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Equation 18 · How AI Inference Serving Actually Works

What does this equation mean?

E[tokens per round]=1−αγ+11−α.\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}.

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Start with1-α^gamma+1
Divide by1-α
This relates toE[tokens per round]
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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E\mathbb{E}

Symbol E

The expected value operator: the probability-weighted average of the quantity inside its brackets.

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αγ+1\alpha^{\gamma+1}

Symbol α^gamma+1

α^gamma+1 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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α\alpha

Symbol α

α is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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addition

addition

Add the term after the plus sign to the term or group before it.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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1−αγ+11-\alpha^{\gamma+1}

Numerator: 1-α^gamma+1

The complete quantity above the fraction bar.

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1−α1-\alpha

Denominator: 1-α

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The size of the win has a clean shape. Model the draft’s acceptance probability as α\alpha per token, roughly constant and independent across the γ\gamma tokens proposed in a round — an idealization real traffic does not fully satisfy, but a useful one for seeing the ceiling. The expected number of tokens accepted per verification round is then E[tokens per round]=1−αγ+11−α\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}. This rises with both α\alpha and γ\gamma , but with steeply diminishing returns in γ\gamma for any α\alpha below one: drafting fifty tokens ahead does not buy anywhere near fifty accepted tokens, because the marginal proposal deep into a long draft is unlikely to be exactly what the target would have generated. The bottleneck decode…
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The size of the win has a clean shape. Model the draft’s acceptance probability as α\alpha per token, roughly constant and independent across the γ\gamma tokens proposed in a round — an idealization real traffic does not fully satisfy, but a useful one for seeing the ceiling. The expected number of tokens accepted per verification round is then E[tokens per round]=1−αγ+11−α\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}. This rises with both α\alpha and γ\gamma , but with steeply diminishing returns in γ\gamma for any α\alpha below one: drafting fifty tokens ahead does not buy anywhere near fifty accepted tokens, because the marginal proposal deep into a long draft is unlikely to be exactly what the target would have generated. The bottleneck decode started with reappears one level down — the draft model is itself a small, serial, memory-bound process, so speculative decoding is not a free lunch; it trades some of decode’s own bandwidth cost for a second model running continuously, plus wasted target-model computation on every rejected proposal. Its win is largest where the roofline argument above says spare compute is most available: at low batch sizes, where the target model is memory-bound and its arithmetic units sit mostly idle during an ordinary decode step anyway. That idle arithmetic verifies the extra proposed tokens at close to no additional cost.

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