Birth process

Properties

As for CTMCs, a birth process has the Markov property. The CTMC definitions for communicating classes, irreducibility and so on apply to birth processes. By the conditions for recurrence and transience of a birth–death process, any birth process is transient. The transition matrices ${\displaystyle ((p_{i,j}(t){)}_{i,j\in \mathbb{N}}),t\ge 0)}$ of a birth process satisfy the Kolmogorov forward and backward equations.

The backwards equations are:

${\displaystyle p_{i,j}^{\prime }(t)={\lambda }_i(p_{i+1,j}(t)-p_{i,j}(t))}$ (for ${\displaystyle i,j\in \mathbb{N}}$)

The forward equations are:

${\displaystyle p_{i,i}^{\prime }(t)=-{\lambda }_i{p}_{i,i}(t)}$ (for ${\displaystyle i\in \mathbb{N}}$)

${\displaystyle p_{i,j}^{\prime }(t)={\lambda }_{j-1}{p}_{i,j-1}(t)-{\lambda }_j{p}_{i,j}(t)}$ (for ${\displaystyle j\ge i+1}$)

From the forward equations it follows that:

${\displaystyle p_{i,i}(t)=e^{-{\lambda }_{i}t}}$ (for ${\displaystyle i\in \mathbb{N}}$)

${\displaystyle p_{i,j}(t)={\lambda }_{j-1}{e}^{-{\lambda }_{j}t}{\int }_0^t{e}^{{\lambda }_{j}s}{p}_{i,j-1}(s)\text{d}s}$ (for ${\displaystyle j\ge i+1}$)

Unlike a Poisson process, a birth process may have infinitely many births in a finite amount of time. We define ${\displaystyle T_{\mathrm{\infty }}=sup\{T_n:n\in \mathbb{N}\}}$ and say that a birth process explodes if ${\displaystyle T_{\mathrm{\infty }}}$ is finite. If ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\lambda }_n}<\mathrm{\infty }}$ then the process is explosive with probability 1; otherwise, it is non-explosive with probability 1 ("honest").

Examples

A Poisson process is a birth process where the birth rates are constant i.e. ${\displaystyle {\lambda }_n=\lambda }$ for some ${\displaystyle \lambda >0}$.

Simple birth process

A simple birth process is a birth process with rates ${\displaystyle {\lambda }_n=n\lambda }$. It models a population in which each individual gives birth repeatedly and independently at rate ${\displaystyle \lambda }$. Udny Yule studied the processes, so they may be known as Yule processes.

The number of births in time ${\displaystyle t}$ from a simple birth process of population ${\displaystyle n}$ is given by:

${\displaystyle p_{n,n+m}(t)=(\genfrac{}{}{0ex}{}{n}{m})(\lambda t{)}^m(1-\lambda t{)}^{n-m}+o(h)}$

In exact form, the number of births is the negative binomial distribution with parameters ${\displaystyle n}$ and ${\displaystyle e^{-\lambda t}}$. For the special case ${\displaystyle n=1}$, this is the geometric distribution with success rate ${\displaystyle e^{-\lambda t}}$.

The expectation of the process grows exponentially; specifically, if ${\displaystyle X_0=1}$ then ${\displaystyle \mathbb{E}(X_t)=e^{\lambda t}}$.

A simple birth process with immigration is a modification of this process with rates ${\displaystyle {\lambda }_n=n\lambda +\nu }$. This models a population with births by each population member in addition to a constant rate of immigration into the system.