meaning of stationary distribution, null-recurrence and transience in queueing theory

Question

I have 3 question about the meaning of the mathematical term in the reality.

1. Let the queue $Q$ have the stationary distribution $\pi_n=(1-\rho)\rho^n$ for $n\geq0$. Does it mean, when $t$ is large, then the probability that $10$ customers are in the queue is $(1-\rho)\rho^{10}$?

2. And if $\Pr(Q(t)=n)\rightarrow 0 \ \forall n$ as $t\rightarrow \infty$. So does this mean for large $t$ I can't predict how much persons are in the queue?

3. Let now $Q$ be a queue that is null-recurrent. Hence the expected amount of time to return to the same state is infinite. This is not so obvious for me. But does this mean that if we have for example 20 persons at time $t$ in the queue, then the event that this will happen again is zero? Hmm...

4. Let the $Q$ be transient. Hence every state will only be visited a finite (random) number of times. But how can I imaginge that witch a queue? I have no idea...

I really hope anybody can help me. It would help me a lot for the mathematical understanding. I appreciate any help!!

Answer 1

1. Yes.

2. No, this means that the probability that the queue is very long goes to one, for every quantification of "very long". This happens when the queue is either null recurrent, see 3., or transient, see 4.

3. No, this means that "this" (whatever "this" is) will happen again, but after a (finite) time whose expectation is infinite. For example, to come back to $0$ starting from $0$ might take a time $T$ with density $1/(1+t)^2$ on $(0,+\infty)$. Then $T$ is finite and $E(T)$ is infinite.

4. Transience means that, for every fixed state, after a while, one never goes back to this state. Equivalently, this means that $Q(t)\to\infty$ almost surely.