# Birth–death process
A birth–death process is a continuous-time [Markov chain](https://www.edgepedia.org/markov-chain) whose state is a count (of particles, individuals, customers, or lineages) and in which the only possible changes are an increase of the state by one, called a birth, or a decrease by one, called a death. When the state is n, births occur at rate λ_n and deaths at rate μ_n, both exponential, so the process jumps only between neighboring states on the non-negative integers.[1](https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/wics.1423) • [2](https://www.sciencedirect.com/topics/mathematics/birth-and-death-process)
| Key fact | Detail |
|---|---|
| Definition | Continuous-time Markov chain on the non-negative integers in which only jumps to adjacent states are allowed; a birth raises the state by 1, a death lowers it by 1[1](https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/wics.1423) |
| Local rates | In a short interval (t, t + Δt), the process in state n transitions to state n + 1 with probability λ_nΔt + o(Δt) and to state n − 1 with probability μ_nΔt + o(Δt)[3](https://academic.oup.com/bib/article/7/1/70/263777) • [7](https://encyclopediaofmath.org/wiki/Birth-and-death_process) |
| Holding time | The sojourn time in state n is exponential with mean 1/(λ_n + μ_n); the next jump is a birth with probability λ_n/(λ_n + μ_n)[3](https://academic.oup.com/bib/article/7/1/70/263777) |
| Linear case | With per-capita rates λ and μ, the extinction probability from one individual is min(1, μ/λ) and the expected population size is e^{(λ−μ)t}[4](https://jovanaeducation.com/glossary/continuous-time-markov-branching) |
| Queueing example | The M/M/1 queue is a birth–death process with λ_n = λ and μ_n = μ; for ρ = λ/μ < 1 the stationary distribution is p_i = (1 − ρ)ρ^i and the mean number in the system is ρ/(1 − ρ)[5](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Discrete_Stochastic_Processes_%28Gallager%29/06%3A_Markov_processes_with_countable_state_spaces/6.05%3A_Birth-Death_Processes) |
| Transience test | An irreducible birth–death process is transient if and only if Σ_{n≥1} (μ_1⋯μ_n)/(λ_1⋯λ_n) < ∞[6](https://www.netlab.tkk.fi/opetus/s38215/k04/Lectures/ch3.pdf) |
| Historical landmarks | Yule (1924) studied the pure birth process λ_n = nλ; Feller (1939) treated constant per-capita rates; Kendall (1948) solved the time-inhomogeneous case[7](https://encyclopediaofmath.org/wiki/Birth-and-death_process) |
## Definition and mechanics
The process is specified entirely by its birth rate function λ_n and death rate function μ_n. Choosing these rates in terms of identifiable parameters of a real system defines the model; analyzing the resulting process is the second step of any application.[3](https://academic.oup.com/bib/article/7/1/70/263777) Because only neighboring states communicate, the generator is tridiagonal: G(n, n+1) = λ_n, G(n, n−1) = μ_n, and G(n, n) = −(λ_n + μ_n).[8](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)
The state probabilities p_n(t) = Pr{X(t) = n} satisfy the Kolmogorov forward equations, a system of differential equations that must be solved subject to an initial condition and boundary conditions.[3](https://academic.oup.com/bib/article/7/1/70/263777) Boundary states matter: in a population model, state 0 with λ_0 = 0 is absorbing and represents extinction, while reflecting boundaries appear in models with mutation or immigration.[3](https://academic.oup.com/bib/article/7/1/70/263777)
## Special cases
**Pure birth and pure death.** If μ_n = 0 for all n the process is a pure birth process; the Poisson process with constant rate λ is the simplest example, and its state probabilities are the [Poisson distribution](https://www.edgepedia.org/poisson-distribution) with parameter λt.[7](https://encyclopediaofmath.org/wiki/Birth-and-death_process) The Yule process, with λ_n = nλ, was first studied by G. Yule in 1924 in connection with the mathematical theory of evolution, modeling the number of species within a genus when new species arise by mutation at a rate proportional to the number present.[7](https://encyclopediaofmath.org/wiki/Birth-and-death_process) • [3](https://academic.oup.com/bib/article/7/1/70/263777)
**The linear process.** When λ_n = nλ and μ_n = nμ, each individual gives birth or dies independently at constant per-capita rates. This linear birth–death process, studied by Feller and solved in generality by David G. Kendall in 1948 for time-dependent rates λ(t) and μ(t), is the classical model of population growth and extinction.