Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010
• 5–9 July, 2010 • Budapest, Hungary
On Robustness Analysis of Large-scale Transportation Networks
Giacomo Como Ketan Savla Daron Acemoglu Munther A. Dahleh Emilio Frazzoli
ISBN 978-963-311-370-7
Abstract —In this paper, we study robustness properties of
transportation networks with respect to its pre-disturbance
equilibrium operating condition and the agents’ response to the
disturbance. We perform the analysis within a dynamical
system framework over a directed acyclic graph between a
single origin-destination pair. The dynamical system is
composed of ordinary differential equations (ODEs), one for
every edge of the graph. Every ODE is a mass balance
equation for the corresponding edge, where the inflow term is a
function of the agents’ route choice behavior and the arrival rate
at the base node of that edge, and the outflow term is function
of the congestion properties of the edge. We consider
disturbances that reduce the maximum flow carrying capacity of
the links and define the margin of stability of the network as the
minimum capacity that needs to be removed from the network
so that the delay on all the edges remain bounded over time.
For a given equilibrium operating condition, we derive upper
bounds on the margin of stability under local information
constraint on the agents’ behavior, and characterize the route
choice functions that yield this bound. We also setup a simple
convex optimization problem to find the most robust operating
condition for the network and determine edge-wise tolls that
yield such an equilibrium operating condition.
I. INTRODUCTION
Social planning for efficient usage of transportation
networks (TNs) is attracting renewed research interest as
transportation demand is fast approaching its
infrastructure capacity. While there exists an abundant
literature on socially optimal traffic assignments, e.g., see
[1], robustness analysis of TNs has received very little
attention. In this paper, we study the relationship of the
robustness properties of a large-scale TN to its
pre-disturbance equilibrium operating condition and the
agents’ response to the disturbance.
We abstract the topology of the transportation network
by a directed acyclic graph between a single
origin-destination pair. For the analysis, we adopt a
dynamical system framework that is composed of
ordinary differential equations (ODEs), one for every
edge of the graph. Every ODE represents a mass
balance equation for the corresponding edge, where the
inflow term is a function of the agents’ route choice
behavior and the arrival rate at the base node of that
edge, and the outflow term is function of the congestion
properties of the edge. We consider a setup where,
before the disturbance, the network is operating at an
equilibrium operating condition and information about this
equilibrium condition is shared by all the agents. Such an
equilibrium condition might be thought of as the outcome
of a slower
G. Como, K. Savla, M. A. Dahleh and E. Frazzoli are with the
Laboratory for Information and Decision Systems at the Massachusetts
Institute of Technology. f giacomo,ksavla,dahleh,frazzoli g @mit.edu.D.
Acemoglu is with the Department of Economics at the Massachusetts
Institute of Technology. daron@mit.edu .
time-scale learning process, e.g., see [2], [3], [4], in
presence of incentive mechanisms such as tolls, e.g.,
see [5], [6]. After the disturbance, we assume that the
global knowledge of the agents remains fixed and that
the agents act by complementing the fixed global
knowledge with real-time local information. Such a
setup is meant is give insight into the evolution of the
network in the immediate aftermath of a disruption
when the availability of accurate global information
about the whole network is sparse or it is too
time-consuming for the agents to incorporate the
real-time information about the whole network
because of the huge computations involved.
We consider disturbances that reduce the maximum
flow carrying capacities of the edges by affecting their
congestion properties. We define the margin of
stability of the TN to be the maximum sum of capacity
losses, under which the traffic densities on all the
edges remain bounded over time. We then prove that,
irrespective of the route choice behavior of the agents,
the margin of stability is upperbounded by the
minimum of all the node cuts of the residual capacities
of the TN. We then characterize the route choice
behaviors that match this upper bound. Finally, we
study the dependence of the margin of stability on the
equilibrium, and formulate a simple optimization
problem for finding the most robust equilibria. This is,
in general, different from the classical socially optimal
equilibrium, as well as from the user-optimal
equilibrium. We also discuss the utility of tolls in
yielding a desired equilibrium operating condition. Our
results provide important guidelines for social
planners in terms of determining robust equilibrium
operating conditions and route choice behaviors for
TNs. Alternate notions of robustness for networks
have been proposed in [7], [8], [9].
