We do a lot with presence, mainly via Beacons. Some of what we've learned you might find useful:
The app responsible for finding nearby beacons should remain in the foreground constantly. iOS limits location services in the background, and you can't reliably count on response times < 15 minutes after a while.
Triangulation is a tricky thing. Beacons are not meant to be super accurate, and with a single device I'd recommend you focus on zones (e.g. Hallway 1, Room #2) instead of triangulation.
The RSSI values you get back will not paint an accurate picture, especially since the beacon is on a person and humans absorb Bluetooth signals like crazy. Unless you have clear line of sight, any number of people, metal objects, and walls will give you bad data.
My two cents? I'd recommend you go for a simpler solution to start. Triangulation will drive you mad, and for very little results. Asking the question "What device is John near right now?" seems like it would be a better fit for you, and not miss out on much accuracy.
Zach is pretty much right about the system. The only thing I'd add to this is that it seems a bit mission critical, so I'm not sure if it's a good idea to implement it at this point.