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Assessing Fixed-Batch Reporting for Deadline-constrained Inference in Intermittently Powered IoT

Onel L. A. López

arXiv:2609.06585v1eess.SYcs.PFcs.SY

TL;DR

Fixed batch size determines the tradeoff between earlier access to partial information and reporting actions or retained state, even when all reports yield the same final posterior. The paper develops an exact framework for comparing timely inference across these configurations, and shows that the preferred batch size depends on inference and operating conditions.

  • Problem

    Batch size trades earlier access to partial information against reporting actions and retained device state, while application value depends on when reports become usable.

    Source: §I · pp. 1–2

  • Method

    The paper combines CRPS-based Bayesian-risk reductions with decision-weighted report availability and exact Markov-reward models for fixed batches under intermittent energy and unreliable delivery.

    Source: §I · p. 2

  • Results

    The preferred batch size can change with the prior, decision-time profile, energy-arrival statistics, link reliability, and accumulation cost.

    Source: §I · p. 2

  • Takeaways & Limitations

    Minimizing report count or deadline-free energy workload need not maximize long-run timely value.

    Source: §I · p. 2

Abstract

from arXiv · show

Energy-harvesting Internet of Things (IoT) devices must decide whether to transmit each observation immediately or accumulate several observations before reporting to the edge. Early reports make information from initial observations available sooner at the edge but require more transmission actions, whereas larger batches save reports while delaying that information and retaining more state. We develop a unified analytical framework to compare this fixed-batching choice in scenarios with repeated, deadline-constrained inference cycles. Every configuration processes the same ordered observations and yields the same final posterior if all reports arrive, enabling a controlled comparison of the persistent state, actions, and energy costs induced by batching. Using reductions in Bayesian risk under the continuous ranked probability score (CRPS), we express long-run timely value exactly as a statistical-value-weighted sum of expected decision-weighted report availabilities. This separates the inferential value of observations from their availability at application-relevant decision times. We construct exact Markov-reward models accounting jointly for intermittent harvesting, finite storage, energy carry-over, unreliable delivery, retransmissions, and deadlines. We prove that the long-run regime is well defined and derive exact batch-size comparisons, timing benchmarks, and a decomposition of harvesting-law effects into within-cycle and carry-over contributions. Directly evaluable Gaussian and Gaussian-mixture belief specializations show that the preferred batch size can change with the prior, the decision-time profile, link reliability, harvested-energy statistics, and accumulation cost.

I. Introduction M

Fixed batching trades earlier access to partial information against reporting actions and retained device state, even when every configuration processes the same observations and yields the same final posterior. The paper compares these architectures under deadline-constrained inference with intermittent energy and derives conditions for their long-run timely value.

I. Introduction M

  • The framework compares fixed batch sizes while holding the ordered observations and final posterior constant if all reports arrive.Batch size is set before deployment, rather than selected as a runtime policy.Source: §I · p. 2
  • Markov-reward models account for stochastic harvesting, finite storage and overflow, batch-dependent costs, unreliable delivery, deadlines, and energy carried across cycles.The analysis separates harvesting-law effects within the current cycle from effects transmitted through carry-over energy.Source: §I · p. 2
  • Exact pairwise comparisons, value-based ordering conditions, and timing benchmarks complement Gaussian and Gaussian-mixture specializations of inference value.The paper identifies prior, decision-time profile, link reliability, harvesting statistics, and accumulation cost as factors that can change the maximizing batch size.Source: §I · p. 2
  • The preferred batch size can vary with the prior, decision-time profile, energy-arrival statistics, link reliability, and accumulation cost.Minimizing report count or deadline-free energy workload need not maximize long-run timely value.Source: §I · p. 2

B. Device Architectures

  • Single-observation, intermediate, and full-cycle batches create distinct device architectures with different persistent state, control logic, and reporting costs.The device may use a one-record outbox, a persistent batch accumulator, or a full-cycle accumulator.Source: B. Device Architectures · p. 3

C. Intermittent Operation

  • The work-conserving operation rule acquires observations when energy and remaining time permit, and retries pending reports until acknowledgment or the deadline.Actions depend on energy, reporting state, and remaining time, not on decision weights or realized observations.Source: C. Intermittent Operation · p. 4

D. Edge Belief

The edge updates its belief as batch reports arrive, so report timing determines which posterior is available at each decision boundary. The section separates each batch’s inferential value from its decision-weighted availability and uses that separation to decompose timely CRPS.

