This is an immigration--death process. New individuals enter independently of the current population at the constant rate

λ

, producing transitions

n \to n + 1

. Each existing individual dies at rate

β

, so the total transition rate

n \to n - 1

β n

ii

Solution

Multiply the master equation by

s^{n}

and sum over

n \geq 0

. For the immigration terms,

\sum_{n \geq 0} s^{n} (p_{n - 1} - p_{n}) = (s - 1) ϕ .

For the death terms, shifting the first sum gives

\sum_{n \geq 0} s^{n} ((n + 1) p_{n + 1} - n p_{n}) = (1 - s) \frac{\partial ϕ}{\partial s} .

\frac{\partial ϕ}{\partial t} = λ (s - 1) ϕ + β (1 - s) \frac{\partial ϕ}{\partial s} .

The normalized ansatz intended in the question is

ϕ (s, t) = exp ((s - 1) f (t)),

for which

ϕ (1, t) = 1

. Substitution into the equation from part (ii) and cancellation of

(s - 1) ϕ

f^{'} = λ - β f .

Thus

f (t) = \frac{λ}{β} + (f (0) - \frac{λ}{β}) e^{- βt} .

f (t)

. Equivalently, differentiating at

s = 1

gives

⟨ n ⟩ = \partial_{s} ϕ (1, t) = f (t), σ^{2} = \partial_{s}^{2} ϕ (1, t) + \partial_{s} ϕ (1, t) - \partial_{s} ϕ (1, t)^{2} = f (t) .

Consequently

⟨ n ⟩ ⟶ \frac{λ}{β}, σ^{2} ⟶ \frac{λ}{β} .

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