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Abstract

Balancing traffic flow by influencing drivers' route choices to alleviate congestion is becoming increasingly more appealing in urban traffic planning. Here, we introduce a discrete dynamical model comprising users who make their own routing choices on the basis of local information and those who consider routing advice based on localized inducement. We identify the formation of traffic patterns, develop a scalable optimization method for identifying control values used for user guidance, and test the effectiveness of these measures on synthetic and real-world road networks.
PHYSICAL REVIEW RESEARCH 2, 032059(R) (2020)
Rapid Communications
Reducing urban traffic congestion due to localized routing decisions
Bo Li ,1,*David Saad ,1,and Andrey Y. Lokhov2,
1Non-linearity and Complexity Research Group, Aston University, Birmingham B4 7ET, United Kingdom
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 25 February 2020; revised 8 July 2020; accepted 20 August 2020; published 2 September 2020)
Balancing traffic flow by influencing drivers’ route choices to alleviate congestion is becoming increasingly
more appealing in urban traffic planning. Here, we introduce a discrete dynamical model comprising users who
make their own routing choices on the basis of local information and those who consider routing advice based
on localized inducement. We identify the formation of traffic patterns, develop a scalable optimization method
for identifying control values used for user guidance, and test the effectiveness of these measures on synthetic
and real-world road networks.
DOI: 10.1103/PhysRevResearch.2.032059
Many of the world’s major cities are increasingly grid-
locked with a staggering estimated annual cost of $166B
in the United States alone [1]. Relentless urban population
growth has created exorbitant traffic demands, which leads to
recurring large-scale traffic jams [24]. Since it is expensive
to satisfy the demand exclusively through further investment
in infrastructure, there is a growing interest in optimizing
transportation systems within the existing infrastructure [57].
Modern information technologies can potentially offer ef-
fective solutions through ride sharing using smart phones
[8], congestion-aware routing schemes [9], and the use of
autonomous vehicles [10,11]. The deployment of smart de-
vices already impacts transportation networks, leading to a
paradigm shift in traffic planning and management. How-
ever, not all these changes are for the better. Most navigation
apps have been designed typically to minimize an individual
driver’s travel time irrespective of street capacity along the
route, safety, or the route choices of other drivers; in many
times this results in traffic chaos [12]. Moreover, recent sim-
ulation results demonstrate the potential of having a mixed
environment, of drivers who make their own route choices en
route and those who follow routing advice that is centrally
optimized, in reducing congestion [13]; this scenario is inher-
ently accommodated within the framework presented here. It
is therefore important to understand the potential and limita-
tions of these technologies and develop scalable algorithmic
tools that would enable their use in real-time settings.
Detailed microscopic modeling of multiagent systems
characterizing the paths of individual users, such as cellular
automata-based simulations [14], model basic traffic systems
*b.li10@aston.ac.uk
d.saad@aston.ac.uk
lokhov@lanl.gov
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
but usually require considerable computational power; it is
also generally difficult to gain insight due to the overwhelming
level of details. On the other hand, models based on traffic
flow, that coarse-grain the behaviors of individual users but
maintain correlations at the network level, are simplistic but
amenable to analysis. Link-based methods have been devel-
oped along this line, mainly for static assignments, selfish
routing, or centralized optimization [15,16]. Such methods
have also been extended to the more difficult dynamic traffic
assignment problem [17]; for instance, the Wardrop’s static
equilibrium principle was extended to dynamic scenarios [18].
In reality, drivers do not have full information of the traffic
flow and unbounded computational capacity to determine the
rational route choices [19,20]. Instead, they typically adjust
their route choice, especially in urban settings, en route ac-
cording to the traffic conditions in downstream junctions,
which has been investigated in some dynamic traffic assign-
ment problems [2123].
