To appear in Proceedings of the 14th IEEE ITS Conference (ITSC 2011), Washington DC, USA, October 2011.
Dynamic Lane Reversal in Trafﬁc Management
Matthew Hausknecht, Tsz-Chiu Au, Peter Stone
School of Civil and Environmental Engineering
{mhauskn,chiu,pstone}@cs.utexas.edu
{davidfajardo2,s.travis.waller}@gmail.com
Abstract— Contraﬂow lane reversal—the reversal of lanes in
order to temporarily increase the capacity of congested roads—can effectively mitigate trafﬁc congestion during rush hourand emergency evacuation. However, contraﬂow lane reversaldeployed in several cities are designed for speciﬁc trafﬁcpatterns at speciﬁc hours, and do not adapt to ﬂuctuationsin actual trafﬁc. Motivated by recent advances in autonomous
An illustration of contraﬂow lane reversal (cars are driving on
vehicle technology, we propose a framework for dynamic lane
the right side of the road). The total capacity of the road is increased by
reversal in which the lane directionality is updated quickly and
approximately 50% by reversing the directionality of a middle lane.
automatically in response to instantaneous trafﬁc conditionsrecorded by trafﬁc sensors. We analyze the conditions under
systems, more aggressive contraﬂow lane reversal strategies
which dynamic lane reversal is effective and propose an integer
can be implemented to improve trafﬁc ﬂow of a city without
linear programming formulation and a bi-level programming
increasing the amount of land dedicated to transportation.
formulation to compute the optimal lane reversal conﬁguration
An important component of implementing dynamic lane
that maximizes the trafﬁc ﬂow. In our experiments, activecontraﬂow increases network efﬁciency by 72%.
reversal is fully understanding the systemwide impact ofincreasing capacity on an individual link. We deﬁne the
objective of contraﬂow as follows: given a road network,a speciﬁcation of vehicles’ locations and destinations, and
Trafﬁc congestion is a major issue in today’s transportation
a method for determining network efﬁciency (such as an
systems. Contraﬂow lane reversal, the reversal of trafﬁc ﬂow
objective function), assign a direction of ﬂow to each lane
along a lane to temporarily increase the capacity of congested
such that network efﬁciency is maximized. To study the
roads at the expense of under-utilized ones, is a method to
network effects of dynamically repurposing lanes, we cast the
increase trafﬁc ﬂow without adding additional roads or lanes.
problem as a maximum multi-commodity ﬂow problem—
On the left of Fig. 1, the top lanes are being more heavily
a version of the maximum ﬂow problem in graph theory
utilized than the bottom ones. On the right, by temporarily
with multiple commodities (or goods) ﬂowing through the
converting a lane to ﬂow in the opposite direction, the
network. Then we propose an integer programming formula-
instantaneous capacity in the left-to-right direction of the
tion and a bi-level programming formulation to compute the
road is increased by 50%. Contraﬂow lane reversal has been
maximum ﬂow in the network. We evaluate our approaches
used routinely in several cities in order to alleviate trafﬁc
in grid-like transportation networks representative of many
during rush hours as well as to reroute trafﬁc around certain
downtown metropolitan areas where it will have the most
areas such as construction sites or stadiums.
Today, contraﬂow lane reversal is used at a macro time
The rest of the paper is organized as follows. In Section II,
scale at rush hour or for quick evacuations from an area. In
we discuss the hardware needed for implementing a dynamic
both cases however, the change in ﬂow must be carefully
lane reversal scheme. In Section III and IV, we analyze under
planned before the event, with little or no room for dynamic
what conditions dynamic lane reversal will be useful for an
changes. Today’s hardware for trafﬁc monitoring is good
individual road and intersection. In Section V and VI, we
enough to gather real-time trafﬁc data. With the help of
introduce both the macroscopic ILP trafﬁc model as well
modern computerized trafﬁc control systems, it is possible
as the bi-level formulation, and investigate the performance
to quickly and dynamically open and close lanes or entire
gains imparted by dynamically reconﬁguring lanes.
roads, or even change the directionality of lanes based onreal-time usage statistics, such that effective capacity of a
road can be dynamically changed based on the demand.
