Source-linked AI summary
Power Allocation Strategies in Energy Harvesting Wireless Cooperative Networks
Zhiguo Ding, Samir M. Perlaza, Inaki Esnaola, H. Vincent Poor
TL;DR
The paper asks how an energy-harvesting relay should distribute accumulated energy among multiple source-destination pairs. It studies individual, centralized, water-filling, and auction-based allocation strategies, finding faster outage decay for centralized methods and a performance-complexity tradeoff for auction allocation.
Problem
The paper addresses how to efficiently distribute energy accumulated by an energy-harvesting relay among multiple communicating user pairs.
Method
The paper analyzes individual transmission, equal-power and sequential water-filling centralized strategies, and an auction-based distributed allocation scheme.
Results
Centralized strategies achieve outage decay at rate 1 over SNR, while individual transmission decays at log SNR over SNR; water filling is optimal for several criteria, and auction allocation approaches its performance.
Takeaways & Limitations
Auction allocation offers a better tradeoff between system performance and complexity than the more complex water-filling strategy.
Abstract
from arXiv · showhide
In this paper, a wireless cooperative network is considered, in which multiple source-destination pairs communicate with each other via an energy harvesting relay. The focus of this paper is on the relay's strategies to distribute the harvested energy among the multiple users and their impact on the system performance. Specifically, a non-cooperative strategy is to use the energy harvested from the i-th source as the relay transmission power to the i-th destination, to which asymptotic results show that its outage performance decays as logSNR over SNR. A faster decaying rate, 1 over SNR, can be achieved by the two centralized strategies proposed this the paper, where the water filling based one can achieve optimal performance with respect to several criteria, with a price of high complexity. An auction based power allocation scheme is also proposed to achieve a better tradeoff between the system performance and complexity. Simulation results are provided to confirm the accuracy of the developed analytical results and facilitate a better performance comparison.
I. INTRODUCTION
The paper studies how an energy-harvesting relay should distribute accumulated energy among multiple source-destination pairs. It compares individual, centralized, water-filling, and auction-based strategies in terms of outage performance, complexity, and channel-information requirements.
- Motivation: Energy harvesting is proposed as a sustainable solution for wireless networks whose battery-powered devices have limited operating lives and costly or impossible battery replacement.The motivation includes sensor networks used for surveillance, environmental monitoring, and health care.
- System model: The considered network has multiple source-destination pairs communicating through a relay that harvests energy from source transmissions over orthogonal channels.The relay can accumulate harvested power because its battery is assumed sufficiently large.
- Individual transmission: The individual transmission benchmark powers destination i using only energy harvested from source i, yielding outage decay at rate log SNR over SNR.The paper derives an exact outage expression for the decode-and-forward strategy and analyzes its high-SNR behavior.
- Centralized allocation: The equal-power strategy improves on individual transmission, while its outage probability decays at rate 1 over SNR.Equal allocation can provide more relay power to user pairs with poor channel conditions.
- Water filling: Sequential water filling serves users with better channels first and achieves optimality for several criteria, including best-user reliability, successful-destination count, and worst-user outage.Its average-outage bounds have the same high-SNR decay rate as the bounds themselves.
- Auction allocation: The auction-based distributed scheme reduces reliance on transmitter CSI and achieves performance close to water filling while outperforming equal-power and individual transmission schemes.The paper presents it as a tradeoff between system performance and complexity in multi-user settings.
III. CENTRALIZED MECHANISMS FOR POWER ALLOCATION
The centralized mechanisms redistribute energy accumulated from reliably decoded sources rather than tying each relay transmission to one source. Equal allocation reduces information requirements, while more sophisticated allocation targets improved outage performance.
- Available harvested power: The relay considers only the N sources whose information it can reliably detect when determining the power available for centralized allocation.N is random because it depends on instantaneous source-relay channel realizations, and source powers are assumed equal for analysis.
- Equal power allocation: Equal power allocation assigns the same relay transmission power to every user and does not require relay-destination CSI.This reduces system overhead, particularly in multi-user systems.
- Equal power allocation: The equal-power scheme has analytically characterized outage probabilities for individual destinations and for users with the best and worst channel conditions.Theorem 1 and Proposition 2 provide these outage characterizations.
