Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies

Abstract: Wireless systems comprised of rechargeable nodes have a significantly prolonged lifetime and are sustainable. A distinct characteristic of these systems is the fact that the nodes can harvest energy throughout the duration in which communication takes place. As such, transmission policies of the nodes need to adapt to these harvested energy arrivals. In this paper, we consider optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of an adaptive directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use stochastic dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies under various different configurations.

• The paper develops optimal energy policies leveraging a directional water-filling algorithm and convex optimization to maximize throughput by a deadline.
• It maps transmission completion time minimization to throughput maximization using the maximum departure curve for efficient performance analysis.
• The research derives online policies via stochastic dynamic programming and proposes heuristic alternatives to address practical energy management challenges.

Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies

Introduction

Energy harvesting in wireless communication systems introduces novel challenges and opportunities for optimization. The paper, "Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies," addresses significant theoretical and practical questions related to these systems. The authors develop optimal policies for data transmission using energy-harvesting transmitters within the constraints of varying energy availability, finite battery capacity, and fading wireless channels. They focus on two key objectives: throughput maximization by a deadline and transmission completion time minimization.

Problem Formulation

The paper considers a point-to-point communication model where a transmitter harvests energy from random sources during data transmission. This harvested energy can then be stored in a finite-capacity battery for future use. The communication happens over a wireless fading channel, further complicating the optimization problem.

Two primary objectives are analyzed:

1. Throughput Maximization by Deadline: The goal is to maximize the number of bits transmitted by a given deadline.
2. Minimizing Transmission Completion Time: The aim here is to minimize the time required to transmit a fixed amount of data.

Offline Policies

For the case of offline knowledge of energy arrivals and channel conditions, the authors present an optimal power allocation strategy employing a directional water-filling algorithm.

Throughput Maximization

The throughput maximization problem is approached by setting up a convex optimization framework that accounts for energy causality and battery constraints. The solution to this yields a monotonic property of the power levels across the transmission epochs, allowing energy transfer only in one direction (forward) due to causality constraints. This is characterized as a directional water-filling approach where water (energy) can flow to the right but is restricted by right-permeable taps.

Transmission Completion Time

This problem is mapped to the throughput maximization problem via a maximum departure curve function. The maximum departure curve represents the upper limit of bits that can be transmitted by a certain deadline given the energy arrivals and channel fading. This relationship allows the optimal transmission completion time to be efficiently determined.

Online Policies

The paper extends the analysis to stochastic settings where only causal (online) information of energy arrivals and channel states is available.

Optimal Online Policy

Using stochastic dynamic programming, the authors derive the optimal policy for online settings. This requires solving a recursive optimization problem, which, while computationally intensive, provides the benchmark for evaluating other heuristic policies.

Proposed Heuristic Policies

The authors also present several heuristic policies that balance performance with computational simplicity:

1. Constant Water Level Policy: Adjusts the power level based on a calculated water level determined by the average energy arrival rate and fading statistics.
2. Energy Adaptive Water-Filling: Adjusts the power level adaptively based on the current battery state.
3. Time-Energy Adaptive Water-Filling: Adjusts power level based on both the current battery state and remaining time to the deadline.

Numerical Results

Numerical simulations reveal insightful comparisons between the proposed policies. The optimal offline policy sets the performance upper bound, demonstrating substantial gains over heuristic approaches in various scenarios. The time-energy adaptive water-filling policy performs admirably in low recharge rate settings, while the constant water level policy shows robustness as deadlines are extended.

Implications and Future Research

The research provides foundational insights into designing transmission policies for energy harvesting systems with finite batteries over fading channels. Practically, the results are applicable to sustainable wireless sensor networks and IoT applications where energy efficiency is paramount. Theoretically, this work bridges gaps in energy management strategies under stochastic constraints, inviting future exploration into more sophisticated models and real-world implementations.

Future directions might involve:

• Developing scalable algorithms for real-time implementation in diverse network configurations.
• Considering multi-user and cooperative communication frameworks.
• Addressing the impact of energy prediction inaccuracies and integrating machine learning for adaptive strategies.

This paper's rigorous approach to energy-aware transmission design sets a high bar for subsequent research, emphasizing the critical interplay between energy availability and communication performance.