# Capacity and idle power limit assigned carbon reductions from daily batch scheduling

## Summary

How much does scheduling a daily batch by historical grid intensity help when computing capacity and idle power are explicit? We evaluate seasonal clock-time profiles trained on 2023 public Great Britain data, with equal work and equal energy in every comparison. At {{R2.y2024.u50.utilization_percent}}% utilization, assigned carbon falls by {{R2.y2024.u50.seasonal_percent}}% in 2024 and {{R2.y2025.u50.seasonal_percent}}% in 2025 relative to uniform processing. Setting idle power to {{R2.idle_example_percent}}% of maximum power reduces these advantages to {{R2.y2024.u50.seasonal_half_idle_percent}}% and {{R2.y2025.u50.seasonal_half_idle_percent}}%. Some dates have higher assigned carbon after shifting. These results describe a fixed historical workload benchmark and an accounting signal; they do not measure avoided emissions.

## Claims

- **C1:** In the frozen Great Britain daily-batch benchmark, seasonal profiles trained on 2023 data reduce assigned carbon relative to uniform processing by {{R2.y2024.u50.seasonal_percent}}% and {{R2.y2025.u50.seasonal_percent}}% at {{R2.y2024.u50.utilization_percent}}% utilization with zero idle power, but by {{R2.y2024.u50.seasonal_half_idle_percent}}% and {{R2.y2025.u50.seasonal_half_idle_percent}}% when idle power is {{R2.idle_example_percent}}% of maximum; at {{R2.y2024.u87_5.utilization_percent}}% utilization with zero idle power, reductions are {{R2.y2024.u87_5.seasonal_percent}}% and {{R2.y2025.u87_5.seasonal_percent}}%, while the half-utilization policy increases assigned carbon on {{R2.y2024.u50.higher_days}} of {{R2.y2024.u50.days}} and {{R2.y2025.u50.higher_days}} of {{R2.y2025.u50.days}} evaluation dates, respectively, conditional on the specified processor model and estimated average intensity data.

## Methods

**Design and scope.** The [registered plan](prereg:88e842c32d3cdb4566093a5280d940df713a3d5bdb86b680533b239d956be6a7) fixes a secondary analysis before these computations. The public records and related earlier analyses had already been inspected. This is not a blind holdout or a replication of an earlier numeric claim. Earlier analyses on the same snapshot considered isolated jobs without shared capacity; this analysis instead gives a single daily batch one capacity budget and a common completion deadline. The entire prespecified grid is reported, with no inferential test or selected significance threshold. There were no analysis deviations.

**Public source.** The [National Energy System Operator Carbon Intensity API](https://carbonintensity.org.uk/) supplies estimated national average carbon intensity in g CO2/kWh. Its field named `actual` is an estimate based on generation information, not a direct measurement of the emissions caused by an extra computing job. The immutable copy under `data/raw/` was retrieved in October 2026 under the provider's CC-BY-4.0 terms. [The source manifest](data/SOURCES.json) records every original public URL, retrieval time, file size and both compressed and uncompressed SHA-256 digests. No new request is needed to reproduce the analysis. Digest verification precedes processing.

We parse UTC intervals, require aligned half-hours of 30 minutes, merge identical duplicates, fail on contradictions, and exclude any date missing a finite, nonnegative numeric `actual` value for any of its 48 slots. No values are interpolated. Training uses complete 2023 dates; evaluation uses complete 2024 and 2025 dates separately. At least 95% of dates must remain in each year. The source audit enumerates every excluded date and invalid value. UTC avoids daylight-saving days with unequal slot counts; it does not describe local-clock contractual deadlines.

**Workload and policies.** A divisible batch becomes available at midnight and must complete before the next midnight on a normalized capacity-one processor. It requires k full-capacity half-hours, for each integer k from 1 to 48. Utilization is $u=k/48$. Preemption has no cost, there are no competing jobs, and no work crosses a day boundary. Capacity and batch size are fixed within each comparison. Profiles rank half-hour slots by mean 2023 intensity, either over the entire year or within the four meteorological seasons. The seasonal policy uses December-February, March-May, June-August, and September-November groups. Every slot of a profile has the same training-date denominator. Earliest UTC time breaks exact ties. The k lowest-ranked slots run at full capacity and all others are idle. No evaluation-day observation enters either trained policy.

Uniform processing, the primary baseline, runs at utilization u in every slot. The immediate baseline processes at full capacity in the first k slots. An unattainable hindsight reference chooses the k lowest actual-intensity slots on each date. Selecting the lowest k values minimizes the linear objective subject to a fixed k-slot budget: exchanging a selected larger value for an unselected smaller value cannot increase cost. This elementary optimizer is not proposed as a new algorithm or a usable forecast.

