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Published equation contexts

E[Cn]≈∑i=1nci1−qi\mathbb{E}[C_n] \approx \sum_{i=1}^{n} \frac{c_i}{1-q_i}

Why this formula appears here

If a step fails and is retried with probability qiq_i , independent of other steps, the expected cost of the trajectory becomes E[Cn]≈∑i=1nci1−qi\mathbb{E}[C_n] \approx \sum_{i=1}^{n} \frac{c_i}{1-q_i}. Holding qiq_i = q constant across the trajectory — a simplification, since real failure rates are neither constant nor independent — the expected extra cost from retries is q1−q\frac{q}{1-q}∑i\sum_i cic_i , and because cic_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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ii

Symbol i

i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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nn

Symbol n

n appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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i=1i=1

Starting index or lower bound: i=1

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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nn

Ending index or upper bound: n

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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E[Cn]≈∑i=1nci1−qi\mathbb{E}[C_n] \approx \sum_{i=1}^{n} \frac{c_i}{1-q_i}

Equation 29 · AI Agents & Systems

The Economics and Physical Limits of Running AI Agents at Scale

This equation gives an approximation: it relates the quantities while allowing an approximation.

If a step fails and is retried with probability qiq_i , independent of other steps, the expected cost of the trajectory becomes E[Cn]≈∑i=1nci1−qi\mathbb{E}[C_n] \approx \sum_{i=1}^{n} \frac{c_i}{1-q_i}. Holding qiq_i = q constant across the trajectory — a simplification, since real failure rates are neither constant nor independent — the expected extra cost from retries is q1−q\frac{q}{1-q}∑i\sum_i cic_i , and because cic_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.

Meanings in this article

  • E\mathbb{E}: The expected value operator: the probability-weighted average of the quantity inside its brackets.
  • qiq_i: the probability.
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