Embedded departure chain of an M-G-1 queue

Let $Q_n$ be the number left immediately after departure $n$, and let $A_n$ be the arrivals during the next service. ThenConditional on service duration $S$, $A_n$ is Poisson with parameter $\lambda S$, so

For a stable first-come first-served M/G/1 queue, write $\rho =\lambda \mathbb{E}S<1$ and $\tilde{S}(s)=\mathbb{E}{e}^{-sS}$. The stationary waiting-time Laplace transform isWith finite second service moment, differentiating at zero gives the Pollaczek-Khinchine mean formula. These are the transform and mean forms of the same queueing identity.

For a stable $M/G/1$ queue with traffic intensity $\rho =\lambda \mathbb{E}S<1$, the mean waiting time before service isEquivalently, the mean number in the system is