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Optimal Status Update for Age of Information Minimization with an Energy Harvesting Source

Xianwen Wu, Jing Yang, Jingxian Wu

arXiv:1706.05773v3cs.IT

TL;DR

The paper asks how an energy-harvesting sensor can minimize long-term average AoI under energy causality. It analyzes online policies for infinite, finite, and one-unit batteries, using uniform, adaptive, and threshold-based strategies. The proposed policies are theoretically optimal or asymptotically optimal in their respective regimes, with simulations supporting these results.

  • Problem

    The paper studies how to minimize long-term average AoI for an energy-harvesting sensor subject to energy causality across different battery sizes.

  • Method

    The paper analyzes online update policies for infinite, finite, and one-unit batteries, including best-effort uniform, energy-aware adaptive, and renewal threshold policies.

  • Results

    The proposed policies are theoretically optimal or asymptotically optimal in their respective battery regimes, with simulations supporting the analytical results.

  • Takeaways & Limitations

    Battery size determines the appropriate AoI-minimizing update structure, ranging from uniform scheduling with infinite storage to threshold-based decisions with one-unit storage.

Abstract

from arXiv · show

In this paper, we consider a scenario where an energy harvesting sensor continuously monitors a system and sends time-stamped status updates to a destination. The destination keeps track of the system status through the received updates. We use the metric Age of Information (AoI), the time that has elapsed since the last received update was generated, to measure the "freshness" of the status information available at the destination. We assume energy arrives randomly at the sensor according to a Poisson process, and each status update consumes one unit of energy. Our objective is to design optimal online status update policies to minimize the long-term average AoI, subject to the energy causality constraint at the sensor. We consider three scenarios, i.e., the battery size is infinite, finite, and one unit only, respectively. For the infinite battery scenario, we adopt a best-effort uniform status update policy and show that it minimizes the long-term average AoI. For the finite battery scenario, we adopt an energy-aware adaptive status update policy, and prove that it is asymptotically optimal when the battery size goes to infinity. For the last scenario where the battery size is one, we first show that within a broadly defined class of online policies, the optimal policy should have a renewal structure, i.e., the status update epochs form a renewal process, and the length of each renewal interval depends on the first energy arrival over that interval only. We then focus on a renewal interval, and prove that if the AoI in the system is below a threshold when the first energy arrives, the sensor should store the energy and hold status update until the AoI reaches the threshold, otherwise, it updates the status immediately. We analytically characterize the long-term average AoI under such a threshold-based policy, and explicitly identify the optimal threshold.

I. INTRODUCTION

The paper studies AoI minimization for energy-harvesting status updates, where limited energy and bandwidth constrain information freshness. It develops online policies across battery regimes, including a threshold policy for a one-unit battery.

  • Motivation: AoI measures status freshness using the elapsed time since the latest received update was generated.It differs from throughput and delay because it accounts for update generation time and can increase when stale packets remain in the network.
  • Related work: Energy harvesting can support extended or perpetual wireless-network operation through transmission scheduling, but AoI optimization in EH networks remains relatively underexplored.Prior work considered only a few specific setups and left the optimal update policy open in some settings.
  • Contribution: The paper formulates optimal online status updating under Poisson energy arrivals, unit energy cost per update, and varying battery sizes.The considered regimes are infinite, finite, and one-unit batteries.
  • Contribution: For infinite and finite batteries, it adopts best-effort uniform and energy-aware adaptive policies, respectively, and establishes their optimality or asymptotic optimality.The policies are motivated by a connection to an optimal sensing problem, with proofs revised for the AoI objective.
  • Contribution: For a one-unit battery, the paper proves a renewal-based threshold policy is optimal within a broadly defined class of online policies.The sensor stores newly arrived energy when AoI is below the threshold and updates immediately otherwise.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models instantaneous status transmission powered by Poisson energy arrivals and formulates long-term average AoI minimization over feasible online update epochs subject to energy causality.

