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A Mathematical Introduction toTraffic Flow Theory
Benjamin Seibold (Temple University)
0
1
2
3
density
Flo
w r
ate
Q (
veh/s
ec)
0 max
Flow rate curve for LWR model
sensor data
flow rate function Q()
September 911, 2015
Tutorials Traffic Flow
Institute for Pure and Applied Mathematics, UCLA
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 1 / 69

References for Further Reading
Some References for Further Reading
general: wikipedia (can be used for almost every technical term)
traffic flow theory: wikibooks, Fundamentals of Transportation/Traffic Flow;Hall, Traffic Stream Characteristics; Immers, Logghe, Traffic Flow Theory
traffic models: review papers by Bellomo, Dogbe, Helbing
traffic phase theory: papers by Kerner, et al.
optimal velocity models and followtheleader models: papers by Bando,Hesebem, Nakayama, Shibata, Sugiyama, Gazis, Herman, Rothery, Helbing, et al.
connections between micro and macro models: papers by Aw, Klar, Materne,Rascle, Greenberg, et al.
firstorder macroscopic models: papers by Lighthill, Whitham, Richards, et al.
secondorder macroscopic models: papers by Whitham, Payne, Aw, Rascle,Zhang, Greenberg, Helbing, et al.
cellular models and cell transmission models: papers by Nagel, Schreckenberg,Daganzo, et al.
numerical methods for hyperbolic conservation laws: books by LeVeque
traffic networks: papers by Holden, Risebro, Piccoli, Herty, Klar, Rascle, et al.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 2 / 69

References for Further Reading
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 3 / 69

Fundamentals of Traffic Flow Theory
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 4 / 69

Fundamentals of Traffic Flow Theory What do we mean by Traffic Flow?
Two extreme situations: traffic flow modeling trivial/easy
Very low density
Negligible interaction between vehicles;everyone travels their desired speed
(Almost) complete stoppage
Flow dynamics irrelevant;simply queueing theory
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 5 / 69

Fundamentals of Traffic Flow Theory What do we mean by Traffic Flow?
Traffic flow modeling of importance, and challenging
Traffic dense but moving
Microdynamics: by which laws do vehicles interact with each other?Laws of flow, e.g.: how does vehicle density relate with flow rate?Macroview: temporal evolution of traffic density? traffic waves? etc.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 6 / 69

Fundamentals of Traffic Flow Theory Fundamental Quantities
Uniform traffic flow
driving direction
vehicle length ` distance d spacing hs
speed
Some fundamental quantities density : number of vehicles per unit length (at
fixed time)
flow rate (throughput) f : number of vehiclespassing fixed position per unit time
speed (velocity) u: distance traveled per unit time time headway ht : time between two vehicles passing
fixed position
space headway (spacing) hs : road length per vehicle occupancy b: percent of time a fixed position is
occupied by a vehicle
Relations
hs = d+`
= #lanes/hsf = #lanes/htf = u
max = #lanes/`
= b maxetc.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 7 / 69

Fundamentals of Traffic Flow Theory Fundamental Quantities
Nonuniform traffic flow
    
     
driving direction
vehicles have different velocities = pattern changes
Fundamental quantities
density : number of vehicles per unitlength (at fixed time) as before
flow rate f : number of vehicles passingfixed position per unit time as before
time mean speed: fixed position, averagevehicle speeds over time
space mean speed: fixed time, averagespeeds over a space interval
bulk velocity: u = f / usually meantin macroscopic perspective
Key questions
Given vehicle positions andvelocities, how do they evolve?
Impossible to predict on level oftrajectories (microscopic).
But it may be possible on levelof density and flow rate fields(macroscopic).
Needed: connection betweenmicro and macro description.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 8 / 69

Fundamentals of Traffic Flow Theory Micro vs. Macro
Microscopic description
individual vehicletrajectories
vehicle position: xj(t)
vehicle velocity:
xj(t) =dxjdt (t)
acceleration: xj(t)
Macroscopic description
field quantities,defined everywhere
vehicle density: (x , t)
velocity field: u(x , t)
flowrate field:f (x , t) = (x , t)u(x , t)
Micro view distinguishes vehicles;macro view does not.
Traffic flow has intrinsic irreproducibility.
No hope to describe/predict precise vehicle
trajectories via models [also, privacy
issues]. But, prediction of largescale field
quantities may be possible.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 9 / 69

Fundamentals of Traffic Flow Theory Micro vs. Macro
Microscopic macroscopic: kernel density estimation (KDE)Some approaches to go from x1, x2, . . . , xN to (x):
piecewise constant: let x1 < x2 < . . . , xN ; for xj < x xj+1, define(x) = 1/(xj+1 xj). no unique density at vehicle positions; does not work for aggregated lanes
moving window: at position x , define (x) as the number of vehiclesin [xh, x+h], divided by the window width 2h. integer nature; discontinuous in time
smooth moving kernels: each vehicle is a smeared Dirac delta;(x) =
j G (x xj); use e.g. Gaussian kernels, G (x) = Z1e(x/h)
2.
Most KDE approaches require a length scale h that must be larger than the typical
vehicle distance, and smaller than the actual length scale of interest (road length).
Macroscopic microscopic: samplingSome approaches to go from (x) to x1, x2, . . . , xN :
sampling via rejection, inverse transform, MCMC, Metropolis, Gibbs, etc.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 10 / 69