The contributions of the paper are as follows: (i) we
formulate a novel dynamical system framework for
robustness analysis of transportation networks, (ii) we
derive an upper bound on the margin of stability of the
network and characterize the features of the agents’
route choice behavior under which this bound is tight,
and (iii) we postulate the notion of robustness price of
anarchy to quantify the loss in robustness due to
sub-optimal equilibrium operating condition of a
network and discuss the use of tolls in removing this
gap.
This technical results of this paper rely on tools from
several disciplines. The upper bounds on the margin
of stability for a given equilibrium operating condition
uses graph theory notions from flow networks, e.g.,
see [10]. The properties of the route choice functions
that give maximum margin of stability are reminiscent
of cooperative dynamics, e.g., see [11]. The problem
of determining tolls for a desired
2399
: R+
! R+
e
e
e
e#0
e
d
e
e
=
d
(e
where
max e
!RE +
E+
ee
e
1e
e
G. Como et al. • On Robustness Analysis of Large-Scale
Transportation Networks
equilibrium condition exploits the fact that the associated
congestion game is a potential game and that the
extremum of the potential function corresponds to the
equilibrium.
The rest of the paper is organized as follows. In
Section II, we describe basic notations and concepts
useful for the paper and formulate the robustness
analysis problem. In Sections III and IV, we derive
bounds on the margin of stability of the network. Section
V discusses the problem of selection of the most robust
equilibrium operating point of the network. Finally, we
conclude in Section VI with remarks on future research
directions.
e
e
T
) :=
(f is Lipschitz continuous, strictly increasing, stri
concave, and such that (0) = 0, lim< +1 and
limsup!+1( ) <+1. Let ( ) be the vector of the e
flow functions
). Let be the set of all functions : Rthat satisfy
condition (A2).
Example: The following flow function belongs to
f 1 e e2E; (2)>0 for all e2E. It follows from assu
(A2) that ( ) admits a continuous inverse ( ). W
then consider the delay functions
e
II. PROBLEM FORMULATION
start by defining a few preliminary notations. Let N
be the set of natural numbers, R be the set of real
numbers, R
be the set of non-negative real numbers and R>0be the
set of positive real numbers. Let 1nbe the n-dimensional
vector, all of whose entries are one. Let jBjbe the
cardinality of set Band let int(B) denote the interior of set
B. Let ✶(x) be the indicator function with respect to A,
i.e., ✶AA(x) = 1 if x2Aand zero otherwise. Given a set A,
let Apcbe its complement. For p2[1;1], k:kis the p-norm.
Specifically, let k:kdenote the Euclidean norm.
A. Physical properties of the network and Wardrop
equilibrium
+ vLet the topology
of the transportation network be described by a directed
graph G= (V;E), where Vis the set of nodes and Eis the
set of edges. An edge e2Efrom u2Vto w2Vis represented
by the ordered set (u;w). Given an edge e= (u;w), uwill be
called the base node of edge e. Let o2Vand d2Vbe the
origin and the destination nodes in G. For every node
v2Vnfdg, we shall denote by E Ethe set of outgoing
edges from v. Similarly, For every node v2Vnfog, we
shall denote by E1 v;v2;:::;v Ethe set of incoming edges to
v. A path from a vertex u2V and w2Vis an ordered set of
vertices u;v;w such that (u;v1);(v1;v2);:::;(vll;w) each
belong to E. The length of such path will be defined to be
l+ 1, i.e., the number of edges in the path. A path is said
to exist between u2Vand w2Eif there exists an ordered
set of nodes connecting uand w.
m
a
x
(A2)
+We
e
max e
+u
max
E 0
p;q2Pf ; (3) i.e., Te(fe) is the time taken to traverse link ewhen e)=f 1 e e(fe
the flow on it is femax e:= limsup e!+1. Following assumption
(A2), let f e( maxe) be the maximum flow capacity of edge
e. Let fmax2RE +be the vector of maximum flow capacities
of the links in E, ordered lexicographically. Let (f) :=
f 2 jlimsup e!+1 e( e) = fmax eE +, put v8e2 Eg. For a vector
f2R+ v(f) := Pfor all v2Vnfog, (f) := P+ ve2Efe ve2Efemaxfor all
v2Vnfdg. For a given Gand f2RE >0, define the set of
admissible flows through Gas
8e2E; (f)
= u(f) + ✶o
) be the interior of F (G;fmax). Throughout this paper,
we will assume that Gand fare such that F (G;f) 6= ;.
