Contemporaneous Failure Batching

• Contemporaneous Failure Batching is a coordination mechanism that groups simultaneous failures to trigger efficient, parallel responses in both equipment replacement and distributed storage systems.
• It employs dynamic programming and simulated moments in econometric models to quantify how local failures reduce continuation utility, with empirical tests showing a reduction of approximately 0.265 utils per neighbor failure.
• In storage applications, specialized FRB and ECBC code constructions enable robust, batch data retrieval and repair even amid multiple node failures, balancing replication overhead and fault tolerance.

Contemporaneous failure batching refers to a class of coordination phenomena and code design principles in both critical infrastructure operations and distributed storage theory, wherein responses (such as repairs or replacements) to multiple simultaneous failures are temporally grouped or "batched." In econometric models of equipment replacement, it quantifies the static complementarity: the utility or incentive for an agent to act is modulated by the contemporaneous failures of spatially proximate peers. In fault-tolerant storage systems, it denotes code constructions that ensure reliable, parallelizable retrieval of data batches even amidst concurrent node failures. Both applications recognize the operational and information-theoretic advantages of aggregating responses to failure events that occur "at the same time," shaping the implementation of dynamic spatial models and fault-tolerant combinatorial codes.

1. Formalization in Dynamic Discrete Choice Models

In spatial structural dynamic discrete choice (SDDC) frameworks, contemporaneous failure batching embodies a direct utility interaction between agents based on current-period local failure observations. Consider the state vector at equipment location $i$ and time $t$: $$s_{it} = (\text{age}_{it}, \text{cage}_i, \text{fail}_{it}, n^{\text{lag}}_{it}, f^{\text{cage}}_{it})$$ where $\text{age}_{it}$ is the equipment age, $\text{cage}_i$ categorizes the (time-invariant) thermal environment, $\text{fail}_{it}$ signals current failure, $n^{\text{lag}}_{it}$ encodes lagged neighbor replacements, and $$f^{\text{cage}}_{it} = \sum_{j \in \mathcal{N}(i)} \text{fail}_{jt}$$ counts concurrent neighbor failures within the same spatial "cage."

The action-specific utility for the "keep" decision incorporates contemporaneous batching: $$u_{\text{keep}}(s_{it}) = \theta_{\text{age}} \cdot \text{age}_{it} + \theta_{\text{cage1}} \cdot 1_{\text{cage}_i=1} + \theta_{\text{cage2}} \cdot 1_{\text{cage}_i=2} + \theta_{\text{fail}} \cdot \text{fail}_{it} + \gamma_{\text{lag}} \cdot n^{\text{lag}}_{it} + \gamma_{\text{fail}} \cdot f^{\text{cage}}_{it}$$ where $\gamma_{\text{fail}}$ quantifies the reduction in flow utility for each contemporaneously failed neighbor, shifting forward-looking incentives through the Bellman recursion. The resulting policy function, solved via nested fixed-point dynamic programming and estimated with a method of simulated moments (NFXP-MSM), links observed batch replacement spikes to local failure clustering.

2. Empirical Manifestations and Estimated Effects

Empirically, contemporaneous failure batching is observed when failures prompt immediate, spatially clustered replacement activity extending beyond failed units to their functional neighbors. In the Oak Ridge National Laboratory's Titan supercomputer dataset (12,915 GPUs), controlling for own-failure, the following effect is isolated: $$m_4 = \mathbb{E}[d_{it} \mid \text{fail}_{it}=0, f^{\text{cage}}_{it} \geq 1] - \mathbb{E}[d_{it} \mid \text{fail}_{it}=0, f^{\text{cage}}_{it}=0]$$ where $d_{it}$ is the replacement decision. The estimated parameter $\gamma_{\text{fail}} = -0.265$ (bootstrap SE $\approx 0.035$) implies that each failed neighbor reduces the continuation utility for keeping a non-failed unit by 0.265 utils—approximately 3.4% of the replacement cost (with $\theta_{\text{replace}}=-7.832$). The probability of proactive replacement for non-failed units rises with the number of simultaneous local failures, especially pronounced in high-risk (hot) cages:

• Cage 0 (cool): $m_4 \approx 0.29$ pp
• Cage 1 (moderate): $m_4 \approx 0.56$ pp
• Cage 2 (hot): $m_4 \approx 4.07$ pp

In hot zones, the contemporaneous batching effect on replacement probabilities exceeds that in cool zones by more than an order of magnitude, confirming thermal heterogeneity in spatial risk propagation. Operators leverage information conveyed by local failure clusters to achieve economies of scale and mitigate common-cause risks in maintenance operations.

3. Comparison to Sequential and Other Coordination Mechanisms

Concurrently, sequential replacement cascades are captured by $\gamma_{\text{lag}} = -0.793$ (SE $\approx 0.106$), modeled as the effect of observing lagged (previous-period) neighbor replacements. Sequential coordination is nearly three times stronger: $$|\gamma_{\text{lag}}|/|\gamma_{\text{fail}}| \approx 2.99$$. This asymmetry points to a greater reliance by operators on revealed neighbor actions (strategic information) than on same-period, possibly coincidental, failure events.

Spatial interdependencies—of which contemporaneous batching is a component—account for 5.3% of the replacement propensity variance unexplained by independent-decision models. Formal likelihood-ratio testing decisively rejects spatial independence ($\chi^2(2)=685.38$, $p<0.001$), establishing the necessity of modeling both immediate and lagged neighbor effects for accurate inference and optimal policy design.

