Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic

Abstract

A coordination of vehicle flows is usually reached by a cyclical operation of traffic lights, and by synchronizing these cycles. The typical conditions, however, for which traffic lights are normally optimized for, never occur exactly. Large fluctuations in the number of vehicles arriving during one cycle time may lead to an inefficient usage of green times, which are often either too short or too long. The method we propose here allows for variable adjustments not only of the duration, but also of the order of green phases, while it reaches at least the same intersection throughput capacity as an optimized fixed-time controller. This is particularly important, when intersections are highly saturated, road networks are heterogeneous, or if public transport is to be prioritized. We reach the stabilization of queues and red-time durations by a decentralized supervision of (potentially unstable) locally optimizing traffic light controllers. The proposed supervisory concept makes sure that all network flows get a green light regularly and long enough. In addition, it enables flexible responses to local fluctuations in the demand, and it favors a self-organized coordination of traffic flows. In a simulation of the city center of Dresden, Germany, we could compare the proposed concept with an adaptive state-of-the-art controller (which has been optimized within the same simulation suite and includes green waves). Results indicate that not only the mean value of travel times is reduced, but also in their variance, which is positive for the reliability of public transport and individual trip scheduling.

Full text

Self-Stabilizing Decentralized Signal Control
of Realistic, Saturated Network Traffic
Stefan L¨ammer
Dresden University of Technology, W¨urzburger Str. 35, 01187 Dresden, Germany
stefan.laemmer@tu-dresden.de
Dirk Helbing
ETH Zurich, CLU E1, Clausiusstrasse 50, 8092, Zurich, Switzerland
Santa Fe Institute, 1399 Hyde Park Road, NM 87501 Santa Fe, USA
dhelbing@ethz.ch
A coordination of vehicle flows is usually reached by a cyclical operation of traffic lights, and by synchronizing
these cycles. The typical conditions, however, for which traffic lights are normally optimized for, never occur
exactly. Large fluctuations in the number of vehicles arriving during one cycle time may lead to an inefficient
usage of green times, which are often either too short or too long. The method we propose here allows for
variable adjustments not only of the duration, but also of the order of green phases, while it reaches at
least the same intersection throughput capacity as an optimized fixed-time controller. This is particularly
important, when intersections are highly saturated, road networks are heterogeneous, or if public transport
is to be prioritized.
We reach the stabilization of queues and red-time durations by a decentralized supervision of (potentially
unstable) locally optimizing traffic light controllers. The proposed supervisory concept makes sure that all
network flows get a green light regularly and long enough. In addition, it enables flexible responses to local
fluctuations in the demand, and it favors a self-organized coordination of traffic flows. In a simulation of the
city center of Dresden, Germany, we could compare the proposed concept with an adaptive state-of-the-art
controller (which has been optimized within the same simulation suite and includes green waves). Results
indicate that not only the mean value of travel times is reduced, but also in their variance, which is positive
for the reliability of public transport and individual trip scheduling.
Key words : traffic signal control; dynamic instability; public transport prioritization; self-organization;
decentralized coordination; intelligent transportation system
1. Introduction
When multi-directional traffic streams need to
cross each other at intersections of a road net-
work, traffic lights can enhance the safety and
throughput, if operated with proper control
strategies.
1.1. Traffic Light Control
Classical control concepts assume a cyclic oper-
ation of the traffic lights, where the flows of
different directions are served periodically. The
cycle time and the parameters defining the
green time splits determine the intersection
throughput, which can be optimized, within
certain bounds, by choosing these parameters
according to the formulas developed in Refs.
Webster (1958), Miller (1963), Little (1966),
Gartner et al. (1975a). A coordination of neigh-
boring intersections can be achieved by oper-
ating them with the same cycle time and by
adjusting the offsets between them appropri-
ately (Gartner et al. (1975b)). If these param-
eters are optimized for an assumed stationary
traffic demand and kept fixed for a certain time
period, e.g. during the morning rush-hour, we
refer to this control principle as “fixed-time con-
trol”.
More advanced concepts aim at an adapta-
tion to slow or systematic variations of the
average traffic demand. SCOOT (Hunt et al.