A reversible lane (or contraﬂow lane) is a lane in which
Rapid changes of lane directions, however, may confuse
trafﬁc may travel in either direction. The common hard-
human drivers. To fully utilize the potential of dynamic lane
ware for creating reversible lanes is overhead trafﬁc lights
reversal, we will need to rely on the upcoming availability
(Fig. 2(a)). In many cities, barrier transfer machines, also
of computer-aided driving systems and fully autonomous
known as zipper machine, are used to relocate the moveable
vehicles that will help vehicles to adjust to the rapid changes
barriers such that the road in one direction can be dynami-
of lane directions. With the help of computerized driving
cally widened at the expense of the other (Fig. 2(b)).
Hardware for controlling contraﬂow lane.
By deﬁnition, the throughput of the road increases after the
The basic requirements to support dynamic lane reversal
are that the reversal has to be done quickly and safely, andthat the drivers must be notiﬁed about the change immedi-
λ(L1,2) + λ(L2,1) < λ (L1,2) + λ (L2,1)
ately. While it’s conceivable to devise a system that satisﬁes
where λ (L1,2) = min{β (I1),c(L1,2) − c(l)}, and λ (L2,1) =
these requirements using the hardware in Fig. 2, there is
min{β (I2), c(L2,1) + c(l)}.
likely to be signiﬁcant cost and risk of driver confusion.
In general, lane reversal is beneﬁcial only when one of the
However once most cars are controlled by computer, these
directions is oversaturated while the other is undersaturated,
costs and risks may be signiﬁcantly reduced by real time up-
as shown in Fig. 1. Formally, we have the following theorem:
dates of lane direction over wireless network communication,
Theorem 1: The throughput of the road R increases after
as computerized driving systems (i.e., autonomous vehicles)
the reversal of a lane l ∈ La,b if and only if La,b is un-
can react to the changes of lane directions much quicker than
dersaturated by δa while Lb,a is oversaturated by δb, where
human drivers. For example, Dresner and Stone proposed an
max{c(l) − δa, 0.0} < δb.
intersection control mechanism called Autonomous Intersec-
Proof Sketch. Due to space limitations, we only consider
tion Management (AIM) that uses a wireless communication
the case in which c(l) > δa and δb < c(l). The reversal of l
protocol to enable ﬁne-grained interleaving of vehicle routes
reduces the effective trafﬁc rate of Lb,a by x = c(l) − δa > 0
through an intersection [1]. With some modiﬁcations to the
while the effective trafﬁc rate of La,b increases by δb. Thus
AIM protocol, autonomous vehicles can be informed about
the throughput of the road increases if and only if δb > x =
the current lane directions as well.
max{c(l) − δa, 0.0}.
We begin by considering, from a theoretical perspective,
Analyzing the change of the intersection throughput is
the effects of lane reversal on a single road. Consider a road
necessary because intersections may potentially be the bot-
between intersections I1 and I2. Let R be the road between
tlenecks of the trafﬁc ﬂow, preventing the adjacent roads
I1 and I2, L1,2 be the set of lanes from I1 to I2, and L2,1
from achieving their maximum throughput as predicted by
be the set of lanes from I2 to I1. As an example, in Fig. 3,
Theorem 1. Estimating the effects on intersection throughput
L1,2 = {l1,l2} and L2,1 = {l3,l4}. The capacity of a lane l,
theoretically, however, can be a challenging task, especially
denoted by c(l), is the maximum rate at which vehicles enter
when vehicles from different roads can enter the intersection
the lane and is measured by the number of vehicles per hour.
at the same time. Therefore, we use empirical methods to see
We assume the capacity of a set L of lanes, denoted by c(L),
whether intersections can handle the increase of the incoming
is the sum of the capacities of all lanes (c(L) = ∑l∈L c(l)).
trafﬁc when the directions of adjacent lanes reverse.
