  We consider an $M/G/\infty$ 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.
