Traffic
Effective Route Guidance
in Traffic Networks
Forecast: Number of cars will increase further
Fact:
Infrastructure will not be enhanced to the same extent
Remedy:
Improve the efficiency of traffic by other means
Lectures developed by
Andreas S. Schulz and Nicolás Stier
May 12, 2003
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2002 Urban Mobility Study
Outline
(http://mobility.tamu.edu/ums)
• Lecture 1
“The bad news is that even if transportation officials do all the right
things, the likely effect is that congestion will continue to grow. . . ”
Route Guidance; User Equilibrium; System
Optimum; User Equilibria in Networks with
Capacities.
• Total congestion “bill” in 2000 was $67.5 billion
• Lecture 2
(= 3.6 billion hours delay + 5.7 billion gallons gas)
time penalty for peak period travelers
1982
2000
16 hours
62 hours
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Constrained System Optimum; Dantzig-Wolfe
Decomposition; Constrained Shortest Paths;
Computational Results.
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Problem
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The Context
• Olaf Jahn (Research Assistant).
• Rolf H. Möhring (Principal Investigator).
People travel (between 6% and 19%) too much because
Collaboration with and support by DaimlerChrysler, Berlin.
of an unfavorable selection of their route.
(Beccaria & Bolelli 1992, Lösch 1995)
• Nicolas Stier (Research Assistant).
• Andreas S. Schulz (Principal Investigator).
Supported by General Motors Innovation Grant and SMA.
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Shortest Path Routing
Potential Remedies
• Toll systems
• Dynamic traffic signal control
• Park and Ride
• Traveller information systems
Improved network performance, but . . .
(Kaufman et al. 1991, Lee 1994)
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Route Guidance
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Shortest Path Routing II
Route Guidance
. . . the same simulations show the performance decreases
as soon as many cars use the system.
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Route Guidance
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Proposed Solutions
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Route Guidance
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In-Car Navigation Systems
• Multiple path routing:
– k shortest paths
– random perturbation
• Feedback control:
– iterative computation of shortest paths
• Traffic assignment:
– user equilibrium
– system optimum
– a new approach
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Route Guidance
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Route Guidance
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Modeling Assumptions
Reality
Our Model
• microscopic
→ individual vehicles
→ exact position, speed
• macroscopic
→ one abstract measure
→ traffic flow
• dynamic
→ consider time
→ on a single point at
any time
• static
→ time independent
→ simultaneously at any
point of the path
• on-line
→ additional input over time
• off-line
→ all data known in advance
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The Traffic Model
selfish users
central planner
the goal
optimize own travel time optimize system welfare
fair, not efficient
efficient, not fair
fair, efficient
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Representation of the Road Network
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Route Guidance
How much can one gain ?
• Study worst-case ratios between guided / unguided traffic
• Without guidance: use game theory to predict traffic
(Wardrop 1952)
• Users’ behavior modeled as user equilibrium (Nash eq.)
• Price of anarchy is a measure of user equilibrium performance
(Papadimitriou 2001)
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The Traffic Model
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Route Guidance
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Flows
• Different drivers have different
origins and destinations
The Traffic Model
• Flows on paths:
fP is the traffic along path P
• Flow on arcs:
fa =
�
fP
P :a∈P
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The Traffic Model
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The Traffic Model
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The Traffic Model
Traversal Time Functions
ta(fa)
• Directed graph G = (V, A),
k demands (oi, di) with rate ri
2
2
0
• Flows on paths fP . Can be non-integral.
• Traversal times: latency functions ta(·)
→ continuous and nondecreasing
→ belong to a given set L (e.g. linear)
0
• The total travel time of a flow is:
�
ta(fa)fa
C(f ) :=
x+1
1
2x
2
0
0
2
• Traversal time of an arc a depends on the flow fa on a
• Dependence captured by the function ta(fa)
2
2 · 4 + 2 · 1 = 10
• Travel time along path P is denoted by tP (f ) =
a∈A
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fa
0
2
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The Traffic Model
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System Optimum
�
a∈P ta(fa)
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The Traffic Model
The Traffic Model
Convex Multicommodity Min-Cost Flow Problem
�
min
• Directed graph G = (V, A),
k demands (oi, di) with rate ri
ta(fa)fa
2
2
a∈A
�
s.t.
for all a ∈ A
fP = ri
for all i = 1, . . . , k
fP ≥ 0
for all P ∈ P
P a
�
• Traversal times: latency functions ta(·)
→ continuous and nondecreasing
→ belong to a given set L (e.g. linear)
P ∈Pi
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The Traffic Model
0
2
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2
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The Traffic Model
(Pigou 1920)
• Directed graph G = (V, A),
k demands (oi, di) with rate ri
1
1
=min fb2 + 1 − fb = 3/4
s.t. 0 ≤ fb ≤ 1
2
2
0
• Flows on paths fP . Can be non-integral.
