= Solution
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 function>
$$
K_\beta(y)=\begin{cases}y^2/(2\lambda),&y\le C,\\+\infty,&y>C.\end{cases}
$$
At $C$ 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>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-26-large-deviation-costs.png]
{title=Exponential-mean minimum costs, moment optimization, and the moderate-deviation rate of clipped Poisson input}
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