macroscopic fundamental diagrams of arterial road networks ...mpetn/files/talks/garoni.pdf ·...

Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems

Macroscopic Fundamental Diagrams ofarterial road networks governed by adaptive

traffic signal systems

Tim Garoni

School of Mathematical SciencesMonash University


Fundamental DiagramI Consider a one-dimensional flow (vehicles along a freeway)I The functional relationship between flow and density is the

fundamental diagram (Greenshields, 1935)

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

density

flow

I Intuitively makes sense to have a unimodal FD in one dimensionI What should happen in a network?I How should one even define network flow?

(No prescribed direction)




0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

density

flow

I Intuitively makes sense to have a unimodal FD in one dimension

I What should happen in a network?I How should one even define network flow?





0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

density

flow

I Intuitively makes sense to have a unimodal FD in one dimensionI What should happen in a network?

I How should one even define network flow?(No prescribed direction)




0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

density

flow

I Intuitively makes sense to have a unimodal FD in one dimensionI What should happen in a network?I How should one even define network flow?



Macroscopic Fundamental DiagramsI Simplest idea: relate arithmetic means of link density and flowI If network has link set Λ:

ρ =1|Λ|

∑λ∈Λ

ρλ, J =1|Λ|

∑λ∈Λ

Jλ

I ρλ is density of link λ and Jλ is its flow

Geroliminis & Daganzo 2008Empirical data from Yokohama

Buisson & Ladier 2009Empirical data from Toulouse


Two Extreme Cases

I Existence of MFDs is trivial:I If all links have the same FDI and if the distribution of congestion is always perfectly uniformI then network MFD coincides with common link FD

I This is not very interesting...I Existence of MFDs is impossible:

I If one has a network and is free to vary the demand on each link inany way imaginable, then no MFD can exist

I e.g. half the links have ρλ = 1 and other half have ρλ = 0, thenρ = 1/2 and J = 0

I e.g. all links have ρλ = 1/2, then ρ = 1/2 but J > 0(could even have J = Jmax)

I Existence of MFDs clearly not independent of demandI MFDs are interesting because there is something in betweenI In practice, on many networks the demand will rise and fall in a

fairly constrained way during a typical day


Two Extreme Cases

I Existence of MFDs is trivial:I If all links have the same FDI and if the distribution of congestion is always perfectly uniformI then network MFD coincides with common link FDI This is not very interesting...

I Existence of MFDs is impossible:I If one has a network and is free to vary the demand on each link in

any way imaginable, then no MFD can existI e.g. half the links have ρλ = 1 and other half have ρλ = 0, thenρ = 1/2 and J = 0





Two Extreme Cases





I Existence of MFDs clearly not independent of demand

I MFDs are interesting because there is something in betweenI In practice, on many networks the demand will rise and fall in a



Two Extreme Cases








What are MFDs?Consider a fixed network with link set ΛFirst of all, one needs to agree on what ρ and J mean.

I ρλ(t) and Jλ(t) are stochastic processesI Aggregate variables

ρ(t) =1|Λ|

∑λ∈Λ

ρλ(t) J(t) =1|Λ|

∑λ∈Λ

Jλ(t)

I MFD is the relationship between EJ(t) and Eρ(t)I Can be interested in instantaneous or stationary MFDs

I “Heterogeneity” is also important

h(t) =

√1|Λ|

∑λ∈Λ

[ρλ(t)− ρ(t)]2

Helbing 2009; Mazloumian, Geroliminis & Helbing 2010;Geroliminis & Sun 2011; de Gier, G & Zhang 2013

I J, ρ, h all stochastic processesI In time dependent context, heterogeneity can explain hysteresis


What are MFDs?Consider a fixed network with link set ΛFirst of all, one needs to agree on what ρ and J mean.

