Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 26 / 3 / d

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- Solution d

Solution

The downstream service rate of a queue now exceeds the upstream cap for all sufficiently large

L

, since

D L^{(1 + γ) /2} > C L^{(1 + β) /2} (γ > β) .

But the bufferless queue output is always at most

u_{L}

. Therefore the downstream queue overflow probability is eventually exactly zero, and

L lim \frac{1}{L ^{γ}} lo g P (overflow) = - \infty.

This is a deterministic cap argument, stronger than merely proving superexponentially small probability.

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