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Equation 19 · Ten Failure Modes That Define Production RAG

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Li≈λi τi ℓi,L_i \approx \lambda_i \, \tau_i \, \ell_i,

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This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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LiL_i

Symbol L_i

LiL_i is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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λi\lambda_i

Symbol lambda_i

lambdaia_i is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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τi\tau_i

Symbol tau_i

tauiu_i is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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ii

Symbol i

i is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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≈

≈

Approximately equal to; the equality is not exact.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

Its accuracy depends on the assumptions and range of use described in the article.

What the article says around this equation

A catalogue is not a triage plan. The ten differ enormously in how much damage accrues before anyone notices, and instrumentation effort should follow that rather than follow which failure is most interesting to write about. For a mode i with arrival rate λi\lambda_i , mean time to detection τi\tau_i , and cost per undetected instance ℓi\ell_i , the loss accrued before intervention is approximately Li≈λi τi ℓiL_i \approx \lambda_i \, \tau_i \, \ell_i. and the marginal value of building a detector is whatever it removes from τi\tau_i . Written this way, the ranking departs sharply from intuition, because τi\tau_i spans several orders of magnitude across the ten while λi\lambda_i and ℓi\ell_i vary far less.

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