PaperlayerView exact revision on arXiv

Source-linked AI summary

Optimal Scheduling and Power Allocation for Two-Hop Energy Harvesting Communication Systems

Yaming Luo, Jun Zhang, Khaled B. Letaief

arXiv:1212.5394v1cs.ITmath.IT

TL;DR

Random harvested energy makes short-term performance difficult to guarantee, and the paper addresses two-hop communication with an EH source and a non-EH relay. It uses single-hop directional water-filling as a design guide; the maximum throughput under the original EH profile matches that under its associated DWF profile.

  • Problem

    Random harvested energy makes satisfactory short-term performance difficult to guarantee in two-hop EH communication systems.

    Source: §I · p. 2

  • Method

    The paper uses single-hop directional water-filling and a proposed DWF EH profile as design tools for the two-hop problems.

    Source: §VI · p. 24

  • Results

    The maximum throughput under the original EH profile is the same as under its associated DWF EH profile.

    Source: C. Achievability of the Throughput… · p. 15

  • Takeaways & Limitations

    The proposed DWF EH profile is a promising design tool for two-hop EH communication systems.

    Source: §VI · p. 24

Abstract

from arXiv · show

Energy harvesting (EH) has recently emerged as a promising technique for green communications. To realize its potential, communication protocols need to be redesigned to combat the randomness of the harvested energy. In this paper, we investigate how to apply relaying to improve the short-term performance of EH communication systems. With an EH source and a non-EH half-duplex relay, we consider two different design objectives: 1) short-term throughput maximization; and 2) transmission completion time minimization. Both problems are joint scheduling and power allocation problems, rendered quite challenging by the half-duplex constraint at the relay. A key finding is that directional water-filling (DWF), which is the optimal power allocation algorithm for the single-hop EH system, can serve as guideline for the design of two-hop communication systems, as it not only determines the value of the optimal performance, but also forms the basis to derive optimal solutions for both design problems. Based on a relaxed energy profile along with the DWF algorithm, we derive key properties of the optimal solutions for both problems and thereafter propose efficient algorithms. Simulation results will show that both scheduling and power allocation optimizations are necessary in two-hop EH communication systems.

I. INTRODUCTION

Random energy arrivals make satisfactory short-term performance challenging, and prior two-hop EH work leaves the general multiple-arrival EH-source case unresolved. The paper studies throughput maximization and completion-time minimization, using single-hop directional water-filling as a design guide for the two-hop system.

I. INTRODUCTION

  • Random, typically small harvested-energy arrivals make satisfactory short-term performance challenging for energy-harvesting communications.The introduction motivates protocol redesign to address this variability.Source: §I · p. 2, A. Related Works and Motivations · p. 2

A. Related Works and Motivations

  • Prior two-hop EH studies cover special cases and throughput objectives, while the optimal policy for a general EH source with multiple arrivals remains unknown.The paper identifies the general source-EH case as more challenging because it affects system design and performance and requires joint scheduling and power allocation.Source: A. Related Works and Motivations · p. 3

B. Contributions

  • The paper considers short-term throughput maximization and transmission completion-time minimization for an EH source assisted by a non-EH relay.Both are joint scheduling and power-allocation problems; the half-duplex relay constraint makes them challenging.Source: B. Contributions · p. 3
  • Single-hop directional water-filling guides the two-hop design by providing a related EH profile and performance upper bound from which optimal solutions can be derived.The paper proposes efficient algorithms to achieve the upper bound and guarantee optimality.Source: B. Contributions · p. 3
  • With a time-varying EH profile, fixed power-allocation and fixed-scheduling policies incur performance loss, especially when source and relay energy levels differ significantly.The simulations are described as demonstrating the importance of adaptive source power allocation and optimal scheduling of source and relay transmission periods.Source: B. Contributions · p. 3

II. SYSTEM MODEL AND PROBLEM FORMULATION

The section models a two-hop, half-duplex decode-and-forward link with an energy-harvesting source and a battery-powered relay. It defines causal energy and data constraints, then describes how harvesting conditions shape alternating source and relay stages.