(1981)) and SCATS (Lowrie (1982)) are promi-
nent examples for this. The on-line (or real-
time) optimization of signal plans, however, still
just varies the paradigm of a mainly cyclic
traffic light operation. Alternative solutions,
in which traffic flows are served in a differ-
ent order with different frequencies, are sim-
ply neglected. However, a cycle-based coordina-
tion does not only restrict the possible solution
space. The traffic lights may also not be able
1

2 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
to respond to random or irregular variations in
demand quickly enough. Due to stochastic fluc-
tuations, even optimized green times are usually
either too short (creating multiple red lights for
queued vehicles that could not be served) or too
long (creating unnecessary delays to vehicles of
other flow directions). Traffic-responsive control
strategies are essential to overcome these prob-
lems and to reduce travel times, fuel consump-
tion, and the emission of pollutants effectively.
1.2. Traffic-responsive Strategies
Also when public transport is prioritized, it is
not sufficient to operate traffic lights according
to pre-specified signal plans. In order to react
to prioritization requests by public transport
vehicles that serve a large number of passen-
gers, traffic control must enable a high degree
of flexibility and responsiveness. According to
the Transit Signal Priority Handbook (Smith
et al. (2005)) and others (Skabardonis (2000)),
a pre-condition for an active and well-balanced
prioritization of public transport is that traffic
lights facilitate variable green times and a vari-
able selection of green phases.
In the last years, a huge variety of traffic-
responsive control strategies has been proposed.
One of the most widely used approaches is
called rolling-horizon optimization, where a
repetitive re-optimization of the signal switch-
ing sequences shall enable an adaptation to
changing traffic conditions. It is applied in
several traffic light controllers, e.g. OPAC
(Gartner (1983)), PRODYN (Henry et al.
(1983)), UTOPIA (Mauro and Taranto (1990)),
CRONOS (Boillot et al. (1992)), ALLONS-D
(Porche et al. (1996)), and RHODES (Mir-
chandani and Head (2001)). They mainly
differ in how the optimization problem is
solved: OPAC basically enumerates the solu-
tion space, PRODYN applies an efficient heuris-
tics, and ALLONS-D searches a complex deci-
sion tree using back-tracking. The associated
dynamic programming techniques could be fur-
ther improved if combined with online learning
techniques, as Cai et al. (2009) propose. The
resulting signal plans are, in general, acyclic
with no fixed order of green phases. Another
real-time strategy is TUC (Papageorgiou et al.
(2003), Aboudolas et al. (2009)), which is based
on a store-and-forward modeling approach and
tries to balance queue lengths by flexible adjust-
ments of green times. Alternative approaches
treat intersections as autonomous, cooperative
agents in a multi-agent system (France and
Ghorbani (2003)), for an overview see Baz-
zan (2009). Other approaches are based on
autonomic and organic computing (Prothmann
et al. (2009)), where a coordination of the
traffic lights is reached by local communica-
tion among neighboring intersections. A simi-
lar synchronization principle, in which intersec-
tions are treated as dynamically coupled oscil-
lators, was proposed by L¨ammer et al. (2006).
Furthermore, fuzzy-logic-based traffic signaling
strategies have been suggested in order to cope
with imprecise, uncertain, or ambiguous infor-
mation (Trabia et al. (1999), Rahman and
Ratrout (2009)). Genetic algorithms are used
to apply multi-goal optimization , focusing on
the minimization of emissions and fuel con-
sumption (Park et al. (2009)) or on a dynamic
re-optimization of green-wave-programs (Braun
et al. (2008)).
Instead of evaluating complex decision trees
and different switching sequences, an alterna-
tive optimization method was recently pro-
posed, which is based on a dynamic prioriti-
zation of traffic flows (L¨ammer and Helbing
(2008)). The formula, according to which the
priority π of each flow is calculated, considers
the queue lengths as well as the anticipated
arrival times of vehicles, which are determined
from the measured inflows into road sections
and the outflows from them (L¨ammer et al.
(2007)). Assuming that there are no other than
the anticipated arrivals, the ranks of the π-
values define the order of switching of the traffic
lights. To minimize the total delay of all vehi-
cles at an intersection, the controller simply has
to give a green light to the flow with the highest
priority.
