A Theory of Probabilistic Power Provisioning for Data Centers with Distributed Energy Storage

Abstract: The growing power demands and variability of AI workloads make electrical power delivery a critical constraint in data-center operation. Distributed energy storage can reduce the power capacity required to support stochastic loads, but its benefits depend fundamentally on the statistics and time scales of demand. This paper develops a probabilistic framework that jointly characterizes provisioned power, energy-storage capacity, and the probability of overdraw. We show that storage-assisted provisioning separates into two operating regimes. In the Small Battery Region, overdraw is dominated by short-lived demand excursions and storage provides nearly linear reductions in the required power margin. In the Large Battery Region, overdraw results from sustained demand fluctuations over longer spans of time, and the required margin exhibits diminishing returns with storage. For this regime we introduce effective power, an analogue of effective bandwidth that captures the temporal statistics of the demand and gives an asymptotically tight characterization of the required power. We further quantify how temporal correlation and spatial aggregation affect storage requirements and statistical multiplexing gains, and extend the analysis to loads with multiple demand time scales. Finally, we evaluate the framework using power-demand traces from three production data centers spanning HPC, GPU-training, and cloud-service workloads. Despite their heterogeneous, cyclo-stationary and multi-modal behavior, the measured workloads exhibit the predicted regimes, and a simple four-parameter two-state model captures the dynamics governing their storage-power tradeoffs. The resulting framework provides both a probabilistic foundation and practical dimensioning principles for storage-assisted power provisioning in next-generation AI data centers.

• The paper develops a closed-form, large-deviations framework showing that small batteries reduce provisioned power nearly linearly, while large batteries yield diminishing 1/b returns governed by effective power and demand correlation.
• The analysis shows that battery scaling determines multiplexing gains: fixed storage is diluted across servers, storage proportional to √n preserves a constant benefit, and storage proportional to n can reduce per-server margins to O(1/n).
• Validation on HPC, GPU-training, and cloud traces shows that a four-parameter two-state Gaussian model predicts required-power curves within 5% for Microsoft Philly, 10% for Alibaba, and up to 17.5% for OSC.

Motivation and problem statement

Data centers provision power for worst-case demand, leaving large margins idle most of the time, while AI workloads exhibit abrupt transitions between high- and low-power regimes and grid interconnection has become a binding constraint on growth. The paper develops a probabilistic framework that jointly characterizes the provisioned power $p_s$, the energy-storage capacity $b$, and the probability of overdraw — the event that demand exceeds supply plus stored power. The central object is the required power

$$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$

with margin $m(b) = p_r(b) - \mu_P$. Prior treatments of storage-assisted provisioning relied on simulation, convex/dynamic optimization, or heuristics without statistical guarantees; this paper supplies a closed-form theory grounded in large-deviations techniques imported from effective-bandwidth analysis of communication networks.

System model

The system is a slotted model of an "energy group" of $n$ servers sharing a constant supply $p_s$ and a battery of power-charge capacity $b$ (energy capacity divided by slot length). The battery evolves as a clamped random walk, $B(t+1) = [p_s + B(t) - P(t)]_0^b$, where $P(t)$ is the aggregate stationary ergodic demand. Demands are not deferrable; storage is used greedily. The framework applies per layer of a hierarchical architecture (racks, rows, PDUs), though cross-layer interactions are explicitly out of scope.

The no-battery baseline

Without storage, overdraw reduces to a single-slot tail event, so $p_r(0)$ is simply the $b$0-quantile of $b$1. For Gaussian demands, Mill's-ratio approximations give $b$2 with $b$3 and $b$4: roughly $b$5 at $b$6 and $b$7 at $b$8. For multi-modal distributions, Chernoff bounds can be loose; when a HIGH state is well separated, the margin is governed by that state's statistics with an adjusted target $b$9. This multi-modal observation underpins both the Markov-modulated analysis and the empirical modeling later in the paper.

The small battery region (SBR)

When the battery is small relative to demand statistics, it is nearly always full, overdraws are driven by single-slot spikes, and the ideal bound $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$0 is tight: each watt of power charge saves exactly one watt of provisioned power. Exact Markov-chain computations for i.i.d. Gaussian demands show the SBR/LBR transition occurs near the midpoint $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$1 across wide parameter ranges — i.e., roughly half of the achievable margin reduction comes from very small batteries. The transition mechanism is intuitive: at the midpoint, the probability of fully recharging in one slot after an overdraw is about $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$2; beyond it, that probability decays exponentially. Positive temporal correlation shrinks the SBR.

Effective power and the large battery region (LBR)

For large batteries, overdraws result from sustained excess demand draining the battery over many slots. The paper introduces effective power, an analogue of effective bandwidth:

$$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$3

where $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$4 is the asymptotic log moment generating function of the demand. Setting $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$5 is sufficient — and asymptotically tight as $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$6 — to meet an overdraw probability $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$7. A second-order expansion yields the practically useful form

$$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$8

where $$p_r(b) = \min\{p_s : \mathbb{P}[\text{overdraw}] \le \varepsilon\},$$9 augments the instantaneous variance with a term capturing temporal correlations. Margin thus decays as $m(b) = p_r(b) - \mu_P$0 rather than linearly, and is relatively insensitive to the overdraw target since $m(b) = p_r(b) - \mu_P$1 enters only logarithmically. The derivation removes finite-battery boundaries, justified because the exponential decay rate of the reflected process matches the unconstrained recursion asymptotically — a standard but nontrivial assumption whose sub-exponential error terms are not quantified.

