On transience of $M/G/\infty$ queues

TL;DR

This paper analyzes the conditions under which an $M/G/\infty$ queue with infinite expected service time exhibits transient or recurrent states, revealing coexistence of these states and providing growth bounds in the transient case.

Contribution

It offers a novel classification of state behavior in $M/G/\infty$ queues with infinite mean service times, including coexistence of recurrent and transient states.

Findings

Identification of transience and recurrence conditions

Existence of coexistence of recurrent and transient states

Lower bound on growth speed in the transient regime

Abstract

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.