Fundamentals of Traffic Flow Theory Continuity Equation
Once we have a macroscopic (i.e. continuum) description
Vehicle density function (x , t); x = position on road; t = time.
Number of vehicles between point a and b:
m(t) =
ba(x , t)dx
Traffic flow rate (flux) is product of density and vehicle velocity u
f = u
Change of number of vehicles equals inflow f (a) minus outflow f (b)
d
dtm(t) =
batdx = f (a) f (b) =
ba
fxdx
Equation holds for any choice of a and b:
t + (u)x = 0 t + fx = 0continuity equation
Note: The continuity equation merely states that no vehicles are lost or created. Thereis no modeling in it. In particular, it is not a closed model (1 equation for 2 unknowns).
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 11 / 69

Fundamentals of Traffic Flow Theory Stochasticity of Traffic Flow
A critical note on stochasticity
It is frequently stated that traffic is stochastic, or traffic is random.
Or even further: traffic flow cannot be described via deterministic models.
Problems with these statements
A stochastic description is only useful if one defines the random variables,and knows something about their (joint) probability distribution.
Certain observables in a stochastic model may evolve deterministically. Itmatters what one is interested in: the evolution of real traffic flow may bereproduced quite well by a model, if one is interested in the density onlength scales much larger than the vehicle spacing. In turn, attempting tomodel the precise trajectories of the vehicles is most likely hopeless.
A better formulation and research goals
Traffic has an intrinsic irreproducibility (that cannot be overcome, say, viamore a careful experiment), due to hidden variables (e.g. drivers moods).
Goals: 1) find models that describe macroscopic variables as well as possible( fundamental insight into mechanics); 2) combine these models withfiltering to incorporate data ( prediction).
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 12 / 69

Fundamentals of Traffic Flow Theory Measurements
Examples of ways to measure traffic variables
human in chair: count passing vehicles ( flow rate)single loop sensor: at fixed position, count passing vehicles (flow rate)and measure occupancy ( density and velocity indirectly, assumingaverage vehicle length)
double loop sensor: as before, but also measure velocity directly
aerial imaging: photo of section of highway; direct measure of density
GPS: precise trajectories of a few vehicles on the road
observers in the traffic stream: as GPS, but also information aboutnearby vehicles
processed camera data: precise trajectories of all vehicles (NGSIMdata set); perfect data, but not possible on large scale
Inferring macroscopic variables (density, flow rate, etc.)
Traditional means of data collection measure them (almost) directly.
Modern data collection measures very different things. Inferringtraffic flow variables (everywhere and at all times) is a key challenge.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 13 / 69

Fundamentals of Traffic Flow Theory First Traffic Measurements and Fundamental Diagram
Bruce Greenshields collecting data (1933)
[This was only 25 years after the first Ford Model T (1908)]
Postulated densityvelocity relationship
Deduced relationship
u = U() = umax(1 /max),max195 veh/mi; umax43 mi/h
Flow ratef = Q() = umax( 2/max)
Contemporary measurements (Q vs. )
0
1
2
3
density
Flo
w r
ate
Q (
veh/s
ec)
0 max
Flow rate curve for LWRQ model
sensor data
flow rate function Q()
[Fundamental Diagram of Traffic Flow]
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 14 / 69

Fundamentals of Traffic Flow Theory Fundamental Diagram
Fundamental Diagram (FD) oftraffic flow (sensor data)
0
1
2
3
density
Flo
w r
ate
Q (
veh/s
ec)
0 max
Flow rate curve for LWRQ model
sensor data
flow rate function Q()
[Greenshields Flux]
0
1
2
3
density
Flo
w r
ate
Q (
veh/s
ec)
0 max
Flow rate curve for LWR model
sensor data
flow rate function Q()
[(smoothed) DaganzoNewell Flux]
Fascinating fact
FDs exhibit the same features, for highwayseverywhere in the world:
f = Q() for small (freeflow) for sufficiently large (congestion),
FD becomes setvalued
setvalued part: f decreases with
Ideas and concepts
ignore spread and define functionf = Q(), e.g.: Greenshields (quadratic),DaganzoNewell (piecewise linear) flux
empirical classification and explanation traffic phase theory [Kerner 19962002]
explanation of spread via imperfections:sensor noise, driver inhomogeneities, etc.
study of models that reproduce spread(esp. secondorder models)
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 15 / 69

Fundamentals of Traffic Flow Theory Fundamental Diagram
Traffic phase theory(here: 2 phases)[Kerner 19962002]
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Free
flow
cur
ve
Synchronized flow
density / max
flow
rat
e: #
veh
icle
s / l
ane
/ sec
.
sensor data
Also: 3phase theory (wide moving jams)
Phase transition induced by trafficwaves in macroscopic model[S, Flynn, Kasimov, Rosales, NHM 8(3):745772, 2013]
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
density / max
flow
rat
e Q
: # v
ehic
les
/ sec
.
equilibrium curvesonic pointsjamiton linesjamiton envelopesmaximal jamitonssensor data
A key challenge in traffic flow theory
Many open questions in explaining (phenomenologically), modeling, andunderstanding the spread observed in FDs.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 16 / 69

Traffic Models An Overview
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 17 / 69

Traffic Models An Overview The Point of Models
One can study traffic flow empirically, i.e., observe and classify what onesees and measures.
So why study traffic models?
Can remove/add specific effects (lane switching, driver/vehicleinhomogeneities, road conditions, etc.) understand which effectsplay which role.
Can study effect of model parameters (driver aggressiveness, etc.).
Can be analyzed theoretically (to a certain extent).
Can use computational resources to simulate.
Yield quantitative predictions ( traffic forecasting).We actually do not really know (exactly) how we drive. Models thatproduce correct emergent phenomena help us understand our drivingbehavior.
Traffic models can be studied purely theoretically (mathematicalproperties). That is fun; but ultimately models must be validatedwith real data to investigate how well they reproduce real traffic flow.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 18 / 69

