung
TL;DR
Wireless backhaul bandwidth allocation in heterogeneous small-cell networks had not been investigated. This paper decomposes the energy-efficiency problem and develops iterative and low-complexity algorithms, with simulations showing the iterative approach improves energy efficiency over the alternatives.
Problem
Wireless backhaul bandwidth allocation in heterogeneous small-cell networks had not been investigated.
Method
The paper decomposes the resource-allocation problem and develops near-optimal iterative and low-complexity suboptimal algorithms.
Results
The proposed iterative optimization algorithm improves energy efficiency more than the low-complexity algorithm and existing schemes.
Takeaways & Limitations
The paper identifies a unique globally optimal energy-efficiency solution and provides an iterative algorithm to obtain this optimum.
Abstract
from arXiv · showhide
The widespread application of wireless services and dense devices access have triggered huge energy consumption. Because of the environmental and financial considerations, energy-efficient design in wireless networks becomes an inevitable trend. To the best of the authors' knowledge, energy-efficient orthogonal frequency division multiple access heterogeneous small cell optimization comprehensively considering energy efficiency maximization, power allocation, wireless backhaul bandwidth allocation, and user Quality of Service is a novel approach and research direction, and it has not been investigated. In this paper, we study the energy-efficient power allocation and wireless backhaul bandwidth allocation in orthogonal frequency division multiple access heterogeneous small cell networks. Different from the existing resource allocation schemes that maximize the throughput, the studied scheme maximizes energy efficiency by allocating both transmit power of each small cell base station to users and bandwidth for backhauling, according to the channel state information and the circuit power consumption. The problem is first formulated as a non-convex nonlinear programming problem and then it is decomposed into two convex subproblems. A near optimal iterative resource allocation algorithm is designed to solve the resource allocation problem. A suboptimal low-complexity approach is also developed by exploring the inherent structure and property of the energy-efficient design. Simulation results demonstrate the effectiveness of the proposed algorithms by comparing with the existing schemes.
I. INTRODUCTION
The paper introduces an energy-efficient OFDMA heterogeneous small-cell optimization that jointly considers power allocation, wireless backhaul bandwidth allocation, and user QoS. It formulates the problem as nonlinear programming, decomposes it into convex subproblems, and develops optimal and suboptimal solution approaches.
- Research problem: The paper addresses the previously uninvestigated joint design of energy-efficient power allocation and wireless backhaul bandwidth allocation in heterogeneous small-cell networks.The small cells use wireless backhauling to maximize the energy efficiency of all small-cell users.
- Paper extension: The journal version extends the conference version with a detailed theorem proof, complexity analysis, and additional simulation results.The paper identifies these additions as differences from the conference version.
- Problem formulation: The proposed OFDMA optimization jointly considers energy-efficiency maximization, transmit power allocation, wireless backhaul bandwidth allocation, and user QoS.The formulation includes maximum transmit-power, small-cell downlink-rate, and minimum user data-rate constraints.
- Solution approach: The resulting nonlinear programming problem is decomposed into two convex subproblems, enabling an algorithm for wireless backhaul bandwidth and power allocation.The formulation is intended to support reliable downlink transmission with low energy consumption for small-cell users.
- Algorithms: The paper obtains an optimum solution for the bandwidth allocation subproblem and develops a suboptimal low-complexity algorithm for power allocation.The low-complexity approach decomposes the power-allocation problem.
II. SYSTEM MODEL
The system model considers a heterogeneous small-cell network with unified wireless-backhaul bandwidth allocation and OFDMA-based downlink power allocation. It formulates energy-efficiency maximization under transmit-power, data-rate, backhaul-capacity, and QoS constraints.
- Network configuration: The network comprises one macro BS, J small cells within its coverage, and K randomly located users per small cell sharing the macrocell spectrum.Each small cell uses a single antenna, while OFDMA supports communication between each small cell and its users.
- Backhaul allocation: A unified bandwidth factor β ∈[0, 1] specifies the fraction of bandwidth allocated to wireless backhauling at all small-cell BSs.All small cells are assumed to use the same bandwidth allocation factor.
- Downlink transmission: The model defines user rates from small-cell transmit powers and channel gains while treating inter-small-cell interference as thermal noise because of wall penetration loss and low transmit power.Macro-BS interference is represented through Ij,k = P0Gj,k.