D. Edge Belief

  • The edge uses the posterior for the first j batches only after their reports arrive, while the sequence of posterior updates reflects the cumulative sample counts.The first j batch means are jointly sufficient for the corresponding observations under the stated Gaussian observation model.Source: D. Edge Belief · p. 4
  • A report’s decision-weighted availability is the total decision weight assigned to boundaries when that report is available before the cycle deadline.Earlier successful reception increases its within-cycle availability; an undelivered report has no timely availability.Source: D. Edge Belief · p. 4

A. Per-Cycle CRPS and Exact Decomposition

  • The decision-weighted timely CRPS decomposes into the prior score and signed CRPS changes from each batch, weighted by how long each gain is available.The pathwise decomposition requires neither stationarity of energy nor averaging across cycles.Source: A. Per-Cycle CRPS and Exact… · p. 5

B. Statistical Value of Observations and Batches

The section separates the statistical value of batches from the energy- and delivery-constrained availability of their reports. It uses that separation to evaluate long-run timely value through repeated-cycle energy dynamics.

B. Statistical Value of Observations and Batches

  • Each batch’s expected value Γb,j is the reduction in Bayesian CRPS risk it supplies, and is nonnegative.Individual realized reductions may be negative, but their expected value is nonnegative.Source: B. Statistical Value of Observations… · p. 5

C. Expected and Long-Run Timely Value

  • Normalized timely value lies between 0 and 1, with the upper endpoint representing all observations available at every positively weighted decision boundary.The lower endpoint corresponds to no report being available at a positively weighted boundary.Source: C. Expected and Long-Run Timely Value · p. 6
  • The long-run evaluation identity separates prior-dependent batch values from architecture- and harvesting-dependent decision-weighted report availabilities.Report availability incorporates delivery reliability, harvesting, finite storage, retransmissions, deadlines, and architecture-dependent costs; report absence or delay is noninformative about the observations.Source: C. Expected and Long-Run Timely Value · pp. 5–6

IV. Report Timeliness

  • Cycle-boundary energy dynamics yield long-run weighted report availabilities; under stated sufficient conditions, the energy law forgets its initialization geometrically fast.A finite reachable energy grid permits exact matrix propagation; otherwise, evaluation requires a measure-valued integral recursion.Source: §IV · p. 6, A. Weighted-Availability Recursion and… · pp. 6–7

B. Long-Run Harvesting-Law Comparison

The harvesting-law comparison separates within-cycle effects from changes in energy carried across cycles. Its example shows that regularity and burstiness can favor different batch architectures, while a fixed-rule comparison has a specific scope boundary.

B. Long-Run Harvesting-Law Comparison

  • Changing the harvesting law has distinct within-cycle and carry-over contributions to long-run value.The within-cycle term holds the cycle-start energy law fixed; the carry-over term captures the induced change in stored energy across cycles.Source: B. Long-Run Harvesting-Law Comparison · p. 7
  • The harvesting-dominance caveat applies to fixed-rule performance; with energy admission, curtailment, or optional idling, a controller can emulate a dominated source.That emulation is not generally possible for crossing harvesting laws.Source: B. Long-Run Harvesting-Law Comparison · p. 7
  • With equal-mean arrivals, deterministic one-unit harvesting yields higher value for A(1), while two-unit bursts can favor A(2) by enabling rapid successive actions.Bursty zero-arrival gaps can instead delay A(1)’s sequence of actions; thus regularity favors frequent low-threshold actions, while burstiness can favor higher-threshold batching.Source: B. Long-Run Harvesting-Law Comparison · p. 8

V. Architecture Comparison

The comparison considers fixed-b architectures at three levels: long-run performance under harvesting, energy-unconstrained timing, and deadline-free energy workload.

V. Architecture Comparison

  • Fixed-b architectures are compared by actual long-run harvesting performance, an energy-unconstrained timing limit, and deadline-free energy workload.The workload benchmark quantifies expected energy to complete observations and reports without accounting for timely value.Source: §V · p. 8

A. Long-Run Ordering under Energy Harvesting

Under harvesting, architecture value is a statistical-value-weighted comparison of report availabilities. The energy-unconstrained benchmark isolates timing and reporting limits, including the effect of deadlines and decision-time weights.