Traffic Models An Overview Types of Traffic Models
Microscopic modelsxj = G(xj+1 xj , uj , uj+1)
0 20 40 60 80 100 120 140 160 180 2000
100
200
300
400
500
600
700
800
900
1000
time t
posi
tions
of v
ehic
les
x(t)
Idea
Describe behavior of individualvehicles (ODE system).
Micro Macro macro = limit of micro when
#vehicles micro = discretization of
macro in Lagrangianvariables
Macroscopic models{t + (u)x =0
(u+h)t +u(u+h)x =1(Uu)
Methodology and role
Describe aggregate/bulkquantities via PDE.
Natural framework formultiscale phenomena,traveling waves, and shocks.
Great framework toincorporate sparse data[Mobile Millennium Project].
Cellular models
Idea
Celltocell propagation(spacetimediscrete).
Cellular Macro macro = limit of cellu
lar when #cells cellular = discreti
zation of macro inEulerian variables
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 19 / 69

Traffic Models An Overview Microscopic Traffic Models
Microscopic Traffic Models Typical Setup
N Vehicles on the road
Position of jth vehicle: xjVelocity of jth vehicle: vjAcceleration of jth vehicle: aj
Physical principles
Velocity is rate of change ofposition: vj = xj
Acceleration is rate of changeof velocity: aj = vj = xj
Follow the leader model
Accelerate/decelerate towardsvelocity of vehicle ahead of you:
aj =vj+1 vjxj+1 xj
Optimal velocity model
Accelerate/decelerate towards anoptimal velocity that depends onyour distance to the vehicle ahead:
aj = V (xj+1 xj) vj
use computersto solve
Combined Model
aj = vj+1 vjxj+1 xj
+ (V (xj+1 xj) vj)
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 20 / 69

Traffic Models An Overview Microscopic Traffic Models
Microscopic Traffic Models Philosophy
Philosophy of microscopic models
Compute trajectories of each vehicle.
Can be extended to multiple lanes (with lane switching), differentvehicle types, etc.
At the core of most microsimulators (e.g., Aimsun, using the Gippsmodel; or Vissim, using the Wiedemann model); usually with adiscrete timestep.
Many other carfollowing models, e.g., the intelligent driver model.
May have many parameters, in particular free parameters that cannotbe measured directly. Hence, calibration required.
In most applications, initial positions of vehicles not known precisely.Ensembles of computations must be run.
Good for simulation (How would a laneclosure affect this highwaysection?); not the best framework for estimation and predictionbased on sparse/noisy data.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 21 / 69

Traffic Models An Overview Macroscopic Traffic Models
Macroscopic Traffic Models Setup
Recall: continuity equation
Vehicle density (x , t). Number of vehicles in [a, b]: m(t) = ba(x , t)dx
Traffic flow rate (flux): f = u
Change of number of vehicles equals inflow f (a) minus outflow f (b):
d
dtm(t) =
ba
tdx = f (a) f (b) = ba
fxdx
Equation holds for any choice of a and b: t + (u)x = 0
Firstorder models (LighthillWhithamRichards)
Model: velocity uniquely given by density, u = U(). Yields flux functionf = Q() = U(). Scalar hyperbolic conservation law.
Secondorder models (e.g., PayneWhitham, AwRascleZhang)
and u are independent quantities; augment continuity equation by a secondequation for velocity field (vehicle acceleration). System of hyperbolicconservation laws.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 22 / 69

Traffic Models An Overview Macroscopic Traffic Models
Macroscopic Traffic Models Philosophy
Philosophy of macroscopic models
Equations for macroscopic traffic variables.
Usually laneaggregated, but multilane models can also beformulated.
Natural framework for multiscale phenomena, traveling waves, shocks.
Established theory of control and coupling conditions for networks.
Nice framework to fill gaps in incorporated measurement data.
Mathematically related with other models, e.g., microscopic models,mesoscopic (kinetic) models, cell transmission models, stochasticmodels.
Good for estimation and prediction, and for mathematical analysis ofemergent features. Not the best framework if vehicle trajectories areof interest. Also, analysis and numerical methods for PDE are morecomplicated than for ODE.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 23 / 69

Traffic Models An Overview Cellular Traffic Models
Cellular Traffic Models Cellular Automata
NagelSchreckenberg model
Cut up road into cells of width h.Choose time step t.
Each cell is either empty or contains asingle car. A car has a discrete speed.
In each timestep, do the following:
1 All cars have speed increased by 1.2 For each car reduce its speed to
the number of empty cells ahead.3 Each car with speed 2 is slowed
down by 1 with probability p.4 Each car is moved forward its
velocity number of cells. Produces traffic jams that resemblequite well real observations.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 24 / 69

Traffic Models An Overview Cellular Traffic Models
Cellular Traffic Models Cell Transmission Models
Cell Transmission Models (CTM)
Cut up road into cells of width h. Choose time step t.
Each cell contains an average density j . In each step, a certain fluxof vehicles, fj+ 1
2, travels from cell j to cell j + 1. The fluxes between
cells change the densities in the cells accordingly.
The flux depends on the adjacent cells: fj+ 12
= F (j , j+1).
The flux function F is based on a sending (demand) function of j ,and a receiving (supply) function of j+1.
The basic CTM is equivalent to a Godunov (finite volume)discretization of the LWR model.
CTMs of secondorder macroscopic models (e.g., ARZ) can beformulated.
Highorder accurate finite volume schemes can be formulated (bothfor LWR and for secondorder models).
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 25 / 69

