M/M/1 queue derivation: how to "recursively solve in dependence on $p_0$"

Question

I want to sketch out the derivation of the equations for an M/M/1 queue for a presentation I'm giving.

I can understand most of the derivation from Willig but I don't understand this section from p10 well enough.

Is there a more fleshed out explanation of this?

Answer 1

Use the first displayed equation to find $p_1$ in terms of $p_0$. Substitute for $p_1$ in the second equation. Then you can find $p_2$ in terms of $p_0$. Do a few more and you can see the pattern, and show it to your audience at the presentation.