macroscopic fundamental diagrams of arterial road networks ...mpetn/files/talks/garoni.pdf ·...
TRANSCRIPT
![Page 1: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/1.jpg)
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
![Page 2: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/2.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
![Page 3: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/3.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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)
![Page 4: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/4.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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)
![Page 5: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/5.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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 dimension
I What should happen in a network?I How should one even define network flow?
(No prescribed direction)
![Page 6: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/6.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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)
![Page 7: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/7.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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)
![Page 8: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/8.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 9: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/9.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 10: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/10.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
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
![Page 11: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/11.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
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 demand
I 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
![Page 12: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/12.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
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
![Page 13: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/13.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 14: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/14.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 15: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/15.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 16: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/16.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 17: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/17.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 18: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/18.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 19: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/19.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 20: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/20.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 21: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/21.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 22: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/22.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 23: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/23.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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 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
![Page 24: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/24.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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:
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
![Page 25: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/25.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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:
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
![Page 26: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/26.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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 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
![Page 27: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/27.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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 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
![Page 28: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/28.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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 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
![Page 29: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/29.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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 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 p
I xn 7→ xn + vn
![Page 30: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/30.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
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 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
![Page 31: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/31.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
1 2 3
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
![Page 32: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/32.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
1 2 3
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
![Page 33: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/33.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
2 3 3
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
![Page 34: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/34.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Nagel-Schreckenberg process
I NaSch generalizes ASEPI Vehicles can have different speeds 0, 1, . . . , vmax
2 3 3
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
![Page 35: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/35.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 36: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/36.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Static demand – Approach to StationarityGenerate MFD by setting αλ = α, βλ = β, γλ = δλ = 0 for all λ ∈ Λ
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
network densityne
twor
k de
nsity
het
erog
enei
ty
SCATS−L: isotropic and time independent rates (γ=0, δ=0), pT = 0.1
1st hr2nd hr3rd hr4th hr5th hr6th hr
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
![Page 37: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/37.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Static demand – Stationary MFDs
I Use MFDs to quantify performance of signal systems
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
network density
netw
ork
flow
Isotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr
SOTLSCATS−LSCATS−F
Isotropic boundary demand
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
network density
netw
ork
flow
Anisotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr
SOTLSCATS−LSCATS−F
Higher demand on west sideI Anisotropic demand can still produce well-defined MFD
![Page 38: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/38.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Static demand – Stationary MFDs
I Use MFDs to quantify performance of signal systems
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
network density
netw
ork
flow
Isotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr
SOTLSCATS−LSCATS−F
Isotropic boundary demand
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
network density
netw
ork
flow
Anisotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr
SOTLSCATS−LSCATS−F
Higher demand on west side
I Anisotropic demand can still produce well-defined MFD
![Page 39: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/39.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Static demand – Stationary MFDs
I Use MFDs to quantify performance of signal systems
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
network density
netw
ork
flow
Isotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr
SOTLSCATS−LSCATS−F
Isotropic boundary demand
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
network density
netw
ork
flow
Anisotropic and time independent rates (γ=0, δ=0) , pT = 0.1 at 6 hr
SOTLSCATS−LSCATS−F
Higher demand on west sideI Anisotropic demand can still produce well-defined MFD
![Page 40: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/40.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
network density
netw
ork
dens
ity h
eter
ogen
eity
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
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
network density
netw
ork
dens
ity h
eter
ogen
eity
SOTL: isotropic and time independent rates (γ=0, δ=0), pT = 0.1
1st hr2nd hr3rd hr4th hr5th hr6th hr
I SOTL has lower heterogeneity than SCATSI Accounts for its better MFD
![Page 41: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/41.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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
![Page 42: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/42.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
Time-dependent demand
I Hysteresis in MFD consequence of heterogeneity
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
network density
den
sity
het
erog
enei
ty
SOTL: time dependent rates (γ=0, δ=0), pT = 0.1
0 ~ 4hr4 ~ 8hr8 ~ 12hr12 ~ 16hr16 ~ 20hr
Zhang, G & de Gier 2013Simulated data
![Page 43: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/43.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 44: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/44.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 45: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/45.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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
![Page 46: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/46.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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?
![Page 47: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/47.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
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?