When applying this local optimizing tech-
nique, signal plans are not based on cyclic con-
trol schemes determined from average traffic
conditions, but they respond instead to actual
real-time detector data. This makes the traf-
fic control more flexible with respect to local

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 3
demands and more robust to perturbations
(such as variations in the arrival flows, turn-
ing rates, accidents, building sites, or special
events).
1.3. The Price of Flexibility
The introduction of flexibility and traffic-
responsiveness, unfortunately, contradicts strate-
gic concepts such as the idea of a network-
wide coordination. Friedrich (2007) calls this
the “jungle-principle”, when each intersection
in the network controls traffic as is seems
best from its local perspective. From a game-
theoretical point of view, selfish optimization
does not necessarily establish the system opti-
mum – it could even create a poor performance
as is known from the “tragedy of the commons”
(L¨ammer and Helbing (2008)). If traffic lights
mainly respond to local demands, each intersec-
tion just reacts to the traffic coming from the
neighboring intersections, which consequently
makes the dynamic inter-dependencies in the
network impossible to predict. We agree with
Papageorgiou et al. (2003) that “the proper-
ties of a completely decentralized operation
(e.g., independent algorithm application at each
intersection) are currently not fully analyzed or
understood”. Generally, cyclic control schemes
tend to lack flexibility, while acyclic schemes
tend to lack coordination, despite best efforts.
Both lead to efficiency losses, which have pre-
vented a widespread practical application of
locally optimizing strategies so far. Without
their appropriate improvement, the inefficient
usage of intersection capacities can cause situ-
ations, in which vehicle queues will eventually
grow longer and longer as depicted in Fig.1.
1.4. Instability Problems
When locally optimizing traffic lights fail to
serve each traffic flow regularly and long
enough, instability problems are likely to occur.
Reasons for an ineffective usage of intersection
capacities are manifold. If traffic lights switch
too often, for example, this may result in a
lower throughput as there is no service during
switching time periods, also called setup times
(Duenyas and van Oyen (1996)). But even if
setup times are not a problem, a locally optimiz-
ing controller could assign too long green times
Figure 1 Illustration of two unstable (i.e. eventu-
ally growing) vehicle queues at a signalized
intersection.
to highly prioritized traffic flows and, even-
tually, marginalize side flows by too long red
times. Moreover, the unpredictability of traffic
can systematically mislead short-sighted opti-
mization algorithms, especially at higher satu-
ration levels or if the intersections. For mate-
rial flow networks, Kumar and Seidman (1990)
have discovered that clearing strategies of dis-
tributed controllers may systematically reduce
the system throughput if demand is high, lead-
ing to a growth of queues and service times. In
the Appendix, we prove this analytically for a
road network with two independently controlled
intersections.
According to the definition of Perkins and
Kumar (1989), a controlled queueing network
is stable, if all queues lengths remain bounded
for all times. Of course, all traffic jams will
resolve sooner or later, e.g. at night or when
drivers choose alternative routes. The notation
of stability, however, allows us to distinguish
between those cases, in which congestion has
a tendency to grow and those cases, in which
the controllers succeed to avoid growing conges-
tion or to dissolve it after a while. Of course,
no controller can prevent queues from growing
if saturation is too high. It would be reason-
able, however, to demand bounded queues in
all situations, in which conventional fixed-time
programs exist for stationary inflows. We will
therefore use the following terminology: As soon
as there are traffic demands, for which the vehi-
cle queues remain bounded with a fixed-time
control, but not if operated with a locally opti-
mizing controller, the latter is “unstable” and
potentially inferior.

4 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
It seems hopeless to identify all possible
sources of instability in a road network of rea-
sonable size, and there is no chance to imple-
ment specific counter-measures for all of them.
Instead, we will seek for a general decentralized
stabilization mechanism that maintains the per-
formance of the road network even when each
intersection is controlled locally.
1.5. Outline
The paper is structured as follows. In Sec. 2
we develop a distributed stabilizing mechanism,
which assists a locally optimizing controller
in providing green times frequently and long
enough. This allows an underlying locally opti-
mizing controller to handle the same amount
of traffic as a designated fixed-time controller,
but also facilitates flexible responses to local
fluctuations in demand. The parameters of the
proposed mechanism are discussed in Sec. 3. In
Sec. 4, we test our stabilization method with a
realistic network simulation of the city center of
Dresden, Germany, and compare it to the cur-
rently implemented state-of-the-art controller.