In this Rapid Communication, we take into account such
behavioral aspects and propose a dynamical model which
includes both impulsive users who make their own decisions
en route, and advice-susceptible users who follow the sug-
gestions given by smart devices. Advice-susceptible users are
incentivized to follow centrally optimized routing suggestions
that benefit traffic globally. Such a strategy may be adopted in
the future to alleviate traffic congestion [13]. In fact, electronic
road pricing already operates successfully in Singapore [24],
and has been recently launched in Israel to motivate drivers
into driving in nonrush hours and carpooling [25]. Our com-
putational model offers complementary insights in support
of such strategies. We focus on scenarios where commuters
travel towards the city center at peak hours, during which
they typically experience severe traffic congestion. A realistic
model of this type is naturally nonlinear, and hard to opti-
mize; one of the contributions of this Rapid Communication
consists in developing a scalable and computationally efficient
optimization method that supports real-time applications. We
analyze the characteristics of emerging traffic patterns, de-
velop an algorithm to determine the optimal incentive, and
investigate their impact on traffic congestion.
2643-1564/2020/2(3)/032059(5) 032059-1 Published by the American Physical Society
BO LI, DAVID SAAD, AND ANDREY Y. LOKHOV PHYSICAL REVIEW RESEARCH 2, 032059(R) (2020)
We model the urban road system as a network, where
intersections are mapped to nodes and roads between them to
edges (or links). We consider a scenario where drivers travel
towards a universal destination D, which is relevant in the
morning rush hour when a large number of people commute to
the city center. The network is depicted as an undirected graph
G(V,E)ofNnodes, where each node iVis connected to ki
neighbors denoted by i, and each edge (i,j)Erepresents
two lanes ijand ji, accommodating noninteracting
traffic from ito jand jto i, respectively. We denote the set of
all lanes as E.
Assume that drivers can be classified into two groups
according to whether they make their own route choices or
follow the advice from navigation devices. In the former, a
user makes routing decisions dynamically, based on her esti-
mated time to destination D. Upon arriving at intersection i
at time t, the user faces a choice between kipossible roads
{ij}ki
j=1. The user first estimates (i) the time it takes to
travel through edge ijas g(ρt
ij) where ρt
ij is the number
of users occupying edge ij(i.e., traffic volume) at that
time and g(ρt
ij) is determined by the Greenshields model [26]
[see also the Supplemental Material (SM) [27]], and (ii) the
remaining time djneeded to travel to Dfrom node j, which
can be taken as the shortest free traveling time or be based
on past experience of the congestion level. Afterwards, their
route choices are made according to the probability
pg,t
ij (ρt)=eβ[g(ρt
ij)+dj]
kieβ[g(ρt
ik )+dk],(1)
where βis a parameter determining the randomness of the
decision-making process. As shown in the SM [27], the de-
pendence on β1 is relatively weak, and hence we choose
β=1 in what follows. Note that we do not limit users from
turning back. The awareness of congestion can be extended
to road segments that are more distant, at the cost of higher
computational complexity. Here, we focus on the one-step
congestion-aware model.
In the latter group, users follow the navigation advice
aimed at improving traffic efficiency. Their route choices at
junction iat time tare determined by the localized probability
pw,t
ij (wt)=ewt
ij
ki
ewt
ik ,(2)
where the weight variables {wt
ij}are optimized centrally. With
the assumption that the fraction of users nwho are susceptible
to routing advice are distributed evenly in the network, on
average the vehicle flow arriving at node iat time twill be
diverted to the adjacent edges {ij}ki
j=1according to the
distribution
pt
ij(ρt,wt)=(1 n)pg,t
ij (ρt)+npw,t
ij (wt).(3)
A similar decision rule has been used to investigate the effect
of altruistic users in the static routing game setting [7,28],
which differs from the current dynamical model.
At each time step ta decision is made to enter edge
ij, the user then spends some time τij traveling on this
edge with distribution P(τij), arriving at the end point at time
t=t+τij. The distribution of time spent can take several
forms, including the typically used discrete Poisson distribu-
tion adopted here [27]. The arrival probability depends on the
traffic volume ρt
ij at the time of entrance t, i.e., P(tt|ρt
ij),
which is a realistic and an important factor in traffic modeling.
