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Concept
Birth–death process
Summary
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.
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:
For recurrence and transience in Markov processes see Section 5.3 from Markov chain.
Conditions for recurrence and transience were established by Samuel Karlin and James McGregor.
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.
Official source
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