Birth–death balance on three states

Topic: Classes, recurrence, and reversibility · more questions on this topic
‹ A walk on a path of four vertices

A birth–death chain on \{0,1,2\} has p_0=1, q_1=1/3, p_1=2/3, and q_2=1. Use detailed balance to find the stationary distribution.

Worked answer (click to show)

Local balance gives \pi_0 p_0=\pi_1 q_1 and \pi_1 p_1=\pi_2 q_2. The first equation is \pi_0=\pi_1/3. The second is (2/3)\pi_1=\pi_2. The weights (1,3,2) satisfy both, and they sum to 6, so \pi=(1,3,2)/6.