As our results indicate, locally optimizing traffic
lights can, if well stabilized, lead to a superior
performance for all modes of transport.
Compared to previous work, L¨ammer and
Helbing (2008), the current paper presents an
extension of the stabilization mechanism as well
as the results of a realistic network simulation
including public transport prioritization and
pedestrian flows.
2. Stabilization Mechanism
When it is possible to operate a network with
stable fixed-time programs, why should it not
be possible to operate the intersections with
properly designed traffic-responsive strategies
and to serve the same amount of vehicles fre-
quently enough? Moreover, why should it not
be possible to utilize excess capacities for local
optimization by means of green time extensions
or additional green phases?
2.1. Demand-Driven Service Concept
For fixed-time controllers, a desired throughput
can be achieved by adjusting cycle times and
splits accordingly. Although offsets and coor-
dination are important for service quality and
waiting times as well, we will neglect these for
the time being. Let us instead discuss the essen-
tials of throughput and stabilization, and derive
an alternative service concept, into which we
will afterwards incorporate aspects of optimiza-
tion.
When the length and frequency of green
times is fixed, but the number of vehicles arriv-
ing during one cycle time is not, the efficiency
of the green times varies randomly. Therefore,
stochastic variations in the arrival and turn-
ing rates can decrease the efficiency of fixed-
time controls considerably (van den Broek et al.
(2006)). To overcome this, we propose to mod-
ify the underlying operation principle as fol-
lows (see Fig. 2): Instead of waiting for a cer-
tain point in time before switching to green,
we now wait for a critical number of vehicles
ready for service at maximum rate which given
by the saturation flow, i.e. the outflow from con-
gested traffic. The critical number of vehicles,
our demand-driven service concept is waiting
for before giving a green light, is specified by
a threshold function. For example, if the criti-
cal number corresponds to be the average num-
ber of vehicles arriving during one cycle time,
the arriving vehicles will accumulate up to this
number and initiate a green light once within a
cycle time on average.
If this threshold is applied, the resulting local
traffic control is capable of serving the same
amount of traffic as the corresponding fixed-
time controlled intersection. Therefore, it will
be stable according to the definition in Sec. 1.4.
The major difference is that both, the green
time as well as the service time interval, are
random variables that can compensate for vari-
ations in the arrival flow. The degree of traffic-
responsiveness can be adjusted by the slope of
the threshold function.
Before we discuss the properties of the flexi-
ble control principle in more detail, let us briefly
explain how the proposed concept can be used
in combination with locally optimizing traffic
light controllers to stabilize them. By giving a
green light after a critical number of vehicles
is waiting for service, red lights are terminated

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 5
Figure 2 For the classical fixed-time service concept (left), the time interval between two consecutive green
times (service interval) for a traffic flow corresponds to the fixed cycle time Z (vertical line), while
the number of vehicles served within a cycle varies due to the randomness of the arrival process (see
diagonal lines). For the proposed demand-driven service concept (right), in contrast, a green time starts
after a critical number of vehicles has accumulated, which can be served during the green time with
maximum rate. Fluctuations in the arrival process are compensated for by variable service intervals (see
projections of the linearly falling threshold function). The efficient usage of green times and the ability
not to exceed maximum red times even in the absence of arrivals (see Sec. 2.3) make the proposed
concept suitable to stabilize locally optimizing strategies.
when this is needed to clear the queues reg-
ularly. In other words, a green light is given
whenever there is a definite demand for it. This
particular property makes the proposed con-
cept suitable as a stabilizing supervisor, which
assists a (possibly unstable) locally optimizing
traffic light controller. The supervisor deter-
mines what traffic flows need to get a green light
before which point in time and, thereby, formu-
lates stabilizing constraints. Stable and regular
behavior is guaranteed, if the local optimization
of green times is subject to these constraints.
Note that the locally optimizing controller is
still free to start green times earlier or to extend
them if possible. As long as it assigns the green
times not later and not shorter than the stabi-
lizing supervisor would assign them, the service
intervals as well as the queue lengths are kept
bounded.