![Page 48: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/48.jpg)
Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin model Open Problems
![Page 49: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/49.jpg)
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
![Page 50: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/50.jpg)
Details of the model
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
![Page 51: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/51.jpg)
Details of the model
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
![Page 52: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/52.jpg)
Details of the model
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
![Page 53: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/53.jpg)
Details of the model
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
![Page 54: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/54.jpg)
Details of the model
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}
![Page 55: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/55.jpg)
Details of the model
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}
![Page 56: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/56.jpg)
Details of the model
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}
![Page 57: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/57.jpg)
Details of the model
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}
![Page 58: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/58.jpg)
Details of the model
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}
![Page 59: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/59.jpg)
Details of the model
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}
![Page 60: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/60.jpg)
Details of the model
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}
![Page 61: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/61.jpg)
Details of the model
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
![Page 62: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/62.jpg)
Details of the model
Lane changing (dynamic)
In order to model freeways or urban networks we need multiple lanesand lane changing
2 3
dn
2 3
d (f )n
vn
I If min(vn + 1,d (f )n , vmax) > min(vn + 1,dn, vmax) the lane change
is desirable
I If d (b)n ≥ v (b)
n the lane change is safeI If desirable and safe accept with probability pchange
I Allow only left→right (right→left) at odd (even) time steps
![Page 63: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/63.jpg)
Details of the model
Lane changing (dynamic)
In order to model freeways or urban networks we need multiple lanesand lane changing
2 3
dn
2
d (b)n
3
d (f )n
vn
v (b)n
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
![Page 64: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/64.jpg)
Details of the model
Lane changing (dynamic)
In order to model freeways or urban networks we need multiple lanesand lane changing
2 3
dn
2
d (b)n
3
d (f )n
vn
v (b)n
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
![Page 65: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/65.jpg)
Details of the model
Lane changing (dynamic)
In order to model freeways or urban networks we need multiple lanesand lane changing
2
3
dn
2
d (b)n
3
d (f )nv (b)
n
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
![Page 66: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/66.jpg)
Details of the model
Lane changing (dynamic)
In order to model freeways or urban networks we need multiple lanesand lane changing
2
3
dn
2
d (b)n
3
d (f )nv (b)
n
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
![Page 67: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/67.jpg)
Details of the model
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
![Page 68: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/68.jpg)
Details of the model
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
![Page 69: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/69.jpg)
Details of the model
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
![Page 70: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/70.jpg)
Details of the model
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
![Page 71: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/71.jpg)
Details of the model
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
![Page 72: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/72.jpg)
Details of the model
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
![Page 73: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/73.jpg)
Details of the model
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
![Page 74: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/74.jpg)
Details of the model
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
![Page 75: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/75.jpg)
Details of the model
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 linksI Each boundary lane λ has a fixed average density ρλ
I This is a boundary condition
![Page 76: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/76.jpg)
Details of the model
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 linksI Each boundary lane λ has a fixed average density ρλI This is a boundary condition
![Page 77: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/77.jpg)
Details of the model
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
![Page 78: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/78.jpg)
Details of the model
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-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
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
![Page 79: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/79.jpg)
Details of the model
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 algorithms
I 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
![Page 80: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/80.jpg)
Details of the model
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 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
![Page 81: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/81.jpg)
Details of the model
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
![Page 82: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/82.jpg)
Details of the model
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 link
I Turning decisions affect lane changing dynamics
![Page 83: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/83.jpg)
Details of the model
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
![Page 84: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/84.jpg)
Details of the model
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
![Page 85: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/85.jpg)
Details of the model
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)
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
![Page 86: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/86.jpg)
Details of the model
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)
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
![Page 87: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/87.jpg)
Details of the model
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)
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
![Page 88: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/88.jpg)
Details of the model
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)
I If there exists P ∈ Pcurrent with:I inlane(P) = λ
I 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
![Page 89: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/89.jpg)
Details of the model
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)
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
![Page 90: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/90.jpg)
Details of the model
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)
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
![Page 91: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/91.jpg)
Details of the model
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)
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
![Page 92: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/92.jpg)
Details of the model
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)
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
![Page 93: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/93.jpg)
Details of the model
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)
![Page 94: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/94.jpg)
Details of the model
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)
![Page 95: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/95.jpg)
Details of the model
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)
![Page 96: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/96.jpg)
Details of the model
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)
![Page 97: Macroscopic Fundamental Diagrams of arterial road networks ...mpetn/files/talks/Garoni.pdf · Fundamental Diagrams Exclusion Processes Simulations Hysteresis & the 2-bin modelOpen](https://reader035.vdocuments.mx/reader035/viewer/2022070111/604d0d5f74befc22074bb433/html5/thumbnails/97.jpg)
Details of the model
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)