I ρλ(t) and Jλ(t) are stochastic processesI Aggregate variables

ρ(t) =1|Λ|

∑λ∈Λ

ρλ(t) J(t) =1|Λ|

∑λ∈Λ

Jλ(t)

I MFD is the relationship between EJ(t) and Eρ(t)I Can be interested in instantaneous or stationary MFDsI “Heterogeneity” is also important

h(t) =

√1|Λ|

∑λ∈Λ

[ρλ(t)− ρ(t)]2

Helbing 2009; Mazloumian, Geroliminis & Helbing 2010;Geroliminis & Sun 2011; de Gier, G & Zhang 2013

I J, ρ, h all stochastic processesI In time dependent context, heterogeneity can explain hysteresis


Asymmetric Simple Exclusion Process (ASEP)“Everything should be made as simple as possible, but not simpler”

(Albert Einstein)I Want an Ising model of traffic flowI One-dimensional stochastic cellular automata very popular in

statistical mechanics starting in 1990sI Such models do a reasonable job of explaining qualitative

behaviour of freeway trafficI “Phantom” jams emerge as consequence of collective behaviour

I Cellular automata are discrete dynamical systemsI Space, time, and state variables are discrete

I ASEP with open boundaries:

I If x1(t) = 0, then with probability α, x1(t + 1) = 1I For each cell i = 1, . . . , L with xi(t) = 1

I If xi+1(t) = 0 then with probability p, xi (t + 1) = 0 and xi+1(t + 1) = 1I Else xi (t + 1) = 1

I If xL(t) = 1, then with probability β, xL(t + 1) = 0


Asymmetric Simple Exclusion Process (ASEP)“Everything should be made as simple as possible, but not simpler”

(Albert Einstein)I Want an Ising model of traffic flowI One-dimensional stochastic cellular automata very popular in

statistical mechanics starting in 1990sI Such models do a reasonable job of explaining qualitative

behaviour of freeway trafficI “Phantom” jams emerge as consequence of collective behaviour

I Cellular automata are discrete dynamical systemsI Space, time, and state variables are discrete

I ASEP with open boundaries:I If x1(t) = 0, then with probability α, x1(t + 1) = 1I For each cell i = 1, . . . , L with xi(t) = 1

I If xi+1(t) = 0 then with probability p, xi (t + 1) = 0 and xi+1(t + 1) = 1I Else xi (t + 1) = 1

I If xL(t) = 1, then with probability β, xL(t + 1) = 0


Nagel-Schreckenberg process

I NaSch generalizes ASEP

I Vehicles can have different speeds 0, 1, . . . , vmax

I Let xn and vn denote the position & speed of the nth vehicleI Let dn denote the gap in front of the nth vehicleI The NaSch rules are as follows:

I vn 7→ min(vn + 1, vmax)I vn 7→ min(vn, dn)I vn 7→ max(vn − 1, 0) with probability pI xn 7→ xn + vn



I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax

3 1 2






3 1 2

I Let xn and vn denote the position & speed of the nth vehicle

I Let dn denote the gap in front of the nth vehicleI The NaSch rules are as follows:





3 1 2

I Let xn and vn denote the position & speed of the nth vehicleI Let dn denote the gap in front of the nth vehicle

I The NaSch rules are as follows:





3 1 2






3 1 2


I vn 7→ min(vn + 1, vmax)

I vn 7→ min(vn, dn)I vn 7→ max(vn − 1, 0) with probability pI xn 7→ xn + vn




3 1 2


I vn 7→ min(vn + 1, vmax)I vn 7→ min(vn, dn)

I vn 7→ max(vn − 1, 0) with probability pI xn 7→ xn + vn




3 1 2


I vn 7→ min(vn + 1, vmax)I vn 7→ min(vn, dn)I vn 7→ max(vn − 1, 0) with probability p

I xn 7→ xn + vn




3 1 2






1 2 3






2 3 3




NetNaSch modelGoal: Minimal stat-mech model that can mimic realistic traffic signals

I Take multiple NaSch models and glue them together

α1,β1

α2,β2

α3,β3 α4,β4

α5,β5

α6,β6

α7,β7α8,β8

γ1 δ1

γ2δ2

γ3δ3

γ4δ4

p1

p2 p3

p4

I Need to include:I Multiple lanes with lane changingI Turning decisions (random)I Input and output

