A birth process with birth rates and initial value is a minimal right-continuous process such that and the interarrival times are independent exponential random variables with parameter.
Infinitesimal definition
A birth process with rates and initial value is a process such that:
is independent of
These conditions ensure that the process starts at, is non-decreasing and has independent single births continuously at rate, when the process has value.
A birth process can be defined as a continuous-time Markov process with the non-zero Q-matrix entries and initial distribution .
Variations
Some authors require that a birth process start from 0 i.e. that, while others allow the initial value to be given by a probability distribution on the natural numbers. The state space can include infinity, in the case of an explosive birth process. The birth rates are also called intensities.
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 of a birth process satisfy the Kolmogorov forward and backward equations. The backwards equations are: The forward equations are: From the forward equations it follows that: Unlike a Poisson process, a birth process may have infinitely many births in a finite amount of time. We define and say that a birth process explodes if is finite. If then the process is explosive with probability 1; otherwise, it is non-explosive with probability 1.
Examples
A Poisson process is a birth process where the birth rates are constant i.e. for some.
Simple birth process
A simple birth process is a birth process with rates. It models a population in which each individual gives birth repeatedly and independently at rate. Udny Yule studied the processes, so they may be known as Yule processes. The number of births in time from a simple birth process of population is given by: In exact form, the number of births is the negative binomial distribution with parameters and. For the special case, this is the geometric distribution with success rate. The expectation of the process grows exponentially; specifically, if then. A simple birth process with immigration is a modification of this process with rates. This models a population with births by each population member in addition to a constant rate of immigration into the system.
