PaperlayerView exact revision on arXiv

Source-linked AI summary

Optimal Energy Management Policies for Energy Harvesting Sensor Nodes

Vinod Sharma, Utpal Mukherji, Vinay Joseph, Shrey Gupta

arXiv:0809.3908v1cs.NI

TL;DR

The paper studies how an energy-harvesting sensor node can store harvested energy and queued sensing packets while transmitting under energy constraints. It derives energy-management policies for throughput optimality and mean-delay minimization, compares implementable alternatives, and identifies a greedy policy with both properties in the low-SNR regime.

  • Problem

    Intermittent harvested energy may be insufficient for regular sensing or transmission, motivating policies that keep the data queue stable and the system operating indefinitely.

  • Method

    The paper models stored harvested energy, queued sensing packets, and energy-dependent transmission, then derives throughput and mean-delay policies and compares suboptimal policies.

  • Results

    The derived policies support energy-neutral operation and optimize throughput or mean delay; a greedy policy is throughput optimal and minimum-delay for linear transmission, especially in low SNR.

  • Takeaways & Limitations

    Energy storage enlarges the stability region and lowers mean delays, while the greedy policy offers an easily implementable optimal policy in the linear-transmission regime.

Abstract

from arXiv · show

We study a sensor node with an energy harvesting source. The generated energy can be stored in a buffer. The sensor node periodically senses a random field and generates a packet. These packets are stored in a queue and transmitted using the energy available at that time. We obtain energy management policies that are throughput optimal, i.e., the data queue stays stable for the largest possible data rate. Next we obtain energy management policies which minimize the mean delay in the queue.We also compare performance of several easily implementable sub-optimal energy management policies. A greedy policy is identified which, in low SNR regime, is throughput optimal and also minimizes mean delay.

I. INTRODUCTION

The paper addresses energy management for sensor nodes that harvest intermittent, rate-limited energy while storing sensed packets for transmission. It develops policies for energy-neutral operation, maximum stable throughput, and minimum mean queueing delay.

  • Motivation: Intermittent and rate-limited harvesting can prevent regular sensing or transmission, so generated energy and node consumption must be matched.Finite storage further requires adapting consumption to the harvesting source and desired operating objectives.
  • Application: Packets generated by periodic random-field sensing are queued when insufficient energy is available and transmitted when energy permits.Generated energy can be stored, possibly in finite storage, for later use.
  • Objectives: The study seeks energy-neutral operation, meaning the system can work forever while maintaining a stable data queue.The policies are designed to support the maximum possible data rate under the model.
  • Objectives: The paper identifies transmission power-control policies that minimize the mean delay of packets in the queue.These policies are obtained alongside throughput-optimal policies.
  • Scope: Routing algorithms for networks of energy-harvesting sensors remain under investigation.The paper focuses on the sensor-node energy-management problem rather than completing the network-routing problem.
  • Contributions: A greedy policy is throughput optimal and has minimum mean delay for linear transmission, while simulations compare it with other implementable policies.The paper also considers nonnegligible sensing and processing energy and fading channels.

II. MODEL AND NOTATION

The model is a slotted single energy-harvesting sensor node with stochastic data and energy processes, finite-rate transmission, and energy storage. It analyzes queue stability using linear or concave nondecreasing transmission functions under initially idealized assumptions.

  • System model: The system is a slotted single sensor node that periodically generates Xk bits and stores them in a queue for transmission.Packets may be arbitrarily fragmented, so the model treats data as fluid bit strings.
  • Transmission model: Using energy Tk in slot k allows the node to transmit g(Tk) bits, where g is nondecreasing and g(0) = 0.For Gaussian channels, g follows Shannon capacity and is concave; at low Tk it is approximately linear.
  • Energy and randomness: The node replenishes its available energy Ek by harvested energy Yk, and the stochastic processes for data and harvesting are initially iid.The model later allows stationary ergodic processes to represent broader traffic and harvesting patterns.
  • Assumptions: The initial model assumes transmission consumes most node energy while data and energy buffers are infinite.When practical buffers are sufficiently large, this is treated as a good approximation; otherwise the results still provide insights and often perform well.
  • Generalizations: Time-varying traffic and harvesting statistics can be approximated by piecewise stationary processes over periods such as one-hour solar intervals.The stated goal is to retain approximate stationarity long enough for the results to apply.
  • Analysis focus: Section III studies queue stability and identifies easily implementable energy-management policies with good performance.This establishes the model’s link to the paper’s throughput analysis.

