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It is a well known fact, the throughput of a given compute pipeline (say, CPU instruction pipeline) is determined by its "slow" segment.

All the resources I've seen so far, demonstrates this fact via describing a concrete pipeline with 2-3 stages.

I'd love to see, more-or-less, some formal proof, the above holds for any in-balanced K-pipeline.

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This is formalized in Amdahl's law, which provides a quantitative version of that claim.

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