What does this equation mean?

Analysis. The following calculation is my own, built to make the MLPerf numbers’ shape explicit rather than to reproduce them exactly. Model checkpoint writes as occurring every T hours, each costing $\tau$ hours of write time, with failures arriving as a Poisson process at fleet-wide rate $\lambda$ (failures per hour). For $\lambda$ T $\ll$ 1 , a failure inside an interval loses on average T/2 hours of recomputation, so the expected fraction of wall-clock time lost to the combination of checkpoint overhead and lost recompute is approximately $f(T) \approx \frac{\tau}{T} + \frac{\lambda T}{2}$. Minimizing over T gives an optimal interval $T^{*}$ = $\sqrt{2\tau/\lambda}$ : the checkpoint interval should shrink as the inverse square root of the fleet failure rate, not linearly with it, so frequency rises roughly as $\sqrt{N}$ purely from the failure-rate side as accelerator count N grows. Comparing that to MLPerf’s own reported tightening — interval falling from 9.3 to 1.5 minutes, a 6.2-times increase in frequency, as scale rises 6.25-times from 16,000 to 100,000 accelerators — the real tightening is close to linear in scale, noticeably steeper than the $\sqrt{N}$ this simple model predicts. That gap is informative rather than a contradiction: it is consistent with operators holding a fixed acceptable progress-loss percentage (MLPerf specifies 5%) rather than freely re-optimizing T against $\tau$ and $\lambda$ jointly, and with $\tau$ itself growing as checkpoints get larger at larger scale — both push the real interval down faster than a pure failure-rate argument would. None of the cited sources explains that gap directly; this derivation is offered to sharpen the open question, not to answer it.