Have a search on the Internet for the DVD title. There may be a known problem with that particular disc. I've heard of similar cases before (can't remember what they were, though).
If the disc isn't showing up at all, they've probably put funny software on it that works on all of their test machines. That would explain why it's drive dependant. Or there is a flaw on the disc that is just pronounced enough to stuff up that one drive but not the other more reliable drives.