- Modeling assumption: The analysis uses a pessimistic assumption that energy is harvested only from sources whose messages are successfully detected at the relay.If detection fails, the model assumes no energy remains for relaying from that source.
- Outage analysis: The centralized allocation analysis derives best- and worst-outage expressions by conditioning on the random number of reliably detected sources and ordering their relay-destination channels.The derivation combines the distribution of N with ordered channel statistics.
B. Sequential water filling based power allocation strategy
Sequential water filling serves destinations in channel-quality order, prioritizing stronger channels and continuing while harvested relay power remains. It optimizes several outage-related criteria, but exact analysis for worst-user performance requires bounds because inverse-exponential sums are difficult to characterize.
- Strategy: The relay serves destinations in descending channel quality, allocating each required power amount before proceeding to the next user.Service continues until all users are served or relay power is exhausted; leftover energy is reserved in an assumed infinite-capacity battery.
- Analysis: The probability of obtaining m successful receivers among n users can be expressed analytically, but averaging over all m and n is challenging.The difficulty mainly arises from the density of sums of inverse exponential variables.
- Performance criteria: The strategy maximizes the number of successful destinations and minimizes outage probability for the user with the best channel conditions.The best-channel user is served first and receives maximal relaying power.
- Performance criteria: Pworst,III = min{Pworst(s), s ∈ S}, so sequential water filling minimizes the worst user outage probability among all possible strategies.The result defines worst-user performance as the maximum outage probability across users.
- Analysis: Exact worst-user outage expressions are difficult because they require inverting a Laplace transform for a sum of inverse exponential variables, whose moments are partly unbounded.The paper therefore develops upper and lower outage bounds for the worst-channel user.
- Analysis: The bounds use a parameter c ∈ [0, M −1] to facilitate asymptotic analysis and ensure a(y)ε approaches zero at high SNR.The upper bound in Proposition 4 is recovered by setting c = 0 in Proposition 5.
IV. ASYMPTOTIC ANALYSIS OF THE OUTAGE PERFORMANCE
The paper develops high-SNR approximations for outage performance using series expansions of Bessel functions. These approximations support asymptotic comparison of the proposed power-allocation strategies.
- Asymptotic method: High-SNR asymptotic analysis uses the series representation of Bessel functions to approximate x^nK_n(x) as x approaches zero.The resulting approximations are used to analyze outage performance at high SNR.
- Asymptotic method: The Bessel-function approximations provide the basis for subsequent high-SNR outage analysis.
A. Averaged outage performance
The averaged outage analysis compares individual transmission, equal power allocation, and water filling at high SNR. Equal power allocation and water filling achieve a 1/SNR decay, faster than the individual strategy’s log SNR-related decay.
- High-SNR comparison: The individual transmission scheme’s averaged outage probability decays as log SNR over SNR at high SNR.
- High-SNR comparison: 1/SNR is the averaged-outage decay rate achieved by equal power allocation, which is faster than the individual transmission scheme’s rate.
- High-SNR comparison: The normalized difference between individual and equal-power outage probabilities can grow without bound as ε approaches zero.The paper notes that the difference can therefore be significant.
- Water filling: 1/SNR is also the averaged-outage decay rate associated with water filling, although the paper does not obtain an explicit expression for that strategy.
B. Best outage performance
The best-user outage analysis shows that equal power allocation and water filling provide similar performance for the user with the strongest channel. Centralized strategies achieve a faster 1/SNR outage decay, while water filling requires global CSI and greater overhead.
- Best-user comparison: Equal power allocation and water filling achieve similar outage performance for the user with the best channel conditions.
- Equal power allocation: The best-user outage under equal power allocation is approximated at high SNR using the derived proposition.
- Worst-user comparison: The individual strategy’s worst outage performance still decays as log SNR over SNR.
- Worst-user comparison: 1/SNR is the outage decay rate achieved by centralized power-allocation strategies, including the equal-power and water-filling approaches.
- Worst-user comparison: The water-filling worst-user decay conclusion follows because both its upper and lower outage bounds decay as 1/SNR.
