Recently I had a conversation with an industrial firm that is facing the following problem. It is a problem that sounds like "solved mathematical proof" so I'm here asking for a reference from anyone who may know.
Problem:
I have an industrial robot that makes things for sale (think 3D printer). It has n (ie 12) buckets that it can hold things, and a queue of people who are ordering things and picking them up from a cell phone application or online. There is a time limit on how long each bucket can be full, and when all n buckets are full no further items can be made.
Given a number m orders every hour, how should the robot optimize the time at which an item is made and stored to minimize the wait time of any one person on their order (time between order and pick up) and also minimize the chance that all n buckets are filled to capacity?
Interesting wrinkle: if there are z types of things that the the robot can make, and multiple people order the same item, can the robot give the first person who shows up the location the first instance of that item, in order to minimize the queue size?
This is the "industrial problem". As stated, I'm not sure I've made a rigorous mathematical statement, apologies it's been a while. I hope I've given the motivation enough to see if anyone has any ways to investigate further. I'm looking for something like "Fundamentals of Queue Theory Featuring the Answer to That One Question" or some such. If anyone has any ideas I would be much appreciative. Thanks!