Traffic Models An Overview Messages
Traffic Models Messages
Messages
Cellular automaton models are easy to code, and resemble realjamming behavior. Not much further focus here.
Celltransmission models will be seen not as separate models, but asfinite volume discretizations of macroscopic models.
Microscopic models converge to macroscopic models as one scalesvehicles smaller and smaller.
Firstorder models (both micro and macro) exhibit shock(compression) waves, but no instabilities.
Secondorder models (both micro and macro) can have unstableuniform flow states (phantom traffic jams) that develop into travelingwaves (traffic waves, jamitons).
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 26 / 69

The LighthillWhithamRichards Model
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 27 / 69

The LighthillWhithamRichards Model Derivation
Continuity equation t + (u)x = 0
One equation, two unknown quantities and u.
Simplest idea: model velocity u as a function of .
(i) alone on the road drive with speed limit: u(0) = umax(ii) bumper to bumper complete clogging: u(max) = 0(iii) in between, use linear function: u() = umax
(1 max
)LighthillWhithamRichards model (1950)
f () = max
(1 max
)umax
A more realistic f ()
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 28 / 69

The LighthillWhithamRichards Model Evolution in Time
Method of characteristics
t + (f ())x = 0
Look at solution along a special curve x(t). At this moving observer:
d
dt(x(t), t) = x x +t = x x (f ())x = x x f
()x =(x f ()
)x
If we choose x = f (), then solution () is constant along the curve.
LWR flux function and information propagation
speed of vehicles speed of information
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 29 / 69

The LighthillWhithamRichards Model Evolution in Time
Solution method
Let the initial traffic density (x , 0) = 0(x) be represented by points(x , 0(x)). Each point evolves according to the characteristic equations{
x = f ()
= 0
Shocks
The method of characteristics eventually creates breaking waves.In practice, a shock (= traveling discontinuity) occurs.Interpretation: Upstream end of a traffic jam.
Note: A shock is a model idealization of a real thin zone of rapid braking.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 30 / 69

The LighthillWhithamRichards Model Weak Solutions
Characteristic form of LWR
LWR model t + f ()x = 0 (1)
in characteristic form: x = f (), = 0.
If initial conditions (x , 0) = 0(x) smooth (C1),
solution becomes nonsmooth at timet = 1infx f (0(x))0(x) .
Reality exists for t > t, but PDE does not makesense anymore (cannot differentiate discont. function).
Weak solution concept
(x , t) is a weak solution if it satisfies 0
t + f ()x dxdt =
[]t=0 dx C 10 test fct., C1 with compact support
(2)
Theorem: If C 1 (classical solution), then (1) (2).Proof: integration by parts.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 31 / 69

The LighthillWhithamRichards Model Weak Solutions
Weak formulation of LWR 0
t + f ()x dxdt =
[]t=0 dx C 10
Every classical (C 1) solution is a weak solution.
In addition, there are discontinuous weak solutions (i.e., with shocks).
Riemann problem (RP)
0(x) =
{L x < 0
R x 0x
a b
L
Rs
Speed of shocks
The weak formulation implies that shocks move with a speed such thatthe number of vehicles is conserved:
RP: (L R) s = ddt ba (x , t) dx = f (L) f (R)
Yields: s = f (R)f (L)RL =[f ()]
[] RankineHugoniot condition
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 32 / 69

The LighthillWhithamRichards Model Entropy Condition
Weak formulation and RankineHugoniot shock condition 0
t + f ()x dxdt =
[]t=0 dx C 10 ; s =[f ()]
[]
Problem
For RP with L > R , many weak solutions for same initial conditions.
One shock
x
6
Two shocks
x
6
Rarefaction fan
x
6

Entropy condition
Single out a unique solution (the dynamically stable one vanishingviscosity limit) via an extra entropy condition:
Characteristics must go into shocks, i.e., f (L) > s > f(R).
For LWR (f () < 0): shocks must satisfy L < R .
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 33 / 69

The LighthillWhithamRichards Model Exercises
Exercises 1
Explain characteristic speed and shock speed graphically in FD plot.
Exercises 2 & 3
LWR model t + f ()x = 0 with Greenshields flux f () = (1 ).
Red light turning green
x
6
0(x) =
{0 x  < 11 x  1
A bottleneck removed
x
6
0(x) =
0.4 x < 10.8 x  10.2 x > 1
Exercise 4
What changes if we use DaganzoNewell flux f () = 12 12 ?
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 34 / 69

The LighthillWhithamRichards Model Limitations
Timeevolution of density for LWR
Result
The LWR model quite nicely explainsthe shape of traffic jams (vehicles runinto a shock).
Shortcomings of LWR
Perturbations never grow(maximum principle).
Thus, phantom traffic jamscannot be explained with LWR.
And neither can multivaluedFD.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 35 / 69

SecondOrder Macroscopic Models
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 36 / 69