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The system uses an energy-harvesting source and a non-harvesting, battery-powered relay to communicate over a two-hop link when the direct link is too weak.Both source-relay and relay-destination channels use a nonnegative, strictly concave, monotonically increasing power-rate function.Source: §II · p. 4

A. Energy Harvesting Model

  • The source’s discrete-time energy-arrival profile is assumed known at the start of the transmission block and can approximate more general profiles.The model assumes static channels and an infinite-capacity source battery; finite capacity is left for future work.Source: A. Energy Harvesting Model · p. 4, MAIN NOTATIONS · p. 5

MAIN NOTATIONS

  • Energy neutrality requires cumulative source energy use not to exceed harvested energy, so future arrivals cannot be spent early but current energy may be saved.The harvested-energy profile defines the feasible energy-consumption domain.Source: MAIN NOTATIONS · p. 5

B. Transmission Policy

  • Half-duplex operation divides transmission into alternating source and relay stages, with each source stage followed by a relay stage in an S-R stage pair.The first stage must be a source stage and the last a relay stage; stage durations generally differ.Source: B. Transmission Policy · pp. 6–7
  • Scheduling allocates transmission time between source and relay, while power allocation sets their instantaneous transmit powers.The paper represents scheduling through the source power-consumption curve; a stage is a period when one node transmits with nonzero power.Source: B. Transmission Policy · pp. 6–7
  • The model treats stage durations as real-valued, while practical symbol durations impose a minimum time unit; rounded results can serve as approximations.A detailed investigation of this practical aspect is left to future work.Source: B. Transmission Policy · p. 6
  • With an energy-harvesting source, a single source-relay stage pair is generally suboptimal because data causality can constrain relay transmission.The optimal schedule may require multiple pairs depending on the energy-harvesting profile, making the problem more challenging than the cited EH-relay setting.Source: B. Transmission Policy · p. 8

C. Problem Formulation

The paper formulates short-term throughput maximization and transmission completion-time minimization for a two-hop EH system. In both, half-duplex scheduling couples the source and relay power-allocation decisions.

C. Problem Formulation

  • The paper considers short-term throughput maximization with a fixed transmission period and completion-time minimization with a fixed data amount.These are denoted RMAX and TMIN, respectively.Source: C. Problem Formulation · pp. 8–9
  • Both problems impose half-duplex operation, source energy-causality and data-causality constraints, and relay power and total-energy constraints.The relay peak-power constraint is tied to linear power-amplifier operation or safety regulation.Source: C. Problem Formulation · p. 8
  • The half-duplex constraint couples scheduling and power allocation: changing the transmission schedule requires adjusting the power allocation, and vice versa.The formulation assumes static source-relay and relay-destination channels.Source: C. Problem Formulation · pp. 8–9
  • An EH source can require multiple source-relay stage pairs depending on its energy profile, so its scheduling problem is more complex than the cited non-EH-source case.The paper therefore seeks an indirect approach to solving the optimization problems.Source: C. Problem Formulation · p. 9

III. SHORT-TERM THROUGHPUT MAXIMIZATION

The paper uses the single-hop directional water-filling structure to guide two-hop EH optimization. A relaxed DWF energy profile provides an upper bound for the original profile.

A. The DWF EH profile

  • The DWF transmit power is nondecreasing, remains constant between energy arrivals, and changes only at arrival epochs where consumed energy equals harvested energy.These properties restrict power changes to selected energy-arrival times.Source: A. The DWF EH profile · pp. 9–10
  • DWF points mark transmit-power changes, and the intervals between adjacent points define the DWF intervals used to organize the two-hop design.The points exclude the origin and include the ending point.Source: A. The DWF EH profile · p. 10
  • The DWF EH profile places energy arrivals at the beginnings of DWF intervals; because it relaxes the original profile and enlarges the feasible domain, it yields a performance upper bound.The profile is uniquely determined by the DWF points, which the paper identifies as important for two-hop optimization.Source: A. The DWF EH profile · p. 10

B. Throughput Upper Bound from the DWF EH profile

For throughput maximization, the paper analyzes optimal transmission under the relaxed DWF EH profile and uses its structure to construct solutions for the original profile. The resulting rules depend on the relay’s energy and peak-power constraints.

