Reversibility of Markov Process and Exponential Distribution of Transition Rates

Question

I am reading the paper Towards Utility-optimal Random Access Without Message Passing by J. Liu, Y. Yi, A. Proutiere, M. Chiang, H. V. Poor. A sentence in Section 3.2 can be paraphrased as follows:

Because a Markov process is reversible, the stationary distribution does not depend on the exponential distribution of the transition probabilities (i.e., the income and the service rates) and it only depends on the mean of the transition rates.

I have two questions:

1. Why we need the exponential distribution assumption at all?
2. Why we can relax the assumption if the process is reversible?

Please describe in detail, since I know almost nothing about Markov processes.