A notable structural result is the typical overdraw duration:

$m(b) = p_r(b) - \mu_P$2

which grows linearly with battery size; for i.i.d. Gaussian demands it simplifies to $m(b) = p_r(b) - \mu_P$3, independent of variance. When provisioning exactly meets the constraint, $m(b) = p_r(b) - \mu_P$4. Comparing rate functions shows LBR overdraws require both larger bursts ($m(b) = p_r(b) - \mu_P$5) and persistence over $m(b) = p_r(b) - \mu_P$6 slots, whereas no-battery overdraws are instantaneous.

Statistical multiplexing

The paper separates spatial multiplexing (pooling servers) from temporal multiplexing (storage averaging over time) and analyzes margin per server $m(b) = p_r(b) - \mu_P$7 under three scaling scenarios. Key findings:

Scenario	Battery scaling	Asymptotic $m(b) = p_r(b) - \mu_P$8
Fixed $m(b) = p_r(b) - \mu_P$9	constant	$n$0 (battery diluted away)
Balanced	$n$1	$n$2 (constant temporal gain)
Proportional	$n$3	$n$4 (both gains contribute)

The natural growth rate of total battery capacity is $n$5 for uncorrelated demands, matching the SBR/LBR transition point; for highly correlated demands it approaches $n$6. Consequently, combining spatial and temporal multiplexing can be additive, neutral, or mildly subtractive depending on how storage scales with group size — a correction to naive expectations that the two gains compound.

Multi-scale correlated loads

Markov-modulated processes capture cyclo-stationary, multi-time-scale behavior. For a symmetric ON/OFF Gaussian model, the margin curve decomposes into four regions across two phases: Phase 1 suppresses ON-state fluctuations (SBR region 1a, then region 1b where margin falls as $n$7 toward the ON mean); Phase 2 reduces power below the ON mean (region 2a with a reduced linear slope $n$8, then region 2b with diminishing returns governed by $n$9). Region 2a ends when the battery can no longer reliably recharge during OFF periods, predicted to occur at roughly 14% below the ON mean for $p_s$0 — a prediction the exact Markov calculations confirm within about 1%. Effective power computed via the Perron–Frobenius eigenvalue of the tilted transition matrix tracks the exact requirement closely except at regime transitions.

Validation on production traces

The framework is evaluated against three production data sets: OSC HPC racks (direct PDU measurements, 30 s resolution), Microsoft Philly GPU-training servers (power estimated from GPU utilization), and Alibaba cloud servers (power estimated from CPU utilization). These span an order of magnitude in mean power (288–2,841 kW), differ in correlation times (127–561 min), and all exhibit strongly multi-modal, highly autocorrelated demand (one-slot correlations 0.949–0.997).

Despite this heterogeneity, empirical simulations reproduce the two-phase, four-region structure. A deliberately simple four-parameter two-state HIGH/LOW Gaussian model predicts the required-power curves well: errors under 5% throughout for Philly, under 10% for Alibaba, and up to about 17.5% mid-region for OSC. At regional transition points, prediction errors are typically ≤5%, with two outliers near 10%.

Two discrepancies deserve emphasis. First, the measured SBR marginal efficiency was roughly half the ideal value (e.g., 53 kW/kWh versus an ideal 120 kW/kWh for OSC), and systems exited the SBR sooner than theory predicts. The authors attribute this to unmodeled intra-HIGH-state temporal correlations, supported by permutation tests that restore i.i.d.-like efficiencies. Second, the analytical transition points had to be replaced by empirically identified ones. Additionally, the OSC simulation could only cover about 46% of potential margin savings because its six-day trace contains too few demand cycles for start-empty and start-full simulations to converge — a data-length limitation rather than a model failure, but one that leaves the deep-LBR prediction for long-dwell workloads only partially validated.

Limitations and open questions

The paper concedes several boundaries. Cross-layer effects — lower-layer storage smoothing higher-layer demand, shared supplies, non-greedy battery policies, and interaction with demand-side orchestration — are excluded. Transition points out of the SBR are characterized analytically only for i.i.d. Gaussian demands; other distributions and the effect of short-range correlation on $p_s$1 remain open. The second-order effective-power approximation requires batteries large enough to average out correlation decay, and the region-2b approximation is accurate only for fractional margins below roughly 20–30%. Empirically, the two-state abstraction cannot capture full workload structure, transition points must be fitted rather than predicted, and the observed halving of SBR efficiency indicates that intra-state temporal correlation is not yet incorporated into the SBR theory. Whether a corrected SBR analysis accounting for such correlations can predict transition points without empirical fitting is a specific question the paper leaves unresolved.

Conclusion

This paper establishes a rigorous probabilistic foundation for storage-assisted power provisioning, showing that the margin-storage tradeoff decomposes into a linear small-battery regime and a diminishing-returns large-battery regime characterized by effective power. It quantifies how temporal correlation inflates requirements through $p_s$2, derives scaling laws for joint spatial-temporal multiplexing, and demonstrates on three production traces spanning HPC, GPU training, and cloud workloads that a four-parameter two-state model captures the dominant storage-power dynamics. The framework converts the supply side from a static design constraint into a statistically dimensioned resource, while leaving the extension to multi-layer hierarchical architectures and correlation-aware SBR analysis as open problems.