The birth–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. 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 or in biology. 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.
Contents
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 birth rates
Examples
A pure birth process is a birth–death process where
A pure death process is a birth–death process where
A (homogeneous) Poisson process is a pure birth process where
M/M/1 model and M/M/c model, both used in queueing theory, are birth–death processes used to describe customers in an infinite queue.
Use in queueing theory
In queueing theory the birth–death process is the most fundamental example of a queueing model, the M/M/C/K/
M/M/1 queue
The M/M/1 is a single server queue with an infinite buffer size. In a non-random environment the birth–death process in queueing models tend to be long-term averages, so the average rate of arrival is given as
The difference equations for the probability that the system is in state k at time t are,
M/M/c queue
The M/M/c is a multi-server queue with C servers and an infinite buffer. This differs from the M/M/1 queue only in the service time, which now becomes
and
with
M/M/1/K queue
The M/M/1/K queue is a single server queue with a buffer of size K. This queue has applications in telecommunications, as well as in biology when a population has a capacity limit. 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 population dies from over population,
The differential equations for the probability that the system is in state k at time t are,
Equilibrium
A queue is said to be in equilibrium if the limit
Using the M/M/1 queue as an example, the steady state (equilibrium) equations are,
If
Limit behaviour
In a small time