Module 1 : Traffic Stream Characteristics

Lecture 02 : Fundamental Relations of Traffic Flow

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	Space mean speed ($v_s$) The space mean speed also averages the spot speed, but spatial weightage is given instead of temporal. This is derived as below. Consider unit length of a road, and let $v_i$ is the spot speed of $i^{th}$ vehicle. Let $t_i$ is the time the vehicle takes to complete unit distance and is given by $\frac{1}{v_i}$. If there are $n$ such vehicles, then the average travel time $t_s$ is given by, $$t_s = \frac{\Sigma {t_i}}{n} = \frac {1}{n}{\Sigma\frac{1}{v_i}}.$$ (1) If $t_{av}$ is the average travel time, then average speed $v_s = \frac{1}{t_s}$. Therefore, from the above equation, $$v_s = \frac{n}{\sum_{i=1}^{n}\frac{1}{v_i}}.$$ (2) This is simply the harmonic mean of the spot speed. If the spot speeds are expressed as a frequency table, then, $$v_s = \frac{\sum_{i=1}^{n}{q_i}}{\sum_{i=1}^{n}{\frac{q_i}{v_i}}}$$ (3) where $q_i$ vehicle will have $v_i$ speed and $n_i$ is the number of such observations.