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Spatial Throughput Maximization of Wireless Powered Communication Networks

Yue Ling Che, Lingjie Duan, Rui Zhang

arXiv:1409.3107v2cs.IT

TL;DR

The paper asks how to optimize energy harvesting and information transmission in large-scale WPCNs with uncertain RF energy and many randomly deployed nodes. It uses stochastic geometry and a harvest-then-transmit protocol to jointly optimize frame partition and transmit power under a transmission-success constraint. Battery-free nodes favor small transmit power, while infinite-capacity batteries make all feasible charging designs optimal and finite-capacity performance lies between the battery-free and infinite-capacity cases.

  • Problem

    Existing work provides limited optimal transmission design for large-scale WPCNs because scalable schemes and random energy-transfer conditions are difficult to analyze.

  • Method

    The paper models dual-function-AP WPCNs with stochastic geometry and optimizes DL/UL frame partition and node transmit power under a successful information-transmission constraint.

  • Results

    Battery-free nodes generally favor small transmit power, while infinite-capacity batteries make all feasible charging designs optimal; finite-capacity performance is bounded by the two cases.

  • Takeaways & Limitations

    Battery storage changes the energy–throughput tradeoff, with finite-capacity networks positioned between battery-free and infinite-capacity performance.

Abstract

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Wireless charging is a promising way to power wireless nodes' transmissions. This paper considers new dual-function access points (APs) which are able to support the energy/information transmission to/from wireless nodes. We focus on a large-scale wireless powered communication network (WPCN), and use stochastic geometry to analyze the wireless nodes' performance tradeoff between energy harvesting and information transmission. We study two cases with battery-free and battery-deployed wireless nodes. For both cases, we consider a harvest-then-transmit protocol by partitioning each time frame into a downlink (DL) phase for energy transfer, and an uplink (UL) phase for information transfer. By jointly optimizing frame partition between the two phases and the wireless nodes' transmit power, we maximize the wireless nodes' spatial throughput subject to a successful information transmission probability constraint. For the battery-free case, we show that the wireless nodes prefer to choose small transmit power to obtain large transmission opportunity. For the battery-deployed case, we first study an ideal infinite-capacity battery scenario for wireless nodes, and show that the optimal charging design is not unique, due to the sufficient energy stored in the battery. We then extend to the practical finite-capacity battery scenario. Although the exact performance is difficult to be obtained analytically, it is shown to be upper and lower bounded by those in the infinite-capacity battery scenario and the battery-free case, respectively. Finally, we provide numerical results to corroborate our study.

I. INTRODUCTION

The paper develops a stochastic-geometry framework for large-scale WPCNs with dual-function APs, optimizing the tradeoff between downlink energy harvesting and uplink information transmission. It analyzes battery-free and battery-deployed nodes under a scalable harvest-then-transmit protocol.

  • I. INTRODUCTION: Large-scale WPCNs with dual-function APs are studied using stochastic geometry to optimize bidirectional energy and information transfer.The framework covers moving nodes with fixed APs and moving APs with fixed nodes, whose DL and UL performance are identically characterized.
  • I. INTRODUCTION: Each frame is partitioned into a DL energy-transfer phase and a UL information-transfer phase in a scalable harvest-then-transmit protocol.The protocol transfers energy from APs to nodes before nodes transmit to associated APs.
  • I. INTRODUCTION: The optimization jointly selects the DL/UL frame partition and wireless-node transmit power subject to a successful information-transmission probability constraint.The formulation targets spatial throughput maximization and is simplified using an equivalent probability constraint.
  • I. INTRODUCTION: Battery-free nodes generally prefer small transmit power because it provides larger transmission opportunity.This result is obtained while studying the effects of AP and wireless-node densities.
  • I. INTRODUCTION: For battery-deployed nodes, all feasible frame partitions and UL transmit powers are optimal with infinite-capacity batteries because stored energy remains sufficient over time.For finite-capacity batteries, the paper proposes a tight lower bound and approximately solves the optimization problem.
  • I. INTRODUCTION: The work differs from prior stochastic-geometry studies by analyzing dual-function APs, DL/UL tradeoffs, battery storage, and long-term average performance.The paper reports tractable DL and UL system performance beyond snapshot-based average analysis.

1) Battery-free Case:

The network model represents APs and wireless nodes as stochastic spatial processes with RF energy harvesting, fading, and path loss. Harvested-energy distributions are characterized analytically, while an independence assumption enables tractable throughput analysis.

