= Poisson large-service overflow asymptotic
{c}
{title2=$\log\mathbb P(X>c)=-c\log(c/\lambda)+c-\lambda+O(\log c)$}
For fixed <Poisson> mean $\lambda>0$ and capacity $c\to\infty$, set $m=\lfloor c\rfloor+1$. The optimized <Chernoff bound> gives $\log\mathbb P(X\ge m)\le-I_P(m)$. The lower bound by $\mathbb P(X=m)$ and <Stirling's formula> differs by only $O(\log m)$. Rounding $m$ to $c$ changes the cost by $O(\log c)$, proving the displayed formula and the leading scale $-c\log c$.
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