Okay, right digital repeaters are a bit different from analogue. With analogue repeaters you basically have a transmitter and a receiver in a box (often these are just two mobile radios), wired together. When the receiver hears a signal, the squelch opens and that keys the transmitter. You can do various neat tricks to share that repeater, like having it decode and retransmit different subaudible signalling tones (although I struggle with the idea of 110.9Hz as "subaudible" since it's about the same frequency as an open A string on a guitar).
Digital works rather differently, mostly because of how the two timeslots work. When your portable transmits, it'll listen to see if the repeater is transmitting first. If it isn't, it'll send a burst of data to wake the repeater up. If you're very out of range of the repeater, you'll see the LED on top blink a couple of times before you get the BEEEEEP tone warning you it can't transmit.
Once the repeater is up, it sends a string of timing pulses (the buzzing noise). When your radio wants to transmit, it picks a pulse and starts working in sync with that. It only transmits on every second pulse, so you get the two timeslots (two "channels" per frequency). Every so often when no-one is using it, the repeater will fire up and send a burst of beacon data so that the radios can check they're still in range. This isn't very useful on a single-repeater system but if you've got several of them it will help your radios pick the strongest signal to listen to.
Now here's where the troubleshooting begins. If you've got a repeater, you've got a strong local signal transmitted from the repeater, and much weaker signals from the portables. If you're having interference from another site, it's far more likely that it's on the frequency that the repeater receives on, than transmits on - it's going to be far easier to step all over the relatively weak signal from the portables, especially if you've got a nice big aerial up high outside.
So to track it down, try programming one of your radios to listen on the same frequency that the repeater does, on analogue, carrier squelch, so that you hear anything that pops up.