# Birth-death process Expected waiting time in when in invariant distribution

Say I have a birth-death process, with birth having Poisson distribution with parameter $$\lambda$$ and death having poisson distribution $$\mu$$. Assuming that both stochastic processes, birth and death, have time homogeneity, then my textbook is telling me that the expected wait time is exponentially distributed: $$\tau ~ \mathcal{E}(\lambda + \mu)$$. Now, I need to find the total expected time when in its invariant distribution.

1) since the sum of two poisson processes has a poisson distribution, with parameter given by the sum of the parameters, (see Poisson Distribution of sum of two random independent variables $$X$$, $$Y$$) Why is it $$\lambda + \mu$$ and not $$\lambda - \mu$$?
2) Obviously, the total epected time is $$\frac{1}{\lambda + \mu}$$, since it is exponentially distributed...but iss this distribution, and hence the expected value going to be different when in its stationary distribution? It seems like not to me, but then I wonder why it is used in the question.