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The passage around this formula

A minimal model makes the difficulty explicit. If each of a fleet’s ranks is independently healthy and idle with probability p at a random instant, the probability that all n ranks a job needs are simultaneously available is $P_{\text{admit}}(n) \approx p^{\,n}$. This is a simplification — real fleets are not independent, and real schedulers do not wait for a spontaneous coincidence — but it explains why gang admission gets combinatorially harder, not linearly harder, as jobs grow, and why every production scheduler in this space is built around avoiding literal simultaneous-availability matching rather than performing it. Google’s Borg, one of the earliest cluster managers to operate at this scale, achieves…

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