> For example, a customer may arrive and observe that the barber is cutting hair, so he goes to the waiting room. While he is on his way, the barber finishes the haircut he is doing and goes to check the waiting room. Since there is no one there (the customer not having arrived yet), he goes back to his chair and sleeps. The barber is now waiting for a customer and the customer is waiting for the barber.

This is the critical part of the analogy, and is why it breaks down. It's such an artificial contrivance to suggest that both the barber and the customer moving to the waiting room won't notice each other - yes, you can come up with theoretical ways to do it, but that's requiring contortions in order to explain what is supposed to be the main point of the analogy.

(it's already a stretch that the barber can't see the waiting clients, as per degenerate's comment above)