Birth–death process

The birth–death process is a special case of continuous-time Markov process where the state transitions are of only two types: "births", which increase the state variable by one and "deaths", which decrease the state by one. The model's name comes from a common application, the use of such models to represent the current size of a population where the transitions are literal births and deaths. Birth–death processes have many applications in demography, queueing theory, performance engineering, epidemiology or in biology. They may be used, for example to study the evolution of bacteria, the number of people with a disease within a population, or the number of customers in line at the supermarket.

When a birth occurs, the process goes from state n to n + 1. When a death occurs, the process goes from state n to state n − 1. The process is specified by birth rates $\{\lambda _{i}\}_{{i=0\dots \infty }}$ and death rates $\{\mu _{i}\}_{{i=1\dots \infty }}$ .

Examples

A pure birth process is a birth–death process where $\mu _{{i}}=0$ for all $i \ge 0$ .

A pure death process is a birth–death process where $\lambda _{{i}}=0$ for all $i \ge 0$ .

A (homogeneous) Poisson process is a pure birth process where $\lambda _{{i}}=\lambda$ for all $i \ge 0$

M/M/1 model and M/M/c model, both used in queueing theory, are birth–death processes used to describe customers in an infinite queue.

Use in queueing theory

In queueing theory the birth–death process is the most fundamental example of a queueing model, the M/M/C/K/ $\infty$ /FIFO (in complete Kendall's notation) queue. This is a queue with Poisson arrivals, drawn from an infinite population, and C servers with exponentially distributed service time with K places in the queue. Despite the assumption of an infinite population this model is a good model for various telecommunication systems.

M/M/1 queue

Main article: M/M/1 queue

The M/M/1 is a single server queue with an infinite buffer size. In a non-random environment the birth–death process in queueing models tend to be long-term averages, so the average rate of arrival is given as $\lambda$ and the average service time as $1/\mu$ . The birth and death process is a M/M/1 queue when,

\lambda _{{i}}=\lambda {\text{ and }}\mu _{{i}}=\mu {\text{ for all }}i.\,

The difference equations for the probability that the system is in state k at time t are,

p_{0}^{\prime }(t)=\mu _{1}p_{1}(t)-\lambda _{0}p_{0}(t)\,

p_{k}^{\prime }(t)=\lambda _{{k-1}}p_{{k-1}}(t)+\mu _{{k+1}}p_{{k+1}}(t)-(\lambda _{k}+\mu _{k})p_{k}(t)\,

M/M/c queue

Main article: M/M/c queue

The M/M/c is a multi-server queue with C servers and an infinite buffer. This differs from the M/M/1 queue only in the service time, which now becomes

\mu _{i}=i\mu {\text{ for }}i\leq C\,

and

\mu _{i}=C\mu {\text{ for }}i\geq C\,

with

\lambda _{i}=\lambda {\text{ for all }}i.\,

M/M/1/K queue

The M/M/1/K queue is a single server queue with a buffer of size K. This queue has applications in telecommunications, as well as in biology when a population has a capacity limit. In telecommunication we again use the parameters from the M/M/1 queue with,

\lambda _{i}=\lambda {\text{ for }}0\leq i<K\,

\lambda _{i}=0{\text{ for }}i\geq K\,

\mu _{i}=\mu {\text{ for }}1\leq i\leq K.\,

In biology, particularly the growth of bacteria, when the population is zero there is no ability to grow so,

\lambda _{0}=0.\,

Additionally if the capacity represents a limit where the population dies from over population,

\mu _{K}=0.\,

The differential equations for the probability that the system is in state k at time t are,

p_{0}^{\prime }(t)=\mu _{1}p_{1}(t)-\lambda _{0}p_{0}(t)

p_{k}^{\prime }(t)=\lambda _{{k-1}}p_{{k-1}}(t)+\mu _{{k+1}}p_{{k+1}}(t)-(\lambda _{k}+\mu _{k})p_{k}(t){\text{ for }}k\leq K\,

p_{k}^{\prime }(t)=0{\text{ for }}k>K\,

Equilibrium

A queue is said to be in equilibrium if the limit $\lim _{{t\to \infty }}p_{k}(t)$ exists. For this to be the case, $p_{k}^{\prime }(t)$ must be zero.

Using the M/M/1 queue as an example, the steady state (equilibrium) equations are,

\lambda _{0}p_{0}(t)=\mu _{1}p_{1}(t)\,

(\lambda _{k}+\mu _{k})p_{k}(t)=\lambda _{{k-1}}p_{{k-1}}(t)+\mu _{{k+1}}p_{{k+1}}(t)\,

If $\lambda _{k}=\lambda$ and $\mu _{k}=\mu$ for all $k$ (the homogeneous case), this can be reduced to

\lambda p_{k}(t)=\mu p_{{k+1}}(t){\text{ for }}k\geq 0.\,

Limit behaviour

In a small time $\Delta t$ , only three types of transitions are possible: one death, or one birth, or no birth nor death. If the rate of occurrences (per unit time) of births is $\lambda$ and that for deaths is $\mu$ , then the probabilities of the above transitions are $\lambda \Delta t$ , $\mu \Delta t$ , and $1-(\lambda +\mu )\Delta t$ respectively. For a population process, "birth" is the transition towards increasing the population by 1 while "death" is the transition towards decreasing the population size by 1.

References

G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 1: Quasi-Birth-and-Death Processes; ASA SIAM, 1999.
M. A. Nowak. Evolutionary Dynamics: Exploring the Equations of Life, Harvard University Press, 2006.
J. Virtamo,"Birth-death processesBirth-death processes", 38.3143 Queueing Theory.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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