= Moderate deviations below a bufferless queue cap
{title2=$\alpha<\beta:\quad S_L\text{ and }\min(S_L,u_L)\text{ are exponentially equivalent at speed }L^\alpha$}
For <Poisson> input and $u_L=L\lambda+C L^{(1+\beta)/2}$ with $C>0$, clipping occurs with logarithmic probability $-C^2L^\beta/(2\lambda)$. At any smaller exponent $0<\alpha<\beta$, this is superexponentially small at speed $L^\alpha$. Dividing the input and output difference by $L^{(1+\alpha)/2}$ therefore gives <exponential equivalence>. More directly, for every threshold $r_L<u_L$, the events $\{\min(S_L,u_L)>r_L\}$ and $\{S_L>r_L\}$ are identical.
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