# Stability of a GI/G/1 queue with $\rho=1$?

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.

• Hi, Daniel. Unfortunately, I don't have it at hand at the moment, but you might look in S. Asmussen, Applied Probability and Queues, 2nd ed., Springer Applications in Mathematics. If I recall, Chapter X discusses (steady state) properties of GI/G/1 and I believe Chapter XII has a brief section on GI/G/m. In both places, some results on $\lambda =\mu$ are provided ($\rho = 1$ in the book's notation). Sorry I can't provide a more precise pointer at the moment. Jun 10, 2014 at 18:53
• Thanks. This seems to be it (although the proofs are tremendously condensed!). Do you want to post this as an answer and I can accept it? Jun 11, 2014 at 11:18

For the Lindley process $Z_{n+1} = \max(0, Z_n+ U_n)$ with $\rho=1$, two results seem to be known:
1. $\frac{Z_n}{\sqrt{n}}$ converges to a normal random variable with mean $0$ and variance $\sigma^2=\mathrm{Var}(U_n)$, where $\sigma\in(0,\infty)$. This is in Billingsley, Chapter 2, cited slightly inaccurately in Asmussen (condition $\sigma>0$ omitted), Proposition X.1.2.