Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 27J / b / i

Table of contents
- Solution i

Solution

With state-dependent admission, the effective birth rate in state

n

is

b_{n} = λ p (n) = \frac{λ}{n + 1},

while the death rate is

μ

for

n \geq 1

. The stationary ratios for an M-M-1 queue with state-dependent admission are therefore

\frac{π _{n + 1}}{π _{n}} = \frac{ρ}{n + 1} .

Induction gives

π_{n} = π_{0} ρ^{n} / n!

, and normalization identifies a Poisson distribution of mean

ρ

:

π_{n} = e^{- ρ} \frac{ρ ^{n}}{n !} .

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