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which is minimized at $T^\star = \sqrt{2\,C\,M}$. Two consequences follow directly, and both are visible in how production systems have actually evolved. First, because M for a job shrinks roughly in proportion to the number of components that can fail — more GPUs, more NICs, more power supplies, more chances for any one of them to fault — the optimal interval $T^\star$ shrinks with cluster scale, roughly as the square root of the failure rate. Second, the only way to hold $T^\star$ low as M falls is to drive C , the checkpoint write cost itself, down; otherwise the optimum forces either constant checkpointing overhead or unacceptable recomputation loss.

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