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Equation 19 · Part 5 · The Physical Plant: Power, Cooling, and Networks in an AI Datacenter

Symbol α

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,
α\alpha

What this part means

the per-hop latency.

Its job in the formula

α is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Where the article explains it

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.

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. And large jobs avoid…

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Learn the underlying idea

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

These citations provide research context; check each source for the exact claim it supports.