Method A

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

Method_A

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

$\langle J \rangle_{\Delta t}=\frac{N^{\Delta t}}{t_N^{\Delta t} - t_1^{\Delta t}}\qquad \text{and} \qquad \langle v \rangle_{\Delta t}=\frac{1}{N^{\Delta t}}\sum_{i=1}^{N^{\Delta t}} v_i(t),$

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 $t$ :

$v_i(t)=\frac{\vec{x_i}(t+\Delta t^\prime/2)-\vec{x_i}(t-\Delta t^\prime/2))}{\Delta t^\prime}.$