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Equation 1 · Part 11 · How AI Memory Systems and the Bandwidth Wall Actually Work

Denominator: bytes moved

I=FLOPs performedbytes moved,Pachievable=min⁡(Ppeak compute,  I×BWpeak memory)I = \frac{\text{FLOPs performed}}{\text{bytes moved}}, \qquad P_{\text{achievable}} = \min\left(P_{\text{peak compute}},\; I \times BW_{\text{peak memory}}\right)
bytes moved\text{bytes moved}

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division. Bytes measure the data moved or stored.

Its job in the formula

bytes moved 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 roofline model, introduced by Williams, Waterman, and Patterson in 2009, gives the concept its formal shape. It plots two independent ceilings on the same chart: the chip’s peak arithmetic throughput (operations per second) on one axis, and its peak memory bandwidth (bytes per second) combined with a workload-specific ratio on the other. That ratio — arithmetic intensity — is defined as the number of floating-point operations performed per byte of data moved between memory and the arithmetic units [ 4 ] . A workload’s achievable throughput is bounded by whichever ceiling it hits first: I=FLOPs performedbytes moved,Pachievable=min⁡(Ppeak compute,  I×BWpeak memory)I = \frac{\text{FLOPs performed}}{\text{bytes moved}}, \qquad P_{\text{achievable}} = \min\left(P_{\text{peak compute}},\; I \times BW_{\text{peak memory}}\right). When a workload’s intensity I is high enough that I ×\times BWpeak memoryW_{\text{peak memory}}…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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