Let Q be a symmetric transition probability matrix on states 1 , . . . , n; that is $q_{i,j} = q_{j,i}$ for all i , j.
Consider a markov chain defined as follows. Whenever the chain is in state i, the value of a random variable $X_i$ with probability mass function $P(X_i = j)= q_{i,j}$ is generated; if $X_i = j,$ then the chain either moves to state j with probability $b_j/(b_i + b_j),$ or it remains in state i otherwise, where $b_1,...,b_n $ are specified positive numbers.
Now how to show that the Markov chain is time reversible with stationary probabilities $$\pi_j = \frac{b_j}{\sum_{i=1}^n b_i}, j = 1,...,n$$
Any computer science help will be accepted.
Note: This question is taken from PROBABILITY MODELS for Computer Science written by Sheldon M. Ross.