# Continuity Problem

Assume a school gym allows entrance of students in blocks:

Block 1: 08:00 AM to 09:00 AM Block 2: 09:00 AM to 10:00 AM ...etc Until 10 PM which is closure.

Assume there's a maximum of users that can be in the gym at any given moment (15). So, to get in, people have to get in line in order to get considered for the next block. If someone arrives and he's number 16, he might as well go home.

Also, a person will stay exactly 1 hour in the gym. (In the future, I'd like to relax this restrain and allow a probability function for the time a person stays).

I want to prove that by allowing the continuous entrance of users (1 out - 1 in, at any given moment), mantaining the 15 users maximum, instead of separating the entrance into 1hr blocks of time, maximizes the average number of users in the gym at any given moment, and minimizes the quantity of people lining up at any given moment.

How can I do this? I've been trying to graph Number of Users (y), against Hour of the Day (x), and make the delta x tend to 0, but it's not working so far.

Can anyone show me the way on this?

Thanks

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