  • 1) Battery-free Case:: The model uses independent homogeneous PPPs for APs and wireless nodes, with either nodes or APs moving independently across frames.Type-I has moving nodes and fixed APs; Type-II has moving APs and fixed nodes.
  • 1) Battery-free Case:: AP transmissions undergo Rayleigh flat fading and path loss, with independently varying fading across time slots.Shadowing is ignored for tractable analysis.
  • 1) Battery-free Case:: The harvested energy is obtained by aggregating RF energy across DL slots, with energy-harvesting efficiency η ∈ (0, 1).The analysis focuses on a typical wireless node at the origin by stationarity.
  • 1) Battery-free Case:: Applying the PPP probability generating functional yields the Laplace transform and CCDF of harvested energy.The resulting characterization applies to both Type-I and Type-II network models.
  • 1) Battery-free Case:: Increasing the number of energy transmitters N increases the harvested-energy CCDF for any fixed z > 0.The analysis also notes an infinite average energy arrival rate under the singular path-loss model, without implying sufficient energy in every frame.
  • 1) Battery-free Case:: Although harvested energies are generally correlated across time and space, the paper assumes they are mutually independent to obtain tractable results.The assumption is motivated by weak correlations and later validated by simulation.

III. PERFORMANCE METRICS AND PROBLEM FORMULATION

The paper derives uplink transmission and success probabilities under stochastic-geometry assumptions, then formulates spatial-throughput maximization over DL allocation and UL transmit power. Lower transmission activity reduces interference and improves successful transmission probability.

  • III. PERFORMANCE METRICS AND PROBLEM FORMULATION: The analysis first characterizes UL information transmission and formulates spatial-throughput maximization using DL-slot allocation N and UL transmit power P_U.The system metric is the successful information-transmission probability of a typical node.
  • A. Successful Information Transmission Probability: Under independent scheduling, each transmitting node selects one of the T − N UL slots, producing identical active-node density across UL slots.This converts transmission activity into a per-slot spatial process.
  • A. Successful Information Transmission Probability: Assumption 1 makes active transmissions in each UL slot a homogeneous PPP with density λ_a.Without that assumption, correlations in harvested energy would generally prevent a PPP model.
  • A. Successful Information Transmission Probability: The typical node is associated with its nearest AP, whose distance has density f_r(r) = 2πλ_AP r e^(-λ_AP πr^2).Stationarity permits placing the associated AP at the origin for UL analysis.
  • A. Successful Information Transmission Probability: The successful information-transmission probability P_suc is derived for a generic transmission probability ρ using the PGFL of a PPP.A closed-form expression is available for α = 4.
  • A. Successful Information Transmission Probability: Decreasing ρ reduces the active-node density and interference, thereby increasing P_suc.The result applies for general α and for the α = 4 specialization.
  • A. Successful Information Transmission Probability: The same throughput-maximization formulation and solutions apply to both network models because their DL and UL performance is identically characterized.The paper therefore does not distinguish Type-I and Type-II models thereafter.

B. Spatial Throughput Maximization Problem

The paper formulates spatial-throughput maximization by balancing DL energy transfer and UL information transmission under a successful-transmission constraint. For battery-free nodes, the resulting optimization is simplified analytically and solved across AP-density regimes.

  • The optimization jointly selects DL slot allocation N and UL transmit power P_U to maximize spatial throughput while satisfying a successful information transmission probability constraint.For α = 4 and small ε, the QoS constraint is replaced by equivalent transmission-probability and minimum-power constraints.
  • Scope: The analysis focuses on α = 4, while the authors state that other fading exponents can be treated using similar modeling methods.The paper also states that α does not affect its main results.
  • Battery-free case: The battery-free optimum depends on AP density relative to wireless-node density, with distinct low-, intermediate-, and high-density regimes.At sufficiently low AP density, the QoS constraint may be infeasible; at sufficiently high density, it is always satisfied for feasible N and P_U.
  • Battery-free case: At sufficiently high AP density, any feasible DL allocation and transmit power satisfying the remaining condition can be optimal, because the transmission-probability constraint is automatically satisfied.The maximum achievable spatial throughput is λ_w log2(1+β) when transmission probability ρ = 1.
  • Battery-free case: For battery-free nodes, the available UL energy varies with DL fading, so optimal operation generally uses the minimum transmit power P_min to increase transmission opportunity.The battery-free problem is solved by deriving transmission probability and spatial throughput, then optimizing N and P_U.
  • Battery-free case: Increasing AP density improves both harvested DL energy and UL desired-signal strength, making the successful-transmission constraint looser.The desired-signal improvement from shorter node–AP distances dominates the increased UL interference in the stated analysis.