2.2. Supervisor
As indicated in Fig. 3, the stabilizing super-
visor can be understood as a separate pro-
cess that observes the current traffic state
and puts constraints on the signal timing of
the locally optimizing controller. Supervisory
control has been established as a concept to
stabilize nonlinear systems. Its purpose is to
identify critical states of the system and to
provide appropriate measures driving the sys-
tem back into a desired state. Previous work
has applied distributed supervisory principles
Figure 3 The stabilizing supervisor observes the sig-
nal timing of a (possibly unstable) locally
optimizing controller as well as the cur-
rent traffic situation. Stabilizing constraints
are imposed, which determine the latest
start and the minimum duration of subse-
quent green phases. As long as the local
optimization is subject to these constraints,
both, red times and queue lengths remain
bounded.
to stabilize material flow networks (for an
overview see Bramson (2008) and references
therein). Note, however, that applying these
principles to traffic control would not limit
red light durations and service intervals, which
makes their direct transfer useless. In contrast,
the supervisory mechanism we propose in this
paper ensures that all traffic flows are served
frequently and long enough.
We formulate the supervisor rules as follows:
1. A traffic flow is identified to be critical,
if its corresponding point in the state space of
Fig. 2 exceeds the threshold function (to be
defined in Sec. 2.3). In that case, the flow is
served as soon as possible.

6 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
2. If there is more than one flow in a critical
state, these flows have to be served in the order
in which they became critical.
3. A traffic flow becomes uncritical again
when it has been served by a green time that an
associated fixed-time program would attribute
to it or when its queue has been cleared before
that.
4. If no flows are in a critical state, the con-
troller performs a local optimization.
A fifth rule is discussed at the end of Sec. 2.3.
These rules are independent of the underly-
ing optimization procedure. Hence, the stabiliz-
ing supervisor can be combined with any local
controller, which can be subject to the con-
straints defined by these rules. Depending on
the particular implementation, signal timings
that lead to critical states can be penalized in
the goal function, for example, or just excluded
from the decision tree. As the points in time,
at which the vehicle queues become critical, can
be anticipated relatively early, the controller
can harmonically impose its constraints on the
optimization procedure. To give an example,
instead of serving a minor flow exactly at the
same time when it becomes critical, the opti-
mization strategy can also choose to schedule it
earlier, if this helps to avoid conflicts with large
vehicle platoons on other streets.
For a large range of traffic demands, we
achieve a load-dependent interplay between
optimization and stabilization. The difference
between the intersection capacity and traffic
demand is not necessarily needed to maintain
stability. This so-called “excess capacity” can
be freely used for green time extensions or addi-
tional switching operations by the locally opti-
mizing controller. If an intersection becomes
more saturated, however, the supervisory mech-
anism steers the optimization strategy tighter.
In case of over-saturation, the optimizing con-
troller has no freedom anymore, and it must
serve the flows according to an associated fixed-
time program.
In summary, if supervised, a locally opti-
mizing controller serves the same amount of
traffic as a fixed-time controller and can, in
addition, minimize travel times and queues due
to flexible adjustments to instantaneous local
Figure 4 Saturation-dependent queue lengths at
an isolated four-armed intersection with
constant inflows under three different
control strategies. In comparison with
fixed-time control, the local optimization
strategy (based on dynamic prioritization
after L¨ammer and Helbing (2008)) per-
forms significantly better at low satura-
tion levels, but becomes unstable at satu-
ration levels above 70%. When stabilized
with the proposed supervisor, however, the
same strategy manages to serve all flows
frequently and long enough, resulting in
lower queue lengths also at higher satura-
tion levels.
demands. The results in Fig. 4 indicate a sig-
nificant improvement of the proposed strategy
compared to fixed-time or non-stabilized con-
trollers.
2.3. Choice of the Threshold Function
The choice of the threshold function in Figs. 2
and 5 has a major impact on the regularity
of service intervals. According to the supervi-
sor rules discussed above, the service of a traf-
fic flow is triggered as soon as its correspond-
ing point in the state space (ˆz, ˆn) exceeds the
threshold function. In order to derive a suit-
able specification of the threshold function, we
first assume that the average arrival rate of that
traffic flow as well as the green times of an asso-
ciated stable fixed-time program are given. We
require that
(i) each traffic flow shall be served once, on
average, within a desired service interval Z, and
(ii) each traffic flow has to be served once, at
least, after a maximum red time period Z
max
.
