Fundamentals of Transportation/Traffic Flow/Solution

Traffic Flow Problem

Four vehicles are traveling at constant speeds between sections X and Y with their positions and speeds observed at an instant in time. An observer at point X observes the four vehicles passing point X during a period of 15 seconds. The speeds of the vehicles are measured as 88, 80, 90, and 72 km/hr respectively. Calculate the flow, density, time mean speed, and space mean speed of the vehicles.

(Note this is a multi-lane section, so calculate flow for the whole section)

File:TrafficFlowProblem.png

Solution

Flow

$q=N\left(\frac{3600}{t_{measured}}\right)=4\left(\frac{3600}{15}\right)=960veh/hr$

Density

$k=\frac{N}{L}=\frac{4}{28}=142veh/km$

Time Mean Speed

$\overline{v_t}=\frac{1}{N}\sum _{n=1}^N{v}_n=\frac{1}{4}\left(72+90+80+88\right)=82.5km/hr$

Space Mean Speed

$\begin{array}{c}\overline{v_s}=\frac{N}{\sum _{n=1}^N\frac{1}{v_i}}=\frac{4}{\frac{1}{72}+\frac{1}{90}+\frac{1}{80}+\frac{1}{88}}=81.86\\ t_i=L/v_i\\ t_A=L/v_A=.028/88=0.000318hr\\ t_B=L/v_B=.028/80=0.000350hr\\ t_C=L/v_C=.028/90=0.000311hr\\ t_D=L/v_D=.028/72=0.000389hr\\ \overline{n_s}=\frac{NL}{\sum _{n=1}^N{t}_n}=\frac{4*0.028}{\left(0.000318+0.000350+0.000311+0.000389\right)}=81.87km/hr\end{array}$