The final theorem in Chapter 19 of Meyn and Tweedie's Markov Chains and Stochastic Stability tells us that if the mean inter-arrival time $\lambda$ of a GI/G/1 queue is greater than its mean service time $\mu$, then the queue is positive Harris recurrent.
Question: What stability results are known for a continuous-time GI/G/1 queue for which $\lambda=\mu$, and in which the variances of the inter-arrival time and service time random variables are positive? Is it known that such a queue cannot be positive Harris recurrent? Regular?
The Lyapunov function used in Meyn-Tweedie, which is an expected hitting time, is not going to work for $\lambda=\mu$.
I searched in Morozov and Delgado's survey paper, which claims that "stability conditions of the classical GI/G/m queue are well known", and in other surveys, but found no mention of the $\lambda=\mu$ case there.