The first criterion shall ensure a desired level
of regularity, and the second one guarantee the
compliance with maximum red light durations
for the sake of road safety.

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 7
The supervisory mechanism needs to antici-
pate the service interval ˆz as well as the num-
ber ˆn of vehicles to be served within a sub-
sequent green time. With suitable procedures,
their values can be anticipated already during
the preceding red time period L¨ammer et al.
(2007), Helbing et al. (2007). Both variables,
ˆz and ˆn, refer to the earliest time point t
0
at
which the corresponding vehicle queue could be
cleared. The service interval ˆz corresponds to
the time interval since the end of the last green
time until this time point t
0
. Likewise, ˆn rep-
resents the number of vehicles that need to be
served until the queue is cleared, considering
new arrivals of vehicles during the service pro-
cess. The interrelation between these two vari-
ables is that ˆn captures the number of vehicles
that arrive during the service interval ˆz. Also
note that the green time required to resolve
the queue increases with every vehicle arriving
during the service process. This implies several
interesting features. The arrival of a vehicle pla-
toon, for example, lets ˆn abruptly jump to a
higher value, since it becomes possible to serve
considerably more vehicles at a maximum rate.
If that jump in ˆn triggers the green light, the
platoon is served without being stopped as it
is intended by “green wave” control. Based on
these anticipative characteristics, the proposed
concept favors flexibly coordinated, platoon-
oriented service patterns.
According to the rules formulated in Sec. 2.2,
a traffic flow should be served by a green light
as soon as there is a critical number n(ˆz) of
vehicles ready to be served at maximum rate,
e.g.
ˆn ≥ n(ˆz). (1)
Herein, n(ˆz) denotes the threshold function
depicted in Fig. 5. Let us now discuss three
important constraints for the choice of n(ˆz).
First of all, it must be ensured that every traffic
flow will eventually become critical as red time
increases, and that it cannot become uncritical
before the vehicle queue is served. Since ˆz and
ˆn are both non-decreasing during red time peri-
ods, the threshold function n(ˆz) must be strictly
decreasing in order to be crossed by the state
trajectory (ˆz, ˆn) exactly once, which implies
dn/dˆz < 0. (2)
Second, the threshold function n(ˆz) shall be
exceeded when there are as many vehicles ready
to be served at maximum rate as are arriving
on average within the desired service interval Z.
In the following, we identify Z with the cycle
time of the associated stable fixed-time pro-
gram. Then the specification
n(Z) = qZ, (3)
where q denotes the average arrival rate, ensures
that the service interval will, on average, be ˆz =
Z as required.
Above we have discussed that it is favorable
to have a variable service interval ˆz in order to
be able to adapt to irregular inflows. For exam-
ple, our mechanism delays the start of a green
time period automatically, if vehicle platoons
have been delayed at an upstream intersection.
In order to ensure that a maximum red time
period Z
max
(with Z
max
> Z) is never exceeded,
even not in the absence or non-detection of
vehicle arrivals, we finally require the threshold
function to meet the condition
n(Z
max
) = 0. (4)
That means, n(ˆz) falls below zero as the service
interval ˆz exceeds Z
max
. This makes Z
max
the
latest possible time point after which a traffic
flow becomes critical and, therefore, is served.
In order to fulfill road safety regulations, one
can extend the previously discussed supervisory
rules by requiring that the service of flows with
ˆz > Z
max
must be prioritized to the service of
other critical flows.
3. Parameter Choice
The parameters of the proposed stabilization
mechanism are the desired service interval Z,
the green times of a stable fixed-time program
with cycle time Z, as well as the maximum red
time period Z
max
.
3.1. Desired Service Interval
The choice of the service interval Z deter-
mines how frequently all traffic flows are served,
even in cases of over-saturation, large variations
in the arrival flows or frequent prioritization
requests of public transport vehicles. In such

8 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
max
^
^
max
^
Figure 5 A well-specified threshold function n(ˆz) is monotonously decreasing and goes through the two marked
points given by Eqs. (3) and (4).
cases, the supervisor will enforce each traffic
flow to be served once, on average, within the
time period Z. The green times and, thus, the
service capacities reserved for the traffic flows
correspond to those of the associated fixed-time
program. Therefore, our decentralized control
approach reaches the same service capacity in
saturated flow conditions.