V. WIRELESS POWERED INFORMATION TRANSMISSION IN BATTERY-DEPLOYED CASE

The battery-deployed analysis separates an ideal infinite-capacity battery case from the practical finite-capacity case. The infinite-capacity scenario provides an upper-performance reference for understanding battery storage.

  • The battery-deployed case first analyzes infinite-capacity batteries and then considers finite-capacity batteries as the practical scenario.The finite-capacity case is described as analytically more difficult and bounded using the ideal and battery-free cases.

A. Infinite-Capacity Battery Scenario (C = ∞)

With infinite battery capacity, harvested energy accumulates across frames and makes RF energy effectively reliable. Consequently, spatial throughput reaches its maximum and multiple charging designs become optimal.

  • With C = ∞, harvested energy from each frame can be stored and used later, alleviating time-varying DL fading effects on UL energy availability.The resulting transmission probability is ρ = 1, so nodes always have sufficient energy to transmit.
  • The infinite-capacity case achieves the maximum spatial throughput R(N,P_U) = λ_w log2(1+β), independent of N and P_U.The optimization therefore degenerates into a feasibility problem rather than a nontrivial throughput search.
  • Unlike the battery-free case, any transmit power P_U ∈ [P_min, P_max] is optimal because stored energy ensures transmission opportunity.The optimal N and P_U can be independently selected within the feasible region.
  • When λ_AP ≥ λ_w/K_ε, every N ∈ {1, ..., T−1} is optimal in the infinite-capacity case.In this high-AP-density regime, the transmission constraint is always satisfied and UL interference is small because associated APs are nearby.

B. Finite-Capacity Battery Scenario (C < ∞)

For finite-capacity batteries, transmission probability increases with capacity but lacks a generally exact analytical expression. The paper therefore uses closed-form bounds and a Markov-chain lower bound to approximate performance and optimize throughput.

  • Closed-form bounds: Finite-capacity transmission probability is bounded below by the battery-free case and above by the infinite-capacity case.The bounds follow because stored energy is capped by C, making battery evolution and transmission probability capacity-dependent.
  • Closed-form bounds: The closed-form bounds are not guaranteed to be tight for arbitrary battery capacities and system parameters.The paper motivates an alternative lower bound because existing bounds may poorly capture the variation with C.
  • Tight lower bound: A finite-state Markov chain yields a numerically computable lower bound that becomes tight as the quantization step approaches zero.The method quantizes battery capacity, harvested energy, and required transmit power, then solves for steady-state probabilities.
  • Tight lower bound: The Markov-chain procedure iteratively reduces the quantization step until the lower-bound error is within a prescribed tolerance.It computes transition probabilities, obtains steady-state probabilities, and returns the resulting lower bound.
  • Tight lower bound: The algorithm is primarily an offline analytical tool, but its complexity can increase with finer quantization and larger transition matrices.This tradeoff makes exact complexity difficult to characterize while preserving practical value for finite-capacity analysis.

3) Spatial Throughput Maximization:

Spatial-throughput optimization differs across battery regimes because transmission probability can equal one or depend on finite storage. In the finite-capacity case, battery constraints can correlate the optimal downlink duration and uplink transmit power.

  • Optimization structure: When transmission probability equals one, spatial throughput becomes constant and the optimization problem reduces to a feasibility problem.This applies in the corresponding battery-deployed regime after substituting ρ = 1.
  • Finite-capacity battery: Finite battery capacity shrinks the feasible region, and the optimal downlink duration and uplink transmit power may become correlated.The added constraint is needed to ensure ρ = 1 in the finite-capacity setting.
  • Infinite-capacity battery: For infinite-capacity batteries, feasible downlink duration and uplink transmit power can be selected independently within the feasible region.Sufficient stored energy makes multiple feasible charging designs optimal.
  • Finite-capacity battery: For finite-capacity batteries without an exact transmission-probability expression, the Markov-chain lower bound supports throughput maximization over candidate frame partitions and transmit powers.The resulting optimized parameters converge to tight approximations as quantization is refined.

VI. NUMERICAL RESULTS

Numerical experiments validate the independence and PPP assumptions used in the analysis, while showing how mobility affects the independence approximation at low AP density.

  • A. Validation of the Analytical Results: The numerical section validates analytical results for both transmission probability and spatial throughput in battery-free and battery-deployed networks.The study also reports similar validation behavior across the considered network cases.
  • A. Validation of the Analytical Results: The independence assumption for harvested-energy variables is well supported, although smaller mobility creates a larger low-density gap between joint and marginal-product probabilities.The gap rapidly decreases as AP density increases.
  • A. Validation of the Analytical Results: The assumed PPP for active UL wireless nodes has void probabilities tightly matching the actual UL point process across interim areas.This validates the PPP approximation used for UL analysis.

