run late

To run late is to be behind the time you should be at. The schedule has slipped and everybody knows it.

It is used about people, trains and appointments, so a clinic running late is the standard experience of any waiting room. It works without an object. The phrase is gentler than being late, since running late describes a process still under way rather than a completed failure, and that is exactly why people use it when they text ahead. Running on time is the happy opposite.