Let e 0 be the toll on edge e2E, and let 2Rbe the
vector of tolls. Assuming a unit dollar value for a unit
amount of delay the utility associated with edge ewhen
the flow on it is feis (Te(fe) + fe). We now recall the
notion of a Wardrop equilibrium [1] that also includes
the effect of tolls. Define P:= fp2Pjfemax>0; 8e2pg. A
Wardrop equilibrium is a vector f2F (G;f), such that, for
all p;q2P,= XTe(fe) + e;
T
e
F (G;fmax) := ff2RE 0 jf f
(u) 8u2Vnfdgg:
Let F (G;fmax
Throughout, we shall assume that (A1) Gis acyclic, and
there exists a path from everyv2Vnfdgto d. Let Pbe the
set of distinct paths from oto din G, and let be the
simplex of probability distributions over P. Traffic arrives
at a unit rate at the node oand is destined for
the node d. We shall denote the traffic density and
flow, on the network at time tby vectors (t);f(t)
2Rrespectively, whose entries, e(t), and feE +(t),
respectively, will denote the traffic density, and flow, on
the link e2Eat time t. Traffic flow and density on a link
are related by the functional dependence
fe
e2E:
); 8e2E; (1) on which we
shall make the following
assumption for all
)+ e eT
e2p=)
X
f p2P
e
the delay function T( ) is continuous, strictly increasing,
and such that Tee(0) >0. Then, the claim follows by
applying Theorems 2.4 and 2.5 from [1].
). Proof It follows from
= e( e
2400
assumption (A2) that,
for all e2E,
=) X
Pr
oc
ee
di
ng
s
of
th
e
19
th
Int
er
na
tio
na
l
Sy
m
po
siu
m
on
M
at
he
m
ati
cal
Th
eo
ry
of
Ne
tw
or
ks
an
d
Sy
st
e
ms
–
M
TN
S
20
10
•
5–
9
Jul
y,
20
10
•
Bu
da
pe
st,
Hu
ng
ar
y
(f u;f
eq) fe;
e(0)
=
1e
+u
e
e
eq;t)k 1
eq
e
eq
limsup
t!+1
e2E,
e
max)
max
max
max
e
<
+1
:
R
e
m
ar
k
2.
3:
E
qu
ati
on
s
(1
)
an
d
(3
)
im
pl
y
th
at,
for
an
y
(t)
!+1i
mpli
= f 2RE +
(feq e
e( e
es
that
T!+1
.
Ther
efor
e,
defin
ition
2.2
impli
es
that
the
tran
spor
tatio
n
netw
ork
is
stabl
e if
the
traffi
c
com
es to
a
stan
dstill
at
one
or
mor
e
locat
ions
in
the
netw
ork.
In
thi
s
pa
pe
r,
w
e
co
ns
id
er
di
st
ur
ba
nc
es
th
at
aff
ec
t
th
e
flo
w
fu
nc
tio
n,
sp
ec
ifi
ca
lly
by
re
du
ci
ng
th
e
m
ax
im
u
m
flo
w
ca
pa
cit
y
of
th
e
ed
ge
s.
W
e
sh
all
m
ea
su
re
th
e
m
ag
nit
ud
e
of
th
e
di
st
ur
ba
nc
e
by
th
e
re
su
lta
nt
re
du
cti
on
in
th
e
flo
w
ca
rry
in
g
ca
pa
cit
y
of
th
e
ne
tw
or
k.
