On transience of M/G/∞ queues

We consider an M/G/infinity 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 irreducible Markov chains, for example) it is possible for recurrent and transient states to coexist. We also prove a lower bound on the growth speed in the transient case.