Birth–death process-Cedar Herald Index

The birth–death process (or birth-and-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. It was introduced by William Feller. 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, biology and other areas. 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.

Definition

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 positive birth rates and positive death rates . Specifically, denote the process by , and . Then for small , the function is assumed to satisfy the following properties:

Birth–death process

Recurrence and transience

For recurrence and transience in Markov processes see Section 5.3 from Markov chain.

Birth–death process

Conditions for recurrence and transience

Conditions for recurrence and transience were established by Samuel Karlin and James McGregor.

Birth–death process

A birth-and-death process is recurrent if and only if

A birth-and-death process is ergodic if and only if

A birth-and-death process is null-recurrent if and only if

By using Extended Bertrand's test (see Section 4.1.4 from Ratio test) the conditions for recurrence, transience, ergodicity and null-recurrence can be derived in a more explicit form.

For integer let denote the th iterate of natural logarithm, i.e. and for any , .

Then, the conditions for recurrence and transience of a birth-and-death process are as follows.

The birth-and-death process is transient if there exist and such that for all

where the empty sum for is assumed to be 0.

The birth-and-death process is recurrent if there exist and such that for all

Wider classes of birth-and-death processes, for which the conditions for recurrence and transience can be established, can be found in.

Application

Consider one-dimensional random walk that is defined as follows. Let , and where takes values , and the distribution of is defined by the following conditions:

where satisfy the condition .

The random walk described here is a discrete time analogue of the birth-and-death process (see Markov chain) with the birth rates

and the death rates

. So, recurrence or transience of the random walk is associated with recurrence or transience of the birth-and-death process.

The random walk is transient if there exist , and such that for all

where the empty sum for is assumed to be zero.

The random walk is recurrent if there exist and such that for all

Stationary solution

If a birth-and-death process is ergodic, then there exists steady-state probabilities where is the probability that the birth-and-death process is in state at time The limit exists, independent of the initial values and is calculated by the relations:

These limiting probabilities are obtained from the infinite system of differential equations for

and the initial condition

In turn, the last system of differential equations is derived from the system of difference equations that describes the dynamic of the system in a small time . During this small time only three types of transitions are considered as one death, or one birth, or no birth nor death. The probability of the first two of these transitions has the order of . Other transitions during this small interval such as more than one birth, or more than one death, or at least one birth and at least one death have the probabilities that are of smaller order than , and hence are negligible in derivations. If the system is in state k, then the probability of birth during an interval is , the probability of death is , and the probability of no birth and no death is . For a population process, "birth" is the transition towards increasing the population size by 1 while "death" is the transition towards decreasing the population size by 1.