On transience of $M\mathrm{/}G\mathrm{/}\mathrm{\infty }$ queues

We consider an $M\mathrm{/}G\mathrm{/}\mathrm{\infty }$ queue with infinite expected service time. We then provide the transience/recurrence classification of the states (the system is said to be at state $n$ if there are $n$ customers being served), observing also that here (unlike e.g. irreducible Markov chains) it is possible for recurrent and transient states to coexist. We also prove a lower bound on the growth speed in the transient case.