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How much convergence is enough for traffic assignments used in feedback?
John Gibb
DKS Associates
For the 14th TRB National Transportation Planning Applications Conference
*Annotated version* - see notes.
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Partners, Sources, Assistance• Sacramento Area Council of Governments• Puget Sound Regional Council• John Bowman, Mark Bradley, RSG• Carson Area Metropolitan Planning Organization• Spokane Regional Transportation Council• Citilabs• PTV America• Caliper• Inro Consultants
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Convergence of traffic assignments: How much is enough?
David Boyce
Biljana Ralevic-Dekic
Hillel Bar-Gera
• Link flow stability – avoid random noise• Relative gap → 0.0001
ASCE Journal of Transportation Engineering 130, 49-55 (2004)
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Skims
Assignment
Link volumes
Demand model
Skims
Assignment
Demand model
Skims
Assignment
Demand model
Skims
Assignment
Demand model
Link timesLink timesLink times
The rest of the assignments
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Skim comparison statistics
Iteration vs. PreceedingIteration vs. Equilibrium
Used paths vs. Shortest path = Relative Gap
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Convergence Progress of a Trip-Based Model – AM Skim Change
2 3 4 5 6 7 8 9 10 11 12 13 14 150.0001
0.001
0.01
0.1
1
RG≈0.05
RG≈0.01
RG≈0.003
RG≈0.001
RG≈0.0004
RG≈0.0002
Iteration (demand model)
Re
lati
ve
Av
era
ge
Sk
im C
ha
ng
e
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Convergence Progress of a Trip-Based Model – AM Displaced Trips
2 3 4 5 6 7 8 9 10 11 12 13 14 150.0001
0.001
0.01
0.1
1
RG≈0.05
RG≈0.01
RG≈0.003
RG≈0.001
RG≈0.0004
RG≈0.0002
Iteration (demand model)
Re
lati
ve
Dis
pla
ce
d T
rip
s
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Convergence Progress of draft Sacramento Activity-Based Model
2 3 4 5 6 7 8 9 100.0001
0.001
0.01
0.1
1
ev2h05h06h07h08h09h14h15h16h17md4ni9Max RelGapRel Gap Criterion
Demand Model Iteration (during newest skim)
Re
lati
ve
Av
g-A
bs
olu
te S
kim
De
lta
fro
m
pre
vio
us
ite
rati
on
;R
ela
tiv
e G
ap
Period of Day
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Skim error study
Well-converged assignment
Less-converged assignment
Skim Skim
Poorly-converged assignment
Skim
Comparison statistics
Comparison statistics
Trip Table & Network
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Skim comparison:unconverged vs. best equilibrium
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
RG 0.016RG 0.0012
Best Equilibrium Skim (min)
Sk
im a
t in
dic
ate
d R
ela
tiv
e G
ap
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Skim Error v. Relative Gap: Sacramento AM (F-W)
0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
r
Average-Absolute
Extreme
RMS
convergence
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0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
rSkim Error v. Relative Gap: Sacramento Mid-Day
Extreme
RMS
Average-Absoluteconvergence
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0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
rSkim Error v. Relative Gap: Reduced congestion (AM)
Extreme
RMS
Average-Absoluteconvergence
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Skim Error v. Relative Gap: Increased congestion (AM)
Extreme
RMS
Average-Absolute
0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
r
convergence
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Skim Error v. Relative Gap: BPR^8(AM)
Extreme
RMS
Average-Absolute
0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
r
convergence
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Skim Error v. Relative Gap: Visum (Spokane)
Extreme
RMS
Average-Absolute
0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
r
convergence
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Skim Error v. Relative Gap: TransCAD (Carson City)
Extreme
RMS
Average-Absolute
0.000001 0.0001 0.01 10.000001
0.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
r
convergence
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Convergence Progress of a Trip-Based Model – AM Average Skim Change
2 3 4 5 6 7 8 9 10 11 12 13 14 150.0001
0.001
0.01
0.1
1
RG≈0.05
RG≈0.01
RG≈0.003
RG≈0.001
RG≈0.0004
RG≈0.0002
Iteration (demand model)
Re
lati
ve
Sk
im C
ha
ng
e
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0.0001 0.001 0.01 0.1 10.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
rSkim Error, AM changes v. Relative Gap
Extreme
RMS
Average-Absolute
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Convergence Progress of a Trip-Based Model – Mid-Day Avg. Skim Change
2 3 4 5 6 7 8 9 10 11 12 13 14 150.00001
0.0001
0.001
0.01
0.1
1
AM 0.05AM 0.01AM 0.003AM 0.001AM 0.0004AM 0.0002RG≈0.05RG≈0.01RG≈0.003RG≈0.001RG≈0.0004RG≈0.0002
Iteration
Re
lati
ve
Sk
im C
ha
ng
e
AM
Mid-Day
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0.00001 0.0001 0.001 0.01 0.1 10.00001
0.0001
0.001
0.01
0.1
1
Relative Gap (posterior)
Re
lati
ve
Sk
im E
rro
rSkim Error, MD changes v. Relative Gap
Extreme
RMS
Average-Absolute
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Conclusions• Relative Gap seems to imply relative skim errors.• Successive skim differences can be misleading – less than skim error implied by RG – esp. in low congestion.
• Choose Relative Gaps to create small skim errors compared to skim changes. • Low RGs don’t seem to accelerate demand convergence,
but high RGs limit it.• Doing so should avoid the misleading-differences problem.
• This is a small study of a few models. Test your own!
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Convergence Progress of draft Sacramento Activity-Based Model
2 3 4 5 6 7 8 9 100.0001
0.001
0.01
0.1
1 ev2
h05
h06
h07
h08
h09
h14
h15
h16
h17
md4
ni9
Max RelGap
Rel Gap Criterion
Expected ErrorDemand Model Iteration (during newest skim)
Re
lati
ve
Ch
an
ge
, E
rro
r, G
ap
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Contact
John Gibb
jag (at) dksassociates (dot) com
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Extra slides
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Example of spurious flow change
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Skim comparison statisticsRelative Gap
(links)
(skims)
= link cost = link flow = virtual shortest-path flow
= demand (trip table) = used-paths average travel time = shortest-path skim travel time
Average Absolute(trip-weighted)
= equilibrium travel time
RMS(trip-weighted)
Max Absolute