Birth-death process

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 application in demography, queueing theory, or in biology, for example to study the evolution of bacteria.

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 $\{lambda\_\{i\}\}\_\{i=0..infty\}$ and death rates $\{mu\_\{i\}\}\_\{i=1..infty\}$.

Examples of birth-death processes

A pure birth process is a birth-death process where $mu\_\{i\}\; =\; 0$ for all $i\; ge\; 0$.

A pure death process is a birth-death process where $lambda\_\{i\}\; =\; 0$ for all $i\; ge\; 0$.

A (homogeneous) Poisson process is a pure birth process where $lambda\_\{i\}\; =\; lambda$ for all $i\; ge\; 0$

A "M/M/1" queue is a birth-death process 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/$infty$/FIF0" (in complete Kendall's notation) queue. This is a queue with Poisson arrivals, drawn from an infinite population, and "C" servers with exponentially distributed service time with "K" places in the queue. Despite the assumption of an infinite population this model is good model for various telecommunciation systems.

The "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 $lambda$ and the average mean service time as $1/mu$. The birth and death process is a "M/M/1" queue when,

:$lambda\_\{i\}=lambda$ and $mu\_\{i\}=mu$ for all $i$.

The differential equations for the probability that the system is in state "k" at time "t" are,

:$p\_0^prime(t)=mu\_1\; p\_1(t)-lambda\_0\; p\_0(t)$:$p\_k^prime(t)=lambda\_\{k-1\}\; p\_\{k-1\}(t)+mu\_\{k+1\}\; p\_\{k+1\}(t)-(lambda\_k\; +mu\_k)\; p\_k(t)$

The "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,

:$mu\_\{i\}=imu$ for $ileq\; C$ and

:$mu\_\{i\}=Cmu$ for $igeq\; C$ with

:$lambda\_\{i\}=lambda$ for all $i$.

The differential equations for the probability that the system is in state "k" at time "t" are,

:$p\_0^prime(t)=mu\; p\_1(t)-lambda\; p\_0(t)$:$p\_k^prime(t)=lambda\; p\_\{k-1\}(t)+(k+1)mu\; p\_\{k+1\}(t)-(lambda\; +kmu)\; p\_k(t)$ for $k\; leq\; C-1$:$p\_k^prime(t)=lambda\; p\_\{k-1\}(t)+Cmu\; p\_\{k+1\}(t)-(lambda\; +Cmu)\; p\_k(t)$ for $k\; geq\; C$

The "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,

:$lambda\_\{i\}=lambda$ for

:$lambda\_\{i\}=0$ for $igeq\; K$

:$mu\_\{i\}=mu$ for $1\; leq\; i\; leq\; K$.

In biology, particularly the growth of bacteria, when the population is zero there is no ability to grow so,

:$lambda\_\{0\}=0$.

Additionally if the capacity represents a limit where the population dies from over population,

:$mu\_\{K\}=0$..

The differential equations for the probability that the system is in state "k" at time "t" are,

:$p\_0^prime(t)=mu\_1\; p\_1(t)-lambda\_0\; p\_0(t)$:$p\_k^prime(t)=lambda\_\{k-1\}\; p\_\{k-1\}(t)+mu\_\{k+1\}\; p\_\{k+1\}(t)-(lambda\_k\; +mu\_k)\; p\_k(t)$ for $k\; leq\; K$:$p\_k^prime(t)=0$ for $k\; >\; K$

Equilibrium

A queue is said to be in equilibrium if the limit $lim\_\{t\; o\; infty\}p\_k(t)$ exists. For this to be the case, $p\_k^prime(t)$ must be zero.

Using the M/M/1 queue as an example, the steady state (equilibrium) equations are,

:$lambda\_0\; p\_0(t)=mu\_1\; p\_1(t)$

:$(lambda\_k\; +mu\_k)\; p\_k(t)=lambda\_\{k-1\}\; p\_\{k-1\}(t)+mu\_\{k+1\}\; p\_\{k+1\}(t)$

If $lambda\_k=lambda$ and $mu\_k=mu$ for all $k$ (the homogenous case), this can be reduced to

:$lambda\; p\_k(t)=mu\; p\_\{k+1\}(t)$ for $kgeq\; 0$

Limit behaviour

In a small time $Delta\; t$, 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 $lambda$ and that for deaths is $mu$, then the probabilities of the above transitions are $lambda\; Delta\; t$, $mu\; Delta\; t$, and $1\; -\; (lambda\; +\; mu)\; Delta\; t$ respectively. For a population process, "birth" is the transition towards increasing the population by 1 while "death" is the transition towards decreasing the population size by 1.

ee also

* Erlang unit
* Queueing theory
* Queueing models
* Quasi-birth-death process

References

* G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 1: Quasi-Birth-and-Death Processes; ASA SIAM, 1999.
* M. A. Nowak. Evolutionary Dynamics: Exploring the Equations of Life, Harvard University Press, 2006.