• Traversal times: latency functions ta(·)
→ continuous and nondecreasing
→ belong to a given set L (e.g. linear)
x
and
fa =
fb =
1
2
1
2
0
x+1
1
2x
• The total travel time of a flow is:
�
ta(fa)fa
C(f ) :=
fa, fb ≥ 0
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0
The Traffic Model
1
s.t. fa + fb = 1
x+1
a∈A
Example of SO
SO =min fa + fb2
0
1
2x
• The total travel time of a flow is:
�
ta(fa)fa
C(f ) :=
Pi : set of paths from oi to di
P = ∪Pi
where
0
• Flows on paths fP . Can be non-integral.
fP = fa
2
2
0
0
2
2
2·1+2·3=8
a∈A
The Traffic Model
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The Traffic Model
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Braess Paradox
• UE non-monotone with network improvements
Example of SO
1
(Braess 1968)
(Pigou 1920)
1/2
1
1
x
1
1
1
1
t
1/2
c
x
x
SO =min fa + fb2
s.t. fa + fb = 1
=min fb2 + 1 − fb = 3/4
s.t. 0 ≤ fb ≤ 1
and
fa =
fb =
1
2
1
2
fa, fb ≥ 0
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The Traffic Model
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Braess Paradox
• UE non-monotone with network improvements
1
1/2
1
1
User Equilibrium
(Braess 1968)
Definition : A flow is a UE iff nobody can switch
to a path with smaller travel time.
• Travel times of users between the same OD-pair are equal
1
x
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The Traffic Model
1
1/2
x
3
1 C(f ) = 2
• UE always exists and is unique
(Beckmann et al. 1956)
1
1
1
x
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The Traffic Model
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Braess Paradox
• UE non-monotone with network improvements
1
1/2
1
1
(Braess 1968)
3
1 C(f ) = 2
Definition : A flow is a UE iff nobody can switch
to a path with smaller travel time.
• UE always exists and is unique
1
1
1
1
x
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(Beckmann et al. 1956)
1
1
0
1
1
1/2
x
x
1
User Equilibrium
• Travel times of users between the same OD-pair are equal
1
x
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The Traffic Model
x
The Traffic Model
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The Traffic Model
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Networks with Capacities
Braess Paradox
• UE non-monotone with network improvements
• Latencies model capacity only implicitly
2
What is the impact of having
explicit capacities on arcs?
0
1
x
0
1
0 cap=2 x
Mx
• Introduce capacities as
hard constraints
0
• Straightforward to define SO
1
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Networks with Capacities
Equilibria in Networks with Capacities
1/2
1
0
1
1
1
1
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Mx
• How good is the best / worst eq. ?
(Papadimitriou 2001)
Price of Anarchy γ := max
2
inst.
• In general, γ unbounded
0
0
0
cap
=2
0
0
2
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The Traffic Model
Price of Anarchy measures impact of
lack of Central Coordination
2
• There may be multiple equilibria
(UE w/o cap. was unique)
1 C(f ) = 2
x
Definition : A flow is a capacitated UE iff nobody can switch
to a path with smaller travel time and residual capacity
• Travel times for users of same demand
may differ (were constant w/o cap.)
3
1 C(f ) = 2
1
x
x
2
1/2
1
1
0
2
• What is now a UE?
(Braess 1968)
2
C(UE)
C(SO)
(Roughgarden & Tardos 2000)
• If latencies are in L, γ ≤ α(L), where α(L) depends only on L
x
In particular, α({linear latencies}) = 4/3
(Roughgarden & Tardos 2000)
(Roughgarden 2002)
• Pigou’s and Braess’ examples are worst possible
2
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Networks with Capacities
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The Traffic Model
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Multiple Equilibria
2
2
2
0
0
2
2
2
2
0
0
Mx
2
2
x
cap=2
2
2
2
1+M
2
2M
1+M
2
1+M
2M
1+M
Networks with Capacities
0
2
2
2
2
4
with costs of:
2
2
4M
2
2
4
M
1+M
Worst UE can be unbounded!
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Networks with Capacities
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Networks with Capacities
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Network with Capacities: Guarantees
Convex Optimization Review
Theorem. For any instance of a network with capacities
with latencies in L, we have
• Let z be a continuously differentiable and convex function on a
convex set.
