Geometric-arrival single-service queue ($X_{n+1}=\max (X_n-1,0)+G_n$)

Let the independent arrivals satisfy $\mathrm{Pr}(G_n=j)=q{p}^j$, with $0<p<1$ and $q=1-p$. During a busy period, each served customer produces $G_n$ further customers, so the total number served has the total-progeny law of a Galton-Watson process with mean $p/q$. Its extinction probability solves $s=q/(1-ps)$ and is the smaller of $1$ and $q/p$. The queue is positive recurrent for $p<1/2$, null recurrent for $p=1/2$, and transient for $p>1/2$. In the positive recurrent case its stationary distribution is ${\pi }_j=(1-p/q)(p/q{)}^j$.