This method calculates the mean value of flow and density over time.

A reference line is taken and studied over a fixed period of time $\Delta {t}$. Using this method we can obtain the pedestrian flow $J$ and the velocity $v_i$ of each pedestrian passing the reference line directly. Thus, the flow over time $\langle J \rangle_{\Delta t}$ and the time mean velocity $\langle v \rangle_{\Delta t}$ can be calculated as

where $N^{\Delta t}$ is the number of persons passing the reference line during the time interval $\Delta {t}$.

$t_N^{\Delta {t}}$ and $t_1^{\Delta {t}}$ are the times when the first and last pedestrians pass the location in $\Delta {t}$.

Note: this time period can be different from $\Delta {t}$.

The time mean velocity $\langle v \rangle_{\Delta t}$ is defined as the mean value of the instantaneous velocities $N^{\Delta t}$ pedestrians.

$v_i(t)$ is calculated by use of the displacement of pedestrian $i$ in a small time interval $\Delta t^\prime$ around $t$: