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Throughput Maximization in Wireless Powered Communication Networks

Hyunsgsik Ju, Rui Zhang

arXiv:1304.7886v4cs.IT

TL;DR

The paper addresses throughput allocation in WPCNs where energy-constrained users rely on an H-AP for communication. It proposes harvest-then-transmit and optimizes sum-throughput, revealing a doubly near-far problem; common-throughput optimization instead enforces equal user rates and is shown effective in simulations.

  • Problem

    WPCNs must allocate wireless energy and uplink transmission time among users without other energy sources while addressing distance-induced unfairness.

  • Method

    The paper proposes harvest-then-transmit and develops time-allocation optimization for sum-throughput and an efficient common-throughput algorithm.

  • Results

    The analysis reveals a doubly near-far problem in which sum-throughput maximization causes notably unfair time and throughput allocation among users.

  • Takeaways & Limitations

    Common-throughput maximization allocates equal rates to all users and is shown effective for addressing the doubly near-far problem in WPCNs.

Abstract

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This paper studies the newly emerging wireless powered communication network (WPCN) in which one hybrid access point (H-AP) with constant power supply coordinates the wireless energy/information transmissions to/from distributed users that do not have energy sources. A "harvest-then-transmit" protocol is proposed where all users first harvest the wireless energy broadcast by the H-AP in the downlink (DL) and then send their independent information to the H-AP in the uplink (UL) by time-division-multiple-access (TDMA). First, we study the sum-throughput maximization of all users by jointly optimizing the time allocation for the DL wireless power transfer versus the users' UL information transmissions given a total time constraint based on the users' DL and UL channels as well as their average harvested energy values. By applying convex optimization techniques, we obtain the closed-form expressions for the optimal time allocations to maximize the sum-throughput. Our solution reveals "doubly near-far" phenomenon due to both the DL and UL distance-dependent signal attenuation, where a far user from the H-AP, which receives less wireless energy than a nearer user in the DL, has to transmit with more power in the UL for reliable information transmission. Consequently, the maximum sum-throughput is achieved by allocating substantially more time to the near users than the far users, thus resulting in unfair rate allocation among different users. To overcome this problem, we furthermore propose a new performance metric so-called common-throughput with the additional constraint that all users should be allocated with an equal rate regardless of their distances to the H-AP. We present an efficient algorithm to solve the common-throughput maximization problem. Simulation results demonstrate the effectiveness of the common-throughput approach for solving the new doubly near-far problem in WPCNs.

I. INTRODUCTION

The paper introduces a WPCN in which an H-AP powers energy-limited users for uplink communication, then optimizes time allocation for throughput and fairness. Its results identify a doubly near-far problem and motivate equal-rate common-throughput maximization.

  • System and protocol: The proposed WPCN uses one constant-power H-AP to coordinate wireless energy and information transmissions for distributed users without other energy sources.Users harvest and store H-AP energy in rechargeable batteries.
  • System and protocol: The harvest-then-transmit protocol has the H-AP broadcast energy in the DL before users send independent information in the UL through TDMA.The protocol separates wireless energy transfer from uplink wireless information transmission.
  • Throughput optimization: Sum-throughput maximization jointly optimizes DL WET and UL WIT time under a total-time constraint using users’ DL and UL channels and harvested energy.The resulting convex problem yields closed-form optimal time allocations.
  • Doubly near-far problem: The doubly near-far phenomenon arises because distance-dependent attenuation affects both DL energy harvesting and UL information transmission.Far users harvest less energy yet require more UL transmit power to achieve the same information rate.
  • Doubly near-far problem: Sum-throughput maximization allocates substantially more time to near users than far users, producing unfair achievable rates.This exposes a throughput-fairness trade-off in the WPCN.
  • Common-throughput optimization: Common-throughput maximization re-optimizes DL and UL time allocation while constraining all users to receive equal rates regardless of distance.The paper presents an efficient algorithm and uses the metric to address the doubly near-far problem.

II. SYSTEM MODEL

The system uses a harvest-then-transmit protocol in which the H-AP broadcasts wireless energy in the DL, followed by TDMA information transmission by energy-constrained users in the UL. User throughput depends on jointly allocated DL harvesting and UL transmission time.