III. STABILITY

The section derives necessary and sufficient stability conditions for energy-harvesting sensor queues and identifies throughput-optimal energy-management policies. It also compares buffer-aware and greedy policies, showing that energy storage enlarges the stability region and can reduce mean delay.

  • Throughput-optimal policy: The policy in (5), denoted TO, is throughput optimal because its sufficient condition approaches the necessary condition as ǫ becomes arbitrarily small.The policy eventually uses E[Y] − ǫ whenever sufficient energy is available, while retaining harvested energy in the buffer when needed.
  • Necessary stability condition: E[X] < g(E[Y]) is necessary for an asymptotically stationary queue when g is nondecreasing and concave.For linear g, this condition matches the corresponding stability requirement; for strictly concave g, using harvested energy immediately can be strictly more restrictive.
  • Throughput-optimal policy: Tk = min(Ek, E[Y] − ǫ) stabilizes the queue whenever E[X] < g(E[Y] − ǫ), yielding a unique stationary ergodic distribution under Theorem 1.Starting from any initial distribution, qk converges in total variation to the stationary distribution.
  • Energy buffering: Tk = Yk has a strictly smaller stability region than TO for strictly concave g, unless Y is identically E[Y].Without an energy buffer, immediate use of harvested energy is unavoidable; storing energy therefore enlarges the stability region and can also lower mean delay.
  • Greedy policy: The greedy policy is stable when E[X] < E[g(Y)], which is optimal for linear g but strictly suboptimal for strictly concave g.For concave g, simulations show lower mean delay than TO at low load, but performance rapidly deteriorates near E[g(Y)] because its stability region is smaller.
  • Finite energy buffers: Finite-buffer greedy control gives an ergodic set when E[X] < E[g(Y)], while the limiting distribution may depend on the initial state in general.Theorem 2 assumes a finite but sufficiently large energy buffer; without irreducibility, multiple ergodic sets can occur.

IV. OPTIMAL POLICIES

The paper formulates mean-delay minimization as a Markov decision problem and establishes optimal stationary policies under finite energy-buffer conditions. For linear transmission functions, the greedy policy is both discounted and average-cost optimal, extending its usefulness to transient delays.

  • Mean-delay optimization: The energy allocation T_k is chosen from the current data-queue and energy-buffer states to optimize queueing cost.The resulting policies are stationary Markov policies.
  • Mean-delay optimization: For 0 < α < 1, an α-discount optimal stationary Markov policy exists under finite energy buffers and continuous g.The discounted cost is minimized by the policy selected through the Markov decision framework.
  • Mean-delay optimization: An average-cost optimal stationary Markov policy exists, and its optimal cost does not depend on the initial state.The result uses finite second moments for arrivals and the throughput-optimal policy’s finite stationary queue mean.
  • Mean-delay optimization: As α approaches 1, optimal α-discount policies tend toward an optimal average-cost policy.This connects discounted optimization with long-term average delay.
  • Greedy policy: For linear g(t) = γt, the Greedy policy is α-discount optimal and average-cost optimal.Its optimality follows because transmitting above the queue requirement wastes energy, while postponing feasible transmission increases cost.
  • Greedy policy: Greedy optimality covers both long-term average delay and transient mean delays.The paper contrasts this with numerical value- or policy-iteration methods, which can be computationally intensive and require traffic and energy statistics.

V. GENERALIZATIONS

The paper extends energy-management analysis to fading channels and to sensing and processing costs. It gives throughput conditions and energy-neutral operation results, while noting that improved or optimal policies for the latter generalized system are not developed.

  • Fading channels: With fading, the channel gain h_k is modeled as stationary, ergodic, and independent of traffic and energy generation.The queue evolution depends on the transmission energy and channel state during each slot.
  • Fading channels: If CSI is unavailable, transmission depends only on the data queue and energy buffer.An unfaded throughput-optimal policy uses T_k = min(E_k, E[Y]−ε), stabilizing the queue when E[X] < E[g(h(E[Y] −ε))].
  • Fading channels: For the Greedy policy in fading channels, the data queue is stable if E[X] < E[g(hY)].This provides a direct stability condition involving the channel-weighted harvested energy.
  • Sensing and processing energy: With sensing and processing costs, the node consumes an additional stationary ergodic energy sequence Z_k while operating in a single energy mode.Lower-energy modes are excluded from the simplified model.
  • Sensing and processing energy: Improved or optimal policies for the sensing-and-processing system are not discussed.The paper limits this extension to an energy-neutral operating policy.
  • Sensing and processing energy: If c + E[Z] < E[Y] −δ, choosing T_k ≡ c supports energy-neutral operation and yields a unique stationary, ergodic queue distribution.The result holds for stationary ergodic triples {(X_k, Y_k, Z_k)}.