**Power and accounting.** Maximum power is normalized to 1 kW. For idle fraction $b\in\{0,0.25,0.5,0.75\}$ and processor utilization $x_t$, power is $P_t=b+(1-b)x_t$. The processor stays powered throughout the day. All policies perform the same work and use the same daily energy, $E=24[b+(1-b)u]$ kWh. We assign carbon by $A=0.5\sum_{t=1}^{48}P_t c_t$, where $c_t$ is frozen grid-average intensity. The annual percentage reduction is $100(1-\sum_d A_{d,\mathrm{policy}}/\sum_d A_{d,\mathrm{baseline}})$; it is not the unweighted mean of daily percentages. Across scenarios, utilization changes workload energy, so the percentages do not identify an optimal hardware size or an optimal absolute-emissions operating point.

The uniform-baseline percentage reduction obeys an algebraic scaling identity under this model. If $S_0$ is the zero-idle result, then $S_b=S_0(1-b)u/[b+(1-b)u]$. The identity isolates the imposed idle-power accounting effect; it is not empirical evidence for an actual server's power curve. For b below one, the sign of each daily difference is unchanged. All differences and tie counts use exact rational arithmetic before display rounding.

**Reporting and verification.** We prespecified highlighted utilizations of 25%, 50%, 75%, 87.5%, and 100%, and report all 48 utilization values, four idle-power cases and three scheduling policies against both baselines. Daily adverse counts include ties in their denominator. Monthly minimum and maximum reductions and nearest-rank daily 10th and 90th percentiles describe variation within the finite record. They are not confidence or prediction intervals. There is no probabilistic uncertainty model for the source estimates or future workloads. Detailed results retain six decimals for auditability; manuscript values use two decimals, without implying environmental accuracy at that precision.

`sh code/run` runs the analysis and a separate validator. The validator does not import the analysis program. It reparses raw records, reconstructs profiles and all daily costs, checks every annual scenario, checks monthly and quantile summaries, and tests energy conservation, the idle scaling identity, constant-signal cases and hindsight bounds. Small-vector optima are also checked by exhaustive enumeration. These are author-side checks, not independent ledger verification.

## Results

The snapshot supplies {{R2.training_days}} complete training dates and {{R2.evaluation_days.2024}} and {{R2.evaluation_days.2025}} evaluation dates. Table 1 gives the primary seasonal-versus-uniform comparison. At half utilization, its reductions are {{R2.y2024.u50.seasonal_percent}}% and {{R2.y2025.u50.seasonal_percent}}% with zero idle power, falling to {{R2.y2024.u50.seasonal_half_idle_percent}}% and {{R2.y2025.u50.seasonal_half_idle_percent}}% in the high-idle example.

**Table 1.** Assigned carbon reduction against uniform processing (%), with fixed seasonal profiles. Idle power in the final columns is {{R2.idle_example_percent}}% of maximum. Full utilization leaves no temporal freedom.

| Utilization (%) | 2024, zero idle | 2025, zero idle | 2024, high idle | 2025, high idle |
|---|---:|---:|---:|---:|
| {{R2.y2024.u25.utilization_percent}} | {{R2.y2024.u25.seasonal_percent}} | {{R2.y2025.u25.seasonal_percent}} | {{R2.y2024.u25.seasonal_half_idle_percent}} | {{R2.y2025.u25.seasonal_half_idle_percent}} |
| {{R2.y2024.u50.utilization_percent}} | {{R2.y2024.u50.seasonal_percent}} | {{R2.y2025.u50.seasonal_percent}} | {{R2.y2024.u50.seasonal_half_idle_percent}} | {{R2.y2025.u50.seasonal_half_idle_percent}} |
| {{R2.y2024.u75.utilization_percent}} | {{R2.y2024.u75.seasonal_percent}} | {{R2.y2025.u75.seasonal_percent}} | {{R2.y2024.u75.seasonal_half_idle_percent}} | {{R2.y2025.u75.seasonal_half_idle_percent}} |
| {{R2.y2024.u87_5.utilization_percent}} | {{R2.y2024.u87_5.seasonal_percent}} | {{R2.y2025.u87_5.seasonal_percent}} | {{R2.y2024.u87_5.seasonal_half_idle_percent}} | {{R2.y2025.u87_5.seasonal_half_idle_percent}} |
| {{R2.y2024.u100.utilization_percent}} | {{R2.y2024.u100.seasonal_percent}} | {{R2.y2025.u100.seasonal_percent}} | {{R2.y2024.u100.seasonal_half_idle_percent}} | {{R2.y2025.u100.seasonal_half_idle_percent}} |

**Adverse dates and temporal variation.** At half utilization, seasonal scheduling increases assigned carbon on {{R2.y2024.u50.higher_days}} of {{R2.y2024.u50.days}} dates in 2024 and {{R2.y2025.u50.higher_days}} of {{R2.y2025.u50.days}} dates in 2025. With zero idle power, the daily lower and upper reported percentiles are {{R2.y2024.u50.daily_p10_percent}}% and {{R2.y2024.u50.daily_p90_percent}}% in 2024, and {{R2.y2025.u50.daily_p10_percent}}% and {{R2.y2025.u50.daily_p90_percent}}% in 2025. The corresponding monthly ranges are {{R2.y2024.u50.monthly_min_percent}}% to {{R2.y2024.u50.monthly_max_percent}}%, and {{R2.y2025.u50.monthly_min_percent}}% to {{R2.y2025.u50.monthly_max_percent}}%. These summarize observed dispersion, not sampling uncertainty.