Le
t
(fj
fg,
w
he
re
th
e
in
eq
ua
lit
y
is
el
e
m
en
t
wi
se
,
de
no
te
th
e
se
t
of
fe
as
ibl
e
di
st
ur
ba
nc
es
.
O
n
ap
pli
ca
tio
n
of
a
di
st
ur
ba
nc
e
2
(
f),
th
e
flo
w
pr
ofi
le
s
wi
tc
he
s
to
2(
f)
w
hi
ch
is
al
so
do
mi
na
te
d
by
th
e
pr
edi
st
ur
ba
nc
e
flo
w
fu
nc
tio
n
by
no
m
or
e
th
an
,
i.e
.,
it
sa
tis
fie
s
e
(e
)
2[
e(
e)
e;
)]
for
all
0
an
d
all
e2
E.
Ex
a
m
pl
e:
Th
e
fol
lo
wi
ng
po
stdi
st
ur
ba
nc
e
m
od
el
is
ad
mi
sIn this paper, we assume that G, and are such that
feq e( ) >0 for every e2E. This is for technical reasons and
we shall address the case when feq eeq( ) = 0 for some e
2Ein a future publication. We shall drop the explicit
dependency of fon , when not required, for brevity in
notation.
B. Dynamic Route Choice Behavior We now describe the
behavioral model of agents’ routing
thro
ugh the network. We envision a very large
population of agents traveling through the
network. Agents enter the network from the
origin node oat a constant unitary rate,
travel through it, and leave the network
from the destination node d. Once left the
n
e
t
w
o
r
k
,
a
g
e
n
t
s
join
a
rese
rvoir
,
from
whic
h
they
will
even
tuall
y
reen
ter
t
h
e
n
e
t
w
o
r
k
.
W
h
il
e inside the network, agents occu
link e2E. The rate at which they le
eis fe= e( e), where eis the curren
occupancy density of e. When lea
e2E+ v v, agents choose which link
join. We shall assume such a cho
depend on two factors: (i) a perso
on what the best path is, which is
at a time scale much slower than
one of the dynamics through the n
(ii) an instantaneous local feedbac
consisting in the observation of th
traffic flows on the outgoing links.
focus of this paper is on robustne
properties when the network is dis
from its equilibrium operating condition, we
study the relationship between the
robustness properties of the network and
the myopic behavior of agents based on
instantaneous local feedback for a constant
belief by the agents on the best path that
also coincides with the equilibrium
operating condition.
More precisely, if feq
e
q
=
1(f
2 F (G;f
e
q
u
+u
+
u
u e(
;feq),
where Gu
u( u;feq
E+ u+
u+
u
ma
x
d
e(t)
= u
d (f)Gu
t e
e(
e
) = (fmax
e)
k
); f= e( ); 8e2E8u2Vnfdg; (4)
DThe stability of the transportation network is f
(
defined
as follows.
G
Definition
2.2: For a given G, flow profiles , equ
; f;t) is stable if, route choice function vector
flow
G
dynamical
system given by Equation (4), or
;
equivalently
D(G;G; ;f
1 eee
;
f
e2E: (5)
e
D. Robustness: Margin of stability We quantify the
robustness of transportation network by
eqdefining a notion of margin of stability as follows. For
a given G, equilibrium flow f, route choice function
vector G, define the set of destabilizing disturbances
as:
;feq ) s.t. D(G;G; ;t) is unstableg: Define the the
margin of stability of the given transporta-
2
(fm
ax
U(G;G;feq
E. Problem statement Given a set of feasible route
choice functions , define
eq) := supthe maximum attainable margin of stability as:
(G; ;fG2 (G;G;feq): (6)
) is the Wardrop equilibrium operating condition of the
sible with respect to the example given in Equation
network then it also constitutes the beliefs shared by all
(2).
the agents on the best path through the network. Let )
be the corresponding equilibrium agent densities on
the links. For a node u2Vnfdg, let be the simplex of
probability distributions over E. We shall assume that
the probability that an agent chooses link e 2Ewhen
traversing node u and observes the actual
density 2Ris given by Gis a Lipschitz continuous
function. Let Gbe the vector of node-wise route choice
functions G. Let be the set of all such Lipschitz
continuous G. In the sequel, we shall consider a
certain subclass of by placing restrictions on the
amount of local information available to the agents.