SecondOrder Macroscopic Models PayneWhitham Model
Modeling shortcoming of the LWR model
Vehicles adjust their velocity u instantaneously to the density .Conjecture: real traffic instabilities are caused by vehicles inertia.
PayneWhitham model (1979)
Vehicles at x(t) moves with velocity x(t) = u(x(t), t). Thus itsacceleration is
d
dtu(x(t), t) = ut + ux x = ut + u ux
Model for acceleration: ut + u ux =1 (U() u)
Here U() desired velocity (as in previous model), and relaxation time.
Full model: {t + (u)x = 0
ut + u ux =(p())x +
1 (U() u)
Here p() traffic pressure (models preventive driving);without this term, vehicles would collide.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 37 / 69

SecondOrder Macroscopic Models PayneWhitham Model
PayneWhitham model{t + (u)x = 0
ut + u ux +(p())x =
1 (U() u)
Examples for traffic pressure:
shallow water type: p() = 22 px = x
singular at m: p() = (+ m log(1 m )
) px =
xm
Model parameters:
relaxation time ( 1 = aggressiveness of drivers).
anticipation rate .
System of hyperbolic conservation laws with relaxation term(u
)t
+
(u
1dpd u
)(u
)x
=
(0
1 (U() u)
)Characteristic velocities: v = u c , with speed of sound: c2 = dpd .
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 38 / 69

SecondOrder Macroscopic Models PW & ARZ
PayneWhitham (PW) Model [Whitham 1974], [Payne: Transp. Res. Rec. 1979]{t + (u)x = 0ut + uux +
1p()x =
1 (U() u)
Parameters: pressure p(); desired velocity function U(); relaxation time
Second order model; vehicle acceleration: ut + uux = p() x +
1 (U() u)
Inhomogeneous AwRascleZhang (ARZ) Model[Aw&Rascle: SIAM J. Appl. Math. 2000], [Zhang: Transp. Res. B 2002]{
t + (u)x = 0(u + h())t + u(u + h())x =
1 (U() u)
Parameters: hesitation function h(); velocity function U(); time scale
Second order model; vehicle acceleration: ut + uux = h()ux +
1 (U() u)
Remark
The ARZ model addresses modeling shortcomings of PW (such as: negativespeeds can arise, shocks may overtake vehicles from behind).
However, the arguments regarding instabilities and traffic waves apply to PW andARZ in the same fashion. Below, we present things for PW (simpler expressions).
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 39 / 69

SecondOrder Macroscopic Models Linear Stability Analysis
System of balance laws(u
)t
+
(u
1dpd u
)(u
)x
=
(0
1 (U() u)
) Eigenvalues{1 = u c2 = u + c
}c2 = dpd
Linear stability analysis
(S) When are constant base statesolutions (x , t) = ,u(x , t) = U() stable (i.e. smallperturbations do not amplify)?
Reduced equation
(R) When do solutions of the 2 2system converge (as ) tosolutions of the reduced equation
t + (U())x = 0 ?
Subcharacteristic condition (Whitham [1959])
(S) (R) 1 2, where = (U())
Stability for PayneWhitham model
U() c() U() + U () U() + c() c() U().
For p() = 22 and U() = um
(1 m
): Stability iff m
mu2m
.
Phase transition: If enough vehicles on the road, perturbations grow.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 40 / 69

SecondOrder Macroscopic Models Detonation Theory
Selfsustained detonation wave
Key Observation: ZND theory [Zeldovich (1940), von Neumann (1942), Doring (1943)] applies tosecond order traffic models, such as PW and ARZ.
Reaction zone travels unchanged with speed of shock.RankineHugoniot conditions one condition short (unknown shock speed).Sonic point is event horizon. It provides missing boundary condition.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 41 / 69

SecondOrder Macroscopic Models Traveling Wave Solution and Jamiton Construction
PW model{t + (u)x = 0
ut + uux +1p()x =
1 (U() u)
Traveling wave ansatz
= (), u = u(), with selfsimilar variable = xst .
Then t = s , x =
1 , ut = s u
, ux =1 u
and px =1 c
2 , c2 = dpd
Continuity equation
t + (u)x = 0
s +
1
(u) = 0
((u s)) = 0
=m
u s =
u su
Momentum equation
ut + uux +px
=1
(U u)
su +
1
uu +
dp
d
=
1
(U u)
(u s)u c2 1u s
u = U u
u =(u s)(U u)(u s)2 c2
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 42 / 69

SecondOrder Macroscopic Models Traveling Wave Solution and Jamiton Construction
Ordinary differential equation for u()
u =(u s)(U() u)(u s)2 c()2
where =m
u swhere
s = travel speed of jamiton
m = mass flux of vehicles through jamiton
Key point
In fact, m and s can not be chosen independently:
Denominator has root at u = s + c . Solution can only pass smoothlythrough this singularity (the sonic point), if u = s + c implies U = u.
Using u = s + m , we obtain for this sonic density S that:{Denominator s + mS = s + c(S) = m = Sc(S)Numerator s + mS = U(S) = s = U(S) c(S)
Algebraic condition (ChapmanJouguet condition [Chapman, Jouguet (1890)]) thatrelates m and s (and S).
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 43 / 69

SecondOrder Macroscopic Models Traveling Wave Solution and Jamiton Construction
Jamiton ODEu =
(u s)(U( mus ) u)(u s)2 c( mus )2
Construction1 Choose m. Obtain s by
matching root of denominatorwith root of numerator.
2 Choose u. Obtain u+ byRankineHugoniot conditions.
3 ODE can be integrated throughsonic point (from u+ to u).
4 Yields length (from shock toshock) of jamiton, , and
number of cars, N =
0 (x)dx .
Periodic case
Inverse constructionIf and N are given, find jamiton by iteration.
jamiton length = O()
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 44 / 69

