The DMV is able to accommodate $P$ people in line in its office. Unfortunate souls come in this line at an Entrance rate $I$ and the DMV processes each of them at a rate $E$ both of which have some unit in the form $\frac{\text{time}}{\text{person}}$. Will the line become full? When exactly does this happen?
To clarify: use discrete time steps where a person will go in/out of line at the end of each time step. As soon as a person exits, a person can come in the line. So if I is 1 min/person and E is 2 min/person then at t=0 there will be no one, and at t=1 there will be exactly one person. at t=2, 2 people, t=3, 2 people (one person leaves and another is immediately put in line).
Intuitively, I though that if $I>E$ the line will become full. However, this is not the case. With P=2, I=1.1 E=1 the line never becomes full.
Then, I tried to think about how many people have exited at each time step and how many have come in and came up with something like $\lfloor\frac{t}{I}\rfloor-\lfloor\frac{t}{E}\rfloor$ to get how many people are in line at any moment. However, the problem with this is that a person only starts exiting after they come into line, so that equation doesn't really apply since it doesn't take into account of that fact.