Bufferless queue output ($Y=\min (A,c),W=(A-c{)}^{+}$)

The work served in one slot is capped at the service rate of a queue. The remainder is lost, so output is input minus its positive part above capacity. Every downstream threshold below the cap has exactly the same overflow event for input and output; thresholds at or above the cap cannot be exceeded by the output.

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.

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.