SecondOrder Macroscopic Models Jamitons in an Experiment
Experiment: Jamitons on circular road [Sugiyama et al.: New J. of Physics 2008]
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SecondOrder Macroscopic Models Jamitons in Numerical Simulations
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SecondOrder Macroscopic Models Jamitons in Numerical Simulations
Infinite road; lead jamiton gives birth to a chain of jamitinos.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 47 / 69

SecondOrder Macroscopic Models Jamitons in the Fundamental Diagram
Recall: continuity equation
t + (u)x = 0
Traveling wave ansatz
= (), u = u(), where = xst ,yields
t + (u)x = 0
s +
1
(u) = 0
((u s)) = 0(u s) = m
q = m + s
Hence: Any traveling wave is a linesegment in the fundamental diagram.
A jamiton in the FD
R
M
S
+
Q(
)
equilibrium curvesonic pointmaximal jamitonactual jamiton
Plot equilibrium curve Q()=U() Chose a S that violates the SCC Mark sonic point (S,Q(S)) (red) Set m=Sc(S) and s =U(S)c(S) Draw maximal jamiton line (blue) Other jamitons with the same m and
s are subsegments (brown)
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 48 / 69

SecondOrder Macroscopic Models Jamitons in the Fundamental Diagram
Construction of jamiton FD
For each sonic density S that violates theSCC: construct maximal jamiton.
Conclusion
Setvalued fundamental diagrams can beexplained via traveling wave solutions insecondorder traffic models.
Temporal aggregation of jamitons
At fixed position x , calculate all possibletemporal average (t = ) densities
= 1t
t0
(x , t)dt .
Average flow rate: q = m + s .
Resulting aggregated jamiton FD is a subsetof the maximal jamiton FD.
Jamiton fundamental diagram
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
Maximal jamitons
density / max
flow
rat
e Q
: # v
ehic
les
/ sec
.
equilibrium curvesonic pointsjamiton linesjamiton envelopes
Emulating sensor aggregation
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
= 1
density / max
flow
rat
e Q
: # v
ehic
les
/ sec
.
equilibrium curvesonic pointsjamiton linesjamiton envelopesmaximal jamitons
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 49 / 69

SecondOrder Macroscopic Models Jamitons in the Fundamental Diagram
Theorem
Jamitons always reduce the flow rate.
Remark
Statement is equivalent to: If t ,(, q) lies below the equilibrium curve:
aggregate(=effective) flow rate of jamitonchain is lower than uniform base state withsame average density.
Good agreement with sensor data
We can reverseengineer model parameters,such that the aggregated jamiton FD showsa good qualitative agreement with sensordata.
Infinite aggregation (t )
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
Effective flow rate
density / max
flow
rat
e Q
: # v
ehic
les
/ sec
.
equilibrium curvesonic pointsjamiton linesjamiton envelopesmaximal jamitons
Sensor data with jamiton FD
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
density / max
flow
rat
e Q
: # v
ehic
les
/ sec
.
equilibrium curvesonic pointsjamiton linesjamiton envelopesmaximal jamitonssensor data
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 50 / 69

SecondOrder Macroscopic Models Validation of SecondOrder Models via Real Traffic Data
LWR modelFirstorder model(does not reflect spread in FD)
t + (U())x = 0 t + Q()x = 0
} Q() = U()
Construct function Q()
. . . via LSQfit to FD data.
Datafitted flux Q()
0
1
2
3
density
Flo
w r
ate
Q (
veh/s
ec)
0 max
Flow rate curve for LWR model
sensor data
flow rate function Q()
Induced velocity curve U()LWR Model
density
velo
city
0 max0
vmax
velocity function u = U()
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 51 / 69

SecondOrder Macroscopic Models Validation of SecondOrder Models via Real Traffic Data
AwRascleZhang (ARZ) model
t + (u)x = 0
(u + h())t + u(u + h())x = 0
where h() > 0 and, WLOG, h(0) = 0.
Equivalent formulation
t + (u)x = 0
wt + uwx = 0
where u = w h()
Interpretation 1: Each vehicle (movingwith velocity u) carries a characteristicvalue, w , which is its emptyroad velocity.The actual velocity u is then: w reducedby the hesitation function h().
Interpretation 2: ARZ is a generalization ofLWR: different drivers have different uw ().
ARZ model velocity curves
ARZ Model
density
velo
city
0 max0
vmax
for different w
family of velocity curves
equilibrium curve U()
oneparameter family of curves:u = uw () = u(w , ) = wh()here: h() = vmax U()equilibrium (i.e., LWR) curve:
U() = u(vmax, )
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 52 / 69

SecondOrder Macroscopic Models Validation of SecondOrder Models via Real Traffic Data
NGSIM (I80, Emeryville, CA; 2005)
three 15 minute intervals precise trajectories of all
vehicles (in 0.1s intervals)
historic FD providedseparately
Approach
Construct macroscopic fields and u fromvehicle positions (via kernel density estimation)
Use data to prescribe i.c. at t = 0 and b.c. atleft and right side of domain
Run PDE model to obtain model(x , t) andumodel(x , t) and error
E(x , t)=data(x,t)model(x,t)
max+udata(x,t)umodel(x,t)
umax
Evaluate model error in a macroscopic (L1)sense:
E =1
TL
T0
L0
E(x , t)dxdt
Spaceandtimeaveraged modelerrors for NGSIM data
Data set LWR ARZ
4:004:15 0.127 (+73%) 0.0735:005:15 0.115 (+36%) 0.0855:155:30 0.124 (+6%) 0.117
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 53 / 69

