AAA does not own tow trucks. They, like all roadside assistance programs (at least that I'm aware of), contract out to numerous garages and tow companies who have the actual equipment. As such, what sort of service you'll be able to get, and how soon it will come, is very much dependent on where you are and how busy the tow companies are...and, of course, on what ones are signed up with AAA or whatever assistance program. It also depends on which AAA chapter you're in the area of, since AAA is a collection of several regional (and fairly autonomous) clubs.
The time I've called AAA for my motorhome (a 32' class C), the service was fine but took a few hours. I didn't need a tow, just a change to the spare tire, which at the time I could not do myself as I did not have a jack for the motorhome.