birthdeath

Birthdeath, also called birth–death process, is a term in probability and stochastic processes that refers to a continuous-time Markov chain on the nonnegative integers, modeling systems in which changes occur by single increments or decrements. From state n, the process transitions to n+1 at rate λ_n (birth) and to n-1 at rate μ_n (death). The boundary at 0 typically has μ_0 = 0, so the state cannot go below zero.

Formally, the generator Q has off-diagonal elements q_{n,n+1} = λ_n and q_{n,n-1} = μ_n, with q_{n,n} = -(λ_n+μ_n). The process is characterized by the pair of rate functions (λ_n) and (μ_n) and the initial distribution over states.

Common special case is homogeneous rates, λ_n = λ and μ_n = μ for all n≥1, with μ_0 = 0. This birth–death process corresponds to an M/M/1 queue. It has a stationary distribution iff λ < μ; then π_n = (1-ρ) ρ^n with ρ = λ/μ. If λ ≥ μ, no normalizable stationary distribution exists and the chain is transient (λ>μ) or null recurrent (λ=μ).

Other important variants include linear birth–death processes with λ_n = λ n and μ_n = μ n, used as branching processes modeling population growth and extinction. In such models, extinction occurs with probability 1 when μ ≥ λ, and the long-run behavior is governed by the relative magnitudes of the rates.

Applications span population biology, ecology, genetics, and queueing theory. Techniques to analyze birth–death processes include solving forward equations, detailed balance conditions, and examining recurrence and stationary distributions. See also: birth–death process, continuous-time Markov chain, M/M/1 queue, branching process.