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Equation 4 · Part 9 · AI Datacenter Systems Engineering in 2035: Scenarios, Signals, and Falsifiable Predictions

Numerator: MTTF_device

MTTFfleet≈1Nλ=MTTFdeviceN.\mathrm{MTTF}_{\mathrm{fleet}} \approx \frac{1}{N\lambda} = \frac{\mathrm{MTTF}_{\mathrm{device}}}{N}.
MTTFdevice\mathrm{MTTF}_{\mathrm{device}}

What this part means

The complete quantity above the fraction bar.

Its job in the formula

MTTFdF_device occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

Analysis. The following is my own structural analysis, not drawn from any single cited source. If a fleet of N identical devices each fails independently at rate λ\lambda , the fleet-wide failure rate is Nλ\lambda , so MTTFfleet≈1Nλ=MTTFdeviceN\mathrm{MTTF}_{\mathrm{fleet}} \approx \frac{1}{N\lambda} = \frac{\mathrm{MTTF}_{\mathrm{device}}}{N}. That relation is what licenses treating fleet-scale failure as a continuous background rate rather than a sequence of independent surprises, and it is the same move Meta’s reliability study makes in fitting a model to project Mean Time to Failure across GPU scales [ 3 ] . But it depends entirely on independence, and the sources above describe two ways that assumption breaks. First, correlated failure: a batch sharing a manufacturing defect or an upstream…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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