 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-26/3/b/solution

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

S_{L} > L λ + C L^{(1 + β) /2}, equivalently Q_{L} > C .

L lim \frac{1}{L ^{β}} lo g P (overflow) = - x > C in f \frac{x ^{2}}{2 λ} = - \frac{C ^{2}}{2 λ} .

The same infimum is obtained for

x \geq C

, so replacing

Q_{L} > C

Q_{L} \geq 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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