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.
[1] 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.
is assumed to satisfy the following properties: This process is represented by the following figure with the states of the process (i.e. the number of individuals in the population) depicted by the circles, and transitions between states indicated by the arrows.
Conditions for recurrence and transience were established by Samuel Karlin and James McGregor.
[2] 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.
Then, the conditions for recurrence and transience of a birth-and-death process are as follows.
Wider classes of birth-and-death processes, for which the conditions for recurrence and transience can be established, can be found in.
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.
If a birth-and-death process is ergodic, then there exists steady-state probabilities
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
If the system is in state k, then the probability of birth during an interval
Birth–death processes are used in phylodynamics as a prior distribution for phylogenies, i.e. a binary tree in which birth events correspond to branches of the tree and death events correspond to leaf nodes.
[5] Notably, they are used in viral phylodynamics[6] to understand the transmission process and how the number of people infected changes through time.
[7] The use of generalized birth-death processes in phylodynamics has stimulated investigations into the degree to which the rates of birth and death can be identified from data.
This is a queue with Poisson arrivals, drawn from an infinite population, and C servers with exponentially distributed service times with K places in the queue.
The M/M/1 is a single server queue with an infinite buffer size.
The birth and death process is an M/M/1 queue when, The differential equations for the probability that the system is in state k at time t are Pure birth process with
We have the following system of differential equations: Under the initial condition
The M/M/C is a multi-server queue with C servers and an infinite buffer.
It characterizes by the following birth and death parameters: and with The system of differential equations in this case has the form: Pure death process with
We have the following system of differential equations: Under the initial condition
we obtain the solution that presents the version of binomial distribution depending on time parameter
In telecommunication we again use the parameters from the M/M/1 queue with, In biology, particularly the growth of bacteria, when the population is zero there is no ability to grow so, Additionally if the capacity represents a limit where the individual dies from over population, The differential equations for the probability that the system is in state k at time t are A queue is said to be in equilibrium if the steady state probabilities
The condition for the existence of these steady-state probabilities in the case of M/M/1 queue is
Using the M/M/1 queue as an example, the steady state equations are This can be reduced to So, taking into account that
, we obtain Bilateral birth-and-death process is defined similarly to that standard one with the only difference that the birth and death rates
[10] Following this, a bilateral birth-and-death process is recurrent if and only if The notions of ergodicity and null-recurrence are defined similarly by extending the corresponding notions of the standard birth-and-death process.