A STRATEGIC USER EQUILIBRIUM FOR INDEPENDENTLY
DISTRIBUTED ORIGIN-DESTINATION DEMANDS
1) It accounts for demand uncertainty in users’
routing mechanism.
2) Network performance measures are provided
analytically, therefore they can be computed
directly from model results.
3) The uniqueness of link flow distribution and
the strategic link choice are guaranteed, which
ensures its applicability in transportation planning
process.
4) A solution algorithm is proposed to compute
the link flow
parameter and
the strategic
link choice.
Tao Wen, Chen Cai, Lauren Gardner, Vinayak Dixit, S. Travis Waller, Fang Chen
The demand volatility
can be represented as a
distribution curve.
Demand volatility
Demand volatility
Probability
Highlights of the model
Demand
What is the Strategic User
Equilibrium model?
Model assumptions
The strategic user equilibrium is defined such
that the expected travel costs are equal on all
used paths, and this commonly expected travel
time is less than the actual expected travel time
on any unused path.
In other words, given user equilibrium expected
path cost, any deviation from the existing
expected path flows cannot reduce the expected
path cost.
Users have the experiences (or are informed) of
the demand distribution, but not the realization on
any given day. Therefore, traditional equilibrium
may not be observed in this model
Two statistical assumptions are made to
ensure a unique solution:
1. The actual OD demand on any given day
is independently distributed and follows a
Poisson distribution.
2. Conditional on the realized demand on any
given day, each user (driver) is assumed to
choose independently between the alternative
paths with a fixed strategic path choice.
Numerical results
link 10
-3
8
x 10
The monte-Carlo
6
simulated results are
4
4
plotted as the green bar
2
charts, the black curve
2
represents the standard
0
0
7700
7900
8100
8300
8500
4800
5000
5200
5400
5600
Poisson distribution with
Flow on link
Flow on link
Simulated
link 31
link 63
the
expected
link
flows
Standard x 10
x 10
8
6
derived from the strategic
6
4
user equilibrium. the
4
standard Poisson
2
2
distribution curve is well
approximated
by
the
0
0
5000
5200
5400
5600
5800
6600
6800
7000
7200
7400
Flow on link
Flow on link
simulated results. This has
Probability distribution of flow on different links
FIGURE2. The probability distribution of flow on Links 5, 10, 31 and 63. validated Assumption 2.
Probability
Probability
x 10
-3
Probability
-3
Probability
Monte Carlo simulation:
i. 100 randomly selected demand scenarios
are sampled from the given demand
distribution.
ii. For each realized demand scenario, 100
sets of multinomially distributed path
flows are sampled based on the strategic
path choices.
iii. As a result, we have 10000 sets of
sampled path flows and 10000 sets of
corresponding sampled link flows, which
represents the unconditional distributions
of link flow. These results are indicated
as simulated results, and those ones
computed from the analytical equations
will be indicated as estimated results.
link 5
-3
6
Implementation algorithms
the standard Poisson distribution curve is well approximated by the
simulated results
In figures 3 and 4:
X axis- Monte-Carlo
simulated results
Y axis- Analytically
computed mean and standard
deviation of link travel time.
Rsquared=0.973
20
0.6
18
0.5
16
Simulated StdTT
Analysis is conducted on this
network.
24 nodes and 76 links.
The BPR parameters α and β
are taken to be 0.15 and 4,
respectively.
Frank-Wolfe relative gap is set
to 1*e-5.
Simulated vs estimated StdTT
0.7
Rsquared=0.993
Simulated ETT
FIGURE1. The Sioux Falls network.
Simulated vs estimated ETT
22
14
12
10
8
0.4
0.3
0.2
6
0.1
4
2
2
4
6
8
10
12
14
16
18
20
Estimated ETT
FIGURE 3. Linear regression
analysis of expected link
travel time.
Possible future research
• Future research will investigate the use of the covariance of loop counts.
• Treat demand proportions as variables and calibrate them in the framework
simultaneously.
Limitations of the model
The assumption of Poisson distributed OD demands forces the expected demand to be equal to variance
of demand, and this assumption may limit the applicability of the model.
22
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
The analytically computed
FIGURE 4. Linear regression results are well approximated
analysis of standard
by simulated results.
Estimated StdTT
deviation of link travel time.
Thank you for your attention!