# Birth-and-death process

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A Markov process with states $0 , 1 \dots$ in which in a time interval $( t , t + h )$ transitions from the state $n$ into the states $n + 1$ and $n - 1$ occur with probabilities $\lambda _ {n} (t) h + o (h)$ and $\mu _ {n} (t) h + o (h)$, respectively, and where the probability of other transitions is $o (h)$. For a special choice of the reproduction coefficients $\lambda _ {n} (t)$ and the death coefficients $\mu _ {n} (t)$ one can obtain particular cases which provide a satisfactory description of various real processes: radioactive transformations, the running of telephone exchanges, the evolution of biological populations, etc. The fact that birth-and-death processes are widely used in applications is due to the simplicity of the equations for the transition probabilities, which often can be obtained explicitly. For instance, in the case of a Poisson process $\lambda _ {n} (t) = \lambda$, $\mu _ {n} (t) = 0$, the probabilities $P _ {n} (t)$( $P _ {n} (t)$ is the probability of being at time $t$ in the state $n$ if the process has started from state $0$) satisfy the system:

$$P _ {0} ^ \prime (t) = - \lambda P _ {0} (t)$$

$$P _ {n} ^ \prime (t) = - \lambda P _ {n} (t) + \lambda P _ {n-1} (t) ,\ n \geq 1 ,$$

where

$$P _ {n} (0) = \left \{ \begin{array}{ll} 1 &\textrm{ if } n = 0, \\ 0 &\textrm{ otherwise } . \\ \end{array} \right .$$

The solution of this system is:

$$P _ {n} (t) = \ \frac{( \lambda t ) ^ {n} }{n ! } e ^ {- \lambda t } ,\ \ n = 0 , 1 ,\dots .$$

A more general process is the one in which $\lambda _ {n} (t) = n \lambda$, $\mu _ {n} (t) = 0$. This type of process was first studied by G. Yule (1924) in connection with the mathematical theory of evolution. A Yule process is a particular case of a pure birth process which is obtained from the general birth-and-death process by assuming $\lambda _ {n} (t) = \lambda _ {n}$, $\mu _ {n} (t) = 0$. If $\lambda _ {n}$ increases very rapidly, then with positive probability one can pass through all states in a finite time and then

$$\sum _ { n=0 } ^ \infty P _ {n} (t) < 1 .$$

The condition

$$\sum _ { n=0 } ^ \infty P _ {n} (t) = 1$$

for a pure birth process is satisfied if and only if the series $\sum 1 / \lambda _ {n}$ diverges. If $\lambda _ {n} (t) = n \lambda + v$, $\nu _ {n} (t) = n \mu$, then the birth-and-death process is a branching process with immigration in which the state $n$ designates the number of particles. Each particle dies during time $( t , t + h )$ with probability $\mu h + o (h)$, splits into two particles with probability $\lambda h + o (h)$ and, moreover, one particle immigrates from the outside with probability $v h + o (h)$. If $v = 0$ then one obtains the simplest branching process without immigration. If $\lambda = 0$ and $v > 0$, then this type of process with immigration can be applied to describe the functioning of a telephone exchange with an infinite number of lines. Here, the state is the number of occupied lines. The reproduction coefficient $\lambda _ {n} (t) = v$ characterizes the rate of incoming telephone calls, and $\mu$ is the expected duration of a conversation.

#### References

 [1] W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1957–1971) [2] B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) [3] R.L. Saaty, "Elements of queueing theory with applications" , McGraw-Hill (1961)