Accordingly, let u be the set of Gsuch that the
node-wise Ghave access only to the densities on the
links E, i.e., if the local density at node uis denoted
by 2R, then such a G) is the route choice function at
node u.
eq)
tion network as:
eq) := inf
k k1:
2 U(G;G;f
(G;G;f
1
C. Dynamical system formulation For a given G, flow 2F (G;f feq
profiles 2 , equilibrium flowmaxeq), route choice
function vector G2 , let D(G;G; ;f;t) be the solution of
the following system of ordinary differential equations:
24 Our primary obje
01 functions Gin the
margin of stabilit
maximum margin
equilibrium flow
max)
and a G2 , consider the initial value problem for
is differentiable
in and @Gv j( v)
+v
every node v2Vnfdg:(d dt j= v(t)Gv j( v1) j( ); j(0) = eq j; Gj( ) = 0; 8j2E
+v
j2Ej+ v: Assume that the input v(t) is continuous, and it
satisfies the following for every node v2Vnfdg: f ; t 0:
(12)
j
1
2 F (G;fmax
1
(A4) Gv j
v(t)
v
X
+
v
max
j
j
Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems –
MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary
< 0; 8j6= k2E: (A5) lim!+1: Assumption (A3) is a
Lemma 4.2 (Local stability and diffusivity
consistency assumption, and ensures
properties): Given a 2 (f
that, if the current flow coincides locally with the agents’e
q
belief, then their route choice coincides with such a
belief. Assumption (A4) instead captures the fact that, if
the flow, and hence the congestion, on a certain link is
increased, then each of the other links is chosen with
higher probability. Let be the subset of whose elements
satisfy assumptions (A3) and (A4).
An example of function Gthat satisfies conditions
(A3)(A5) when f) is the i-logit route choice with noise
level >0, defined by
)
j : 8J E
(t))fThen, for all v2Vnfdgand t 0, we h
X
+v
v
such that j= eq j for some j2E + ; e
v
v j( v) >Gv j
eq(
feq e)Gj( )
Gj( v(t)) j( j
e) >( X
G
v
e
(
v
e
q
e
)
=
f
e
x
p
(
(
e
(
e
e
q
e
)
);
.f+ We now present a local result for the behaviour of
v
the
j
2
E
fe
q
j
e
x
p
(
(
)
)
P
f
o
r
a
l
l
v
2
V
n
f
d
g
,
a
n
d
e
2
E
+
v
j
(
j
eq e;
j
8e6= j2E+ v
( eq j) 0:
eq j)
v
:
eq j
e
q
j
system on the set of outgoing edges from a given node.Second, if both
We start with the following result ensuring existence of a
local equilibrium.
Le
mma 4.1: For every node v2Vnfdg, and 0 + ve2Efmax e,
there exists an equilibrium point < Psuch that vGv e( v)
= e( e); 8e2E+ v: Proof We state the proof for the case
when jE+ vvv2RE+ v+ vj= 2. The proof for the general case
follows along similar lines. Let E= fe;jg. First, if both vmax
e f, and vmax j, then the map g(fe) = vGv e(fe; vfe f)
continuously maps the closed interval [ vmax jf;fmax e] in
itself, and existence of a fixed point is guaranteed by
Brouwer’s fixed point theorem.
e
fe)) fe
1 j( v
m
a
x
max j
v >fmax e,
and
v
>fmax j
vGv e( 1 e(fe);
, one has that
(t)Gv j( v)
j( j)
n<0; 8 v 2@R; t 0: It fo
v
e
; lim
e
v
e
e
q
e
e(t)
such that h(f e) = 0. Setting f j concludes the proof.
max e
~
6=
) = , then consider the function h(f. Then, h is
continuous over the nonempty interval ( ;f f), and
satisfies limh(f
fe! v
max jf
e ) = fmax j
fe max e!f h(fe max e) = f :
max j
j
; 8e2E; 8t 0: (14) On the other hand, thanks to
Lemma 4.1, there exists
ef = v Therefore, by the mean-value theorem, ther
some f e
= j for some j2E;
e
v
e)
= vGv e( 1 e(fe);
h(f ) >0; lim
e
e
h(f
;
v j(
R
fe!f
v
j
) <G
j
(
(
Gv j( v)
j( j)
n<0; 8 v
2@R
g;
!
