These sorts of services do seem very dependent on the details of how they are organized.

I’ve had experience with two university libraries with 3d printers. They both advertise them similarly, and they were nominally similar services, ostensibly letting students and staff both 3d print and learn about 3d printing.

At one, the arrangement was that they’d show you around the machines, give you a link to a list of notes and rules, and then you could come in and use the printers. If you wanted to do something unusual or use an exorbitant amount of filament, they asked that you talk to them first. That service is what initially introduced me to 3d printing.

At the other, the library staff decided they’d rather handle everything themselves. You’d submit an stl, then they’d print it at some point, potentially weeks later. Random color pla only, no slicing and providing gcode or even requests for settings. In practice staff decided they would only accept links to popular stls online; submitting your own stl would be rejected. They printed at such bad settings that everything would come out horribly. The service was worse than useless, taught nothing, and may well have turned many students off 3d printing entirely, if they thought the results were indicative of what 3d printing could do. We essentially have to warn students that the service is not practically usable.

But both, of course, say they have 3d printers.

In the equation there seems to be a typo;

103 - number of ppl in queue, 4 - up to X weeks per person, 2 - number of machines 12 - ??

Maybe you initially wanted to use full months for how long a person can hold an item, but then switched to weeks, and accidently still used number of months to get the number of years?

Anyway, for an imprecise number, you can do with months - 104*1/2/12 ~4.3y.

For more precise result, use seconds, as that's the unit used for the precise length of the year. Year is not 365 days. It's actually longer, quoting Wikipedia for (tropical) year,

> Approximately 365 solar days, 5 hours, 48 minutes, 45 seconds

That gives

  104 * (4 * 7 * 24 * 60 * 60) /2 /(((365 * 24 + 5) * 60 + 48) * 60 + 45)

Which results in 3.986 years. At maximum. Much less than 17!

Edit: getting asterikses * right

That theoretical wait time doesn’t usually end up being so long. Between borrowers returning things early, people on the wait list giving up and most importantly, the library deciding that the current inventory is insufficient, the wait times usually are much less than that (I’ve observed this with books and other materials at my local library and the wait on in-demand times is never as long as the queue would imply).

While the wait time to rent a machine may be long (I didn't even know Denver had this,) Denver also has the ideaLAB maker spaces at five branches, all of which are equipped with sewing machines to just drop in and use.

Some ideaLAB locations (like my local one) have fancy machines like embroidery, quilting, and industrial sewing machines. There's also lots of other tools, from basic hand tools to laser cutters and 3d printers (again, location dependent.) There's always staff on hand to help during open lab hours. All free to use. Really an impressive system that I use frequently.

https://denverlibrary.org/idealab

That doesn’t seem right - off the top of my head I think it’s more like 4 years? maybe it should be “103*4/2/52” (not “/12”) ?

… hopefully the effective wait time is considerably shorter if a long line is taken as a demand signal, leading them to buy more.