A. Long-Run Ordering under Energy Harvesting

  • Long-run architecture gaps are determined by statistical-value-weighted differences in expected decision-weighted report availabilities.A sufficient model-independent ordering condition is that one architecture’s availabilities are no lower for every observation.Source: A. Long-Run Ordering under Energy… · p. 8
  • For every nonzero diminishing marginal-value profile, the stated prefix condition gives robust architecture ordering, while the exact weighted test can be less restrictive.Some report-availability disadvantages may be offset by advantages accumulated over other observations.Source: A. Long-Run Ordering under Energy… · pp. 8–9

B. Energy-Unconstrained Timing Benchmark

  • Even with unlimited energy and a reliable link, report j cannot arrive on time when j exceeds T/(b + 1); delivering all observations on time requires T ≥ N + N/b.Source: B. Energy-Unconstrained Timing Benchmark · p. 9
  • With unlimited energy, the gap from oracle value reflects sequential acquisition, batching and report-attempt time, link failures, and deadline truncation; the remaining gap to harvested performance isolates energy blocking.Batch-dependent costs affect that latter gap through the state distribution and actual report availabilities.Source: B. Energy-Unconstrained Timing Benchmark · p. 9
  • The reliable-link comparison depends on exact integer decision boundaries; a continuous relaxation can incorrectly favor larger batches.Source: B. Energy-Unconstrained Timing Benchmark · p. 9
  • With reliable communication, unlimited energy, and enough time to deliver all observations, batching changes early availability but not terminal inference.Source: B. Energy-Unconstrained Timing Benchmark · p. 10

C. Deadline-Free Energy Workload

The deadline-free benchmark measures the energy needed to complete a full sensing/reporting sequence beyond the cycle deadline. Its workload comparison separates observation costs from repeated-report costs, whose impact grows as link reliability falls.

C. Deadline-Free Energy Workload

  • This benchmark asks how much energy completes the entire sensing/reporting sequence when it may continue beyond the cycle deadline.It is distinct from the deadline- and decision-time-constrained benchmark.Source: C. Deadline-Free Energy Workload · p. 10
  • A full N-observation sequence requires N observation actions and, in expectation, N/(b pR) report attempts.Each of the N/b reports is attempted until delivery; the expected number of attempts per report is 1/pR.Source: C. Deadline-Free Energy Workload · p. 10
  • The deadline-free workload includes observation energy and expected report energy; Ework_b ≤ T E[Zt] is necessary, but not sufficient, for completing one full workload per cycle.The repeated-reporting term is amplified by 1/pR relative to the observation term.Source: C. Deadline-Free Energy Workload · p. 10
  • For b < b′, the reporting-cost difference is positive and grows as pR falls, though a lower observation cost for b can offset it.When the observation-cost difference and reporting-term numerator have the same weak sign, workload ordering is fixed over pR ∈ (0, 1]; with opposite signs, their sum determines a crossover.Source: C. Deadline-Free Energy Workload · p. 10

VI. Prior-Specific Statistical-Value Profiles

The prior determines the statistical value assigned to observations, while decision timing and energy and communication dynamics determine report availabilities. Gaussian and Gaussian-mixture specializations show how prior information and decision-time profiles shape batch-size comparisons.

Prior-specific value profiles

  • The framework separates prior-dependent statistical weights from decision-weighted report availabilities, which remain unchanged when the prior changes under the value-independent policy.The availabilities reflect the fixed decision-time profile and energy and communication dynamics.Source: §VI · p. 10
  • The examples use a Gaussian prior and a two-component Gaussian mixture that can represent a nominal regime and an unobserved shifted fault, overload, or blocked-channel regime.For the mixture, the regime must be inferred jointly with the continuous within-regime condition.Source: §VI · p. 10, A. Gaussian Prior · p. 11

Information limits

  • For either prior, the first report carries a value fraction tending to b/N in the weak-information limit and to one in the high-information limit.With uniform decision-time weights, the smaller batch is preferred in the weak-information limit when bb′ > N, the larger batch when bb′ < N, and they tie when bb′ = N; b = 1 is the unique high-information optimizer.Source: §VI · pp. 10–11

Gaussian prior

  • For the Gaussian prior, successive observations have strictly diminishing expected CRPS value, so small batches expose more valuable early evidence sooner while larger batches save report transactions.Source: A. Gaussian Prior · p. 11
  • For Gaussian weights, if the decision-weighted availability advantage is non-increasing across observations, the pairwise value difference is nondecreasing in κ and the preferred architecture changes exactly once when the endpoint condition holds.The condition makes the availability advantage strongest for early observations, whose normalized statistical value rises with κ.Source: A. Gaussian Prior · p. 11
  • If the Gaussian availability-difference sequence is not non-increasing, the exact comparison remains available, but monotonicity in κ is not guaranteed.Source: A. Gaussian Prior · p. 11

B. Gaussian-Mixture Prior: Regime Uncertainty

The mixture-prior analysis links observation-value timing to batch boundaries, while the numerical comparisons show that preferred batch size varies with the prior, delivery reliability, and accumulation cost.

