Module 2 : Markov Processes and Markov Chains

Lecture 2 : Markov Chains, Birth-Death Processes, Flow Balance

In general, the equations expressing flow balance in a Birth-Death Chain of this type will be -

Global Balance Equations

Closed boundary encircling each state j


Detailed Balance Equations

Equates flows between states i and j, in a pair-wise fashion.

Boundary between states i and j, closed at + ∞ and - ∞



Conditions for Existence of Solution for Birth-Death Chain


(a) All states are transient , if and only if =∞, ß<∞

(b) All states recurrent null , if and only if α=∞, ß=∞

(c) All states ergodic , if and only if α < ∞ , ß = ∞
 

Equilibrium State Distribution will exist only for the case where all the states are ergodic and hence, the chain itself is also ergodic

Note that equilibrium solutions to a Birth-Death Chain exist only when condition (c) given above is satisfied. An alternative (but equivalent) statement is to say that, in order to have an equilibrium solution, the chain must have some state K such that λk/μk+1<1 for all k>K.