0
; 8j2E
1.
2403
Finally, consider the case
<
, while f
when f
1 j( v
f )) fmax e max e
f
X such
that (
:= f
:
e;
8e6= j2E: Assumption (A4) implies that G), and
then (t)Gv j( v) j( j) < Gv j( ) j j) = 0: Therefore,
e
6=
;
where
+v
f
j);
8j2E
j
he case when the roles of eand j are reversed can be
treated by a symmetric argument). Then, the function
h(fis continuous and decreasing over the interval (0;f),
and satisfies limmax e) = f
and existence of a zero is again guaranteed by the
meanvalue theorem.
We now prove an important
lemma on the node-wise stability and diffusivity
properties of G2
(t such that
v
Then, for all
e
v
G. Como et al. • On Robustness Analysis of Large-Scale
Transportation Networks
and nis the outward-pointing normal vector to its
boundary. This proves that
e2E
~ e(t)
e; 8e2E; 8t 0; from which, in
e(~ ee2E(t))
X X
particular, it follows that
A.
R
ob
us
t
eq
uil
ibr
iu
m
se
le
cti
on
as
an
op
ti
mi
za
tio
n
pr
ob
le
m
Th
e
ro
bu
st
eq
uil
ibr
iu
m
se
le
cti
on
pr
ob
le
m
ca
n
be
po
se
V. ROBUST EQUILIBRIUM SELECTION AND OPTIMAL
TOLL SELECTION
e(
e)
=
: (15)
In the previous sections, we studied robustness
properties of a transportation network around a g
equilibrium point. We now return to our secondar
objective of identifying the most robust equilibrium
operating point for the network.
d
as
f
eqa
); subj. to f
1
Pj=2J j(~ Now, combining (15) with (14),
one gets that, for all t 0 P v(t))
Pj=2J jj( eq j) =
+ Pj2J f + Pj=2J feq j ( eq
j2J
j
j)
+ P)
e +
;
and
j2J
maximize (G;
j(~ j(t))
v
e
q
2F (G;f
The solution to this optimization problem can help a
system planner evaluate the distribution of traffic flow
that is most robust to disruptions and can implement
this distribution using, for example, using tolls , e.g.,
see [5]. Similar optimization problems and their
solution methodologies have been widely studied in
context of traffic assignment in [1].
qj
j
j
~j
+v
ma
x
j
j
e
e
v j(~ v)
G v j( v
u2Vnf
(fmax e=
dg e2E+ u
Gv j(~ v) j(~ j)
(t)Gv j( v) j( j) = ddt
j
2404
eq e)
8G2
e
qas
stat
)traffic fl
respon
=and the
anarch
Pmeasu
e
2respec
Esocieta
by the
transpo
societa
thereb
anarch
1:
d
t
~
j
where . To complete the proof, it remains to show that
(t) . In order to see this, first observe that ~ (t); t 0:
(16) for all j2E(0) = (0). Moreover, if ~ j(t) = (t) for
some j, and ~ (t) (t) for all other e, then Assumption
(A4) guarantees that G), and, as a consequence, d
j j2E
+
v
=P
v(feq
=
v
;feq. Therefore the optimization problem stated in
;
Equation (18) is equivalent to minimizing a convex
function over a convex polytope. However, note that
the objective function, (G; ) is non-smooth and one
needs to use non-smooth convex optimization
techniques, e.g., see [12], to solve this problem.