Finite Volume and CellTransmission Models
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 54 / 69

Finite Volume and CellTransmission Models Godunovs Method
Initial condition
x
6
Cellaveraged initial cond.
x
6
Godunovs method
REA = reconstructevolveaverage
1 Divide road into cells of width h.
2 On each cell, store the average density j .
3 Assume solution is constant in each cell.
4 Evolve this piecewise constant solutionexactly from t to t + t.
5 Average over each cell to obtain apwconst. sol. again.
6 Go to step 4.
Solution evolved exactly
x
6
Cellaveraged evolved solution
x
6
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 55 / 69

Finite Volume and CellTransmission Models Godunovs Method
Godunovs method
4 Evolve pwconst. sol. exactly from t to t+t.
5 Average over each cell.
6 Go to step 4.
Key Points
If we choose t < h2 max f  (CFL condition),waves starting at neighboring cell interfacesnever interact. Thus, can be solved as localRiemann problems.
Because the exactly evolved solution isaveraged again, all that matters for the changej(t) j(t+t) are the fluxes through thecell boundaries:
j(t+t) = j(t) +th (Fj 12
Fj+ 12)
Solution at time t
x
6 j
Solution evolved exactly
x
6

Fj 12

Fj+ 12
Solution at time t + t
x
6 j
?
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 56 / 69

Finite Volume and CellTransmission Models Godunovs Method
Godunovs method
j(t+t) = j(t) +th (Fj 12
Fj+ 12)
rightgoing shock or rarefaction:Fj+ 1
2= f (j)
leftgoing shock or rarefaction:Fj+ 1
2= f (j+1)
transsonic rarefaction:Fj+ 1
2= f (c) = max f ()
Equivalent formulation of fluxes CTMFj+ 1
2= min{D(j), S(j+1)}
where:
demand function: D() = f (min(, c))
supply function: S() = f (max(, c))
Solution at time t
x
6 j
Solution evolved exactly
x
6

Fj 12

Fj+ 12
Solution at time t + t
x
6 j
?
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 57 / 69

Finite Volume and CellTransmission Models CellTransmission Model
Cell transmission model (CTM)
j(t+t) = j(t) +th (Fj 12
Fj+ 12)
Flux between cells:
Fj+ 12
= min{D(j),S(j+1)}
demand function: D() = f (min(, c))supply function: S() = f (max(, c))
Principles
The flux of vehicles between cells j and j + 1cannot exceed:
a) the demand (on road capacity) that thesending cell j requires;
b) the supply (of road capacity) that thereceiving cell j + 1 provides.
The actual flux is the maximum flux thatsatisfies these constraints.
Demand function

6f
c max
Supply function

6f
c max
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 58 / 69

Finite Volume and CellTransmission Models Convergence Results
Convergence of the Godunov method (or CTM)
As t = Ch 0, the sequence of approximate solutions converges(in L1) to the unique weak entropy solution of the LWR modelt + f ()x = 0.
For any fixed t and h, the temporal evolution that Godunov=CTMprovides is only an approximation to the LWR model.
To leading order, Godunov=CTM solutions behave like solutions ofthe modified equation t + f ()x = chxx .
Hence, the error between Godunov=CTM and LWR scales like O(h);firstorder numerical scheme.
Also, shocks get smeared out over several cells.
Other, more accurate (highorder) numerical schemes for LWR exist(MUSCL, discontinuous Galerkin, etc.). These are more complicated.
CTMs can also be cast for secondorder traffic models.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 59 / 69

Traffic Networks
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 60 / 69

Traffic Networks CellTransmission Model
The simplest network: 1 inroad; 1 outroad
Example: Greenshields flux
road 1 has 4 lanes and speed limit 2:f1() = 2(1 /4)road 2 has 3 lanes and speed limit 4:f2() = 4(1 /3)
Cell transmission model (CTM)
On each road segment, apply CTM as before.
Interface between last cell of road 1 and firstcell of road 2: apply same principle as well.
Froad 1road 2 = min{D1(1),S2(2)}
demand function: D1() = f1(min(, 1c))
supply function: S2() = f2(max(, 2c))
Demand function

6f
2
2 4
Supply function

6f
3
1.5 3
Nice
As simple to code as thebasic CTM.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 61 / 69

Traffic Networks Generalized Riemann Problem
Generalized Riemann problem
Given state 1 on road 1, and state 2 onroad 2, determine new states 1 and 2 thatsolution will assume.
1 Calculate demand and supply:D1(1) and S2(2).
2 Maximum flux that does not exceeddemand and supply:F = min{D1(1),S2(2)}.
3 Choose new states so that no wavestravel into the interface:
1 =
{1 F = f1(1)
f 1,1 (F ) F 6= f1(1)
2 =
{2 F = f2(2)
f 1,2 (F ) F 6= f2(2)
Demand function

6f
2
2 4
Supply function

6f
3
1.5 3
Exercise
1 = 2.5 and 2 = 2.5D1 = 2 and S2 = 5/3
1 2.8 and 2 = 2.5
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 62 / 69

