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!
Since you bring it up, the problem doesn't specify the distribution of
the duration (and finishing) of the meals. All it states is that
the average meal duration is one hour.
It is not at all likely that meals will take less than 5 minutes, nor
is it likely that a meal will take more than 5 hours, but each is
possible. The distribution is critical to solve this problem, and
without it, one is making assumptions.
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 holds true to the wording of the problem, but the chance that you will have to wait less than an 70 minutes is zero.
(We can make similar extreme distributions with the meals already having started, but I wish to make the point clear.)
Many queue simulations use an independent Poisson distribution, because
it often mirrors reality quite well. But a beta, bell curve, or a
uniform or other distribution could be used as well. They will
all give different answers to this problem even though their average
meals will all equal one hour.
In short, there is not enough information to solve this problem.
a thought