- Energy-efficiency metric: Energy efficiency accounts for both transmit power pj,k and circuit power PC, with each small-cell link consuming total power PC + pj,k.Circuit power is independent of transmission state and represents device-electronics consumption during transmission.
- Optimization problem: The optimization maximizes energy efficiency through power and unified backhaul-bandwidth allocation subject to power, downlink-rate, backhaul, and heterogeneous QoS requirements.The resulting downlink formulation is a nonlinear programming problem.
III. ENERGY-EFFICIENT RESOURCE ALLOCATION AND BACKHAULING
The formulated energy-efficient resource allocation problem is non-convex, but the separability of β and p_j,k enables its decomposition into two convex subproblems for power allocation and wireless backhaul bandwidth allocation.
- The formulated optimization problem is non-convex.
- The continuous variables β and p_j,k are separable in the formulation, enabling a decomposition approach.
- The problem is decomposed into two convex subproblems: energy-efficient power allocation and energy-efficient wireless backhaul bandwidth allocation.
A. Energy-Efficient Power Allocation
Given a unified wireless backhaul bandwidth allocation value β, the power-allocation subproblem maximizes energy efficiency. Strict concavity and quasiconcavity establish a unique globally optimal transmission-rate vector, solvable by theorem-based or low-complexity iterative methods.
- The optimal energy-efficient power allocation achieves the maximum energy efficiency.
- If rj,k(pj,k) is strictly concave, each Uj,k(pj,k) is strictly quasiconcave.
- A unique globally optimal transmission rate vector exists when rj,k(pj,k) is strictly concave.
- The power-allocation problem can be solved using Theorem 1 or low-complexity iterative algorithms based on GABS.
B. Energy-Efficient Wireless Backhaul Bandwidth Allocation
The wireless backhaul bandwidth allocation is formulated through subproblem P1.3, with the original problem solved by iterating P1.2 and P1.3 until convergence. Because the objective decreases monotonically with β, P1.3 becomes a feasibility problem seeking the smallest feasible β.
- Bandwidth allocation: Subproblem P1.3 addresses unified wireless backhaul bandwidth allocation using functions evaluated at the optimal power allocation P∗.Rj(β, P∗) and Cj(β, P∗) denote the corresponding function values evaluated at P∗.
- Iterative solution: The original problem is solved by iteratively solving subproblems P1.2 and P1.3 until convergence.
- Feasibility formulation: P1.3 reduces to a feasibility problem because its objective is equivalent to maximizing (1 −β), a monotonically decreasing function of β.Its solution is the smallest feasible value of β satisfying constraints (34).
IV. ALGORITHM DESIGN
The algorithm design builds on the analysis of power allocation and wireless backhaul bandwidth allocation. It proposes an iterative optimization algorithm alongside a suboptimal low-complexity approach.
- The design follows the analysis of power allocation and wireless backhaul bandwidth allocation.
- An iterative optimization algorithm is proposed for the resource-allocation problem.
- A suboptimal low-complexity approach is also developed.
A. Iterative Resource Allocation Algorithm · B. Low-Complexity Optimization Algorithm
The iterative algorithm alternates backhaul bandwidth and user power allocation, with the macro base station selecting and broadcasting the optimal bandwidth factor. A lower-complexity variant fixes β from equal-power allocation before computing power allocation.
- A. Iterative Resource Allocation Algorithm: Algorithm 1 is presented as the proposed iterative resource allocation algorithm for jointly processing the resource-allocation steps.The algorithm advances through iterations by updating l.
- A. Iterative Resource Allocation Algorithm: Algorithm 1 initializes each small cell base station with equal positive transmit power per user and iterates across resource-allocation steps.The iteration begins with pj,k > 0 and l = 1, then processes backhaul bandwidth allocation and small-cell users.
- A. Iterative Resource Allocation Algorithm: The iterative procedure computes optimum β according to (37), broadcasts the updated backhaul allocation factor, and checks power constraints during per-user processing.These operations connect macro-level backhaul coordination with small-cell user-level power allocation.
- A. Iterative Resource Allocation Algorithm: Each small cell base station calculates φj, sends it to the macro base station, and the macro base station selects the maximum φj as β.The selected β is broadcast to all small cell base stations.
- B. Low-Complexity Optimization Algorithm: To reduce Algorithm 1 complexity, the paper proposes a low-complexity optimization algorithm shown in Algorithm 2.This variant is explicitly introduced as a complexity-reduction approach.
