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!

(In reply to a thought by SilverKnight)

"For example, If I suggest that the distribution of the 40 tables inside is that: 10 will free up in 30 minutes, but 30 will free up in 70 minutes, then the average of those 40 tables will free up in 60 minutes."

This doesn't correctly characterize what is being said in the problem.  It's not that the average will free up 60 minutes from now, but rather that the average diner takes 60 minutes to complete dinner.  Not only is there a mixture of durations of dining, but also random starting times.  If indeed 10 are 30-minute eaters and 30 are 70-minute eaters, the 70-minute eaters will free up 30*60/70 = 25.71... tables per hour or one every 2+1/3 minutes, and the 30-minute eaters will free up 10*60/30 = 20 tables per hour or one every 3 minutes.  Combined they free up 45.71... per hour or about 1 every 1.3125 minutes.

Posted by Charlie on 2004-08-09 16:19:49