To express the dynamics we introduce the time-dependent
flux ft
ij arriving at the end point jof the edge ijat time
t. Assuming users enter the road system at time t=0 with
initial volume ρ0, the dynamics of the traffic volume and flux
on edge ij(i= D) are governed by the discrete forward
dynamics
ρt
ij =pt1
ij
ki,k=D
ft1
ki +ρt1
ij ft1
ij ,(4)
ft
ij =
t
t=1ρt
ij ρt1
ij ft1
ij Pttρt
ij+ρ0
ijPtρ0
ij.
(5)
Equation (4) describes the traffic volume at edge ijat time
t; it is composed of the newly joined users who selected this
junction at node iat time t1, and users who were already
traveling through this edge but have not yet reached the end
point j. Equation (5) states that the vehicles flux at the edge
ijend point at time tcomprises the fraction of traffic
volume ρt
ij (ρt1
ij ft1
ij ) entering the road segment at t,
who have completed the trip on this road segment within a du-
ration ttas dictated by the probability P(tt|ρt
ij), which
is defined such that the mean traveling time follows the Green-
shields model [26,27]. The resulting model bears similarity to
certain link-based models of dynamic traffic assignment [17].
We assume that no vehicles leave the destination node, i.e.,
the destination Dis an absorbing boundary which satisfies
ρt
Dj =ft
Dj =0,jD.
The model is simulated for a fixed time window T.To
evaluate the efficiency of the system, we measure the average
time to destination Dahead of T,
O=1
eEρ0
e
T
t=1
(Tt)
jD
ft
jD,(6)
and use it as the main performance measure. Other measures
can be easily accommodated within the same framework but
will not be considered here.
We perform numerical experiments on both generated and
realistic road networks. The former are constructed by ran-
domly rewiring a planar square lattice with shortcut edges,
which is motivated by the recent observation that high-speed
urban roads constitute effective long-range connections and
render the system to exhibit small-world characteristics [29].
The realistic road network used is extracted from the Open-
StreetMap data set [30], and converted to a network format
by using the GIS F2Esoftware [31]. Two examples of the net-
works considered are shown in Fig. 1. Details of the network
generation are described in the SM [27].
Model characterization without control. The initial traffic
volume is assigned independently and identically at random
as ρ0
iusers departing from each node i, which can be pro-
portional to the population at that node; users rest on the
node’s neighboring edges {ij|ji}with equal proba-
bility ρ0
ij =ρ0
i/ki, constituting the initial traffic volume {ρ0
ij}.
032059-2
REDUCING URBAN TRAFFIC CONGESTION DUE TO PHYSICAL REVIEW RESEARCH 2, 032059(R) (2020)
(a) (b)
FIG. 1. (a) A small-world network generated by rewiring a 21 ×
21 square lattice with shortcut links with rewiring probability pr=
0.05. (b) The Birmingham road network composed of major roads
in the city of Birmingham, U.K. We define the city center as the the
region enclosed by ring road A4540. In both cases, the red nodes
constitute the city center and determine the destination in the model.
After entering the system at time t=0, all users drive towards
the center node Daccording to the instantaneous decision-
making rule of Eq. (1), i.e., n=0inEq.(3). Clearly, the
same framework can accommodate users entering the network
at any time. It leads to macroscopic dynamical traffic pat-
terns governed by Eqs. (4) and (5). We define the traffic load
level as
L=eEρ0
e
eEρjam
e
.(7)
The load level Lis similar to the demand-to-supply ratio in-
troduced in Ref. [7], which is suggested to be a good predictor
of the congestion level.
