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.

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

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

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).

How were the models developed? Theoretical derivation Fitting the data

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

Greenshields’ Model k u uf kj k v vf kj q u uf um um qmax k q kj k q km

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

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

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

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

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

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

Drew’s Models

Drew’s Generalized formulation

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.

Car Following Theory and Macro Models on m and l Matrix

Car-Following Models

Non-Integer Exponents of m and l

Non-Integer Exponents of m and l

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

Field Observations 2 regimes

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

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

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

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

Two Regime Car-Following Models

Multiregime Models

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

More Speed-Flow Field Observations 4-lane

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

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

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

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

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

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

Flow-Speed Models

Flow-Speed Models

Flow-Speed Models HCM 2000 Freeway Basic Segment

Travel Time Models Haase’s Model

Haase’s Travel Time Model

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

Other Travel Time Models

3 D Models

3 D Models Letter standing for different speeds

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

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. http://www.tfhrc.gov/its/tft/tft.htm