For simplicity, we ignore the effect of lane changing which
We experiment with the intersection in Fig. 4, which has
potentially reduces the capacity of L.
six lanes on each incident road. Initially, 3 lanes are incoming
Assume both I1 and I2 are sources at which vehicles are
lanes and 3 lanes are outgoing lanes. We set the target trafﬁc
“generated” to travel along R at the target trafﬁc rates β (I1)
rate of the eastbound road be 5500 vehicles per hour, the
and β (I2) respectively. But the effective trafﬁc rates λ (L1,2)
target trafﬁc rate of westbound road be 1100 vehicles per
and λ (L2,1) at which vehicles actually enter the road are lim-
hour, and the traget trafﬁc rates of both northbound and
ited by the capacity of the lanes. More precisely, λ (L1,2) =
southbound roads are 1650 vehicles per hour. Thus, the trafﬁc
min{β (I1), c(L1,2)} and λ (L2,1) = min{β (I2),c(L2,1)}. If
on the eastbound road is several times higher than other
λ(L1,2) = c(λ(L1,2)), we say L1,2 is saturated. If β(I1) >
roads, causing trafﬁc congestion on the eastbound road. We
c(L1,2), L1,2 is oversaturated by an amount of β (I1)−c(L1,2).
check whether reversing the direction of two lanes on the
Clearly, if L1,2 is oversaturated, L1,2 is saturated. L1,2 is
westbound road can help to increase the throughput of the
undersaturated by an amount of c(L1,2) − β (I1) if β (I1) <
eastbound road as well as the intersection throughput (the
c(L1,2). Clearly, if L1,2 is undersaturated, L1,2 is not saturated.
number of vehicles entering the intersection per hour). The
The saturation of L2,1 is deﬁned in the same manner.
new lane conﬁguration is shown on the right side in Fig. 4.
The throughput of the road R is the sum of the effective
We repeated the experiment 30 times and in each run
trafﬁc rates of the lanes (i.e., λ (L1,2) + λ (L2,1)). Now con-
we measured 1) the total number of vehicles entering the
sider what happens if the direction of l ∈ L1,2 is reversed.
intersection during the 1-hour period, and 2) the number of
The multi-commodity ﬂow problem is a generalization
of the well-known max ﬂow problem in which multiplecommodities or goods ﬂow through the network, each withdifferent source and sink nodes. Modeling a road networkat the macroscopic level allows us to map the well-studiedproblem of multi-commodity ﬂow directly onto our problem
Fig. 4. The reversal of two lanes on the westbound road of an intersection.
of dynamically reconﬁguring lanes. In order to solve this
problem we utilize the the mathematical machinery of linear
A Linear Program contains a linear function to be max-
imized over a set of variables, subject to constraints. Innormal linear programs these variables are allowed to as-
vehicles entering the intersection from each road during a 1-
sume fractional values, but since all of our ﬂow demands
hour period. The average of the number of vehicles and the
are required to be integer-valued, we must approach this
95% conﬁdence intervals are shown in Table I. The results
problem as an Integer Linear Program. Unfortunately, the
show that the throughput of the intersection increased by 6%,
multicommodity ﬂow problem has long been known to be
and this is mainly due to the increase of incoming trafﬁc
NP-complete when dealing with integer ﬂows, even for only
from the eastbound road whose throughput is increased by
13%. Note that both increases are statistically signiﬁcant.
We deﬁne the following Integer Linear Program: Given a
After lane reversal, the eastbound road’s trafﬁc rate is much
graph G = {V, E}, each edge (u, v) has some integer capacity
closer to the target trafﬁc rate, and this means that the
c(u, v) representing the total number of lanes present on
intersection successfully handled the increase of the trafﬁc
that road. There are k distinct commodities (trafﬁc ﬂows)
coming from the eastbound road. The lane reversal has only
K1, ., Kk where each commodity Ki = (si,ti, di) has an as-
minor detrimental effects on other roads, because they are
sociated source si, destination ti, and demand di. Flow of
undersaturated and the lane reversal does not reduce the
commodity i over edge (u,v) is denoted fi(u, v).
capacity of these roads below their target trafﬁc rate.
