Markov chains - compulsory test

1. (A version of 3.2)

Consider a birth-death process with three states, i=1,2,3. The transition rates i->i-1, i=2,3 are mu, the transition rates i->i+1, i=1,2 are lambda.

a) Give the state transition matrix, Q.

b) Give the matrix equation of the stationary solution, and calculate the stationary state distribution.

c) Show that the time spent in state 2 is exponentially distributed with mean 1/(lambda+mu).