X(t) is an unknown function that gives the number of people in a system that can serve 1 person at once and takes 1 minute to do so.

At a given time t, what defines how many people are there?

1 person may be being served.  If so, then any number of people may be waiting in line.  People arrive in ordered delays.

X(t) ~= (number of people who have arrived between 0 and t) - (number of people who have been fully served between 0 and t)