Our objective is to ﬁnd an assignment of ﬂows which
satisfy the following three constraints: The capacity con-
straint, shown in Equation 2, speciﬁes that the total amountof ﬂow (in both directions) over a given (u, v) edge must
While the ability to improve throughput on individual
not exceed the capacity of that edge (note c(u, v) = c(v, u) in
roads and at individual intersections are important proofs
the undirected case). The conservation constraint, Equation
of concept, the true question is whether (and how much)
3, ensures that for all non sink/source vertices the amount of
dynamic lane reversal can help on a full road network. To
inﬂow of a given commodity equals the amount of outﬂow.
address this question we model a road network as a graph
Finally Equation 4 speciﬁes that the ﬂow of each commodity
consisting of vertices and edges. Each node of the graph
must meet or exceed the demand for that commodity.
represents an intersection and each edge represents a road
between intersections. Additionally, each u, v edge has an
∀(u,v)∈E ∑(fi(u,v)+ fi(v,u)) ≤ c(u,v)
associated capacity c(u, v) which constrains the maximum
amount of trafﬁc that road can handle (in this section we
∀i∈1.k,v∈V−{s,t}[ ∑ fi(u,v) = ∑ fi(v,w)]
model intersections as having inﬁnite capacity).
Trafﬁc in the network is modeled in terms of aggregate
∀i∈1.k ∑ fi(si,w) = ∑ fi(w,ti) ≥ di
demand. Speciﬁcally we consider a ﬁnite number of ﬂows,
where each ﬂow has an associated source vertex, destination
The goal of the ILP solver is to ﬁnd an assignment of
vertex, and integer-valued demand. For example, a parking
directionality to each lane which maximizes the objective
garage at a mall could be a source, and a bridge at the
function (Equation 5) subject to the constraints speciﬁed
edge of the network could be a destination. The demand
above. This is done by assigning integer values to individual
represents the instantaneous number of vehicles that want to
ﬂows. We choose to use the maximum multi-commodity
travel between these two points. Vehicles may take any path
objective function in which the objective is to reconﬁgure
between the source and destination so long as they do not
the network to maximize the sum of all commodity ﬂows:
violate capacity constraints of roads. We seek to determine
1) whether or not the lanes of a given road network can be
maximize ∑ ∑ fi(si,w)
dynamically reconﬁgured in order to accommodate a given
set of trafﬁc ﬂows and 2) what is the maximum demand a
While multiple possible objective functions could meet our
criterion of ﬁnding a lane conﬁguration capable of handling a
given set of ﬂows, we choose the maximum multicommodityobjective because it forces the ILP solver not only to satisfy
the ﬂow demands, but also to ﬁnd the absolute maximum
amount of trafﬁc a network can handle. In contrast, thefollowing section explores an alternative, least-cost objectivefunction. C. Bi-Level Programming Formulation
The multicommodity ﬂow formulation of our problem
is convenient in that its solution—the maximum ﬂow—is
unique and independent of vehicles’ behavior. However it
ignores the fact that drivers are self-interested—they are
A example generated graph with two incident S, T ﬂows. Thick
concerned about their own travel times and have no incentive
lines represents highways, medium lines represent arterial roads, and thin
to cooperate to achieve the maximum ﬂow of the network.
Therefore, we consider an alternative formulation for the
adaptive capacity problem using a bi-level approach, where
ROAD TYPES COMPOSING THE RANDOMLY GENERATED NETWORKS.