B. Throughput Upper Bound from the DWF EH profile

MAIN NOTATIONS RELATED TO THE DWF EH PROFILE

  • The stated optimal-policy properties apply to both relay energy constraints and relay power constraints.The paper treats the combined-constraint case separately with Algorithm 1.Source: MAIN NOTATIONS RELATED TO THE DWF EH… · pp. 12–13
  • With an energy-constrained relay, the optimal DWF-profile policy uses a single S-R stage pair in each DWF interval.The source and relay stages and their powers are specified for each interval.Source: MAIN NOTATIONS RELATED TO THE DWF EH… · p. 12
  • With a power-constrained relay, its transmit power in an optimal RMAX policy can only be zero or the peak power.The paper gives this as a stronger property than the general RMAX structure.Source: MAIN NOTATIONS RELATED TO THE DWF EH… · pp. 11, 13
  • For a relay subject to both energy and power constraints, Algorithm 1 constructs an optimal RMAX solution using the energy- and power-constrained corollaries.Its procedure checks relay-power feasibility across DWF intervals and adjusts the interval solutions.Source: MAIN NOTATIONS RELATED TO THE DWF EH… · pp. 13–14

C. Achievability of the Throughput Upper Bound with the Original EH Profile

The DWF EH profile supplies an upper bound and a temporary solution; modifying its schedule for the original EH profile can attain the same maximum throughput.

C. Achievability of the Throughput Upper Bound with the Original EH Profile

  • The original EH profile attains the DWF-profile maximum throughput by modifying the temporary solution when it is infeasible.The modification divides a DWF interval into multiple source-relay stage pairs while preserving the critical parameters and respecting data and energy constraints.Source: C. Achievability of the Throughput… · p. 15
  • Within each DWF interval, the original and DWF profiles share the source and relay powers and total source period of the optimal RMAX solution.Solutions with these same parameters achieve the same throughput.Source: C. Achievability of the Throughput… · p. 15
  • Algorithm 2 uses the DWF-profile solution as a temporary solution, then adjusts scheduling to obtain a feasible solution for the original profile.Power allocation is completed in the first step; the second step determines scheduling.Source: C. Achievability of the Throughput… · pp. 15, 17
  • DWF power allocation guides the two-hop design by determining optimal throughput and enabling joint scheduling and power allocation to be decoupled.The paper says the method also applies to more general EH profiles when their cumulative EH curve lies between the DWF EH profile and DWF power-allocation curve.Source: C. Achievability of the Throughput… · p. 17

IV. TRANSMISSION COMPLETION TIME MINIMIZATION

Transmission completion time minimization (TMIN) seeks the shortest time to send a given amount of data. The single-hop DWF result does not supply the EH profile needed for the two-hop problem, so the required DWF points must be found first.

IV. TRANSMISSION COMPLETION TIME MINIMIZATION

  • TMIN minimizes total transmission time for a given amount of data, making it the dual of throughput maximization with a given deadline.Source: §IV · p. 17
  • Unlike in RMAX, single-hop DWF power allocation for TMIN does not provide the DWF EH profile needed in the two-hop case.Source: §IV · p. 17
  • The TMIN approach must obtain the required DWF points before applying the RMAX scheduling and power-allocation algorithm.Source: §IV · p. 17

A. Obtaining All the DWF Points with a Power Constrained Relay

For a power-constrained relay, the paper finds the DWF points iteratively without requiring the unknown completion time in advance. Algorithm 3 also yields the completion time and temporary DWF-profile power allocation.

A. Obtaining All the DWF Points with a Power Constrained Relay

  • Lemma 2’s condition requires the completion time T, which is unknown before optimization and therefore imposes a non-causal requirement.The proposed approach removes this requirement by using an earlier deadline for the first DWF point and iteratively updating the remaining energy and data.Source: A. Obtaining All the DWF Points with a… · p. 18
  • Algorithm 3 finds all DWF points for a power-constrained relay, and Theorem 3 establishes that they satisfy Lemma 2’s conditions.Source: A. Obtaining All the DWF Points with a… · p. 18

N D = k, the transmission completion time is T = tD

B. Obtaining All the DWF Points with an Energy Constrained Relay

For an energy-constrained relay, the paper adapts the DWF-point search because the relay cannot always transmit at constant power. The resulting DWF-profile solution is extended to the original EH profile; simulations also show a throughput gap over fixed-policy baselines.

