Highway Engineering

where the traffic density,k, is a measure of the number of vehicles,n, occupy-

ing a length of roadway,l.

For a given section of road containing kvehicles per unit length l, the average

speed of the kvehicles is termed the space mean speed u(the average speed for

all vehicles in a given space at a given discrete point in time).

Therefore:

(4.3)

where liis the length of road used for measuring the speed of the ith vehicle.

It can be seen that if the expression for q is divided by the expression for k, the

expression for uis obtained:

(4.4)

Thus, the three parameters u,kand qare directly related under stable traffic

conditions:

(4.5)

This constitutes the basic relationship between traffic flow, space mean speed

and density.

4.2.1 Speed-density relationship

In a situation where only one car is travelling along a stretch of highway, den-

sities (in vehicles per kilometre) will by definition be near to zero and the speed

at which the car can be driven is determined solely by the geometric design and

layout of the road; such a speed is termed free-flow speed as it is in no way hin-

dered by the presence of other vehicles on the highway. As more vehicles use

the section of highway, the density of the flow will increase and their speed will

decrease from their maximum free-flow value (uf) as they are increasingly more

inhibited by the driving manoeuvres of others. If traffic volumes continue to

increase, a point is reached where traffic will be brought to a stop, thus speeds

will equal zero (u=0), with the density at its maximum point as cars are jammed

bumper to bumper (termed jam density,kj).

Thus, the limiting values of the relationship between speed and density are as

follows:

When k=0,u=uf

When u =0,k=kj.

Various attempts have been made to describe the relationship between speed

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74 Highway Engineering