Intuition of the Mean wait time in queuing system

Question

In queuing theory, (with a single queue and a single server) writing $A$ for the service rate (of customers) and $B$ for the arrival rate (of customers) we know that the average time a customer waits in the system is given by,

$$W=\frac{1}{A-B}$$

What is the intuitive interpretation of this equation?

What I understand from basic intuition is that $A$ customers gets service in 1 sec (as $A$ is the service rate) so, one customer should get service in $1/A$ seconds.

Now, $B$ means, $B$ customers comes in 1 second (the arrival might be bursty or not depending on the probability distribution). So, $(A-B)$ is , how many more customers can the server serve per second, right?

So, why is the waiting time $W=1/(A-B)$?

How can I understand this relationship intuitively?

Answer 1

I am canoeing up a river at a speed (relative to the water) of $A$ miles per hour, and the current in the other direction is $B$ miles per hour. If $A\gt B$, my speed relative to the shore line is $A-B$, so it will take me $\frac{1}{A-B}$ hours to travel $1$ mile.

Or if you prefer I am running up a down escalator.