Module 1 : Traffic Stream Characteristics

Lecture 02 : Fundamental Relations of Traffic Flow

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	Illustration of mean speeds In order to understand the concept of time mean speed and space mean speed, following illustration will help. Let there be a road stretch having two sets of vehicle as in figure 1. Figure 1: Illustration of relation between time mean speed and space mean speed The first vehicle is traveling at 10m/s with 50 m spacing, and the second set at 20m/s with 100 m spacing. Therefore, the headway of the slow vehicle $h_s$ will be 50 m divided by 10 m/s which is 5 sec. Therefore, the number of slow moving vehicles observed at A in one hour $n_s$ will be 60/5 = 12 vehicles. The density $K$ is the number of vehicles in 1 km, and is the inverse of spacing. Therefore, $K_s = 1000/50$ = 20 vehicles/km. Therefore, by definition, time mean speed $v_t$ is given by $v_t = \frac{12\times10+12\times20}{24}~=~15~m/s$. Similarly, by definition, space mean speed is the mean of vehicle speeds over time. Therefore, $v_s = \frac{20\times10 + 10\times20}{30}~=~13.3~m/s.$ This is same as the harmonic mean of spot speeds obtained at location A; ie $v_s = \frac{24}{12\times\frac{1}{10}+12\times\frac{1}{20}} = 13.3~m/s.$ It may be noted that since harmonic mean is always lower than the arithmetic mean, and also as observed, space mean speed is always lower than the time mean speed. In other words, space mean speed weights slower vehicles more heavily as they occupy the road stretch for longer duration of time. For this reason, in many fundamental traffic equations, space mean speed is preferred over time mean speed.