- Complexity trade-off: Global CSI is required for centralized water filling, creating significant overhead in systems with many users and motivating distributed auction-based allocation.
A. Power auction game
The power auction game distributes relay transmission power through user bids and payments, using an iterative best-response process. For prices above a threshold, the game has a unique Nash equilibrium and the updating algorithm converges to it.
- Game formulation: Each destination submits a nonnegative bid, and the relay allocates transmission power and charges payment proportional to that allocation.Only destinations whose source messages can be reliably decoded at the relay participate in the game.
- Game formulation: Users choose bids to maximize payoff based on achievable data rate minus the price paid for allocated relay power.The data rate is expressed as Rd,i = 1/2 log(1 + Pri|gi|^2), while payment is Ci = πPri.
- Equilibrium analysis: The addressed power auction game has a unique Nash equilibrium when the price π exceeds a threshold; otherwise, infinitely many equilibria exist.The proposition also provides a unique best-response function for each player in the threshold-price regime.
- Equilibrium analysis: The best-response function is a contraction mapping when the price exceeds the threshold, so iterative bidding converges to the unique fixed point representing the Nash equilibrium.This uniqueness proof avoids the nonnegative matrix theory used in earlier work.
- Implementation: The direct iterative update requires users to know other players’ actions, but an equivalent update allows bidding from local information alone.Users can use their previous allocated power and previous bid without observing other users’ actions.
VI. NUMERICAL RESULTS
Numerical results validate the analytical outage expressions and compare power-allocation strategies under several outage criteria. Water filling generally performs best, while auction-based allocation approaches it with lower complexity.
- Analytical validation: The analytical outage results exactly match simulations for individual transmission and equal power allocation at R = 2 BPCU.The comparison uses complex Gaussian channels, η = 1, and outage probability versus SNR.
- Individual versus equal power: 1 × 10^-2 versus 3 × 10^-3 outage probability at 40 dB shows equal power allocation outperforming individual transmission.The equal power scheme also has the faster asymptotic decay rate, 1/SNR, compared with the individual scheme's 1/[SNR(1 + 2 ln SNR)].
- Water filling: The water filling scheme yields the best outage performance across the studied criteria, with tight analytical bounds.The bounds are tight because the largest ordered variable dominates the relevant sum.
- Worst-user outage: The water filling scheme is optimal for the worst-channel user's outage, while auction allocation performs close to this optimum.The worst-channel comparison uses R = 0.5 BPCU and 20 user pairs under Rayleigh fading with path loss.
- Average outage: For average outage, water filling performs best and auction allocation remains close, whereas the centralized worst-user strategy loses performance under the average criterion.The centralized strategy matches water filling for worst-user outage but is weaker for averaged outage.
- Successful receivers: Two more successful receivers are achieved by auction allocation than equal power from 10 dB to 25 dB.The strategy maximizing worst-user outage can nevertheless yield fewer successful receivers than individual transmission.
APPENDIX
The appendix derives outage expressions and bounds for the proposed allocation strategies, then establishes properties of the worst-user optimization and auction game. It uses conditioning, transformed channel distributions, optimization conditions, and contraction arguments.
- Outage derivations: The outage analysis partitions destinations according to whether their source messages are reliably decoded at the relay.S1 contains unsuccessful source-relay links, while S2 contains the remaining destinations with |S2| = N.
- Distributional analysis: Conditioned channel sums are analyzed through Laplace transforms and order-statistics densities to obtain outage expressions and bounds.The derivation includes sums of exponentially distributed channel coefficients and the largest ordered variables.
- Worst-user optimization: The worst-user power-allocation problem is formulated by maximizing the minimum destination rate under a total-power constraint.An auxiliary parameter converts the optimization into an equivalent form, and Karush-Kuhn-Tucker conditions yield a closed-form solution.
- Worst-user optimality: The optimized worst-user strategy and water filling achieve the same worst outage performance.The proof compares the resulting worst-user outage expressions after accounting for source decoding and relay-power insufficiency events.
- Auction equilibrium: The auction power-allocation game has at least one Nash equilibrium because each payoff is strictly quasi-concave in the player's bid.A unique equilibrium follows when the best-response mapping becomes a contraction above a pricing threshold.