[9](https://www.columbia.edu/cu/simontavare/STpapers-pdf/BDP.pdf) Its expected size grows exponentially, N(t) = N_0 e^{(λ−μ)t}, with net growth rate λ − μ.[10](https://cmp.phys.ufl.edu/PHZ4710/files/unit3/birth-death.html) From a single founder, extinction occurs with probability min(1, μ/λ); when λ > μ the population grows without bound with the complementary probability.[4](https://jovanaeducation.com/glossary/continuous-time-markov-branching)
**Immigration.** Adding a constant immigration rate ν gives λ_n = nλ + ν, a [branching process](https://www.edgepedia.org/branching-process) with immigration. With λ = 0 and ν > 0 the model describes a telephone exchange with infinitely many lines, where ν is the rate of incoming calls and μ the expected conversation duration.[7](https://encyclopediaofmath.org/wiki/Birth-and-death_process) The immigration–death process with constant birth rate c and per-particle death rate b is positive recurrent (returns to a state and has a steady distribution) with a stationary Poisson distribution with parameter c/b.[8](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)
## Long-run behavior
Whether a birth–death process settles down, wanders forever, or escapes to infinity is governed by products of rate ratios. An irreducible birth–death process is transient if and only if the series Σ_{n=1}^{∞} (μ_1⋯μ_n)/(λ_1⋯λ_n) converges.[6](https://www.netlab.tkk.fi/opetus/s38215/k04/Lectures/ch3.pdf) When the process is positive recurrent, its invariant distribution π satisfies the balance equations π_n λ_n = π_{n+1} μ_{n+1}, giving π_n = π_0 ∏_{j=0}^{n−1} λ_j/μ_{j+1}.[5](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Discrete_Stochastic_Processes_%28Gallager%29/06%3A_Markov_processes_with_countable_state_spaces/6.05%3A_Birth-Death_Processes) Positive recurrent birth–death chains on the non-negative integers are reversible with respect to this distribution.[8](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)
A deterministic model and its stochastic counterpart can disagree qualitatively. A branching process whose mean offspring number equals one goes extinct with certainty, while the corresponding deterministic model describes a population of constant size; similarly, in stochastic logistic models extinction at the absorbing state 0 is certain even though the deterministic solution approaches a positive carrying capacity.[3](https://academic.oup.com/bib/article/7/1/70/263777)
## Applications
The neighbor-only jump structure simplifies the mathematics while remaining applicable to many real systems, which is why birth–death processes have become a standard tool for queueing, reliability, and inventory systems; they also model flows of radioactive and cosmic particles in physics and the development of enterprise numbers and manpower in economics.[11](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)
**Queueing.** The M/M/1 queue, with Poisson arrivals at rate λ and exponential service at rate μ, is the canonical birth–death process: λ_n = λ for all n and μ_n = μ for n ≥ 1. For traffic intensity ρ = λ/μ < 1 the stationary distribution is geometric, p_i = (1 − ρ)ρ^i, the mean number in the system is ρ/(1 − ρ), and the mean time a customer spends in the system is 1/(μ − λ). At ρ = 1 the chain becomes null recurrent and no steady state exists.[5](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Discrete_Stochastic_Processes_%28Gallager%29/06%3A_Markov_processes_with_countable_state_spaces/6.05%3A_Birth-Death_Processes) Multi-server M/M/m systems, in which the death rate is iμ for i < m and mμ for i ≥ m, are analyzed by the same balance equations.[5](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Discrete_Stochastic_Processes_%28Gallager%29/06%3A_Markov_processes_with_countable_state_spaces/6.05%3A_Birth-Death_Processes)
**Biology and epidemiology.** In phylogenetics and population genetics, linear birth–death processes with λ_n = nλ(t) and μ_n = nμ(t) describe the number of species in a genus, individuals in a population, or copies of an allele; the underlying tree structure, pruned of lineages that do not survive to the present, yields the reconstructed phylogenetic trees used to estimate divergence times and speciation and extinction rates.