CHAPTER 31.

FUNDAMENTAL RELATIONS OF TRAFFIC FLOW

NPTEL May 3, 2007

10 m/s 10 m/s 10 m/s 10 m/s 10 ms

50 50 50 50
20 m/s 20 mvs 20 m/s

100 100

h=30/20 =5sec n=G0/5 = 12 k = 1000/30 =20

hr= 100/20 = 5sec nf=60/5 =12 kf= 1000/100 = 10

Figure 31:1: Illustration of relation between tine mean speed and space mean
speed
hs will be 50 m divided by l0 m/s which is 5 sec. Therefore. the number of slow moving
vehicles observed at A
in one hour n, will be 60/5 = 12 vehicles. The density K is the number of vehicles in l km,
and is the inverse
of spacing. Therefore, K, = 1000/50 = 20 vehicles/km. Therefore, by definition, time mean speed v is given
by v = 12x10+12x20
24 15 m/s. Similarly, by definition, space mean speed is the mean of vehicle speeds over
tine. Therefore, vs = 20x10i0x20 = 13.3 m/s This is same as the harmonic mean of spot
24
speeds obtained at
location A; ie v, = 12xth12x=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 oceupy 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.

31.5 Relation between time mean speed and space mean speed
The relation between time mean speed and space mean speed can be derived as below. Consider a stream of
vehicles with a set of substream flow q1,q2, ..qi, ...qn having speed v1,02, . . . Un: The fundamental
relation between flow(g), density(k) and mean speed vs is,

q=k X Us (31.6)
Therefore for any substream qi. the following relationship willbe valid.
qi = k; X Ui (31.7)
The summation of all substream fAows will give the total fow q.
Lq; = q (31.8)
Similarly the summation of all substream density will give the total density k.
Ek; = k (31.9)
Let fi denote the proportion of substream density k; tothe total density k.

(31.10)
Introduction to Transportation Engineering 31.3 Tom V. Mathew and KVKrishna Rao
 CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007
Space mean speed averages the speed over space. Therefore, if kË vehieles has , speed, then space men speed
is given by,

(3111)
Time nean speed averages the speed over time.Therefore,

(31.12)
Substituting in 31.7 v, can be written as,
Skv2
(31.13)
Rewriting the above equation and substituting 31.11, and then
substituting 31.6, we get,

By adding and subtracting v, and doing algebraic

manipulations, Up can be written as.
U =
Efi(v, + (u v.))²
(31.14)
Ef(v.)? + (u; . ) + 2.v, . (v; v)
(31.15)
Efv,? 2.0,.Efi(v-v.)
+
(31.16)
The third term of the equation will be zero
The numerator of the second term gives the
because fi (v; Us) will be zero, since U, is the mean speed of
v,.
standard deviation of v;. Ef; by definition is 1.Therefore.
= U,Ef: + +0
(31.17)

Hence, time mnean speed is space mean speed plus (31.18)

mean speed. Time mean speed will be always standard deviation of the spot speed divided by the space
be negative. If all the speed of the greater than space mean speed since standard
vehicles are the same, then spot speed, timne mean deviation cannot
speed will also be same. speed and space mean

31.6
Fundamental relations of traffic flow
The relationship between the fundamental variables of traffic flow,
the fundamental relations of traffic flow. This namely speed, volume,
and density is called
can be derived by a
simple concept. Let there be a road with
length v km, and assume all the vehicles are
moving with v km/hr. (Fig 31:2). Let the
number of vehicles
Introduction to Transportation Engin eering 31.4 Tom V. Mathew and
KVKrishna Rao
 CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007

vkm
B

876 514 3121I

Figure 31:2: llustration of relation between fundamental parameters of trafie tlow

counted by an observer at A for one hour be nË. By defnition, the number of vehicles conted in one hour Is
flow(q). Therefore,
nË =q
(31.19)

Similarly, by definition,density is the nunber of vehicles in unit distance. Therefore number of vehicles n2
in aroad stretch of distance U1 will be density x distance. Therefore,
n2 = k xv (31.20)

Since all the vehicles have speed v, the number of vehicles counted in 1 hour and the number of vehieles in the
stretch of distance v will also be same. (ie n1 =n2). Therefore,
q=k xv (31.21)

This is the fundamental equation of traffic flow. Please note that, v in the above equation refers to the space
mean speed.

31.7 Fundamental diagrams of traffic flow

The relation between fow and density, density and speed, speed and flow, can be represented with the help of
some curves. They are referred to as the fundamental diagrams of traffie flow. They will be explained in detail
one by one below.

31.7.1 Flow density curve

The fow and density varies with time and location. The relation between the density and the corresponding ftow
on a given stretch of road /s referred to as one of the fumdamental diagram of traffie flow. Some characteristics
of an ideal flow-density relatiÍnship is listed below:
1. When the density is zero, 'oy will also be zero,since there is no vehicles on the road.
2. When the number of vehicles gradsally increases the density as well as flow increases.
3. When more and more vehicles are added it reaches a situation where vehicles can't move. This is referred
to as the jam density or the maximum density. At jam density, fow will be zero becnuse the vehicles are
not moving.
4. There will be sone density between zero density åd jam density, when the fow is maximum. The
relationship is normally represented by a parabolic cure as shown in figure 31:3

Introduction to Thansportation Engineering 31.5 Tom V. Mathew and K V Krishna Rao

 ethee he4d vh ensth v kH and assHme all he
vehicles are moving with v km/hr. (Fig
t he Aabee nf vehicex
bee vounted hy an abserver at Afor one hour be n). By
ehilx vounted in one hour is low (q), Therefore, definition, the

Vkm

aULeL5LALa21E

Fiure 2: llustration ofrelation

between tinme mean speed and space nean speed
Similarty, by detiniion, density is the number of vehicles in
vehicleN t in aYad streteh of distance v l will be density x unit distance. Therefore number of
distance. Therefore,
Sinee all the vehicles have speed v, the
vehicles in the stretch of distancevwill number of vehicles counted in I hour and the
also be sanme.(ie nn2). number of
Therefore,
q-kxV ******.(A)
Thix is the
to the spce fundamental
mean speed.
equation of trafie tlow. Please note that, v in the
above equation refers

FUNDAMENTAL DIAGRAMS OF TRAFFIC FLOW:

The relation between flow and
with the help of some curves. density, density and speed, speed and flow, can be
They are referred to as the represented
fundamental diagrams of traffic flow.
 with The to
This
FUNDAMENTAL the
vehiclesSince vehicles
Similarly,
the
relation
space is number Let Derivation:
Let This
help the all
inthe the there
n Figure can
of
between fundamental
mean the of
number
vehicles in by bebe
some stretch detinition,a vehicles
speed. road 2: deriveda
1llustration road
curves.
DIAGRAMS
flow of
have
of stretch counted
equation velhicles with by
and distance density 87
speed a
They lengtth
simple
density, of of in
counted
are of distance isrelation one
OF traflic will vthev, the 65 concept.v
referredTRAFFIC n hour km,
density also
number number by
vl between and
flow. k
x will is an
be low assume
to V observer
km
same.(ie of v beof
asandFLOW: Please time ().
the vehicles
density 4
speed, vehicles
Therefore, all
fundamental note mean L3 at the
n)
x A
speed that, counted
n2). distance. in speed vehicles
for
unit 21
Therefore, one
and v (A).... distance. and
diagrams in in hour are
the Therefore. space
flow,
hour I moving
be
above A
n.
Therefore mean
of
can and
traffic be equation By with
the speed
represented definition, v
number number km/hr.
flow.
refers
of of the (Pig