The two-clock problem
Every appointment system has two clocks. One is the schedule, agreed in advance. The other is the room, which is always drifting. A design that only represents the first will tell a visitor their appointment is at 11:00 while they sit in a waiting area watching 11:20 go past.
Representing both honestly — you are booked for 11:00, you are third in the queue — is more work and considerably more useful.