1. Traffic Stream Models
Traffic stream models provide the fundamental relationships of macroscopic traffic stream characteristics for uninterrupted flow situations.
Traffic flow models describe the relationship among traffic variables such as speed, flow, and density (or occupancy).
Why we need traffic models?
With the knowledge of some traffic characteristics, such as volume, we can use the models to predict traffic flow performance, such as speed, density, capacity, etc.

2. Traffic Flow Models Macroscopic model (Stream models)
Flow-speed-density relationship
Single Regime
Multi-Regime
Microscopic Model
Car-following
Lane-changing/gap acceptance
Mesoscopic

3. Stream Model Types Speed-Density Flow-Density Flow-Speed
Most fundamental
Driver adjust speed according to density
Flow-Density
Very useful for traffic control
Unifies various theoretical ideas
Flow-Speed
Very useful for operational analysis
HCM curve is of this type
“Concentration” is used for density in older literature

4. Stream Flow Fundamentals
Basic flow variables:
Flow (q). Maximum flow is defined as capacity
Speed (u).
Free-flow speed (uf)is defined as the speed which corresponds to flows approaching zero,
Optimum speed (uo) is defined as the speed which corresponds to maximum flows (capacity).
Density (k).
Jam density(Kj) is defined as the density that corresponds to flow and speed approaching zero
Optimum density (Ko) is defined as the density that corresponds to maximum flows (capacity).

5. How were the models developed?
Theoretical derivation
Fitting the data

6. Greenshields’ Model Linear relationship bet. u and k
Boundary conditions:
Flow is zero at zero density
Flow is zero at max. density (kj)
Free-flow speed (uf) at zero density
Flow-density curves are convex k u uf kj

7. Greenshields’ Model k u uf kj k v vf kj q u uf um um qmax k q kj k qkm

8. Greensberg’s Model u Uo: speed at max. flow Problem: infinite FFS kj k
Useful for high density, not for low density kj k

9. Underwood’s Model Based on data on Merritt Parkway in Connecticut
Corrected the problem of infinite ffs in Greensberg’s model
ko is optimum density, which is hard to observe
Problem: speed never goes to zero (at jam density)
Useful for low density, not for high density

10. Northwestern Model Developed at Northwestern University
Based on observations that speed-density curves appear to be s-shaped
ko is optimum density, which is hard to observe
Similar to Underwood’s in ffs and jam density speed

11. Single regime models A-D superimposed on freeway data
Source: May 1990

12. Pipes-Munjal Models n is a real number greater than 0
n=1reduces to Greenshields’

13. Drew’s Models From a general equation of n is a real number
n=-1  Greenberg’s
n=1  Greenshields’

14. Drew’s Models

15. Drew’s Generalized formulation

16. Car-Following Models
Speed-Density models can be derived from the car-following models
Different l, m parameter values lead to different models, some of which are popular ones that have been discussed.

17. Car Following Theory and Macro Models on m and l Matrix

18. Car-Following Models

19. Non-Integer Exponents of m and l

20. Non-Integer Exponents of m and l

22. Need for Multi-Regime Models
Some single regime models only useful for certain density range
Greenberg’s: high density
Underwood’s: low density
Field data show different relationship in different density ranges

23. Field Observations
2 regimes

24. Payne’s observation also indicated the limitation of single regime models

25. Eddie’s Model Composite two regime model
Low density regime: Underwood’s
High density regime: Greenberg’s

26. Underwood’s Two-Regime Model
Modified high density regime
Modification relatively arbitrary

27. Dick’s Two-Regime Model
Assuming a fixed upper limit for speed
Combined with Greenshields’ model

28. Two Regime Car-Following Models

29. Multiregime Models

30. Multi-regime models A-D superimposed on freeway data
Source: May 1990

31. More Speed-Flow Field Observations
4-lane

34. Flow-Density Models
Zero density => Zero flow => Curve must pass origin
Possible to have high density with zero flow
There must be one or more points of maximum flow
q-k curve not necessarily continuous

35. Parabolic q-k Model From Greenshields’ km=kj/2, and um=uf/2 q qmax kj

36. Logarithmic q-k Model
From Greensberg's
km=kj/e

37. Discontinuous q-k Model
Eddie pointed out traffic behavior to be different at high density and at low density
Two speed-density models lead to two flow-density curves

38. q-k Models for Different Flows
Example: Flow at Bottleneck, flow in different lanes

39. q-k Model Applications
Bottleneck Analysis/Shockwave
Freeway control (based on occupancy instead of density)

40. Flow-Speed Models

41. Flow-Speed Models

42. Flow-Speed Models
HCM 2000 Freeway Basic Segment

43. Travel Time Models
Haase’s Model

44. Haase’s Travel Time Model

45. Other Travel Time Models
Planning models
BPR Function
To,Tmin: FF travel time

46. Other Travel Time Models

47. 3 D Models

48. 3 D Models
Letter standing for different speeds

49. 3 D Models
One of the conclusions drawn by Gilchrist and Hall was that "conventional theory is insufficient to explain the data", and that the data were more nearly consistent with an alternative model based on catastrophe theory

50. Recommended References
Adolf D. May, Traffic Flow Fundamentals, Prentice-Hall, Inc, 1990.
Donald R. Drew, Traffic Flow Theory and Control, McGraw-Hill Book Company, 1968.
TRB Special Report 165, Traffic Flow Theory, A Monography, Transportation Research Board, 1975.
Highway Capacity Manual, Transportation Research Board, 2000.
Revised Monograph on Traffic Flow. Henry Lieu, editor.