Equation 31 · Part 6 · The Economics and Physical Limits of Running AI Agents at Scale

Denominator: 1-q

\frac{q}{1-q}\sum_i c_i

1-q

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

1-q occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Full expressionDenominator: 1-qArticle meaning

The passage around this formula

Holding $q_i$ = q constant across the trajectory — a simplification, since real failure rates are neither constant nor independent — the expected extra cost from retries is $\frac{q}{1-q}$ $\sum_i$ $c_i$ , and because $c_i$ increases with i , that extra cost is weighted toward the trajectory’s most expensive, latest steps rather than distributed evenly across it. A one-percent per-step retry rate is a rounding error at step five. Applied at step two hundred of an uncached or lightly cached loop, it is a rounding error on a much larger number.

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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.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

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