Highway Engineering

and density between these two limiting points. Greenshields (1934) proposed the
simplest representation between the two variables, assuming a linear relation-
ship between the two (see Fig. 4.1).
In mathematical terms, this linear relationship gives rise to the following
equation:

(4.6)

This assumption of linearity allows a direct mathematical linkage to be formed
between the speed, flow and density of a stream of traffic.
This linear relationship between speed and density, put forward by
Greenshields (1934), leads to a set of mathematical relationships between
speed, flow and density as outlined in the next section. The general form of
Greenshields’ speed-density relationship can be expressed as:

(4.7)

where c 1 and c 2 are constants.

However, certain researchers (Pipes, 1967; Greenberg, 1959) have observed
non-linear behaviour at each extreme of the speed-density relationship, i.e. near
the free-flow and jam density conditions. Underwood (1961) proposed an ex-
ponential relationship of the following form:

(4.8)

Using this expression, the boundary conditions are:

When density equals zero, the free flow speed equals c 1

When speed equals zero, jam density equals infinity.

The simple linear relationship between speed and density will be assumed in all
the analyses below.

uc=-^12 exp()ck

uc ck=+ 12

uu

k

f kj

=-ÊËÁ 1 ˆ ̄ ̃

Basic Elements of Highway Traffic Analysis 75

uf

kj

0

0

Speed (km/h)

Density (veh/km)

Figure 4.1

Illustration of speed-

density relationship.