# 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 Xk and pk be the birth and death rates in state k, respectively. Then, the probabilities of birth and death in the interval At are XkAt and pkAt for the system in state k, respectively:

• 1. Pjstate k to state k + 1 in time At} = Xk(At)
• 2. Pjstate k to state k- 1 in time At} = pk(At)
• 3. Pjstate k to state k in time At} = 1 - (Xk + pk)(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,