= Solution
The downstream <service rate of a queue> now exceeds the upstream cap for all sufficiently large $L$, since
$$
D L^{(1+\gamma)/2}>C L^{(1+\beta)/2}\qquad(\gamma>\beta).
$$
But the <bufferless queue output> is always at most $u_L$. Therefore the downstream <queue overflow> probability is eventually exactly zero, and
$$
\boxed{\lim_L\frac1{L^\gamma}\log\mathbb P(\mathrm{overflow})=-\infty.}
$$
This is a deterministic cap argument, stronger than merely proving superexponentially small probability.
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