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

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- Solution b

Solution

In this bufferless queue, queue overflow is the event

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

The Poisson moderate deviation principle therefore gives

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

by

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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