Birth process#Simple birth process

A birth process with birth rates $(\backslash lambda\_n,\; n\backslash in\; \backslash mathbb\{N\})$ and initial value $k\backslash in\; \backslash mathbb\{N\}$ is a minimal right-continuous process $(X\_t,\; t\backslash ge\; 0)$ such that $X\_0=k$ and the interarrival times $T\_i\; =\; \backslash inf\backslash \{t\backslash ge\; 0:\; X\_t=i+1\backslash \}\; -\; \backslash inf\backslash \{t\backslash ge\; 0:\; X\_t=i\backslash \}$ are independent exponential random variables with parameter $\backslash lambda\_i$.{{sfnp|Norris|1997|p=81}}

A birth process with rates $(\backslash lambda\_n,\; n\backslash in\; \backslash mathbb\{N\})$ and initial value $k\backslash in\; \backslash mathbb\{N\}$ is a process $(X\_t,\; t\backslash ge\; 0)$ such that:

• $X\_0=k$
• $\backslash mathbb\{P\}(X\_\{t+h\}=X\_t+1)=\backslash lambda\_\{X\_t\}h+o(h)$
• $\backslash mathbb\{P\}(X\_\{t+h\}=X\_t)=o(h)$

(The third and fourth conditions use little o notation.)

These conditions ensure that the process starts at $i$, is non-decreasing and has independent single births continuously at rate $\backslash lambda\_n$, when the process has value $n$.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

A birth process can be defined as a continuous-time Markov process (CTMC) $(X\_t,\; t\backslash ge\; 0)$ with the non-zero Q-matrix entries $q\_\{n,n+1\}=\backslash lambda\_n=-q\_\{n,n\}$ and initial distribution $i$ (the random variable which takes value $i$ with probability 1).{{sfnp|Norris|1997|p=81–82}}

$Q=\backslash begin\{pmatrix\}$

-\lambda_0 & \lambda_0 & 0 & 0 & \cdots \\

0 & -\lambda_1 & \lambda_1 & 0 & \cdots \\

0 & 0 & -\lambda_2 & \lambda_2 & \cdots\\

\vdots & \vdots & \vdots & & \vdots \ddots

\end{pmatrix}

Some authors require that a birth process start from 0 i.e. that $X\_0=0$,{{sfnp|Grimmett|Stirzaker|1992|p=232}} while others allow the initial value to be given by a probability distribution on the natural numbers.{{sfnp|Norris|1997|p=81}} The state space can include infinity, in the case of an explosive birth process.{{sfnp|Norris|1997|p=81}} The birth rates are also called intensities.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

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,{{sfnp|Karlin|McGregor|1957}} any birth process is transient. The transition matrices $((p\_\{i,j\}(t))\_\{i,j\backslash in\backslash mathbb\{N\}\}),\; t\backslash ge\; 0)$ of a birth process satisfy the Kolmogorov forward and backward equations.

The backwards equations are:{{sfnp|Ross|2010|p=386}}

:$p\text{'}\_\{i,j\}(t)=\backslash lambda\_i\; (p\_\{i+1,j\}(t)-p\_\{i,j\}(t))$ (for $i,j\backslash in\backslash mathbb\{N\}$)

The forward equations are:{{sfnp|Ross|2010|p=389}}

:$p\text{'}\_\{i,i\}(t)=-\backslash lambda\_i\; p\_\{i,i\}(t)$ (for $i\backslash in\backslash mathbb\{N\}$)

:$p\text{'}\_\{i,j\}(t)=\backslash lambda\_\{j-1\}p\_\{i,j-1\}(t)-\backslash lambda\_j\; p\_\{i,j\}(t)$ (for $j\backslash ge\; i+1$)

From the forward equations it follows that:{{sfnp|Ross|2010|p=389}}

:$p\_\{i,i\}(t)=e^\{-\backslash lambda\_i\; t\}$ (for $i\backslash in\backslash mathbb\{N\}$)

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

Unlike a Poisson process, a birth process may have infinitely many births in a finite amount of time. We define $T\_\backslash infty=\backslash sup\; \backslash \{T\_n:n\backslash in\backslash mathbb\{N\}\backslash \}$ and say that a birth process explodes if $T\_\backslash infty$ is finite. If then the process is explosive with probability 1; otherwise, it is non-explosive with probability 1 ("honest").{{sfnp|Norris|1997|p=83}}{{sfnp|Grimmett|Stirzaker|1992|p=234}}

File:Poisson process.svg is a special case of a birth process.]]

A Poisson process is a birth process where the birth rates are constant i.e. $\backslash lambda\_n=\backslash lambda$ for some $\backslash lambda>0$.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

File:Simple birth process.svg

A simple birth process is a birth process with rates $\backslash lambda\_n=n\backslash lambda$.{{sfnp|Norris|1997|p=82}} It models a population in which each individual gives birth repeatedly and independently at rate $\backslash lambda$. Udny Yule studied the processes, so they may be known as Yule processes.{{sfnp|Ross|2010|p=375}}

The number of births in time $t$ from a simple birth process of population $n$ is given by:{{sfnp|Grimmett|Stirzaker|1992|p=232}}

:$p\_\{n,n+m\}(t)=\backslash binom\{n\}\{m\}(\backslash lambda\; t)^m(1-\backslash lambda\; t)^\{n-m\}+o(h)$

In exact form, the number of births is the negative binomial distribution with parameters $n$ and $e^\{-\backslash lambda\; t\}$. For the special case $n=1$, this is the geometric distribution with success rate $e^\{-\backslash lambda\; t\}$.{{sfnp|Ross|2010|p=383}}

The expectation of the process grows exponentially; specifically, if $X\_0=1$ then $\backslash mathbb\{E\}(X\_t)=e^\{\backslash lambda\; t\}$.{{sfnp|Norris|1997|p=82}}

A simple birth process with immigration is a modification of this process with rates $\backslash lambda\_n=n\backslash lambda+\backslash nu$. This models a population with births by each population member in addition to a constant rate of immigration into the system.{{sfnp|Grimmett|Stirzaker|1992|p=232}}

{{reflist}}

• {{cite book | title=Probability and Random Processes | first2=D. R. | last2=Stirzaker | first1=G. R. | last1=Grimmett | author-link=Geoffrey Grimmett | year=1992 |edition=second | publisher=Oxford University Press|ISBN=0198572220 }}
• {{cite journal| url=https://www.ams.org/journals/tran/1957-086-02/S0002-9947-1957-0094854-8/S0002-9947-1957-0094854-8.pdf

| title=The classification of birth and death processes | last1=Karlin |first1= Samuel | last2= McGregor |first2= James | year=1957 | journal=Transactions of the American Mathematical Society | volume=86 | issue=2 | pages=366-400 | author1-link= Samuel Karlin}}

• {{cite book |last=Norris |first=J.R. |title=Markov Chains |year=1997 |publisher=Cambridge University Press |isbn=9780511810633}}
• {{cite book |last=Ross |first=Sheldon M. |title=Introduction to Probability Models |edition=tenth |year=2010 |publisher=Academic Press |isbn=9780123756862}}
• {{cite book |last=Upton |first=G. |last2=Cook |first2=I. |year=2014 |title=A Dictionary of Statistics |edition=third |isbn=9780191758317}}

{{Stochastic processes}}

Category:Markov processes

Category:Poisson point processes