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Equation 19 · Part 4 · The Physical Plant: Power, Cooling, and Networks in an AI Datacenter

Symbol B

Tring≈2 p−1p⋅NB+2 (p−1) α,T_{\mathrm{ring}} \approx 2\,\frac{p-1}{p}\cdot\frac{N}{B} + 2\,(p-1)\,\alpha,
BB

What this part means

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

Its job in the formula

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

The passage around this formula

The cost structure of a collective is arithmetic, not policy. For a ring AllReduce over p ranks reducing N bytes at per-link bandwidth B with per-hop latency α\alpha , each rank moves Tring≈2 p−1p⋅NB+2 (p−1) αT_{\mathrm{ring}} \approx 2\,\frac{p-1}{p}\cdot\frac{N}{B} + 2\,(p-1)\,\alpha. so bandwidth cost saturates near 2N/B while the latency term grows linearly in p . Two consequences follow. The slowest link sets the pace for every rank, because the operation does not complete until all ranks have contributed.…

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A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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