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Equation 1 · Part 6 · How AI Memory Systems and the Bandwidth Wall Actually Work

fraction

P  ≤  min⁡ ⁣(Pmax⁡,  I⋅B),I=FLOPsbytes movedP \;\le\; \min\!\left(P_{\max},\; I \cdot B\right), \qquad I = \frac{\text{FLOPs}}{\text{bytes moved}}
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

The passage around this formula

Given a memory system built this way, the question for any specific piece of computation is simple to state and consequential to answer: for this kernel, is the bottleneck the arithmetic units or the memory system that feeds them? The quantity that answers it is arithmetic intensity , defined as the number of floating-point operations a kernel performs per byte it moves across the memory boundary that matters. Williams, Waterman and Patterson formalized the relationship between intensity and attainable performance as the roofline model: P  ≤  min⁡ ⁣(Pmax⁡,  I⋅B),I=FLOPsbytes movedP \;\le\; \min\!\left(P_{\max},\; I \cdot B\right), \qquad I = \frac{\text{FLOPs}}{\text{bytes moved}}. with Pmax⁡P_{\max} the peak arithmetic rate of the device, B the achievable bandwidth of the memory tier supplying the operands, and P the…

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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

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