= Solution
In this <bufferless queue>, <queue overflow> is the event
$$
S_L>L\lambda+C L^{(1+\beta)/2},\quad\text{equivalently }Q_L>C.
$$
The <Poisson moderate deviation principle> therefore gives
$$
\boxed{\lim_L\frac1{L^\beta}\log\mathbb P(\mathrm{overflow})
=-\inf_{x>C}\frac{x^2}{2\lambda}=-\frac{C^2}{2\lambda}.}
$$
The same infimum is obtained for $x\ge C$, so replacing $Q_L>C$ by $Q_L\ge C$ gives the same logarithmic exponent. Any integer rounding of the <service rate of a queue> changes the scaled threshold by a quantity tending to zero.
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