I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct):

${\bf W} =\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0\\ \mu & -\mu & 0 & 0 & 0\\ 0 & \mu & -\mu & 0 & 0 \\ 0 & 0 & \mu & -\mu & \dots \end{array} \right)$

But the question I have is that I am unclear how to solve this Markov chain. That is, I'm looking for an analytic solution to

$\bf pW=0$

I think $\bf p$ should look something like

${\bf p} = [1−ρ,…],$

but again, I am unclear how to solve these problems.

Thanks for help in these matters.