B. Obtaining All the DWF Points with an Energy Constrained Relay

  • With an energy-constrained relay, constant relay power may not be feasible throughout transmission, so Algorithm 3 cannot be applied directly.The proposed procedure first obtains temporary DWF points, then searches for the completion time between an upper bound and the beginning time using bisection.Source: B. Obtaining All the DWF Points with… · pp. 19–20

C. Optimal Transmission Policy

A. Short-term Throughput Maximization

  • Simulations report a large throughput gap between the proposed optimal solution and two baseline policies, with some loss relative to the upper bound under random EH.The fixed-scheduling baseline cannot adapt stage lengths to source EH levels, while the other baseline does not adapt power allocation to the source EH profile.Source: A. Short-term Throughput Maximization · p. 22

B. Transmission Completion Time Minimization

For completion-time minimization, the paper reports substantial gains from jointly optimizing scheduling and power allocation. Its conclusions identify DWF as a useful design tool for two-hop energy-harvesting systems.

B. Transmission Completion Time Minimization

  • The completion-time simulations use 1,000 random energy-harvesting realizations and a 20-kbit transmission requirement.Figure 6 plots completion time against source average harvesting rate and relay peak power.Source: B. Transmission Completion Time… · p. 23
  • Optimal scheduling and power allocation yield great performance improvement over two baseline policies for completion-time minimization.The authors also emphasize that both optimizations are necessary.Source: B. Transmission Completion Time… · p. 23

VI. CONCLUSIONS

  • The study minimizes completion time for a fixed data amount, alongside throughput maximization for a fixed transmission period.The conclusions frame these as the two objectives for a two-hop system with an energy-harvesting source and a non-energy-harvesting relay.Source: §VI · p. 24
  • The paper identifies single-hop directional water-filling as a useful design tool for two-hop energy-harvesting communication systems.The conclusion says the DWF power-allocation algorithm provides useful design insights.Source: §VI · p. 24

APPENDIX 1 PROOF OF THEOREM 1

The appendices prove structural properties used to simplify the throughput-maximization problem, then establish necessary and sufficient equations for its reduced form. The proof uses equivalent rescheduling arguments and uniqueness results.

APPENDIX 1 PROOF OF THEOREM 1

  • Equivalent reschedulings establish relay data equality by contradiction while preserving transmit powers and using a constant-power replacement to improve throughput.The argument rearranges source and relay stages across DWF intervals; the proof also states that the power-and-energy-constrained relay case follows similarly.Source: APPENDIX 1 PROOF OF THEOREM 1 · pp. 24–25
  • The source energy equality follows by moving source transmission between DWF intervals and delaying the relay without violating energy or data causality.The rescheduling preserves throughput; the resulting violation of data equality permits replacement by a higher-throughput policy.Source: APPENDIX 1 PROOF OF THEOREM 1 · p. 25
  • When relay power changes in the second DWF interval, optimality requires the relay to exhaust its energy or have transmitted all source data.For a non-energy-harvesting relay, the proof uses the energy-buffer condition to conclude that the relay must empty its data buffer.Source: APPENDIX 1 PROOF OF THEOREM 1 · p. 25
  • The case where neither source nor relay power changes in the second DWF interval is ruled out as infeasible by the source energy-causality condition.The proof compares average source power with the harvested energy available at the first DWF point.Source: APPENDIX 1 PROOF OF THEOREM 1 · pp. 25–26

APPENDIX 2 PROOF OF COROLLARY 1

  • Theorem 1 reduces Problem RMAX to equality constraints, and the resulting equation set is both necessary and sufficient with a unique solution.The proof establishes uniqueness for the component equations and for the whole equation set.Source: APPENDIX 2 PROOF OF COROLLARY 1 · pp. 26–27

APPENDIX 3 PROOF OF THEOREM 3

The proof of Theorem 3 uses a contradiction based on a modified energy-harvesting profile and a monotonicity-based uniqueness argument. It also states that Algorithm 3's DWF points satisfy the remaining conditions.

APPENDIX 3 PROOF OF THEOREM 3

  • The authors state that DWF points obtained through Algorithm 3 satisfy the remaining parts of the required conditions.Source: APPENDIX 3 PROOF OF THEOREM 3 · p. 27
  • The proof assumes an index inequality, constructs a new energy-harvesting profile, and uses a unique-solution argument to contradict that assumption.The modified profile retains the original through a specified interval and has no energy arrival in the interval under examination.Source: APPENDIX 3 PROOF OF THEOREM 3 · p. 28
  • The monotonicity argument places the calculated time on the opposite side of the assumed bound, ruling out the case for both profiles.The proof attributes the ordering to the calculation procedure and uniqueness of the equation's solution.Source: APPENDIX 3 PROOF OF THEOREM 3 · p. 28
Loading 1212.5394v1…