which in turn can be shown to imply (16). Remark 4.3:
Lemma 4.2 implies that for a G2 , the effect of a
B. The robustness price of anarchy Conventionally,
disturbance diffuses away from the location where it is
transportation networks have been viewed
applied. Specifically, the increase in the flow on all the
It is worth noting that, for a parallel topology, we have
edges downstream from a node whose outgoing edges
that (G; 1;feq) = (G; ;feqf1 for all feq) = 0 for all feq. That is,
are affected by a disturbance, is less than the magnitude
the margin of stability is independent of the equilibrium
of the disturbance.
operating condition and hence, for a parallel topology,
One can exploit the local stability and diffusivity
P(G; ;f. However, for a general topology and a general
property of Gin from Lemma 4.2 along with a standard
equilibrium, this quantity is non-zero. In the next section,
induction associated with a topological ordering of an
we discuss the use of tolls to yield a robust equilibrium
acyclic graph to prove the following theorem.
point for a given topology, i.e., the one for which the
robustness price of anarchy is zero.
Equation (17) shows
a minimum of a set o
is concave in f
1
X
f
(17)
Note that the second part of the Theorem 4.4 follows
by
combining the first part and Theorem 3.2. Remark 4.5:
Theorem 4.4 implies that, for a given feq, the route
choice functions Gin 1give the maximum possible
margin of stability.
1
eq
)
(G;G;f
and hence,
(G;
minu2Vnfdg
eq) = min
X
1;f
eq
(fmax +ue2E
e
eq ef)
8G2
1;
P(G; ;f
eq)
=
(G; ) (G; ;f
eq):
Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 •
Budapest, Hungary
) Te(feq(0))
VI. CONCLUSION
a desired equilibrium operating condition, feq( ), for the
network.
form
Remark 5.2: The set of tolls
equilibrium operating condi
any toll of the
as the equilibrium condition. Proposition 5.1 gives just would yield f
one such set of tolls.
Proposition 5.1: Given a graph Gsatisfying (A1), 2 ,
and a desired equilibrium f 2 F (G;fmax), a set of
tolls 2RE +that yield this desired equilibrium is given
by
e2E (1)
= max
);
TeTe(feq(0)) T(feq(0)) T(f
e2E
= T(f
eq(0))T(f
), with
maxe2E Te(f
0
where feq
fe
V
(
e
2
E
=
T
)
f
+
=
X
Z
fe
X
Z
Te(z)dz; (19)
(0) is the Wardrop equilibrium for
tolls set to zero. Proof Let be a
simplex of dimension P(number of
paths in Gbetween oand d).
Consider the function V : !R that
serves as a potential function for the
congestion game at hand:
where f = A T
eq
Te(z)dz;
V( ) = (A )Te2E + XZ0(AT )e
In this paper, we studie
transportation networks
pre-disturbance equilib
agents’ response to the
disturbances that reduc
capacities of the edges
properties. We define t
network to be the maxi
under which the traffic
bounded over time. We
choice functions that yi
stability for a given equ
also the formulated an
most robust equilibrium
use of tolls in yielding a
condition.
ACKNOWLEDGEMENT
This work was supported in part by NSF EFRI-ARES
grant number 0735956. Any opinions, findings, and
conclusions or recommendations expressed in this
publication are those of the authors and do not
necessarily reflect the views of the supporting
organizations.
(
2
minimize V( ); subj. to 2 :
0
)
, with A2f0;1gP Ep;ebeing the path-link incidence matrix, Following assumption (A2) and the discussion
i.e., for all e2Eand p2P, A= 1 if elies in path pand zero thereafter, it is easy to see that V( ) is convex in . It is
otherwise. Equation (19) can be rewritten as
known, e.g., see Theorem 2.1 in [1], that the (unique)
Wardrop equilibrium f( ) is equivalent to the first order
optimality condition of the following optimization
problem:
In future, we plan to extend the research in several
robustness analysis in a probabilistic framework versus
topologies, e.g., graphs h
directions. First, we plan to study the dependence of the min-max framework of this paper, possibly
origindestination pairs et
the margin on stability on the amount of information
considering other general models for disturbances.
available to the agents. We also plan to perform
Finally, we also plan to consider more general graph
Let 2R be the Lagrange multiplier corresponding to
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