C(BUE) ≤ α(L) C(SO)
• Then x∗ is a global minimum of z iff
In particular, if latencies are linear, C(BUE) ≤ 43 C(SO)
the gradient along all feasible directions is non-negative
Guarantee does not change with
introduction of capacities
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Networks with Capacities
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Beckmann UE
Proof (for the Linear Case)
• Assume ta(fa) = qafa + ra for all a and let f = BUE
�
• Let C f (x) = a xa ta(fa)
• Space of UE non-convex: Difficult to get Best UE
• Instead, Beckmann UE (BUE) is
• For all flows x: C(f ) ≤ C f (x)
�
(same as a(xa − fa)ta(fa) ≥ 0, the condition for )
�
�
�
• C f (x) = a xa(qafa + ra) ≤ a xa(qaxa + ra) + 14 a fa2 qa
min
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�
0
fa
ta(x)dx
subject to f feasible flow
capacity constraints
because (x − f /2) ≥ 0
�
�
• Last, a xa(qaxa + ra) + 14 a fa2 qa ≤ C(x) + 14 C(f )
⇒
�
a∈A
2
3
4
• Opt. Cond. BUE: g feasible direction ⇒
�
a gata(fa)
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Networks with Capacities
• Assume ta(fa) = qafa + ra for all a and let f = BUE
�
• Let C f (x) = a xa ta(fa)
Lemma. f is a BUE ⇒ f is a UE
• For all flows x: C(f ) ≤ C f (x)
�
(same as a(xa − fa)ta(fa) ≥ 0, the condition for )
�
�
�
• C f (x) = a xa(qafa + ra) ≤ a xa(qaxa + ra) + 14 a fa2 qa
• Suppose f is not a UE ⇒ ∃ two paths P , Q s.t.
flow can be re-routed from P to Q and tP (f ) > tQ(f )
Proof :
• P and Q define a circulation g which is a feasible direction
because (x − f /2)2 ≥ 0
�
�
• Last, a xa(qaxa + ra) + 14 a fa2 qa ≤ C(x) + 14 C(f )
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•
�
a gata(fa)
<0
⇒
f is not a BUE
But BUE is not necessarily the best equilibrium
C(f ) ≤ C(x)
Networks with Capacities
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Beckmann UE is an Equilibrium
Proof (for the Linear Case)
3
4
≥0
C(f ) ≤ C(x)
Networks with Capacities
⇒
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Networks with Capacities
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Networks with Capacities
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Non-convexity of UE
2
Proof (for the Linear Case)
2
• Assume ta(fa) = qafa + ra for all a and let f = BUE
�
• Let C f (x) = a xa ta(fa)
w
0
1
1
2
0
SO
cap
=2
x
1
0
because (x − f /2)2 ≥ 0
�
�
• Last, a xa(qaxa + ra) + 14 a fa2 qa ≤ C(x) + 14 C(f )
feasible flows
0
0
1
0
2
UE
BUE
w
z
• For all flows x: C(f ) ≤ C f (x)
�
(same as a(xa − fa)ta(fa) ≥ 0, the condition for BUE)
�
�
�
• C f (x) = a xa(qafa + ra) ≤ a xa(qaxa + ra) + 14 a fa2 qa
2
z
⇒
2
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Networks with Capacities
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Best vs. Beckmann
0
C(BUE) = 7
0
1
x+1
2
and
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cap
=1
Networks with Capacities
• For all flows x: C(f ) ≤ C f (x)
�
(same as a(xa − fa)ta(fa) ≥ 0, the condition for BUE)
�
�
�
• C f (x) = a xa(qafa + ra) ≤ a xa(qaxa + ra) + 14 a fa2 qa
x
because (x − f /2)2 ≥ 0
�
�
• Last, a xa(qaxa + ra) + 14 a fa2 qa ≤ C(x) + 14 C(f )
0
2
C(best UE) = C(SO) = 6.875
Networks with Capacities
⇒
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Review
No capacities
With capacities
UE unique
Set of UE
may be non-convex
UE/SO ≥ α(L)
UE/SO unbounded
UE/SO ≤ α(L)
BUE/SO ≤ α(L)
Networks with Capacities
3
4
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C(f ) ≤ C(x)
Networks with Capacities
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Proof (for the Linear Case)
• Assume ta(fa) = qafa + ra for all a and let f = BUE
�
• Let C f (x) = a xa ta(fa)
• For all flows x: C(f ) ≤ C f (x)
�
(same as a(xa − fa)ta(fa) ≥ 0, the condition for BUE)
�
�
�
• C f (x) = a xa(qafa + ra) ≤ a xa(qaxa + ra) + 14 a fa2 qa
because (x − f /2)2 ≥ 0
�
�
• Last, a xa(qaxa + ra) + 14 a fa2 qa ≤ C(x) + 14 C(f )
⇒
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• Assume ta(fa) = qafa + ra for all a and let f = BUE
�
• Let C f (x) = a xa ta(fa)
2
0
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C(f ) ≤ C(x)
Proof (for the Linear Case)
• The BUE does not need to be best UE:
2
3
4
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3
4
C(f ) ≤ C(x)
Networks with Capacities
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