VI. SIMULATIONS

The simulations compare throughput- and delay-oriented energy policies across fading, distributional, and utility-function settings. Greedy performs well at low loads, while throughput-optimal policies generally support larger stability regions, especially under nonlinear utility and fading.

  • Simulation setup: The simulations compare Greedy, throughput-optimal, modified throughput-optimal, unbuffered, and fading-aware policies using finite, quantized buffers and several distributions.The numerical examples use finite data and energy buffers of size 50; the distributions include truncated Poisson, exponential, uniform, Erlang, and hyperexponential cases.
  • No fading: Greedy has negligible or lower mean queue length at low loads, but under nonlinear g it becomes unstable before throughput-optimal policies.For truncated-Poisson inputs, Greedy’s mean queue length rises rapidly after E[X] = 0.8; with nonlinear g, its stability thresholds are E[g(Y)] = 2.01 or 2.32 versus 2.40 for throughput-optimal policies.
  • No fading: With linear g, all considered policies are throughput optimal, but their delay performances differ substantially.The unbuffered policy performs worst, while comparative delay performance varies with the input distribution; exponential inputs are especially worse for unbuffered and throughput-optimal policies than uniform inputs.
  • Fading: Fading-aware throughput optimization expands the stability region but can increase mean queue length because transmission is restricted to rare favorable channel states.For Erlang inputs with linear g, fading TO is stable for E[X] < 22.0, versus E[X] < 10 for the other three algorithms, while transmitting only at h = 2.2 occurs with probability 0.2.
  • Fading: Under nonlinear fading, unbuffered and Greedy policies have the smallest stability regions, while WF and modified WF provide the largest; modified policies improve mean queue lengths.WF and modified WF are stable for E[X] < 0.70 in the reported fading cases, and the stability-region gap is smaller for Erlang than for hyperexponential inputs.

VII. CONCLUSIONS

The paper identifies throughput-optimal and mean-delay-optimal energy-management policies for energy-harvesting sensor nodes. Simulations compare them with simpler policies, finding that energy storage enlarges the stability region and lowers mean delay, with extensions to fading and sensing or processing consumption.

  • Contributions: The paper studies an energy-harvesting sensor node deployed for random-field estimation and identifies throughput-optimal and mean-delay-optimal energy-management policies.The policies are designed for operation in which the system can remain energy neutral.
  • Contributions: Simulations compare the proposed policies’ mean delays with those of other suboptimal policies.The comparison covers the policy families evaluated throughout the simulation section, including fading-channel cases.
  • Conclusions: Energy storage allows a larger stability region and lower mean delays.This is the paper’s central simulation-based conclusion about storing harvested energy for later use.
  • Extensions: The analysis is extended to fading channels and to energy consumed by sensing and data processing.The paper also notes that leakage or wastage during storage and extraction can be incorporated.

IX. APPENDIX

The appendix establishes finite-moment and drift-based conditions used to prove recurrence properties of the queue-energy process. It shows that suitable conditions allow the process to reach a full-energy, empty-queue state with finite moments.

  • Return-time result: Combining finite-time reachability with a standard coin-tossing argument yields finite α-th moments for the return time τ.The appendix states E[τ^α] < ∞ for any initial state (q,e).
  • Drift argument: The appendix uses a drift condition for h(q,e) = q to establish return-time properties of the queue-energy Markov process.The required condition is a negative expected one-step drift outside a designated set A.
  • Drift argument: The proof applies the drift argument to a sampled Markov chain over M steps rather than directly to the one-step chain.M is chosen as an appropriately large positive integer.
  • Conditions: Under E[X] < g(E[Y] − ǫ) and E[X^α] < ∞ for some α ≥ 1, the stated drift requirement can hold for suitable δ > 0.The appendix derives this condition through a sufficiently large β and an appropriate sampling horizon M.
  • Reachability: From any state in A, the process can reach the empty-queue, full-energy state with positive probability in finitely many steps.The construction uses positive-probability events involving zero arrivals and sufficiently large energy arrivals.
Loading 0809.3908v1…