  • Network and assumptions: The network contains one H-AP and K users without embedded energy sources, so users must harvest energy from received DL signals.The H-AP and user terminals are each equipped with one antenna.
  • Protocol: Each transmission block begins with DL wireless energy transfer from the H-AP and continues with UL information transmission by users using TDMA.The DL occupies τ0T, while user Ui receives τiT for UL transmission.
  • Energy model: User Ui harvests energy according to Ei = ζiPAhiτ0 and uses a fixed portion ηi of that energy for UL information transmission.The paper subsequently assumes equal harvesting efficiencies and ηi = 1 for all users.
  • Throughput model: The achievable UL throughput Ri depends on DL and UL time allocation, with Ri increasing in either τ0 or τi when the other is fixed.The two time allocations cannot both increase under the total block-time constraint.

III. SUM-THROUGHPUT MAXIMIZATION

The paper formulates sum-throughput maximization over DL and UL time allocations and solves it through convex optimization, obtaining an optimal allocation characterized by a unique scalar solution. The resulting allocation favors near users and exposes a doubly near-far problem that produces highly unequal throughputs.

  • Problem formulation: The sum-throughput maximization problem optimizes DL WET and users’ UL WIT times under the total time constraint.The individual throughput functions are concave, making their sum suitable for convex optimization techniques.
  • Allocation behavior: Greater channel power gains increase the UL WIT allocations and reduce the DL WET time needed for sum-throughput maximization.Larger γi values imply less energy is required for a given transmission rate, allowing users to harvest sufficient energy with less DL time.
  • Doubly near-far problem: The solution allocates more time to near users than far users, creating an unfair allocation called the doubly near-far problem.The effect reflects distance-dependent attenuation in both the DL energy-transfer and UL information-transmission channels.
  • Numerical illustration: τ* = [0.2441, 0.7114, 0.0445] and Rsum(τ*) = 4.58 bps/Hz in the illustrated two-user network.At this allocation, R1(τ*) = 4.13 bps/Hz and R2(τ*) = 0.45 bps/Hz, demonstrating highly unequal user throughputs.
  • Comparison with conventional TDMA: The WPCN’s throughput ratio between the two users decreases twice faster than the corresponding ratio in conventional TDMA on the logarithm scale.The comparison uses conventional TDMA with equal energy supply and attributes the stronger disparity to the doubly near-far problem.

IV. COMMON-THROUGHPUT MAXIMIZATION

The paper formulates common-throughput maximization to allocate equal rates despite users’ unequal channels and develops an efficient convex-optimization-based algorithm. Simulations show that this approach improves fairness by shifting transmission time toward far users, while tracing the throughput–fairness trade-off.

  • Problem formulation: Common-throughput maximization allocates equal throughput to all users regardless of their distances from the H-AP.The formulation targets the worst-channel user and enforces equal-rate allocation.
  • Problem formulation: The common-throughput problem is formulated by maximizing a feasible common rate subject to users’ rate inequalities and time-allocation constraints.For a given candidate common rate, the paper first solves a convex feasibility problem.
  • Solution method: For a given dual vector, the weighted sum-throughput subproblem is convex and yields an optimal time allocation through equations involving a positive constant.The algorithm iteratively updates per-user variables and the scalar parameter until convergence.
  • Solution method: The algorithm tests feasibility using the dual function, updates dual variables with a subgradient method, and searches over the common rate by bisection.If the optimized dual function is positive, the candidate common rate is infeasible; otherwise the rate can be increased.
  • Numerical illustration: In the two-user example, the optimal allocation is τ* = [0.3683, 0.1386, 0.4932] and the maximum common-throughput is 1.46 bps/Hz for both users.Compared with sum-throughput maximization, common-throughput reduces near-user time and increases far-user time to equalize rates.
  • Fairness trade-off: The weighted-throughput formulation characterizes the achievable throughput region and includes maximum sum-throughput and maximum common-throughput as extreme fairness trade-offs.The common-throughput approach is also identified as a special case of the rate-profile method.

V. SIMULATION RESULT

Simulations compare sum-throughput, common-throughput, and heuristic equal-time allocation under varying transmit power, pathloss exponent, and user count. The results show that common-throughput substantially improves fairness but can reduce aggregate throughput, especially as pathloss increases.

  • Transmit power: When sum-throughput is maximized, the near user dominates the far user, producing notably unfair rate allocation in the two-user WPCN.The comparison uses K = 2, distances of 5 m and 10 m, α = 2, and averages over 1000 fading realizations.
  • Transmit power: Maximum common-throughput is smaller than normalized maximum sum-throughput by the number of users, representing the cost of strict fairness.For K = 2, equal-rate allocation is enforced regardless of user distance from the H-AP.
  • Pathloss exponent: As the pathloss exponent increases, the near user approaches the maximum sum-throughput while the far user’s throughput approaches zero under sum-throughput maximization.The resulting rate imbalance becomes more severe because of the doubly near-far problem.
  • Pathloss exponent: Maximum common-throughput decreases faster with increasing pathloss exponent than normalized maximum sum-throughput.Common-throughput allocates more time to the far user as the channel-gain ratio between near and far users increases.
  • Number of users: Both normalized maximum sum-throughput and maximum common-throughput decrease as the number of users increases.The proposed optimizations outperform the corresponding heuristic equal-time allocation benchmarks.

