Fundamentals of Transportation/Traffic Flow/Solution

TProblem

Problem:

Four vehicles are traveling at constant speeds between sections X and Y (280 meters apart) 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.

Example

Solution:

Flow

$q = N (\frac{3600}{t_{m e a s u r e d}}) = 4 (\frac{3600}{15}) = 960 v e h / h r$

Density

$k = \frac{N}{L} = \frac{4 * 1000}{280} = 14.2 v e h / k m$

Time Mean Speed

$\overline{v_{t}} = \frac{1}{N} \sum_{n = 1}^{N} v_{n} = \frac{1}{4} (72 + 90 + 80 + 88) = 82.5 k m / h r$

Space Mean Speed

$\begin{matrix} \overline{v_{s}} = \frac{N}{\sum_{n = 1}^{N} \frac{1}{v_{i}}} = \frac{4}{\frac{1}{72} + \frac{1}{90} + \frac{1}{80} + \frac{1}{88}} = 81.86 \\ t_{i} = L / v_{i} \\ t_{A} = L / v_{A} = 0.28 / 88 = 0.00318 h r \\ t_{B} = L / v_{B} = 0.28 / 80 = 0.00350 h r \\ t_{C} = L / v_{C} = 0.28 / 90 = 0.00311 h r \\ t_{D} = L / v_{D} = 0.28 / 72 = 0.00389 h r \\ \overline{v_{s}} = \frac{N L}{\sum_{n = 1}^{N} t_{n}} = \frac{4 * 0.28}{(0.00318 + 0.00350 + 0.00311 + 0.00389)} = 81.87 k m / h r \end{matrix}$