[12](https://pmc.ncbi.nlm.nih.gov/articles/PMC11867164/) In phylodynamic inference, births represent transmission events and deaths represent cessation of infectiousness, so a birth–death model fitted to pathogen sequence data estimates the rate of spread of an epidemic.[13](https://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1009805) In population genetics, the Moran model, a birth–death process with nonlinear rates on a finite state space, gives exact results for fixation probabilities of mutants, recovering Kimura's classical diffusion formula as an approximation.[3](https://academic.oup.com/bib/article/7/1/70/263777)
## Statistical inference
Although the theoretical properties of general birth–death processes are well understood, statistical work has traditionally been limited to the simple linear (Kendall) process, which is analytically tractable. For a discretely observed general process, the likelihood usually cannot be written in closed form, because the state path between observations is unobserved; this long hindered estimation. Robust computational methods have narrowed the gap: finite-time transition probabilities, first-passage times, and equilibrium probabilities can now be evaluated numerically, often via continued-fraction representations of Laplace transforms, and expectation–maximization algorithms exploiting the connection between continuously and discretely observed processes deliver maximum likelihood estimates for previously intractable models.[1](https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/wics.1423) • [14](https://pmc.ncbi.nlm.nih.gov/articles/PMC4196218/) Regression approaches analogous to [Poisson regression](https://www.edgepedia.org/poisson-regression) follow straightforwardly from these likelihood tools.[1](https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/wics.1423)
## References
1. [Forrest W. Crawford, Lam Si Tung Ho, Marc A. Suchard (2018). Computational methods for birth-death processes. WIREs Computational Statistics 10:e1423.](https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/wics.1423)
2. [Birth and Death Process: Overview. ScienceDirect Topics.](https://www.sciencedirect.com/topics/mathematics/birth-and-death-process)
3. [N. Kareva et al. Biological applications of the theory of birth-and-death processes. Briefings in Bioinformatics 7(1):70.](https://academic.oup.com/bib/article/7/1/70/263777)
4. [Continuous-time Markov branching, JOVANA Education glossary.](https://jovanaeducation.com/glossary/continuous-time-markov-branching)
5. [6.5: Birth-Death Processes, Discrete Stochastic Processes (Gallager), Engineering LibreTexts.](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Discrete_Stochastic_Processes_%28Gallager%29/06%3A_Markov_processes_with_countable_state_spaces/6.05%3A_Birth-Death_Processes)
6. [Chapter 3: Continuous-Time Markov Chains, lecture notes, Helsinki University of Technology.](https://www.netlab.tkk.fi/opetus/s38215/k04/Lectures/ch3.pdf)
7. [Birth-and-death process. Encyclopedia of Mathematics (V.P. Chistyakov).](https://encyclopediaofmath.org/wiki/Birth-and-death_process)
8. [16.21: Continuous-Time Birth-Death Chains (Kyle Siegrist), Statistics LibreTexts.](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)
9. [Simon Tavaré (2018). The linear birth–death process: an inferential retrospective. Applied Probability Trust.](https://www.columbia.edu/cu/simontavare/STpapers-pdf/BDP.pdf)
10. [Birth-Death Process, PHZ4710 Introduction to Biological Physics, University of Florida.](https://cmp.phys.ufl.edu/PHZ4710/files/unit3/birth-death.html)
11. [Frank Beichelt. Stochastic Processes (Chapter 5, Continuous-Time Markov Processes), Yale University reading.](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)
12. [Simon Tavaré (2025). Birth and death processes in phylogenetics and population genetics. Philosophical Transactions of the Royal Society B 380:20230300.](https://pmc.ncbi.nlm.nih.gov/articles/PMC11867164/)
13. [A computationally tractable birth-death model that combines phylogenetic and epidemiological data. PLOS Computational Biology (2022).](https://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1009805)
14. [Forrest W. Crawford, Marc A. Suchard (2012). Estimation for general birth-death processes. Annals of Applied Statistics / PMC.](https://pmc.ncbi.nlm.nih.gov/articles/PMC4196218/)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Point, renewal, and branching processes › General and continuous-time branching processes*
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