VI. CONCLUSION

The paper studies WPCNs using harvest-then-transmit TDMA and identifies a doubly near-far problem caused by attenuation in both downlink energy transfer and uplink information transfer. It proposes common-throughput maximization to equalize user rates, which is effective but reduces sum-throughput.

  • System and protocol: The WPCN uses a harvest-then-transmit protocol in which the H-AP broadcasts energy downlink before users transmit information uplink by TDMA.Users do not have independent energy sources.
  • Main finding: The doubly near-far problem results from signal attenuation in both downlink WET and uplink WIT.Maximizing conventional sum-throughput consequently creates unfair time and throughput allocation among users.
  • Proposed remedy: Common-throughput maximization allocates equal rates to users regardless of their distances from the H-AP.The proposed allocation assigns transmission time inversely proportional to users’ distances.
  • Conclusion: Simulation results show that common-throughput is effective for addressing the doubly near-far problem, but at the cost of sum-throughput degradation.

PROOF OF LEMMA 3.1

The proof establishes that each user’s throughput is concave in the time-allocation vector by showing that its Hessian is negative semidefinite.

  • Hessian analysis: The Hessian entries of each user-throughput function are derived from the throughput expression, including diagonal and off-diagonal terms.
  • Concavity conclusion: Because the time allocations satisfy τ_i ≥ 0, the Hessian of each throughput function is negative semidefinite.Therefore, each user-throughput function is concave in the time-allocation vector.

PROOF OF LEMMA 3.2

The proof establishes that f(z) is convex, minimized at z = 1, and nonnegative and increasing for z ≥ 1. Consequently, f(z) = A has a unique solution z∗ > 1 when A > 0.

  • f(z) is convex for z ≥ 0 and attains its minimum at z = 1, where f(1) = 0.
  • For z ≥ 1, f(z) is nonnegative and monotonically increasing.
  • When A > 0, the equation f(z) = A has a unique solution z∗ > 1.

PROOF OF PROPOSITION 3.1

The proof applies convex optimization, strong duality, and KKT conditions to characterize the optimal time allocation for problem (P1). Monotonicity arguments then establish the relevant allocation expressions and complete Proposition 3.1.

  • Problem (P1) is convex, and Slater’s condition ensures strong duality.
  • Because strong duality holds, the KKT conditions are necessary and sufficient for global optimality.
  • The function t(x) is monotonically increasing for x ≥ 0 because its derivative is nonnegative.
  • Under the stated conditions, there is a unique z∗ > 1 solving (38), which determines the optimal DL WET time allocation.
  • The optimal UL WIT time allocations follow from equations (36) and (39), completing the proposition.

PROOF OF COROLLARY 3.1

The proof shows that the optimal DL WET time decreases monotonically as A increases, based on the increasing behavior of the associated solution z∗.

  • As A increases, z∗ increases, causing both z∗ln z∗ and z∗ − 1 to increase.
  • Because z∗ln z∗ increases faster than z∗ − 1, the optimal DL WET time decreases monotonically with A.
  • This monotonicity result completes Corollary 3.1.

PROOF OF LEMMA 4.1

The proof establishes a feasibility characterization for problem (12): feasibility is equivalent to G(λ) ≤ 0 for every λ ≥ 0. It then uses convexity, KKT conditions, and variable transformations to derive the associated optimality relations.

  • If a feasible allocation satisfies R_i(τ′) ≥ R̄ for every user, then G(λ) ≤ 0 for all λ ≥ 0.
  • Thus, problem (12) is feasible if and only if G(λ) ≤ 0 for every λ ≥ 0.
  • Conversely, if problem (12) is feasible, assuming G(λ′′) > 0 for some λ′′ ≥ 0 leads to a contradiction.
  • Problem (15) is convex with zero duality gap, so its optimal primal and dual solutions satisfy the KKT conditions.
  • The variables z_1, …, z_K and μ∗ are uniquely determined by K + 1 independent equations because the relevant logarithmic function is monotonically increasing.
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