- B. Low-Complexity Optimization Algorithm: The low-complexity algorithm calculates the bandwidth allocation factor from equal power allocation, fixes β, and then calculates power allocation using the Section IV scheme.Its procedure also initializes equal positive transmit power per user and checks the power constraint.
- B. Low-Complexity Optimization Algorithm: In Algorithm 2, the macro base station broadcasts the backhaul bandwidth allocation factor to all small cell base stations before processing each small cell user.The listed steps include backhaul bandwidth allocation, macro broadcasting, and nested small-cell base-station and user loops.
C. Complexity Analysis
Because the original nonconvex problem requires exhaustion for an optimal solution, its complexity grows exponentially with the numbers of small cells and users. The proposed algorithms are less complex: Algorithm 1 has total complexity O(JKΩ∆), while Algorithm 2 has O(JKΩ) because it omits iteration.
- Complexity of the method of exhaustion: The nonconvex formulation requires the method of exhaustion to obtain an optimal solution.The method evaluates power-allocation and bandwidth-allocation choices under the constraints.
- Complexity of the proposed algorithms: Algorithm 1 has total complexity O(JKΩ∆), where ∆ is the number of iterations required for convergence.Each iteration includes bandwidth-allocation and per-user power-allocation computations.
- Complexity of the method of exhaustion: The method of exhaustion has exponentially increasing complexity as J and K increase, making it much more complex than both proposed algorithms.J denotes the number of small cells, and K denotes the number of users in each small cell.
V. SIMULATION RESULTS
Simulations show that the proposed algorithms converge and consistently outperform the comparison methods in energy efficiency and downlink capacity, with gains increasing as users or small cells increase. Algorithm 1 approaches the optimal solution closely, while larger power constraints and user populations improve energy efficiency.
- Convergence: Algorithm 1 converges to stable energy-efficiency solutions in nearly 16 iterations.The evaluation uses J = 5, Rt = 0.01 bps/Hz, and Pmax = 20 dBm.
- Energy efficiency versus users: Algorithm 1 delivers 20% higher energy efficiency than Algorithm 2, and energy efficiency improves as the number of users per small cell increases.The comparison varies users per small cell from 2 to 10.
VI. CONCLUSION
The paper investigates energy-efficient wireless backhaul bandwidth and power allocation in heterogeneous small cell networks, establishing a unique globally optimal solution and algorithms for obtaining or approximating it. Simulations show improved energy efficiency and capacity with more small cells and users, while the low-complexity approach offers a performance–complexity tradeoff.
- The study addresses energy-efficient wireless backhaul bandwidth allocation and power allocation in heterogeneous small cell networks.
- A unique globally optimal energy efficiency solution exists, and an iterative algorithm obtains this optimum.
- For downlink operation, the authors develop a near optimal energy-efficient resource allocation approach and a low-complexity suboptimal algorithm based on the objective function’s structure.
- Increasing the number of small cells and users per small cell improves both energy efficiency and capacity.
- The iterative optimization algorithm improves energy efficiency over the low-complexity algorithm and existing schemes, while the low-complexity algorithms offer a promising performance–complexity tradeoff.
- Future work will investigate nonunified backhaul bandwidth allocation and inter-small-cell interference in heterogeneous small cell networks.
APPENDIX A PROOF OF LEMMA 1
The appendix proves that each user-level energy-efficiency function is strictly quasiconcave and increases before strictly decreasing, establishing Lemma 1. The proof uses strict concavity of the rate function and monotonicity of an associated derivative expression.
- Strict quasiconcavity: Uj,k(pj,k) is strictly quasiconcave because its upper-level sets Sα are strictly convex for all α.For α ≤ 0, convexity follows from the contour structure; for α > 0, strict concavity of rj,k(pj,k) makes −rj,k(pj,k) strictly convex.
- Strict quasiconcavity: Strict quasiconcavity of Uj,k(pj,k) follows from the strict convexity of Sα.The proof explicitly concludes strict quasiconcavity after establishing strict convexity of the upper-level sets.
- Derivative analysis: Because rj,k(pj,k) is strictly concave, f′(pj,k) < 0, so f(pj,k) is strictly decreasing.Here f(pj,k) = (PC + pj,k)r′j,k(pj,k) − rj,k(pj,k).
- Derivative analysis: Uj,k(pj,k) is first strictly increasing and then strictly decreasing in pj,k, yielding the stated Lemma 1.The appendix concludes the lemma after analyzing the derivative behavior and the existence of the relevant optimum.