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A queue which retains no unfinished work between slots. Given nonnegative input $A$ and capacity $c$, its bufferless queue output is $\min (A,c)$ and its lost work is $(A-c{)}^{+}$.

The bufferless queue output isPut $r_L=L\lambda +B{L}^{(1+\alpha )/2}$. Since $\alpha <\beta$, we have $r_L<u_L$ for all sufficiently large $L$. Consequently the two queue overflow events are exactly equal eventually:Apply the Poisson moderate deviation principle with exponent $\alpha$ to obtainThere is also an exponential equivalence proof, which describes the whole output law at this smaller scale. For each $\epsilon >0$,The right side has strictly negative logarithmic exponent at speed $L^{\beta }$. Dividing instead by $L^{\alpha }$ multiplies that negative exponent by $L^{\beta -\alpha }\to \infty$, giving $-\infty$. Thus the centered, scaled input and output are exponentially equivalent and have the same good rate function. This is the moderate deviations below a bufferless queue cap.

The downstream service rate of a queue now exceeds the upstream cap for all sufficiently large $L$, sinceBut the bufferless queue output is always at most $u_L$. Therefore the downstream queue overflow probability is eventually exactly zero, andThis is a deterministic cap argument, stronger than merely proving superexponentially small probability.

The input admits positive moderate deviations on every scale $L^{(1+\delta )/2}$ with $0<\delta <1$. The bufferless queue output retains exactly the same overflow events at thresholds below the cap, so clipping does not improve the exponents on the smaller $\alpha$ scale. It removes every positive burst at thresholds on the larger $\gamma$ scale.

At the cap's own scale $\beta$, apply the contraction principle for large deviations to the map $x\mapsto \min (x,C)$. The output has good rate functionAt $C$ the cost is ${\inf }_{x\ge C}{x}^2/(2\lambda )=C^2/(2\lambda )$. Thus the burstiness of queue input has been reduced in its sufficiently large positive tail, while its smaller-scale rare fluctuations persist. These are the moderate deviation scales of a clipped Poisson aggregate.