You are outside a well known restaurant, waiting in queue, with 18 couples in front of you. You know there are forty tables inside, and you think an average meal will take one hour.

How long will you have to wait, on average?

PS. This problem comes from queueing theory, but you don't have to know anything about it to find the answer!

Well, I don’t know if this is the right way of thinking or not, but here goes. Let’s say that the 40 tables had been seated in even spaced time during the hour beforehand (starting with the first table being sat at time –60). So a table got seated every 60/40 minutes = 1.5 minutes. So in theory, once you get into the line (time 0), one of the 40 tables will be done, and one of the 18couples will be seated. At time 1.5, another table will be freed, and one of the 17 couples will be seated.

Continuing in this way, we see that it will take 18*1.5 minutes until it is your turn to be seated. And that’s 27 minutes.

I don’t know if maybe I did stuff a little goofy, and the first couple actually needs to wait 1.5 minutes before being seated. In that case your wait would be 28.5 minutes.

Another way to think about it is this. What fraction of a table cycle to you need to wait for? For example, if there were 40 couples in front of you, you would need to wait a whole table cycle (all 40 tables would have to finish and be replaced before you got your turn). So you have to wait 18/40 of a table cycle. Since the average meal takes 1 hour, then your wait is 18/40*60 minutes = 27 minutes.

I have no idea if either of these ways is the correct way of thinking, though.

Posted by nikki on 2004-08-11 16:52:24