Moderate deviation scales of a clipped Poisson aggregate ($K_{\beta }(y)=y^2/(2\lambda )(y\le C),K_{\beta }(y)=\infty (y>C)$)

Clip Poisson input of mean $L\lambda$ at $L\lambda +C{L}^{(1+\beta )/2}$. The Poisson moderate deviation principle and contraction principle for large deviations give the displayed output rate at the clipping scale. At smaller exponents the output retains the original rate by moderate deviations below a bufferless queue cap. At larger positive-deviation scales every overflow event is eventually impossible because the requested threshold exceeds the cap.