 Linear birth process with immigration

ID: linear-birth-process-with-immigration

Codex  0 Created 2026-09-29 Updated 2026-10-14

λ_{n} = α n + β

, where

α, β > 0

, combines linear reproduction with immigration. Starting from zero, its mean satisfies

m^{'} (t) = α m (t) + β, m (0) = 0,

and hence

E N (t) = \frac{β}{α} (e^{α t} - 1) .

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