Birth-Death Process

A Birth-Death Process is a special homogenous, aperiodic, irreducible (discrete-time or continuous-time) Markov Chain where state changes can only happen between neighbouring states.

If the current state (at time instant n ) isX_n = i, then the state at the next instant can only be X_n+1 = (i +1), i or (i -1) . This implies that, for a Birth-Death Process, the state changes that can happen between one time instant and the next are only the ones which can either increase or decrease the state by unity or keeps the system in the same state.

Note that states are represented as integer-valued (i.e. 0, 1, 2, ........., ) without loss of generality.

A Pure Birth Process is a special type of Birth-Death Process where decreasing the state is not allowed (i.e. the system state can only go up by one or stay the same from one time instant to the next). On the other hand, for a Pure Death Process, the system starts from a non-zero state and can either decrease by one from one time instant to the next or stay in the same state. This is illustrated below.

Pure Birth	Pure Death
No decrements, only increments	No increments, only decrements

Continuous Time, Birth-Death Markov Chain

Let	λ_k be the birth rate in state k µ_k be the death rate in state k

Then	P{state k to state k+1 in time Δt }= λ_k Δ_t	Δt→0
	P{state k to state k-1 in timeΔt } = µ_k Δ_t
	P{state k to state k in time Δ t } = 1- λ_k Δ_t - µ_k Δ _t
	P{other transitions in time Δ t }= 0

System State X(t) = Number in the system at time t
= Total Births - Total Deaths in time interval (0, t)

The initial condition will not matter when we are only interested in the equilibrium state distribution. It will matter if we want the transient solution for the queue state as then we need to consider the starting state of the queue.