B. Gaussian-Mixture Prior: Regime Uncertainty

  • Mixture-risk evaluation separates residual within-regime uncertainty from additional regime uncertainty, uses one-dimensional quadrature for each R_m, and reduces to the single-Gaussian risk when the components coincide.A delayed value peak can arise when several observations are needed to identify a rare regime; its diagnostic increases with regime rarity, observation noise, and component overlap, and decreases with component separation.Source: B. Gaussian-Mixture Prior: Regime… · p. 11
  • For a delayed mixture-value profile, the favored batch size need not be full batching: the boundary criterion is minimized at b = 1 for r = 1, b = N for r = N, and uniquely at b = r when interior r divides N.Under terminal-only weighting, architectures tie when they deliver the relevant observation by the deadline; otherwise, architectures that miss it have zero value.Source: B. Gaussian-Mixture Prior: Regime… · p. 12

VII. Numerical Architecture Comparisons

  • The numerical comparisons use prior-dependent normalized marginal values and combine them with decision-weighted report availabilities; the default setup uses N = 4, T = 8, B = 8, cR = 2, pR = 0.8, and uniform weighting.The larger-accumulator observation-cost premium is ξacc = cO(N)/cO(1) − 1; the model reflects the extra persistent state needed for a wider running sum and counter.Source: §VII · p. 12

A. Prior-dependent batching regimes

  • The Gaussian profiles shift normalized statistical value from front-loaded at higher κ to more evenly distributed at lower κ, while the rare-mixture profile is back-loaded; under uniform weighting, the A(1)-to-A(2) crossover is κ⋆ ≃ 0.69155.The rare-mixture profile’s delayed regime-identification diagnostic is m† ≈ 11.4 > N.Source: A. Prior-dependent batching regimes · pp. 12–13
  • High- and low-information Gaussian profiles prefer A(1) and A(2), respectively, while the rare Gaussian mixture prefers A(4), whose timely value is V⋆(4) = 1/2 for every profile.A(2) and A(3) have the same report count and average observation lifetime, but expose different observations earlier, so their ordering depends on the statistical-value profile.Source: A. Prior-dependent batching regimes · p. 13

B. Link reliability and action-cost effects

  • With unreliable links, retransmissions favor fewer reports, while improving reliability restores the value of early reporting; under uniform weighting, A(2) crosses to A(1) at pR ≃ 0.793.Under terminal-only weighting, A(3) is preferred below its crossing with A(4) at pR ≃ 0.578, A(4) above it, and all architectures tie at pR = 1.Source: B. Link reliability and action-cost… · p. 13
  • At ξacc = 0.5, the deadline-free workload screen selects b = 4, but timely value selects b = 2 for the Gaussian profile and b = 4 for the mixture.As accumulation cost rises, the workload selector changes from A(4) to A(2) at ξacc = 1.25; the Gaussian and mixture timely-value selectors change within ξacc ∈ [0.860, 0.865] and ξacc ∈ [0.975, 1.000], respectively.Source: B. Link reliability and action-cost… · p. 14

C. Harvesting level and burstiness

The preferred fixed batch size varies with energy abundance and burstiness, the statistical-value profile, and the application’s decision-time weighting. Across these comparisons, the harvesting conditions and the timing of observation value jointly shape the timely-value advantage of batching.

C. Harvesting level and burstiness

  • Under the mixture profile, the preferred batch grows from A(1) to A(4) as mean harvested energy rises, while the Gaussian profile alternates between A(1) and A(2).For the mixture, A(1) is preferred for z0 ≤ 0.4, A(2) for 0.5–0.7, A(3) for 0.8–0.9, and A(4) for z0 ≥ 1; the Gaussian profile favors A(1) through z0 = 1.1, A(2) for 1.2–1.4, and A(1) again from 1.5.Source: C. Harvesting level and burstiness · p. 14
  • At fixed mean energy, A(1) remains preferred under the Gaussian profile, while the mixture favors A(4) from zpk = 1.25 onward.For A(4), the harvesting-law gain is approximately (0.003, 0.007) in direct and carry-over terms; the larger carry-over gain reverses the A(3)–A(4) ordering only for the late-valued mixture profile.Source: C. Harvesting level and burstiness · pp. 14–15

D. Decision-time weighting

  • Shifting decision weight toward the terminal boundary favors larger batches, but harvesting conditions determine how far the preferred batch size increases.Under the Gaussian profile, scarcity preserves A(1), while greater energy abundance permits stepwise increases; the mixture favors moderate batching under scarcity and full batching otherwise, with little sensitivity to the weighting shift.Source: D. Decision-time weighting · pp. 15–16