  • System model: AoI is the destination’s elapsed time since the generation timestamp of its latest received status update.The time-average AoI over [0,T] is represented as accumulated age R(T) divided by T.
  • System model: Status measurement and transmission are assumed instantaneous relative to inter-update delays, eliminating unnecessary queueing delay.A status update can be generated and transmitted at any time when sufficient energy is available.
  • Energy model: Each update consumes one normalized energy unit, and energy arrives according to a Poisson process with parameter λ, with λ = 1 used for exposition.The battery capacity is B ≥ 1, and B = ∞ denotes the infinite-battery case.
  • Problem formulation: The optimization selects update epochs S1, S2, … to minimize time-average AoI subject to energy causality.Online decisions use update history, observed energy arrivals, and the energy-harvesting statistics.
  • Problem formulation: The resulting stochastic optimization is difficult because its expectation spans all energy-harvesting sample paths and generally lacks an MDP formulation.The paper therefore seeks explicit feasible online policies rather than relying on a general dynamic-programming formulation.

III. OPTIMAL STATUS UPDATING WHEN B IS LARGE

For large batteries, the paper connects AoI minimization to an optimal sensing-scheduling problem and adapts its uniform and energy-aware policies despite a missing technical condition.

  • Connection to sensing: The sensing results require an increasing convex cost, increasing f(x)/x, and an upper-bounded f(x)/x condition.For AoI, the first two properties hold, but the upper-boundedness condition does not.
  • Connection to sensing: The AoI objective can be treated as a special sensing-scheduling case with inter-update cost f(x) = x^2/2.This cost captures the accumulated age over an inter-update interval.
  • Adaptation: The paper revises the proofs to re-establish optimality or asymptotic optimality for AoI minimization.It adapts the major policies and results from the sensing problem while supplying new proofs where necessary.
  • Adaptation: The infinite-battery analysis begins by seeking an AoI lower bound and a feasible policy attaining it through a uniform updating structure.This structure motivates the best-effort uniform policy developed in the next section.

A. Status Update with Infinite Battery

With an infinite battery, the maximum sustainable average update rate is one per unit time, yielding a 1/2 AoI lower bound; the feasible best-effort uniform policy attains optimality.

  • Lower bound: 1/2 is the lower bound on long-term average AoI obtained by replacing energy causality with the maximum average rate of one update per unit time.Uniform updates once per unit time achieve this relaxed lower bound, though they may violate energy causality.
  • Best-effort policy: The best-effort uniform policy schedules updates at sn = n and transmits only when at least one energy unit is available.If the scheduled epoch is infeasible, the sensor remains silent until the next scheduled epoch.
  • Best-effort policy: Scheduled epochs and actual update epochs can differ because some scheduled updates lack sufficient energy.This distinction preserves feasibility while retaining the uniform schedule.
  • Optimality: Theorem 1 states that the best-effort uniform status update policy is optimal when the battery size is infinite.With infinite storage, long-run averaging removes the effect of energy-harvesting fluctuations asymptotically.

B. Status Update with Finite Battery

For finite batteries, status updating must balance energy overflow against the need for temporally uniform updates. An energy-aware adaptive policy adjusts its update rate to battery level and becomes asymptotically optimal as battery size grows.

  • Policy design: Finite-battery AoI minimization requires balancing battery-overflow prevention with temporally uniform status updates.Wasted energy degrades performance, while excessive timing jitter increases time-average AoI.
  • Policy design: The energy-aware adaptive policy updates more frequently at high battery levels and less frequently at low levels.This reduces overflow when energy is abundant and avoids infeasible scheduled updates when energy is scarce.
  • Asymptotic behavior: As battery size B increases, the adaptive policy converges to the best-effort uniform policy and its long-term average AoI approaches 1/2.The parameter β tends to zero for fixed k as B approaches infinity.
  • Asymptotic behavior: Theorem 2 characterizes the adaptive policy's gap between long-term average AoI and the infinite-battery lower bound 1/2.This establishes asymptotic optimality as the battery size grows.

