Poisson large-service overflow asymptotic ($\log \mathbb{P}(X>c)=-c\log (c/\lambda )+c-\lambda +O(\log c)$)

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 $-c\log c$.