(endogenous/exogenous)I Appropriate rules for how vehicles

traverse intersections

Varying all the αλ, βλ, γλ, δλ,pn... cannot give an MFDVarying a lower-dimensional space of parameters can


Static demand – Approach to StationarityGenerate MFD by setting αλ = α, βλ = β, γλ = δλ = 0 for all λ ∈ Λ

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network density

netw

ork

flow

SCATS−L: isotropic and time independent rates (γ=0, δ=0), pT = 0.1

1st hr2nd hr3rd hr4th hr5th hr6th hr

0 0.2 0.4 0.6 0.8 10

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network densityne

twor

k de

nsity

het

erog

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I Intersections governed by model of SCATS with adaptive linkingI Instantaneous MFD converges to stationary curveI Although there is uniform boundary demand, the density

distribution in the network is not homogeneous


Static demand – Stationary MFDs

I Use MFDs to quantify performance of signal systems

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netw

ork

flow

Isotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr

SOTLSCATS−LSCATS−F

Isotropic boundary demand

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network density

netw

ork

flow

Anisotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr


Higher demand on west sideI Anisotropic demand can still produce well-defined MFD




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network density

netw

ork

flow




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network density

netw

ork

flow



Higher demand on west side

I Anisotropic demand can still produce well-defined MFD




0 0.2 0.4 0.6 0.8 10

0.05

0.1

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network density

netw

ork

flow




0 0.2 0.4 0.6 0.8 10

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network density

netw

ork

flow



Higher demand on west sideI Anisotropic demand can still produce well-defined MFD


Self-organizing traffic lights

I SOTL is a toy model of a highly adaptive acyclic signal systemI Always gives green to phase with the highest demand

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network density

netw

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dens

ity h

eter

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eity



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dens

ity h

eter

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SOTL: isotropic and time independent rates (γ=0, δ=0), pT = 0.1


I SOTL has lower heterogeneity than SCATSI Accounts for its better MFD


Time-dependent demandI Vary α, β over 24 hours to mimic am/pm peaksI Hysteresis observed - clockwise and anticlockwise

Buisson & Ladier 2009Empirical data from Toulouse

0 0.2 0.4 0.6 0.8 10

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network density

netw

ork

flow

SOTL: time dependent rates, pT = 0.1(γ = 0, δ = 0)

0 ~ 4hr4 ~ 8hr8 ~ 12hr12 ~ 16hr16 ~ 20hr

Zhang, G & de Gier 2013Simulated data


Time-dependent demand

I Hysteresis in MFD consequence of heterogeneity

0 0.2 0.4 0.6 0.8 10

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0.1

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0.2

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0.35

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0.45

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network density

netw

ork

flow

SOTL: time dependent rates, pT = 0.1(γ = 0, δ = 0)

0 ~ 4hr4 ~ 8hr8 ~ 12hr12 ~ 16hr16 ~ 20hr


0 0.2 0.4 0.6 0.8 10

0.05

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network density

den

sity

het

erog

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ty

SOTL: time dependent rates (γ=0, δ=0), pT = 0.1

0 ~ 4hr4 ~ 8hr8 ~ 12hr12 ~ 16hr16 ~ 20hr



Two-bin modelI Consider two adjacent networks (bins) exchanging vehiclesI Each bin has same well-defined MFD J(ρ)

dρ1

dt=

a1 − b1J(ρ1) + p2J(ρ2)− p1J(ρ1)

L1

dρ2

dt=

a2 − b2J(ρ2) + p1J(ρ1)− p2J(ρ2)

L2

I Let bin 1 be boundary layer, bin 2 the interior

Loading Recovery


Two-bin modelI Consider two adjacent networks (bins) exchanging vehiclesI Each bin has same well-defined MFD J(ρ)

dρ1

dt=

a1 − b1J(ρ1) + p2J(ρ2)− p1J(ρ1)

L1

dρ2

dt=

a2 − b2J(ρ2) + p1J(ρ1)− p2J(ρ2)

