UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Erfassung von Verkehrskenngrößen der freien Strecke Detection of roadside traffic data direction of traffic flow measurement crosssection traffic flow time gap local speed density headways momentary speed

Dynamische Modelle Dynamic modells UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV)

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) car following models vehicle position

Psycho-physical headway model UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) distance domain without reaction distance decays perception threshold for distance perception threshold for closing up distance growths domain with reaction difference speed distance decays

conservation law traffic flow as forward difference continuum approximation summary i-1i i+1

where is the substantial derivative and is the equilibrium speed- density relation from the fundamental diagram For the speed variation we take a relaxation ansatz: The summary can be transformed into a new conservation law with

The new conservation law and the relaxation ansatz can be put together: (1)+(2) is the optimum velocity model after Bando et al.*) or simply the “Bando model“ *)Bando, M., et al.: Phys. Rev. E Vol.5, pp. 1035(1995)

Linear Stability Analysis of the Continuum Traffic Flow Model after Bando et al. allows the decomposition: with and gives in first order: Introducing an operating point :

Selecting an operating point in very dense traffic

Linear Stability Analysis of the Continuum Traffic Flow Model after Bando et al. Scaling of x, t gives in matrix notation with

Stability Analysis of the Continuum Traffic Flow Model (cont‘d) gives the eigenvalue equation = 0 ansatz with the explicite form decomposition into real part and imaginary part

Stability Analysis of the Continuum Traffic Flow Model (cont‘d) Stability for Re(ω) < 0 or α=a(1+a)- ν < 0

Stability Anaysis of the Continuum Traffic Flow Model (cont‘d) Paramter plane for stability regime Re(ω)<0 or α<0

Dispersion relation ω( κ) for and UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Re( ω ( κ ) ) κ

Special Case Δx=0 decomposition gives in first order for Δx=0: Scaling of x, t gives eigen- value equation = 0

Special Case Δx=0 (cont‘d) explicite eigenvalue equation decomposition into real part and imaginary part Stability i.e. Re(ω) < 0 can not be achieved for Δx=0 !

Special Case τ →0 For τ →0 the density k follows instantaneously V opt (k) the Bando model then reads decomposition gives in first order:

Special Case τ →0 (cont‘d) Scaling of x, t ansatz gives eigenvalue equation = 0 Stability i.e. Re(ω) < 0 is always achieved for τ →0 !

Optimum velocity model after Bando et al. and energy flow analysis conservative force derived from a potential U(Δx) F diss = disssipative force reflecting vehicles as active particles

the optimum velocity function must fulfill 1) V opt =0 for Δx=D „bumper to bumper“ 2) V opt =v f for free flow traffic · simplified approach · van Aerde approach

van Aerde simplified approach Δx=ℓ o givesΔx=D gives fixed point normalized force F nomalized spacing Δx

van Aerde simplified model y ξ y ξ

Introduction of the potential for the conservative force and mutiplying with gives for the optimum velocity model or time simulation with periodicboundary conditons

probabilistic capacity interpretation

Text UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Weiterer Text

Auto film

Decomposition of a traffic stream into freely driving and congested vehicles

free flow traffic cluster adhesion rate: inverse time gap q discharge rate: Traffic breakdown description from a queueing theory standpoint

Messungen HN Measurements NH UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV)

Breakdown possibility as measured on the highway 401 in Toronto during 16 days UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) (nach Persaud et. al., 1998) Zusammenbruchswahrscheinlichkeit 0 0,2 0,4 0,6 0, traffic flow Pcu/h/lane. breakdown probability

Probability of no instability within one hour UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) traffic volume

Capacity of nonsignalized Intersecions three leg intersections and generalizations

3 leg intersection

3 leg intersection situation plan

Time Gap Terminology limiting accepted time gap follow-up time gap accepted time gap rejected time gap

Time gap distribution and definition of limiting time gap after Greenshields

Poisson distribution – assumptions UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Warteschlangentheorie, Markovprozesse 1 1. probability of 1 arrival between t and t +  t: 2.probability of more than 1 arrival between t and t +  t: 3.consecutive arrivals are stochasticaly independent

derivation of the Poisson- distribution UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Warteschlangentheorie, Markovprozesse 1

successive solution of the differential- difference- equation for the Poisson- distribution UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Warteschlangentheorie, Markovprozesse 1 Poisson distribution

Poisson- distribution UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) for different values of source: Sachs, Angewandte Statistik, 8. Auflage, Springer Verlag, 1996 Warteschlangentheorie, Markovprozesse 1 n n n

Poisson distribution and time gap distribution p 0 (T) is the probability of no arrival within 0 ≤ t ≤ T that is equivalent to a time gap t g ≥ T p 0 (T) is therefore identical with the time gap distribution

time gap distribution and merging

Listing of all turning right situations of the merging lane (t g =limiting time gap, t f = follow-up time gap) t ≤ t g no vehicle turning right t g ≤ t ≤ t g +t f 1 vehicle turning right t g +t f ≤ t ≤ t g +2t f 2 vehicles turning right …… probability of n vehicles turning right, during a time gap t= prob{ t ≥ t g +(n-1)t f and t ≤ t g +nt f } → p n =e -q(t g +(n-1)t f ) - e -q(t g +nt f ) the amount of such time gaps is given by the traffic volume q of the main stream

capacity for a merging lane of a 3 leg intersection with limiting accepted time gap t g and follow-up time gap t f C m = Σ q n p n

Final result for the capacity of merging traffic into a 3-leg intersection

generalizations

generalizations (cont´d)

Warteschlangentheorie, Vorlesung, Einführung Waiting time at traffic signals

UNIVERSITÄT STUTTGART INSTITUT FÜR STRASSEN- UND VERKEHRSWESEN (ISV) LEHRSTUHL VERKEHRSPLANUNG UND VERKEHRSLEITTECHNIK (VuV) Time gaps when discharging at traffic signals after (1974) Δt [s] = 2,10 / n+1,47 after (1987) Δt [s] = 2,03 / n+1,60 Time difference [s] Vehicle -Position

= inflow vehicle/s = max. discharge vehicle/s (= 0.5 vehicle/s) Number of lined up passenger cars time

Warteschlangentheorie, Vorlesung, Einführung Total waiting time during red Waiting time at a traffic signal as queueing problem with random inflow and deterministic discharge

Total waiting time during green from queueing theory Webster-Formula for total waiting time at a traffic signal Waiting time at a traffic signal as queueing problem with random inflow and deterministic discharge

Comparison between Webster- Formula and Simulation Results