4. Contemporaneous Failure Batching in Storage Code Design

In distributed storage theory, contemporaneous failure batching is addressed through code families that guarantee robust, parallel batch retrieval under simultaneous node failures. Two main constructions are utilized: Fractional Repetition Batch (FRB) codes and Erasure Combinatorial Batch Codes (ECBCs) (Silberstein, 2014).

• FRB codes combine the uncoded repair property of fractional-repetition codes (FR) with the parallel-read and batch properties of combinatorial batch codes (CBCs).
• ECBCs are uniform codes with parameters $\rho$ (replication degree), $t$ (batch size), $n$ (number of nodes), and $f$ (failure tolerance), designed so that any subset of $t$ symbols can be retrieved by accessing at most one per surviving node, even with $f$ failures.

Construction is characterized via incidence matrices $A = (a_{i,j})$:

• Each row: storage node.
• Each column: encoded symbol.
• $a_{i,j}=1$ iff node $i$ stores symbol $j$.
• Hall-type conditions (minimum neighborhood coverage) ensure that, for any set of $t$ symbols and any possible $f$ node failures, simultaneous batch retrieval is feasible.

Explicit families are constructed from combinatorial objects such as resolvable transversal designs ($\mathrm{TD}(\ell, h)$) and affine planes $A(q)$; the parameters (node count, symbol count, redundancy, batch size, failure tolerance) are summarized in the following table.

Construction	Nodes ($n$)	Replication ($\rho$)	Failure Tolerance ($f$)	Max Batch ($t$)
$\mathrm{TD}(2,h)$	$2h$	$2$	$1$	$3$
$\mathrm{TD}(3,h)$	$3h$	$3$	$2$	$4 \leq t \leq 2h-2$
$A(q)$	$q^2$	$q$	$q-1$	$\frac{q^2-q+2}{2} \leq t \leq q^2-q$

The incidence-matrix criteria enforce that for all requested symbol sets of size $\leq t$, the joint neighborhood covers at least $t+f$ nodes, guaranteeing contemporaneous failure batch tolerance.

5. Bounds, Trade-Offs, and Performance Metrics

Elementary necessary conditions on parameters for contemporaneous failure batch codes include:

• $\rho \geq f+1$
• $n \geq t+f$
• For batch codes: $n \geq t$, $\theta \geq t$, $M \leq k\alpha$

These bounds reflect the fundamental tension between storage overhead (replication), batch size, and failure tolerance. For example, the overhead required in ECBCs is fixed by symbol replication ($N = n\alpha = \rho\theta$), while in FRB codes, additional overhead ensures uncoded repair and batch-read capabilities.

Performance metrics relevant to practical deployments:

• Repair bandwidth in FRB codes: $\beta = 1$ symbol per helper node (table-based, uncoded)
• Batch retrieval efficiency: Any batch of size $t$ can be served in a single parallel round, reading at most one symbol per node—even after $f$ failures in ECBCs, provided $n-f \ge t$.
• Comparison: CBCs with $f=0$ admit more symbols for fixed $(n, \rho, t)$ but lack fault tolerance; FRB codes integrate repair and batch capabilities but with increased overhead.

ECBCs and FRB codes thus generalize classical codes, allowing robust batched responses to contemporaneous node failures without compromising parallelism or repair efficiency. This design paradigm links closely to the operational insights from dynamic replacement models, where batching responses to same-period failures exploits “economies of scale” and mitigates the impact of correlated risks.

6. Economic and Operational Implications

The presence of contemporaneous failure batching mechanisms has decisive operational and economic consequences. In critical infrastructure management, it reflects—and rationalizes—policies of proactive, clustered intervention following local failure surges. These strategies exploit observable externalities, enabling operators to economize on maintenance and replacement resources, particularly under common-cause threat profiles (e.g., thermal runaway in hot cages).

The distinct but weaker role of contemporaneous failure batching compared to sequential cascades indicates that operators react more strongly to neighbors’ actual replacement decisions than to mere co-occurrence of failures, emphasizing the information value of observed actions over state indicators.

Formally, failing to account for spatial coordination—of which contemporaneous batching is a key component—leads to systematic mistiming of interventions and loss of potential coordination gains. This substantiates the need for spatially-extended, forward-looking models in both policy analysis and the design of automated maintenance protocols (Diamond et al., 5 Nov 2025).

7. Open Problems and Extensions

Substantive questions remain regarding the optimization and generalization of contemporaneous failure batching mechanisms:

• Determination of tight bounds on batch-size ($t$) and file-size ($M$) for general FRB code families with given parameters.
• Construction of explicit FRB and ECBC codes achieving maximal values for multiple constrained parameters.
• Characterization of batch parameters for transversal-design and affine-plane constructions—current results provide only bounds.
• Extensions to asynchronous batch retrieval, support for non-binary alphabets, and models incorporating weighted or fractional erasure patterns.
• Integration of locality for multiple concurrent repairs and scalability for emerging distributed and cyber-physical systems.

Resolution is anticipated via advances in combinatorial design theory (e.g., higher-order block designs, expanders with two-phase expansion), promising further improvements in the trade-off surface for batch, repair, and erasure-resilience in the presence of contemporaneous failures (Silberstein, 2014).