Traffic Networks Bifurcation or OffRamp
1in2out node
Inroad: f1(1); outroads: f2(2) and f3(3).
Additional info: split ratios 2 + 3 = 1 (how many cars go where).
Generalized Riemann problem
1 Calculate demand and supplies: D1(1) and S2(2), S3(3).
2 Flux is maximum flux that does not exceed demand and supply.
Constraints: F D1(1); 2F S2(2); 3F S3(3).Yields: F = min{D1(1), S2(2)2 ,
S3(3)3}.
3 If all we want is CTM, we are done (F1 = F ; F2 = 2F ; F3 = 3F ).
4 If we need new states: no waves travel into the interface.
1 =
{1 F1 = f1(1)
f 1,1 (F1) F1 6= f1(1)
2 =
{2 F2 = f2(2)
f 1,2 (F2) F2 6= f2(2)3 =
{3 F3 = f3(3)
f 1,3 (F3) F3 6= f3(3)
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 63 / 69

Traffic Networks Bifurcation or OffRamp
Generalized Riemann problem for offramp
1 Calculate demand and supplies: D1(1) and S2(2), S3(3).
2 Flux is maximum flux that does not exceed demand and supply.
Constraints: F D1(1); 2F S2(2); 3F S3(3).Yields: F = min{D1(1), S2(2)2 ,
S3(3)3}.
Wait a minute. . .
1 If the offramp is clogged (3 = max3 ), this produces F = 0, i.e., no
traffic passes on along the highway.
2 No contradiction, because we required F2 and F3 to be distributedaccording to split ratios: FIFO (firstinfirstout) model.
3 One can also formulate nonFIFO models. Problem there:consequences of queuing of vehicles that want to exit are neglected.
4 Fix of these issues: [J. Somers: Macroscopic Coupling Conditions with Partial Blocking for HighwayRamps, MS Thesis, Temple University, 2015].
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 64 / 69

Traffic Networks Confluence or OnRamp
2in1out node
Inroads: f1(1) and f2(2); outroad: f3(3).
Generalized Riemann problem
1 Calculate demands and supply: D1(1),D2(2) and S3(3).
2 Now we have two fluxes to bedetermined: F1 and F2.
3 Constraints on flux vector ~F = (F1,F2):
0 F1 D1(1); 0 F2 D2(2);F1 + F2 S3(3).
4 Maximize flux F1 + F2.
5 If that nonunique, use additionalinformation: merge ratio F1/F2 = .
6 Use fluxes for CTM, or obtain newstates, as before.
All traffic can get through
F1
6F2
F 1
F 2
D1
D2
@@@@@S3
o
Not all traffic gets through
F1
6F2
F 1
F 2
D1
[email protected]@@@@@@S3
mergeratio
o
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 65 / 69

Traffic Networks General Networks
General Networks
The presented 1in1out, 1in2out, and 2in1out nodes alreadyenable us to put together large road networks.
Can use CTM to compute LWR solutions on such a network.
Even for the simple on and offramp case, there is still somemodeling and validation work to be done with regards to the questionwhich choices of fluxes are most realistic.
The general ninmout case can be treated similarly. However, theconstrained flux maximization leads to a linear program that mayneed to be solved numerically (can be costly).
Secondorder models: coupling conditions for the ARZ model havebeen developed, albeit not for the most general case.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 66 / 69

Microscopic Traffic Models
Overview
1 Fundamentals of Traffic Flow Theory
2 Traffic Models An Overview
3 The LighthillWhithamRichards Model
4 SecondOrder Macroscopic Models
5 Finite Volume and CellTransmission Models
6 Traffic Networks
7 Microscopic Traffic Models
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 67 / 69

Microscopic Traffic Models
Microscopic traffic models
Vehicle trajectories: x1(t), x2(t), . . . , xN(t).
Popular framework: carfollowing models.
Firstorder models: xj = V (xj+1 xj).Secondorder models: xj = G (xj+1 xj , xj , xj+1).Scaling: introduce reference distance X , scale space headway byxj+1xj
X , and place N 1
X on the roadway (spaced X ).Firstorder micro model converges (as X 0) to LWR with samevelocity function.
Certain secondorder microscopic models converge to different typesof secondorder macroscopic models.
Example: xj = Xxj+1xjxj+1xj + (V (
xj+1xjX ) xj)
converges to ARZ model with logarithmic pressure.
Due to this connection, secondorder microscopic models can exhibitunstable uniform flow (phantom traffic jams) and traffic waves.
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 68 / 69

Microscopic Traffic Models
Car following model with parameters determined so that traffic waves ofringroad experiment are reproduced in quantitative agreement
Benjamin Seibold (Temple University) Mathematical Intro to Traffic Flow Theory 09/0911/2015, IPAM Tutorials 69 / 69
Fundamentals of Traffic Flow TheoryWhat do we mean by ``Traffic Flow''?Fundamental QuantitiesMicro vs. MacroContinuity EquationStochasticity of Traffic FlowMeasurementsFirst Traffic Measurements and Fundamental DiagramFundamental Diagram
Traffic Models An OverviewThe Point of ModelsTypes of Traffic ModelsMicroscopic Traffic ModelsMacroscopic Traffic ModelsCellular Traffic ModelsMessages
The LighthillWhithamRichards ModelDerivationEvolution in TimeWeak SolutionsEntropy ConditionExercisesLimitations
SecondOrder Macroscopic ModelsPayneWhitham ModelPW & ARZLinear Stability AnalysisDetonation TheoryTraveling Wave Solution and Jamiton ConstructionJamitons in an ExperimentJamitons in Numerical SimulationsJamitons in the Fundamental DiagramValidation of SecondOrder Models via Real Traffic Data
Finite Volume and CellTransmission ModelsGodunov's MethodCellTransmission ModelConvergence Results
Traffic NetworksCellTransmission ModelGeneralized Riemann ProblemBifurcation or OffRampConfluence or OnRampGeneral Networks
Microscopic Traffic Models