We first study emerging traffic patterns in the absence of
routing advice, n=0. The movement of traffic mass can be
visualized by contrasting the traffic volume ρeto the distance
to destination of each lane. To this end, we define the distance
dist(e,D) of lane e=(ij) to destination Das the shortest
free traveling time from the midpoint of the lane to destina-
tion dist(e,D)=dj+tfree
ij /2. Figure 2demonstrates how the
average traffic volumes ρt
eat specific distances change over
time under two different load levels. At the low load regime
L=0.1, shown in Fig. 2(a), the vehicles are able to move
fairly quickly towards the destination Dfrom the initial posi-
tions at t=0tot=25. The roads near the city center become
congested, leading to a slow clearance of traffic from t=50
(a) (b)
FIG. 2. Average volume ρt
evs average distance to destination
dist(e,D)ina21×21 small-world network. Maximal time is T=
100 and users are unaided n=0. The road sections are first binned
into groups according to dist(e,D) in the interval of 10, after which
ρt
eand dist(e,D) are averaged within each group. (a) Load level L=
0.1. (b) Load level L=0.4.
(a) (b)
FIG. 3. Relative frequency of fractional change of the objective
function Oafter adding a link to the existing network, defined as the
performance change O=Oadd-link Oorigin divided by the objec-
tive function before adding a link Oorigin. The results are aggregated
from ten networks generated from the small-world network model
of size 21 ×21 with rewiring probability pr=0.05. The parameters
are T=100, n=0. One end of the new link is randomly chosen
from the top five sites with the highest population, while the other
end is randomly chosen from the nearest neighbors of the destination
node. (a) Load level L=0.1. (b) Load level L=0.5.
to t=100, which indicates that the limited connectivity of
the city center is a bottleneck of the traffic system. At high
loads L=0.4, shown in Fig. 2(b), the traffic volumes at large
distance to destination decrease, while those at short distances
increase over time, but at a much slower rate compared to the
case of a small load L=0.1. It indicates that the excessive
demand creates congestion in the transportation network and
leads to an increase in travel time. More details of the system
efficiency as a function of load are depicted in the SM [27].
The simplest measure to reduce congestion is to improve
the infrastructure, e.g., by building new roads or by increasing
the capacity of existing ones. In particular, increasing the
number of possible routes to the city center/destination node
can significantly enhance the traffic clearance rate, yet it is
rarely possible to do so due to the limited land availability. To
examine the effect of network extension, we perform experi-
ments by adding links from sites with the largest populations
to nearest neighbors of the destination node. From the rela-
tive frequency of the fractional change of objective function
shown in Fig. 3, it is surprising to observe that the majority of
link additions lead to a decrease in system performance. It sug-
gests that newly introduced shortcuts, being attractive to users,
create congestion in the shortcut edges and nearby areas. The
phenomenon is reminiscent of Braess’s paradox in the static
routing game [32] and other complex systems [3335], where
adding resources can possibly lead to a degradation of system
performance. In our model, drivers have limited knowledge
and are unaware of the long-distance traffic condition, so
that the myopic decisions make the system more prone to
congestion. If users are aware of more global information as
in routing game scenarios, it is possible that they may adapt,
in a repeated game scenario, to avoid the already congested
shortcuts, such that the probability of performance decrease
becomes smaller.
Nevertheless, there is a small likelihood that adding a new
link would lead to a significant improvement of the objective
function O, which can be up to 10% for L=0.1 and 20% for
L=0.5. Such an improvement is more commonly observed
in higher loads, but in the majority of cases the improvement
is marginal. In either case, it is crucial to select the correct
032059-3
BO LI, DAVID SAAD, AND ANDREY Y. LOKHOV PHYSICAL REVIEW RESEARCH 2, 032059(R) (2020)
(a) (b)
FIG. 4. (a) Fractional change in objective function O(defined
as the performance change O=Ooptimized Oorigin divided by
the objective function without advice-susceptible users Oorigin)asa
function of the fraction of advice-susceptible users n. The Birming-
ham road network (BHM) and a small-world network (SW) of size
21 ×21 are considered. (b) Time evolution of the fraction of traffic
volume eρt
e/eρ0
eremaining on the Birmingham road network;
at time t=0, the system load is L=0.1.
shortcut to invest in, which becomes a difficult task when the
demand profile is fluctuating.
