= Solution
Use the stated <continuity> of the infinite-buffer and finite-buffer <queue workload> maps on the input path space. The <contraction principle for large deviations> gives, at the same speed as the input sequence,
$$
\boxed{J(x)=\inf\{I(a):q(a)=x\},\qquad\bar J(x)=\inf\{I(a):\bar q(a)=x\}.}
$$
Empty fibers have <infimum> infinity. Both rates are good because they are continuous contractions of the <good rate function> I. In particular, at speed L the infinite-buffer <queue workload> satisfies, for every Borel set E,
$$
-\inf_{E^\circ}J\leq\liminf_L L^{-1}\log\mathbb P(q(A^L)\in E)\leq\limsup_L L^{-1}\log\mathbb P(q(A^L)\in E)\leq-\inf_{\overline E}J,
$$
and the identical bounds with bars give the finite-buffer <queue workload> LDP. This states the entire principles, not just an informal tail approximation.
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