Batch-birth linear-death process

A batch-birth linear-death process has transitions

n ⟶ n + k at rate λ, n ⟶ n - 1 at rate β n .

P_{n} (t) = P (N_{t} = n)

and taking

P_{j} = 0

for

j < 0

, its master equation is

\dot{P}_{n} = λ P_{n - k} + β (n + 1) P_{n + 1} - (λ + β n) P_{n} .

Table of contents
- Moments of a batch-birth linear-death process Batch-birth linear-death process

Moments of a batch-birth linear-death process

For

m = E N

and

v = var N

, the Markov jump-process generator applied to

n

and

n^{2}

gives

m^{'} = kλ - β m, v^{'} = k^{2} λ + β m - 2 β v .

Consequently

m ⟶ \frac{kλ}{β}, v ⟶ \frac{k ( k + 1 ) λ}{2 β} .

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 6C / Solution

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