Birth–death process

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.

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 $\{\lambda_i\}_{i=0\dots\infty}$ and death rates $\{\mu_i\}_{i=1\dots\infty}$ .

State diagram of a birth-death process

Examples

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 $i \ge 0$

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/ $\infty$ /FIFO (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 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 $\lambda$ and the average service time as $1/\mu$ . The birth and death process is a M/M/1 queue when,

\lambda_{i}=\lambda\text{ and }\mu_{i}=\mu\text{ for all }i. \,

The difference 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) \,

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

\mu_i = i\mu\text{ for }i\leq C \,

and

\mu_i = C\mu\text{ for }i\geq C \,

with

\lambda_i = \lambda\text{ for all }i. \,

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\text{ for }0 \leq i < K \,

\lambda_i=0\text{ for }i\geq K \,

\mu_i=\mu\text{ 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)\text{ for }k \leq K \,

p_k^\prime(t)=0\text{ for }k > K \,

Equilibrium

A queue is said to be in equilibrium if the limit $\lim_{t \to \infty}p_k(t)$ 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 homogeneous case), this can be reduced to

\lambda p_k(t)=\mu p_{k+1}(t)\text{ for }k\geq 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.

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.
J. Virtamo,"Birth-death processesBirth-death processes"[1], 38.3143 Queueing Theory.

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Birth–death process

Contents

Examples

Use in queueing theory

M/M/1 queue

M/M/c queue

M/M/1/K queue

Equilibrium

Limit behaviour

See also

References

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