IV. A SPECIAL CASE: B = 1

With a one-unit battery, energy causality and Poisson arrivals motivate renewal policies whose intervals depend only on the first energy arrival. The optimal renewal policy uses a threshold rule with threshold 0.9012 and average AoI 0.9012.

  • One-unit battery setting: When B = 1, the battery is either empty or full, so a newly arriving energy unit creates the decision of when to update.No update can be scheduled while the battery is empty.
  • One-unit battery setting: For any feasible online policy, each inter-update delay Xi is at least the first post-update energy-arrival delay Γi, whose distribution is exp(1) and is i.i.d.The bound follows from energy causality and the memoryless Poisson-arrival property.
  • Renewal structure: Within uniformly bounded policies, replacing a policy with a renewal policy cannot worsen time-average AoI, and each renewal interval depends only on Γi.The renewal construction averages over status-update sample paths and then over time.
  • Threshold policy: The optimal renewal policy sets Xi to τ0 when Γi ≤ τ0 and to Γi otherwise, with τ0 = 0.9012 and long-term average AoI equal to τ0.Thus, a low AoI at energy arrival leads to storing energy until the threshold, whereas a higher AoI leads to immediate updating.

V. SIMULATION RESULTS

Simulations evaluate the proposed policies under Poisson and Markovian energy harvesting, across infinite, finite, and unit battery sizes. The results support optimality for uniform updating, convergence toward the infinite-battery lower bound with adaptive updating, and superiority of threshold updating when B = 1.

  • Infinite battery: The best-effort uniform policy's time-average AoI approaches the lower bound 0.5×10^2 s as the simulation horizon increases.At T = 500×10^2 s, the simulation differs only slightly from the analytical lower bound.
  • Finite battery: For finite batteries, the gap between adaptive-policy AoI and the lower bound decreases monotonically as B increases for each fixed k.The simulations use T = 10^5 time units and average over 10^3 sample paths.
  • Unit battery: When B = 1, the optimized best-effort uniform policy updates once every 0.4302×10^2 seconds.The adaptive policy is modified for unit capacity and optimized numerically before comparison.
  • Unit battery: When B = 1, the threshold-based policy outperforms the other two policies in both sample-path and sample-path-average comparisons.Its performance approaches its limiting value as the simulation horizon becomes sufficiently large.
  • Markovian energy harvesting: Under Markovian energy harvesting, AoI increases as the harvesting process becomes more bursty, while the proposed policies remain close to 0.5 for uniform harvesting.For B = ∞ and p0 = p1 = 1.0, the average AoI is exactly 0.5 unit time; for B = 10, adaptive updating shows the same monotonicity.
  • Markovian energy harvesting: The simulations suggest that the Poisson assumption may not be critical for policy optimality when B = ∞ or when finite B is large.Performance under more general energy-harvesting processes remains theoretically uncharacterized.

VI. CONCLUSIONS

The paper investigates optimal online status-update policies for energy-harvesting sources across infinite, finite-but-large, and one-unit battery sizes. It proposes policies for all three cases and establishes their optimality through theoretical analysis, supported by simulations.

  • The study covers infinite, finite but large, and one-unit battery cases for an energy-harvesting source.
  • Three status-updating policies are proposed and their optimality is established through theoretical analysis.
  • The proposed policies are also evaluated through simulation results.
  • The infinite- and finite-battery policies are closely related to an earlier optimal sensing problem based on long-term average sensing performance.

A. Proof of Theorem 1

The proof analyzes the slot structure induced by the uniform best-effort policy and uses Poisson arrivals to characterize infeasible intervals. Geometric interval moments then yield the limiting average-AoI expression.

