# Birth-Death Processes

A birth-death process is a Markov process (a random process whose future probabilities are determined by its most recent values) with the following properties:

- 1. It is a discrete state space.
- 2. The states of which can be enumerated with index
*i*= 0, 1, 2, ... - 3. The state transitions can occur only between neighbouring states,
*i ^ i*+ 1 or*i ^ i*- 1.

Consider and p, where *i* = 1, 2, ..., *n* be the birth and death rates, respectively. Then, the transition rates are defined in the following way (Figure 10.1):

Let *X _{k}* and p

_{k}be the birth and death rates in state k, respectively. Then, the probabilities of birth and death in the interval At are

*X*At and p

_{k}_{k}At for the system in state k, respectively:

- 1. Pjstate
*k*to state*k*+ 1 in time At} =*X*_{k}(At) - 2. Pjstate
*k*to state*k*- 1 in time At} = p_{k}(At) - 3. Pjstate
*k*to state*k*in time At} = 1 -*(X*+ p_{k}_{k})(At) - 4. Pjother transitions in At} = 0

We have the system state *X(t)* at time *t* = [total births - total deaths] in (0 , t) assuming system states from state 0 at *t* = 0.

FIGURE 10.1

Transition of states.

Hence,