Proving that an M/M/1 queue is at stationarity

Question

I'm pondering the following:

Suppose that an M/M/1 queue with arrival rate $\lambda$ becomes stationary, and immediately after this happens, person A enters the queue. That is, person A sees the queue at stationarity.

Is it true that the next person to enter the system, person B say, will also see the queue at stationarity?

My thinking is that they will, but this is only my intuition.

Can anyone provide any rigour to this, or explain what's wrong with my intuition?

Answer 1

I guess this is a rather philosophical question, and I am certainly not the most fit to answer, nor is there probably one correct answer.

Suppose that an $M/M/1$ queue with arrival rate $λ$ becomes stationary.

This means that we are modelling a queue of which we know nothing with some probability. Our model tells us that after some time the probability distribution of the length of the queue does not change: it becomes stationary.

Person A enters the queue. That is, person A sees the queue at stationarity.

This is not possible in the world we are used to, since we do not see superpositions of events. If we see a queue, this queue will a definite length, not a length distributed according to a probability distribution.

But then, how to interpret the first statement, namely that in our model the queue is stationary?

1. First, one could say that person A only sees a realization of our model. Suppose that we are writing a computer program which simulates the queue (assuming that we are able to simulate exactly the random variables we need): then every time we run the program some random numbers are drawn, but what we see is always only a realization of the model. Truth is, we are not able to simulate exactly random variables: we can only build deterministic objects that look very much like random variables.

2. Our model is just a model. We can't know what is governing the real queue, but we can do some statistical studies to see whether our model seems reasonable. So the model makes only sense when we consider a large number of persons entering and leaving the queue. We can't really say anything about person A.

3. The queue is really behaving randomly until the point at which a person A oberves it. At this point the queue collapses to having just the one length the person A sees. The (stationary) probability distribution represents the real state of the queue prior to an observation.

I guess there are many more points of view on this. See for example here or here.

To come to your question

The next person to enter the system, person B say, will also see the queue at stationarity?

In view of the above, the answer is again no. You can imagine a queue that grows with random structure, then A comes and everything collapses to one realization. Then the queue grows randomly once again starting from the length A has observed, until B arrives. If B arrives a long time after A it will have a probability distributed like the stationary measure of seeing the queue at a certain length. But as soon as he sees the queue he will see only one length.