This is an example from notes 
I have worked out questions from i to iii, no problems with that. But I don't know how to answer question iv, the note says that "The system reaches an equilibrium for all λ and μ except the trivial case of non-zero arrivals and zero removals". But how come it could reach an equilibrium for all λ and μ?
To show that the system reaches an equilibrium distribution for all $\mu, \lambda$ you must show that $P_i(t) \to P_i(\infty)$ for $i$. This has to be done from the set of differential equations for the $P_i$.
Start with $P_0$ and show that it satisfies $P_0' = \mu(1-P_0) - \lambda P_0$. Find the solution and show that it has a limit.
Then look at $P_1$. It satisfies $P_1'=-(\mu + \lambda)P_1 + \lambda P_0$. Using the limiting behavior of $P_0$, show that $P_1$ has a limit as well.
Then turn this argument into an induction proof.
This will also give you an explicit formula for the limiting distribution. It's a zero-inflated geometric distribution if I am not mistaken.