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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-31/4/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 4 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

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,

J (x) = in f {I (a) : q (a) = x}, \overset{ˉ}{J} (x) = in f {I (a) : \overset{q}{ˉ} (a) = x} .

(1)

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,

- in f_{E^{\circ}} J \leq lim inf_{L} L^{- 1} lo g P (q (A^{L}) \in E) \leq lim sup_{L} L^{- 1} lo g P (q (A^{L}) \in E) \leq - in f_{\overline{E}} J,

(2)

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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