the objective is to set link capacities such that, as ﬂows are
determined by User Equilibrium behavior, the total system
travel time, i.e. the sum of the travel times of all users,
is minimized. By modeling the route choice user behavior
as a travel cost minimization, we can more accuratelycharacterize the behavior of users in a trafﬁc context. The
allocated to a link (i, j) that is taken from the reverse link
mathematical formulation is shown in Equations 6–8. The
( j,i). The ﬁtness function used is the total system travel
upper level problem includes the allocation of capacity x to
time in the underlying UE problem given a decision vector
each of the links, while the lower level problem is the classic
User Equilibrium model presented by Wardrop [3]:
min f (x) s.t. − ci j < xi j < c ji
In this section we use the ILP solver to empirically
compare different trafﬁc management systems – those which
can reverse lane directions quickly and those which can
reverse slowly or not at all. We hypothesize that trafﬁc
∑vodji bod(i) ∀i,od.vij ≥ 0 ∀i, j (8) managers which have the ability to quickly reverse the ﬂow
of trafﬁc along lanes will achieve higher throughputs than
where c is the capacity vector, vod is the ﬂow vector for
each OD, x is the dynamic lane capacity allocation vector, t fis the free ﬂow speed vector, and α and β are parameters, and
bod(i) is equal to the node supply/demand for each OD pair
We automatically generate graphs qualitatively similar to
od. As bi-level problems such as this are difﬁcult to solve
downtown regions of many cities. Each graph takes the form
exactly, we present a Genetic Algorithms based solution
of a connected planar grid. To determine the capacities for
each road, we randomly select from one of the three roadtypes shown in Table II with the associated probabilities. D. Genetic Algorithm Solution Method
Flows are generated by selecting a random source, sink
Genetic algorithms (GAs) is a global search heuristic that
vertex pair from the graph. An example road network is
uses techniques inspired by evolutionary biology ([4], [5]).
GAs are based on the assumption that the best solution isfound in regions of solution space having a high proportion
of good solutions. GAs explore the solution domain to
To measure the performance difference between trafﬁc
identify the promising region and then search the promising
control systems with the ability to quickly reverse lanes (such
regions more intensely. GAs start with a population of ran-
as the AIM protocol) and those that can only slowly reverse
domly regenerated individuals that evolves with generations
lanes (such as zipper machines), we evaluate each trafﬁc
based on the principle of survival of the ﬁttest. Unlike
management system for 10 hours. Each hour a new set of
classical methods, GAs work with a population of points.
random ﬂows is spawned, and it is the job of the trafﬁc
Therefore, the chances of getting trapped at local optima
manager to accommodate this trafﬁc as well as possible by
are reduced. Moreover, many variations of GAs are suitable
reversing lane directionality. However not all trafﬁc managers
for handling complex problems involving discontinuities,
will be able to reverse lanes every hour. Trafﬁc management
disjoint feasible spaces, and noisy function evaluation [6],
systems differ in their reconﬁguration period – the amount
[7], [8]. In our formulation, a gene represents the capacity
of time that must elapse between lane reconﬁgurations. For
example, a system with reconﬁguration period of 2 will
reconﬁgure lane directions every other hour while a system
with reconﬁguration period of 5 will only reconﬁgure everyﬁfth hour. If a trafﬁc manager is unable to reconﬁgure lanes
for a given hour, the network throughput is computed by
ﬁnding the maximum multicommodity ﬂow over the cur-
rent lane conﬁguration (e.g. directed multicommodity ﬂow)
without allowing the directions of any lanes to change. On
the other hand, if a trafﬁc manager is able to reconﬁgure
lanes for a given hour, we compute throughput by ﬁnding the
maximum multicommodity ﬂow over the network consistingentirely of undirected edges. This gives managers with lowreconﬁguration period the ability to more fully adapt to
An example run proceeds as follows: we ﬁrst randomly
Total network throughput (vehicles) as a function of DLR period.
generate a road network using the procedure described above.
A reconﬁguration period of 1 means that the network was reconﬁguredeach timestep; 2 means every other timestep; ∞ means the network was
Initially this network is conﬁgured in a balanced manner
never reconﬁgured (always in balanced, directed conﬁguration). Error bars
in which the capacity of each road is divided as evenly as
denote 95% conﬁdence intervals. Period 1,2,3 are statistically signiﬁcant
possible between lanes ﬂowing in one direction and those
with respect to period 0 as well as each other.
in the opposite. Next, random ﬂows are generated and the
technology which can adapt quickly to changing trafﬁc ﬂows
ILP solver is used to compute the network throughput for
that hour either on the current directed conﬁguration (if themanager cannot reconﬁgure this hour) or on the undirected
conﬁguration (if the manager can reconﬁgure this hour). Any
This section shows preliminary results of the implemen-
changes made to the directionality of lanes carry over into
tation of the bi-level formulation presented in Section V-C.
the next hour when new ﬂows are generated.
