Embedded departure chain of an M-G-1 queue
= 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. Then
$$
Q_{n+1}=Q_n-\mathbf1_{\{Q_n>0\}}+A_n.
$$
Conditional on service duration $S$, $A_n$ is Poisson with parameter $\lambda S$, so
$$
\mathbb EA_n=\lambda\mathbb ES,
\qquad
\mathbb EA_n^2=\lambda\mathbb ES+\lambda^2\mathbb ES^2.
$$