L2

I Let bin 1 be boundary layer, bin 2 the interior

00

ρ1

ρ 2

a1 = 0.1, b

1 = 0, p

1 = 0.08, p

2 = 0.02

ρj

ρj

ρc

ρc

Loading

00

ρ1

ρ 2

a1 = 0, b

1 = 0.1, p

1 = 0.08, p

2 = 0.02

ρj

ρj

ρc

ρc

Recovery

00

a1 = 0.1, b

1 = 0.1, p

1 = 0.08, p

2 = 0.02

Jc

ρj

ρc

Instantaneous MFD


Open Problems

I Can we observe anticlockwise hysteresis empirically?I Can we understand cross-correlations between flow, density and

density heterogeneity?I How does driver adaptivity affect the shape of MFDs?

I How should one partition networks in order to producewell-defined MFDs?

I Several groups are attempting to use MFDs as a basis forperimeter control?


Open Problems

I Can we observe anticlockwise hysteresis empirically?I Can we understand cross-correlations between flow, density and

density heterogeneity?I How does driver adaptivity affect the shape of MFDs?I How should one partition networks in order to produce

well-defined MFDs?I Several groups are attempting to use MFDs as a basis for

perimeter control?

Details of the model

Paths

Consider a particular node n in a traffic network

n

mn

nm′

DefinitionA path P is an ordered pair of lanes(λ, λ′) with λ ∈ mn and λ′ ∈ nm′

I Vehicles can only move from onelink to another along paths

I Ignore the actual dynamicsthrough the intersection

I No cells in the intersection – weuse paths to glue the CA onadjacent links together


Paths

Consider a particular node n in a traffic network

n

mn

nm′

P

λ

λ′ DefinitionA path P is an ordered pair of lanes(λ, λ′) with λ ∈ mn and λ′ ∈ nm′

I Vehicles can only move from onelink to another along paths

I Ignore the actual dynamicsthrough the intersection

I No cells in the intersection – weuse paths to glue the CA onadjacent links together


Phases

We can’t simply let all paths be traversed at once – vehicles wouldcrash inside the intersection

nP1

P2 P3

P4

P5

P6P7

P8

DefinitionA phase P of node n is a subset of thepaths belonging to n

I At each instant node n has acurrent phase Pcurrent

I Only paths in Pcurrent may betraversed

I Implement traffic signals usingphases

I Time t :Pcurrent = P1 = {P1, . . . ,P8}


Phases

We can’t simply let all paths be traversed at once – vehicles wouldcrash inside the intersection

n

P9

P10

P11

P12P13

P14P15

P16

DefinitionA phase P of node n is a subset of thepaths belonging to n

I At each instant node n has acurrent phase Pcurrent

I Only paths in Pcurrent may betraversed

I Implement traffic signals usingphases

I Time t + ∆t :Pcurrent = P2 = {P9, . . . ,P16}


Lane changing (dynamic)

In order to model freeways or urban networks we need multiple lanesand lane changing

2 3

2 3

I If min(vn + 1,d (f )n , vmax) > min(vn + 1,dn, vmax) the lane change

is desirableI If d (b)

n ≥ v (b)n the lane change is safe

I If desirable and safe accept with probability pchange

I Allow only left→right (right→left) at odd (even) time steps




2 3

dn

2 3

d (f )n

vn


is desirable

I If d (b)n ≥ v (b)

n the lane change is safeI If desirable and safe accept with probability pchange





2 3

dn

2

d (b)n

3

d (f )n

vn

v (b)n









2

3

dn

2

d (b)n

3

d (f )nv (b)

n







Lane changing (topological)

l′l′′ l′′′l

l ′

l ′′

l ′′′

I Red car:not needednot allowed

I Blue car:not neededis allowed

I Green car:needed

I Each vehicle wants to be in a lane for which there exists a pathconsistent with its desired turn

I Only allow dynamical lane changing if it doesn’t contradicttopological lane changing – only blue car can


Boundaries

I We must consider open systemsI So some links only have one endpoint in the network

