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Equation 17 · AI Inference Economics in Practice: An Advanced Technical Guide

What does this equation mean?

u∗=PrQmax⁡ codu^{*} = \frac{P_r}{Q_{\max}\, c_{\mathrm{od}}}

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withP_r
Divide byQ_max c_od
This relates tou^*
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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u∗u^{*}

Symbol u^*

u∗u^* is the quantity selected or evaluated by the optimization written on the right.

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PrP_r

Symbol P_r

the reserved capacity’s price per hour.

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Qmax⁡Q_{\max}

Symbol Q_max

the maximum throughput it sustains per hour at full utilization.

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codc_{\mathrm{od}}

Symbol c_od

the effective on-demand cost of the equivalent work.

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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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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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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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Qmax⁡ codQ_{\max}\, c_{\mathrm{od}}

Denominator: Q_max c_od

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 arithmetic is simple once both prices are in hand, and generalizes across any reserved-capacity product structured this way. Let PrP_r be the reserved capacity’s price per hour, Qmax⁡Q_{\max} the maximum throughput it sustains per hour at full utilization, and codc_{\mathrm{od}} the effective on-demand cost of the equivalent work. At actual utilization u , the effective cost per unit under reservation is PrP_r / (u \, Qmax⁡Q_{\max}) , and reservation wins exactly when u∗=PrQmax⁡ codu^{*} = \frac{P_r}{Q_{\max}\, c_{\mathrm{od}}}. is cleared. Utilization above u∗u^* favours reservation; below it favours paying as you go. The formula’s value is what it forces a team to measure honestly before signing: not peak throughput or a favourable week’s…
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The arithmetic is simple once both prices are in hand, and generalizes across any reserved-capacity product structured this way. Let PrP_r be the reserved capacity’s price per hour, Qmax⁡Q_{\max} the maximum throughput it sustains per hour at full utilization, and codc_{\mathrm{od}} the effective on-demand cost of the equivalent work. At actual utilization u , the effective cost per unit under reservation is PrP_r / (u \, Qmax⁡Q_{\max}) , and reservation wins exactly when u∗=PrQmax⁡ codu^{*} = \frac{P_r}{Q_{\max}\, c_{\mathrm{od}}}. is cleared. Utilization above u∗u^* favours reservation; below it favours paying as you go. The formula’s value is what it forces a team to measure honestly before signing: not peak throughput or a favourable week’s average, but utilization sustained across the full commitment term including the slow periods — capacity sized to a launch-week peak and left running at a quarter of that traffic for the following eleven months has pre-paid idle time at the reserved rate.

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