  • The uniform best-effort policy partitions time into unit-length slots and groups them into intervals where scheduled updates are infeasible or feasible.Intervals u_i begin with an empty battery and end when energy becomes available; intervals v_i begin with positive energy and end when it is depleted.
  • Poisson energy arrivals make each empty-battery interval length u_i an independent geometric random variable.Its distribution is P[u_i = k] = e^{-(k−1)}(1−e^{-1}) for k = 1, 2, ....
  • The aggregate AoI over feasible intervals is expressed using f(x)=x^2/2, while the count of empty-battery intervals grows sublinearly in time.
  • The number of feasible and infeasible interval pairs over [0,T] differs by at most one.The number of v_i intervals is either K(T) or K(T)+1, depending on whether time T−1 is a feasible update epoch.
  • The normalized sums of interval lengths and squared lengths converge to finite first and second moments of the geometric distribution.These moment limits are used to show that the relevant normalized aggregate converges almost surely to 1/2.

B. Proof of Theorem 2

The proof tracks scheduled epochs, battery overflows, and infeasible updates under the adaptive policy. It compares scheduled delays with actual feasible delays using a transformed geometric tail and asymptotic sample-path arguments.

  • Scheduled update epochs are classified by interval durations 1/(1−β), 1, and 1/(1+β), together with battery overflows and infeasible epochs.
  • The scheduled time of the nth update is determined by the counts of the three interval types.Specifically, T_n = n_+/(1−β) + n_0 + n_-/(1+β).
  • Poisson arrivals allow the energy-arrival count A(T_n) to be related to scheduled updates, enabling normalization by n and passage to an asymptotic limit.
  • Actual inter-update delays differ from scheduled delays only when battery outage occurs.The proof isolates the affected delays through the count of X_n values exceeding 1/(1−β).
  • For X_n ≥ 1/(1−β), the transformed delay X̃_n = X_n(1−β) follows a geometric distribution with parameter p = 1−e^{-1/(1−β)}.Its second moment is used in the asymptotic analysis.

C. Proof of Theorem 3

The proof shows that uniformly bounded online policies can be replaced by renewal policies whose inter-update delays depend only on the first subsequent energy-arrival delay. The constructed renewal policy satisfies energy causality and achieves no larger long-term average AoI.

  • The proof uses Poisson memorylessness to relate the update process to the first energy-arrival delay.
  • Energy causality bounds the number of status updates by the total number of energy arrivals.
  • For a uniformly bounded policy, renewal policies are constructed by grouping sample paths according to the first energy-arrival delay Γ_i.
  • The constructed conditional average delay depends only on Γ_i and is at least Γ_i, ensuring feasibility under energy causality.
  • The resulting inter-update delays are independent and identically distributed, so the constructed policy is renewal over [0,T].
  • The constructed renewal policy always outperforms the original uniformly bounded policy in terms of AoI.The comparison is obtained through conditional-expectation bounds, Jensen’s inequality, and the minimum over feasible renewal policies whose delays depend only on Γ_i.

D. Proof of Theorem 4

The proof reduces the renewal-policy optimization to a parameterized problem whose constrained solution has a threshold form. Solving the resulting scalar condition yields the unique optimal threshold λ∗ = 0.9012.

  • Optimization reformulation: The proof reformulates long-term average AoI minimization as a parameterized optimization over X(τ) subject to X(τ) ≥ τ.The parameterized problem is solved for fixed λ before imposing the scalar optimality condition.
  • Scalar optimality condition: Proposition 3 selects the optimal solution as the smallest nonnegative λ∗ satisfying p(λ∗) = 0.The proof solves the fixed-λ problem, substitutes its solution into p(λ), and chooses the minimum root.
  • Threshold structure: The Lagrangian introduces a nonnegative multiplier function µ(τ), and complementary slackness yields two cases for the optimal X(τ).The cases are X(τ) ≠ τ, which implies X(τ) = λ, and X(τ) = τ, which requires τ ≥ λ.
  • Threshold structure: Combining the cases gives an X(τ) structure corresponding to a threshold policy.The resulting policy switches between waiting until a threshold and updating according to the observed energy-arrival interval.
  • Optimal threshold: The scalar equation p(λ) = 0 has the unique solution λ∗ = 0.9012, which achieves the minimum AoI of the optimization problem.The identified λ∗ is the optimal threshold parameter for the proved policy.
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