VIII. Conclusions and Outlook

  • The framework compares fixed-batch architectures using the same ordered observations and final posterior, while accounting for each architecture’s persistent state, actions, and energy costs.Its timely-value metric combines CRPS-based Bayesian-risk reductions with decision-weighted report availability, and its Markov-reward models include harvesting, finite storage, carry-over, unreliable delivery, retransmissions, and deadlines.Source: §VIII · p. 16
  • Early informative observations and reliable, inexpensive communication generally favor smaller batches when earlier availability is rewarded; later-concentrated value and costly reporting can favor larger batches.Accumulation costs and energy-arrival timing can alter these tendencies, and burstiness can change the preferred batch size even at the same average harvested energy.Source: §VIII · p. 16
  • Minimizing report count or energy use without a deadline need not maximize long-run timely value because neither criterion captures when informative reports arrive.The supported conclusion is to consider inferential value, decision-time profile, communication reliability, deadlines, device costs, and the full energy-arrival process together.Source: §VIII · p. 16

Appendix

The appendix establishes risk-improvement identities, conditions for a well-defined long-run energy regime, and formulas for report availability. Its proofs use conditional-expectation arguments, Markov-chain contraction, and delivery-attempt calculations.

B. Proof of Lemma 1

  • Nested conditional expectations yield nonnegative CRPS risk reductions for each successive observation block, and summing them telescopes from prior risk to final posterior risk.Strict reduction follows when the observations are dependent on the hidden variable, so the posterior differs from the prior with positive probability.Source: B. Proof of Lemma 1 · p. 18

C. Proof of Proposition 2

  • A finite-step minorization of the cycle-energy transition kernel gives strict total-variation contraction and a unique invariant energy law.The proof establishes the minorization under each of its stated conditions; one sufficient route bounds the workload and allows energy to accumulate during favorable harvest-and-delivery cycles.Source: C. Proof of Proposition 2 · p. 18

D. Proof of Proposition 3

  • The harvesting-law effect is expressed using direct and carry-over terms, with the carry-over contribution represented through the cycle-start energy distribution.The proof obtains these expressions by applying an integral identity to the relevant energy-value functions.Source: D. Proof of Proposition 3 · p. 18
  • Expected decision-weighted report availability is obtained from report arrival boundaries; with retransmissions, the attempts needed for j deliveries follow a negative-binomial distribution.For a reliable link, every report takes one attempt, giving report boundaries at j(b + 1).Source: F. Proof of Proposition 5 · p. 18

H. Proof of Proposition 6

This proof characterizes Bayesian-risk behavior in low- and high-information limits. For Gaussian priors and observations, the coefficient governing the low-information expansion is positive and finite.

H. Proof of Proposition 6

  • For the stated Gaussian cases, the coefficient CH is positive and finite, and the low-information risk expansion is Rm = R0 − mCH/σ² + o(σ⁻²).The proof establishes the coefficient’s bounds using Gaussian-tail integrability and nondegeneracy.Source: H. Proof of Proposition 6 · p. 19
  • As observation noise tends to zero, posterior risk Rm tends to zero for every m ≥ 1, while q1 tends to R0 and qm tends to zero for m ≥ 2.The proof uses these limits to obtain the corresponding limiting value expressions.Source: H. Proof of Proposition 6 · p. 19

I. Proof of Corollary 2

The proof uses information-limit behavior and decision-weight comparisons to characterize architecture differences. It also derives a Gaussian-mixture CRPS expression and a monotonicity result that constrains possible sign changes.

I. Proof of Corollary 2

  • In the high-information limit, ω1 tends to 1 and ωm tends to 0 for m ≥ 2; the proof substitutes the low- and high-information limits into the reliable-link comparison identity.The supplied proof passage identifies these substitutions but does not state the resulting comparison explicitly.Source: §I · p. 19
  • Under uniform decision weights, the reliable-link comparison reduces to the stated uniform-weight expression; under terminal-only weights, all observations have equal availability.The proof substitutes the corresponding availability weights into the architecture-difference identity.Source: §I · p. 19
  • For ordered architecture differences, the comparison is monotone, and opposite endpoint signs imply exactly one zero crossing.The stated monotonicity is strict unless all the corresponding differences are equal.Source: K. Proof of Corollary 3 · p. 19
  • For the Gaussian-mixture belief, conditioning on the sample mean gives a two-component Gaussian-mixture posterior, whose conditional Bayes risk is computed from two independent posterior draws.Averaging the absolute difference between those draws over component indices and the sample mean yields the CRPS expression.Source: L. Proof of Proposition 8 · pp. 19–20
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