Model characterization with control. It becomes increas-
ingly more appealing and cost effective to influence the route
choice of drivers in order to reduce congestion. We examine
the particular type of instantaneous advice in the form of
Eq. (2), which is adapted such that the objective function
Oof Eq. (6) is maximized. The resulting highly nonlinear
optimization problem is nonconvex and suffers from multiple
local maxima.
To solve this difficult optimization problem, we adopt an
optimal control framework [36,37], whereby the dynamics,
Eqs. (4) and (5), is enforced as constraints in the Lagrangian
formulation. The optimality conditions lead to a set of coupled
nonlinear equations solved by forward-backward iterations.
To suppress divergent behavior due to radical changes of
the control parameters [36,38], we employ a gradient as-
cent in the updates of the control parameters [27]. Our
method achieves similar objective function values to state-
of-the-art constrained optimization approaches, while offering
significant stability and scalability advantages. This point is
illustrated in the SM through a benchmarking comparison
to the state-of-the nonlinear programming solver IPOPT [27].
The results shown in Fig. 4demonstrate that the optimization
algorithm successfully improves the system performance, as
indicated by the fractional increase in the objective function
Ocompared to the value Oorigin without advice-susceptible
users. The maximal average improvements are remarkably
significant and range from 7% to 14%, depending on the
network structure and load level. Naively, one would expect
for the objective function to monotonically increase with
n. However, it seems not to be the case in the experiment
shown in Fig. 4(a) and a slight decrease in performance is
shown close to n=1. For n<1, the mixture probability
pt
ij =(1 n)pg,t
ij +npw,t
ij (wt) includes information on the un-
guided users and effective distance to destination through
pg,t
ij , which facilitates the search for an optimal solution. This
information is gradually lost at high nvalues, resulting in
a less pronounced increased performance, compared to the
maximally achieved level of gain.
In Fig. 4(b), we demonstrate the evolution of the fraction
of traffic eρt
e/eρ0
eremaining on the Birmingham road
network at a given time as a function of the guided-users
fraction n. One can observe a faster rate of traffic decrease
when nincreases from 0 to 0.8, suggesting more users can
reach the destination within the same time period with the
increase in the number of advice-susceptible users.
Modeling the dynamics of a transportation network that
accommodates different driver behaviors facilitates a greater
understanding of the emerging traffic patterns in a regime that
is of great interest and relevance [13], while the suggested
optimization scheme provides a scalable and efficient way to
implement it, providing better performance than state-of-the-
art continuous optimization solvers and offering significant
advantages in the online setting, where sudden changes in
traffic conditions can be adjusted by a few update steps to
obtain a quality approximate solution. We demonstrate how
extending the network may result in increased congestion
and a degradation in system performance, highlighting the
importance of a careful selection of the most beneficial roads
to add, which will be the subject of future research. Balanc-
ing the traffic flow by influencing user route choices offers
a less costly and more flexible solution to the congestion
problem. Our experiments on macroscopic traffic-flow opti-
mization by giving instantaneous and localized routing advice
demonstrates its potential for improvements in system perfor-
mance. The framework also allows for the study of balancing
demand by scheduling departure times, which could be in-
tegrated into our optimization framework; this is one of the
future directions for a follow-up study. These extensions can
be tested at a low computational cost using our model and
optimization method without the need for expensive large-
scale agent-based simulations. Other possible generalizations
include the introduction of a spill-back mechanism, the in-
tegration of more nonlocal traffic condition information, and
cases of multiple destinations.
Acknowledgments. The map data are copyrighted by Open-
StreetMap contributors and is available from [41]. B.L.
and D.S. acknowledge support from the Leverhulme Trust
(RPG-2018-092), European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-
Curie Grant Agreement No. 835913. D.S. acknowledges
support from the EPSRC programme grant TRANSNET
(EP/R035342/1). A.Y.L. acknowledges support from the
Laboratory Directed Research and Development program
of Los Alamos National Laboratory under Projects No.
20190059DR and No. 20200121ER.
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