Let be a Markov chain with transition probabilities .
E.g. A sample realization is 1, 2, 3, 4, 5
Let be the same sequence in reverse, i.e.
Let be the transition probabilities of the reversed process. That is
In the steady state, assuming the limiting probabilities exist, we have
The above equation is saying that the rate of transmissions from i to j in the reversed chain is equal to the rate of transitions from j to i in the forward chain.
A DTMC is time reversible if