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The passage around this formula

The authors frame that tradeoff with the classical Young–Daly optimal-checkpoint-interval model, which balances the fixed cost of writing a checkpoint against the expected cost of losing work to a failure: $\Delta t_{cp}^{*} = \sqrt{\frac{2\,w_{cp}}{N_{\text{nodes}}\,r_f}}$. where $w_{cp}$ is the time to write a checkpoint, $N_{\text{nodes}}$ is the job’s node count, and $r_f$ is the per-node failure rate. As $N_{\text{nodes}}$ grows, the failure rate the job experiences as a whole scales with it, which pulls the optimal interval shorter — meaning larger jobs must checkpoint more often even as each individual checkpoint write competes with the same limited storage bandwidth every other job on the cluster is also using. Reporting a real operating…

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