**Baselines and information.** At half utilization with zero idle power, the annual-profile reductions against uniform processing are {{R2.y2024.u50.annual_percent}}% and {{R2.y2025.u50.annual_percent}}%; the hindsight references are {{R2.y2024.u50.hindsight_percent}}% and {{R2.y2025.u50.hindsight_percent}}%. Using immediate processing as the baseline instead gives seasonal reductions of {{R2.y2024.u50.against_immediate_percent}}% and {{R2.y2025.u50.against_immediate_percent}}%. Every alternative, including unfavorable dates, is in [the full grid](results/R1.json) and [daily table](results/daily.csv.gz).

The validator reconstructs {{validation.raw_reconstructed_daily_rows}} daily policy-budget rows, {{validation.annual_scenarios_checked}} annual scenarios, {{validation.monthly_values_checked}} monthly values and {{validation.daily_quantiles_checked}} daily quantiles. It also passes {{validation.exact_idle_scaling_checks}} exact scaling checks, {{validation.exhaustive_small_vector_optima}} exhaustive small-vector optimum checks, and {{validation.work_energy_constant_signal_fixtures}} energy and constant-signal fixtures. [The source audit](results/source-audit.json) and [training profiles](results/profiles.json) expose the remaining processing decisions.

## Discussion

The benchmark quantifies two distinct constraints. A higher daily capacity occupation leaves fewer low-intensity slots to select. Idle power adds a component that the tested scheduler cannot move, diluting the benefit attributed to flexible computation. The latter effect follows directly from the assumed power curve, so these results should not be interpreted as a measurement of idle losses in deployed hardware. The fact that the high-idle percentages at quarter utilization are smaller than at half utilization also warns against assuming that a lower workload fraction always raises the percentage improvement in total assigned carbon.

[Wiesner et al. (2021)](arxiv:2110.13234) examine temporal shifting under time constraints, scheduling strategies and forecast accuracy across several regions. [Sukprasert et al. (2024)](arxiv:2306.06502) quantify ideal and practical shifting limits and explicitly examine spare capacity in spatial migration. Neither the general value of flexibility nor the elementary scheduling rule is new. The added result here is a fully auditable daily capacity and idle-power sensitivity surface for the specified newer Great Britain snapshot, with fixed training profiles, two baselines and adverse dates retained. It does not reproduce their global numerical estimates.

[Sukprasert et al. (2024), on intensity signals](doi:10.1145/3632775.3661953), show why average and marginal carbon objectives can disagree. Our accounting reduction therefore supports no claim about the generators that would respond to shifted demand. A useful deployment test would measure actual batch arrivals and processor power, archive predictions when available, and evaluate scheduling against a consequential electricity model. It would also compare the gain with delay costs and possible shutdown strategies.

## Limitations

The workload is fully divisible, all work is available at midnight, switching is free, and a single processor runs without other demand. Real jobs may be nonpreemptible, have heterogeneous power, arrive throughout the day, or contend for memory and network capacity. The uniform baseline assumes useful work can be spread fractionally at proportional throughput; the immediate baseline tests only one other operating convention. The model has no sleep state, queue uncertainty, startup loss, storage, migration, cooling variation, embodied carbon or capacity investment. The idle fractions are scenarios, not hardware measurements.

National intensity aggregates local differences and may have been revised before retrieval. Past-only calendar profiles prevent direct look-ahead into the evaluation dates but do not recreate the data vintage visible in real time. The related raw data had been seen before registration; no untouched test-set claim is made. Excluding incomplete days can alter representativeness. Only one national grid and two evaluation years are studied, and neither future performance nor a universal utilization trend is inferred. Rational arithmetic establishes the computations conditional on the snapshot and assumptions, not the accuracy of those assumptions.

## Provenance

A GPT-family model, gpt-6, designed this secondary analysis, wrote the code and manuscript, inspected the literature, and performed the validation. Public CC-BY-4.0 NESO raw responses and source metadata were reused unchanged from an earlier GPT-family analysis; no earlier analytical code or results are required. New code uses the Python standard library in a pinned official container. No person supplied prose, code, private records or unpublished data. The bundle contains original analysis text and code, public numerical data with attribution, and citations rather than copies of research articles.