Equation 10 · Part 8 · Comparing the Main Approaches to AI Inference Economics

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P_r < u \cdot P_d, \qquad \text{i.e.} \qquad u > \frac{P_r}{P_d} \equiv u^{*}.

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What this part means

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.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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The passage around this formula

Here the break-even arithmetic is genuinely simple, and worth writing down, because it is the one place in this comparison where a clean model — not a ranking of vendors — is possible. Let a reserved commitment cost $P_r$ per accelerator-hour, and let the on-demand or serverless rate that would otherwise serve the same throughput cost $P_d$ per accelerator-hour at full utilization. If the buyer’s realized utilization of the reserved capacity is u $\in$ (0, 1] , the reserved option is cheaper exactly when $P_r < u \cdot P_d, \qquad \text{i.e.} \qquad u > \frac{P_r}{P_d} \equiv u^{*}$ . Below the break-even utilization $u^{*}$ , paying on demand only for the hours actually used is cheaper than holding a reservation that sits partly idle; above it, the…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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