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.
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
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)
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
Skim comparison statistics
Iteration vs. PreceedingIteration vs. Equilibrium
Used paths vs. Shortest path = Relative Gap
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
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
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
Skim error study
Well-converged assignment
Less-converged assignment
Skim Skim
Poorly-converged assignment
Skim
Comparison statistics
Comparison statistics
Trip Table & Network
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
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
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
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
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
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
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
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
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
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
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
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
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!
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
Contact
John Gibb
jag (at) dksassociates (dot) com
Extra slides
Example of spurious flow change
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