I Do not model traffic flow on boundary links

I Each boundary lane λ has a fixed average density ρλI This is a boundary condition


Boundaries


I Do not model traffic flow on boundary linksI Each boundary lane λ has a fixed average density ρλ

I This is a boundary condition


Boundaries


I Do not model traffic flow on boundary linksI Each boundary lane λ has a fixed average density ρλI This is a boundary condition


Turning decisions

I Each vehicle should know which link it wants to turn into when itreaches the end of its current link

I In this sense the model should be agent-basedI A sophisticated approach would use origin-destination data and

route planning algorithmsI We take a simple approachI For each node n, inlink l = mn, & outlink l ′ = nm′, we input

P(l → l ′) = P(vehicle on link l wants to turn into link l ′)I Turning decision made when vehicle first enters a linkI Turning decisions affect lane changing dynamics


Turning decisions


I In this sense the model should be agent-based

I A sophisticated approach would use origin-destination data androute planning algorithms

I We take a simple approachI For each node n, inlink l = mn, & outlink l ′ = nm′, we input



Turning decisions



route planning algorithms

I We take a simple approachI For each node n, inlink l = mn, & outlink l ′ = nm′, we input



Turning decisions



route planning algorithmsI We take a simple approach

I For each node n, inlink l = mn, & outlink l ′ = nm′, we inputP(l → l ′) = P(vehicle on link l wants to turn into link l ′)

I Turning decision made when vehicle first enters a linkI Turning decisions affect lane changing dynamics


Turning decisions




P(l → l ′) = P(vehicle on link l wants to turn into link l ′)

I Turning decision made when vehicle first enters a linkI Turning decisions affect lane changing dynamics


Turning decisions




P(l → l ′) = P(vehicle on link l wants to turn into link l ′)I Turning decision made when vehicle first enters a link

I Turning decisions affect lane changing dynamics


Turning decisions






Mark paths

l′,3l

λl ′

I Consider each lane λ of each link l

I Let v be the last vehicle on λI Suppose x(v) + v(v) > length(λ)

I If there exists P ∈ Pcurrent with:I inlane(P) = λI outlane(P) has unoccupied first cellI outlink(P) = turn(v)

I Then associate v↔ P (in this case we say P is marked)I Else stop v at the end of λ


Mark paths

l′,3l

λl ′

I Consider each lane λ of each link lI Let v be the last vehicle on λ

I Suppose x(v) + v(v) > length(λ)




Mark paths

l′,3l

λl ′

I Consider each lane λ of each link lI Let v be the last vehicle on λI Suppose x(v) + v(v) > length(λ)




Mark paths

l′,3l

λl ′


I If there exists P ∈ Pcurrent with:

I inlane(P) = λI outlane(P) has unoccupied first cellI outlink(P) = turn(v)



Mark paths

l′,3l

λl ′


I If there exists P ∈ Pcurrent with:I inlane(P) = λ

I outlane(P) has unoccupied first cellI outlink(P) = turn(v)



Mark paths

l′,3l

λl ′


I If there exists P ∈ Pcurrent with:I inlane(P) = λI outlane(P) has unoccupied first cell

I outlink(P) = turn(v)I Then associate v↔ P (in this case we say P is marked)I Else stop v at the end of λ


Mark paths

l′,3l

λl ′





Mark paths

l′,3l

λl ′



I Then associate v↔ P (in this case we say P is marked)

I Else stop v at the end of λ


Mark paths

l′,3l

λl ′





Clear pathsConsider each marked path P of each node n

I If P must give way to another marked path P ′ of n

I Stop the vehicle v↔ P on the last cell of inlane(P)

I Else move the vehicle v↔ P to the first cell of outlane(P)


Clear pathsConsider each marked path P of each node n

I If P must give way to another marked path P ′ of nI Stop the vehicle v↔ P on the last cell of inlane(P)

I Else move the vehicle v↔ P to the first cell of outlane(P)

macroscopic fundamental diagrams of arterial road networks ...mpetn/files/talks/garoni.pdf ·...

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