Moderate deviations below a bufferless queue cap ($\alpha <\beta :S_L\text{ and }\min (S_L,u_L)\text{ are exponentially equivalent at speed }{L}^{\alpha }$)

For Poisson input and $u_L=L\lambda +C{L}^{(1+\beta )/2}$ with $C>0$, clipping occurs with logarithmic probability $-C^2{L}^{\beta }/(2\lambda )$. 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.