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

Method A: Illustration of the measurement line.

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 \(\Delta {t}\).

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 \(t\):

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