B. Study on Transmission Probability and Spatial Throughput

The numerical study examines transmission probability and spatial throughput across battery capacities, AP densities, and finite-battery design choices. Results show increasing battery capacity improves transmission probability, while throughput optimization favors low transmit power and suitable phase allocation.

  • 1) Transmission Probability: As battery capacity increases, the tight lower bound approaches the upper bound ρ = 1 and exceeds the battery-free transmission probability.The tight lower bound becomes close to ρ = 1 for large capacity.
  • 2) Spatial Throughput: In the battery-free case, maximized spatial throughput reaches λw log2(1 + β) after a threshold AP density and remains constant thereafter.The maximum is achieved in the medium AP-density regime before the high-density regime.
  • 2) Spatial Throughput: For finite-capacity batteries, the throughput approximation is maximized at N*=14 and P_U*=P_min=0.0055W under the stated parameter settings.The selected design assigns 86 of 100 slots to the UL phase.
  • 2) Spatial Throughput: Lower transmit power increases transmission probability and can reduce the required DL duration, while longer UL allocation helps reduce interference through independent scheduling.The feasible region in N becomes smaller as P_U decreases.

APPENDIX A PROOF OF PROPOSITION 3.2

The appendix proves Proposition 3.2 by relating monotonic auxiliary functions to the successful information transmission constraint and an equivalent transmission-probability condition.

  • APPENDIX A PROOF OF PROPOSITION 3.2: The proof uses monotonicity of y0(x) and y1(x) to establish how q compares with g across the threshold g0.The comparison changes from q < g when g < g0 to q > g when g > g0.
  • APPENDIX A PROOF OF PROPOSITION 3.2: The auxiliary relation y0(x) ≥ y1(x), with convergence as x increases, supports the case distinctions used in Lemma A.2.The appendix illustrates these cases through Fig. 8.
  • APPENDIX A PROOF OF PROPOSITION 3.2: Lemma A.3 supplies the limiting relation needed to connect the auxiliary variables in the proposition’s constraint transformation.The result is obtained in the limit as ǫ approaches zero.
  • APPENDIX A PROOF OF PROPOSITION 3.2: Proposition 3.2 follows by converting the successful information transmission requirement into a transmission-probability constraint with a minimum transmit-power condition.The derivation uses Lemmas A.1–A.3 and yields the condition P_U ≥ g2.

APPENDIX B PROOF OF THEOREM 4.1

Theorem 4.1 is proved by partitioning AP density into high, medium, and low regimes, characterizing feasibility in each regime, and searching over feasible frame lengths and transmit powers.

  • APPENDIX B PROOF OF THEOREM 4.1: The optimization is solved by comparing AP-density regimes, with feasibility determined by the relation between λw and KǫλAP(T−N).The proof identifies high, medium, and low AP-density cases.
  • APPENDIX B PROOF OF THEOREM 4.1: In the high-density regime, any feasible N and P_U can satisfy the constraint, while the maximum throughput is λw log(1 + β) when the full-success condition holds.Otherwise, the optimum uses N*=T−1 and P_U*=P_min with throughput below that upper value.
  • APPENDIX B PROOF OF THEOREM 4.1: In the medium-density regime, feasible frame lengths satisfy N ≤ N0, and the optimum is found by selecting the best feasible N and corresponding transmit power.The threshold N0 exists uniquely in this regime.
  • APPENDIX B PROOF OF THEOREM 4.1: In the low-density regime, no feasible solution exists when λw exceeds KǫλAP(T−1); otherwise, Algorithm 1 searches all N values and runs in O(T) complexity.For each feasible N, the algorithm selects P_U=max(P_S,P_min).

APPENDIX C PROOF OF PROPOSITION 5.1

The proof of Proposition 5.1 uses the distribution of ZF and i.i.d. PPP structure across downlink frames to establish the result. It shows that ρ=1.

  • Using the distribution of ZF yields a simpler proof than the alternative random-walk proof in.
  • Under Assumption 1, the point processes at the ends of downlink phases are i.i.d. PPPs with density λAP.
  • The harvested energy summed over F frames is equivalent to harvested energy from a PPP with density FλAP.
  • The proof concludes that ρ=1, establishing Proposition 5.1.
  • The proof derives Q using the Laplace transform of ZF and bounds ρ between L and 1 before concluding equality.
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