The
birth–death process is a special case of
continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths. Birth–death processes have many applications in
demographyDemography is the statistical study of human population. It can be a very general science that can be applied to any kind of dynamic human population, that is, one that changes over time or space...
,
queueing theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
,
performance engineeringPerformance engineering within systems engineering, encompasses the set of roles, skills, activities, practices, tools, and deliverables applied at every phase of the Systems Development Life Cycle which ensures that a solution will be designed, implemented, and operationally supported to meet the...
, or in
biologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
, for example to study the evolution of
bacteriaBacteria are a large domain of prokaryotic microorganisms. Typically a few micrometres in length, bacteria have a wide range of shapes, ranging from spheres to rods and spirals...
.
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
and death rates
.
Examples
A
pure birth process is a birth–death process where
for all
.
A
pure death process is a birth–death process where
for all
.
A (homogeneous)
Poisson processA Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
is a pure birth process where
for all
M/M/1 modelIn queueing theory, a discipline within the mathematical theory of probability, a M/M/1 queue represents the queue length in a system having a single server, where arrivals are detemined by a Poisson process and job service times have an exponential distribution. The model name is written in...
and
M/M/c modelIn the mathematical theory of random processes, the M/M/c queue is a multi-server queue model. It is a generalisation of the M/M/1 queue.Following Kendall's notation it indicates a system where:*Arrivals are a Poisson process...
, both used in
queueing theoryQueueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
, 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 modelIn queueing theory, a queueing model is used to approximate a real queueing situation or system, so the queueing behaviour can be analysed mathematically...
, the
M/M/C/K/
/FIFO (in complete
Kendall's notationIn queueing theory, Kendall's notation is the standard system used to describe and classify the queueing model that a queueing system corresponds to. First suggested by D. G...
) queue. This is a queue with Poisson arrivals, drawn from an infinite population, and
C servers with
exponentially distributedIn probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
service time with
K places in the queue. Despite the assumption of an infinite population this model is a good model for various telecommunication systems.
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
and the average service time as
. The birth and death process is a
M/M/1 queue when,
The difference equations for the
probabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that the system is in state
k at time
t are,
M/M/C queue
The
M/M/C is 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
exists. For this to be the case,
must be zero.
Using the M/M/1 queue as an example, the steady state (equilibrium) equations are,
If
and
for all
(the homogenous case), this can be reduced to
Limit behaviour
In a small time
, only three types of transitions are possible: one death, or one birth, or no birth nor death. If the rate of occurrences (per unit time) of births is
and that for deaths is
, then the probabilities of the above transitions are
,
, and
respectively. For a population process, "birth" is the transition towards increasing the population by 1 while "death" is the transition towards decreasing the
population sizeIn population genetics and population ecology, population size is the number of individual organisms in a population.The effective population size is defined as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies under...
by 1.
See also
- Erlang unit
The erlang is a dimensionless unit that is used in telephony as a statistical measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. It is named after the Danish telephone engineer A. K...
- Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
- Queueing models
- Quasi-birth–death process
- Moran process
A Moran process, named after Patrick Moran, is a stochastic process used in biology to describe finite populations. It can be used to model variety-increasing processes such as mutation as well as variety-reducing effects such as genetic drift and natural selection...