The objective of these numerical results is to show that the
In our experiments we evaluated networks of size 100
algorithm is implementable for modest sized problems.
(10x10) for 10 hours. Each hour contained a set of 4
We solve a problem on a 10x10 grid network, where the
randomly generated ﬂows. Demands for these ﬂows were
number of lanes varies for each link, the capacity is 1800
set to 0 to ensure that the ILP solver would both reach a
vehicles/hour/lane, α = 0.1, β = 4, and t f is 25 mph, 35 mph
valid solution and maximize the achievable throughput.1 The
and 55 mph depending on the number of available lanes.
throughput of each trafﬁc manager was evaluated at each
For the GA, the size of the population and the number of
timestep and the total throughput for a trafﬁc manager with a
generations were both set to 30, the mutation probability was
given reconﬁguration period was the sum of the throughputs
set to 0.002, and the cross-over probability was set to 0.75.
it achieved over all 10 hours. We evaluated the performance
Based on these parameters, the model was solved for each
of each trafﬁc manager over 34 different networks (34 trials)
of 10 time periods. In order to compare the behavior of the
GA based solution method with the ILP solver, we examine
Figure 6 shows the total network throughput achieved by
the resulting lane conﬁgurations after each time period and
trafﬁc managers with different reconﬁguration periods. The
calculate the correlation coefﬁcient of the solution vectors
results show a signiﬁcant increase in performance of trafﬁc
for the ILP and GA. The correlations in lane conﬁguration
managers who have some form of lane reconﬁguration in
for each of the 10 hours are shown in Table III. As shown by
comparison to no lane reconﬁguration. This is not surpris-
the resulting correlation coefﬁcients, the two solution vectors
ing since the beneﬁts of contraﬂow are well established.
are quite different, which conforms to the idea that increased
However, we seek to address the question of how much
accountability in the driver route choice response process
performance increase is bestowed by frequent rather than
will lead to signiﬁcantly different solution strategies.
infrequent lane reconﬁguration. Infrequent reconﬁguration
While the bi-level and ILP solutions show signiﬁcant
(reconﬁguration period = 3,4,5) shows only modest im-
differences, we expect that both solutions are valid under
provements over the static conﬁguration, approximately 11%
the different assumptions each model makes. Because of the
throughput gain. However, decreasing our reconﬁguration pe-
highly redundant grid based network topology, we expect that
riod to 2, we see a 32% performance gain over the static case.
(source, destination) demands may be satisﬁed by multiple
Finally, the fully dynamic reconﬁguration period 1 trafﬁc
possible paths. We hypothesize that the discrepancy in the
manager provides a 72% increase in throughput compared
solutions results from different paths being utilized for the
to the static network. This trend suggests that the large
same set of demands. However, while both models are ﬁnd-
gains in trafﬁc efﬁciency are achievable with reconﬁguration
ing valid solutions given their constraints, we should expectthe bi-level formulation’s solution to be more realistic in an
1Incorporating non-zero demands is straightforward to implement but
could have resulted in the ILP solver being unable to ﬁnd valid solutions
actual trafﬁc network because the bi-level model incorporates
many aspects of real trafﬁc such as congestion, equilibrium
agenda includes designing car control and intersection con-
CORRELATION COEFFICIENTS OF ILP AND GA SOLUTION VECTORS
trol policies for dynamic lane reversal.
Acknowledgments. This work has taken place in part in theLearning Agents Research Group (LARG) at UT Austin. LARG
and speed limits. On the other hand, the bi-level model can
research is supported in part by NSF (IIS-0917122), ONR (N00014-
guarantee only approximate solutions while the ILP solver
09-1-0658), and the FHWA (DTFH61-07-H-00030).
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