1. Fundamental relations of traffic flow
Transportation Systems Engineering
Transportation Systems Engineering Lecture Notes by Prof. Tom V Mathew

2. Fundamental relations of traffic flow
Introduction
Fundamental relations
Time and space mean speed
Fundamental equation (q, k, v)
Fundamental diagrams (q, k, v)
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Fundamental relations of traffic flow

3. Fundamental relations of traffic flow
Mean speeds
Time mean speed
average of all vehicles passing a point over a duration of time
It is the simple average of spot speed
Expression for vt
vi spot speed of ith vehicle
n number of observations
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Fundamental relations of traffic flow

4. Fundamental relations of traffic flow
Mean speeds
Time mean speed
Speeds may be in the form of frequency table
then vt
qi number of vehicles having speed vi
n number of such speed categories
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Fundamental relations of traffic flow

5. Fundamental relations of traffic flow
Mean speeds
Space mean speed
average speed in a stretch at an instant
It also averages the spot speed
But spatial weightage instead of temporal
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Fundamental relations of traffic flow

6. Fundamental relations of traffic flow
Mean speeds
Space mean speed - derivation
Consider unit length of a road
let vi is the spot speed of ith vehicle
let ti is the time taken to complete unit distance
ti=1/ vi
If there are n such vehicles, then the average travel time ts is given by
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Fundamental relations of traffic flow

7. Fundamental relations of traffic flow
Mean speeds
Space mean speed - derivation
If average travel time is ts then
average speed vs is 1/ts
the harmonic mean of the spot speed
If speeds are in a frequency table
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Fundamental relations of traffic flow

8. Fundamental relations of traffic flow
Mean speeds
Space mean speed
If speeds are in a frequency table
then vs
qi number of vehicles having speed vi
n number of such speed categories
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Fundamental relations of traffic flow

9. Fundamental relations of traffic flow
Mean speeds
Relation between Vt and Vs
If SMS is vs TMS is vt and the standard deviation of speed is σ
Then
vt > vs sine SD cannot be negative
If all the speed are same, then vs = vt
Derivation (assignment: Refer notes)
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Fundamental relations of traffic flow

10. Mean speeds: Illustration
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Fundamental relations of traffic flow

11. Fundamental relations of traffic flow
Mean speeds
Example 1
If the spot speeds are 50, 40, 60,54 and 45, then find the TMS and SMS
Example 2
The results of a speed study is given in the form of a frequency distribution table. Find the TMS and SMS
speed range
frequency
2-5 1
6-9 4
10-13
14-17 7
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Fundamental relations of traffic flow

12. Fundamental relations
Relationship between q, k, v
Let there be a road with length v km
assume all vehicles are moving with v km/hr
Number of vehicles counted by an observer at A for one hour be n1
By definition, number of vehicles counted in one hour is flow (q)
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Fundamental relations of traffic flow

13. Fundamental relations
Relationship between q, k, v
n1=q
Density is the number of vehicles in unit distance
n2=kv
But, n1=n2
since all vehicle have speed v and the distance is v
Therefore q=kv
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Fundamental relations of traffic flow

14. Fundamental relations of traffic flow
Fundamental diagrams
Follows fundamental relations
Also phenomenological
Flow-density (qk)
Speed-density (kv)
Speed-flow (qv)
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15. Fundamental relations of traffic flow
Fundamental diagrams
Flow-density (q k) curve
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16. Fundamental relations of traffic flow
Fundamental diagrams
Flow-density (q k) curve
The relationship is normally represented by a parabolic curve
At jam density, flow will be zero because the vehicles are not moving.
There will be some density between zero density and jam density, when the flow is maximum.
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Fundamental relations of traffic flow

17. Fundamental relations of traffic flow
Fundamental diagrams
Speed-density (v k) curve
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18. Fundamental relations of traffic flow
Fundamental diagrams
Speed-density (v k) curve
Max. speed is free flow speed
Max. density is jam density
At zero density, speed is free flow speed
At jam density, speed becomes zero
Most simple assumption is a linear
Non-linear relationships also possible
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19. Fundamental relations of traffic flow
Fundamental diagrams
Speed-flow (v q) curve
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Fundamental relations of traffic flow

20. Fundamental relations of traffic flow
Fundamental diagrams
Speed-flow (v q) curve
Flow is zero either because there is no vehicles or there are too many vehicles so that they cannot move
At maximum flow, the speed will be in between zero and free flow speed
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Fundamental relations of traffic flow

21. Fundamental relations of traffic flow
Fundamental diagrams
Combined
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22. Relation between Vt and Vs
Derivation
Consider a stream of vehicles with a set of sub-stream flow q1, q2,… ... qi ,... qn having speed v1, v2 , ... vi, ... vn.
Fundamental relation between flow (q), density (k) and mean speed (vs)
Therefore for any sub-stream qi, the relationship is
Summation of all sub-stream flows is total flow q
Summation of all sub-stream density is total density k
Let f i denote the proportion of sub-stream density k i to the total density k
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Fundamental relations of traffic flow

23. Relation between Vt and Vs
Derivation
Consider a stream of vehicles with a set of sub-stream flow q1, q2,… ... qi ,... qn having speed v1, v2 , ... vi, ... vn.
Fundamental relation between flow (q), density (k) and mean speed (vs)
Therefore for any sub-stream qi, the relationship is
Summation of all sub-stream flows is total flow q
Summation of all sub-stream density is total density k
Let f i denote the proportion of sub-stream density k i to the total density k
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24. Fundamental relations of traffic flow
Space mean speed averages the speed over space.
If ki vehicles have vi speed, then space mean speed is
Time mean speed averages the speed over time,
Substituting in 
Rewriting the above equation and substituting in  , and then substituting in
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25. Fundamental relations of traffic flow
By adding and subtracting vs and doing algebraic manipulations, vt can be written as,
The third term of the equation will be zero because will be zero, since vs is the mean speed of vi .
The numerator of the second term gives the standard deviation of vi.
by definition is 1, Therefore
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26. Thank You