3.2. Green Time Durations
Modern urban planning aims at a bundling of
traffic on arterial roads and to restrict traffic in
residential areas, which is classically reached by
a centrally imposed allocation or restriction of
road capacities through green times (see Wong
and Yang (1997), Yang et al. (2000)). However,
the same goal can be achieved by decentral-
ized control concepts, if the supervisory mech-
anism is parameterized as follows: Minor flows
are given green times just long enough to sat-
isfy the corresponding average demand, while
main flows with a large importance for the over-
all network traffic are given all the remaining
green time. An example of how this helps to
prevent spill-overs on main roads and to resolve
their queues more efficiently is given in Fig.6.
Note that the difference between Z and the
minimum possible cycle time is the amount of
time the locally optimizing controller can use
for travel time optimization, e.g. by green time
extensions or additional switches. If the traffic
demand is not too high, this may lead to short,
but frequent green phases. An illustration of
how a supervised locally optimizing controller
utilizes excess capacities for flexible switching
sequences is shown in Fig. 7.
3
4
1
2
Figure 7 Periodic green time sequences emerging
at a four-armed intersection with mutually
conflicting traffic flows for different levels of
saturation, assuming constant arrival rates.
At low saturation levels, the local opti-
mization (based on dynamic prioritization
after L¨ammer and Helbing (2008)) serves
the traffic flows as frequently as possible.
When the saturation level increases, how-
ever, the multi-lane flows 1 and 2 with a
higher outflow capacity are selected more
often than the minor flows 3 and 4. Note
that the desired service interval Z is never
exceeded. The service of the minor traf-
fic flows is initiated by the proposed super-
visory mechanism, if needed (indicated by
bold frames). At maximum saturation, the
flows are selected similar to a fixed-time
control with cycle time Z.
3.3. Maximum Red Time Periods
The maximum red time period Z
max
applies in
cases in which neither the optimizing controller
nor the Z-mechanism of the supervisor selects
each traffic flows frequently and long enough for
service. This, typically happens when arrivals
are absent or if they are not detected. Then
the supervisor initiates the minimum admis-
sible green time to the neglected traffic flow
after a maximum red time period of Z
max
. Fur-

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 9
2
1
2
1
Side street
2
Main street 1
2
1
2
1
Figure 6 Temporal evolution of queue lengths (left) and corresponding state trajectory (right) at a two-armed
intersection with constant arrival rates. Initially, both queues are critical (bold lines), but the stabilizing
supervisor manages to reduce the queues until local optimization can take over (thin lines). Top: If the
green times are assigned proportionally to the saturation of the flows, both queues resolve equally fast.
Bottom: The concept we propose allows for a distinction between streets of different importance for
the overall network flows. By assigning only the minimum stabilizing green time to the less important
side street (dashed line) and all the remaining green time to the more important two-lane main street
(continuous line), the queue in the side street is not resolved before the situation has relaxed on the
main street. The used parameters are: Z = 90 s, Z
max
= 120 s, saturation flow rate S = 1/1.8 s per lane,
constant arrival rate q
1
= q
2
= 0.4 S per lane, and setup time τ = 5 s.
thermore, the closer the values of Z and Z
max
are, the steeper declines the threshold func-
tion. Therefore, in the limiting case Z → Z
max
,
the service concept becomes analogous to a
fixed-time controller, compare Figs. 2 and 5.
In contrast, when choosing very large values of
Z
max
, the service concept becomes purely traf-
fic responsive. In order to reach a maximum
level of traffic-responsiveness, we recommend to
set Z
max
to the largest acceptable value that is
compatible with regulations regarding the max-
imum red times.
The decentralized stabilizing mechanism pre-
sented above stabilizes queue lengths and ser-
vice intervals at all intersections in a network.
While the intersections may still be dynamically
coupled by the traffic flows between them, the
supervisory control principle guarantees that it
does not lead to a loss of throughput capacity
(see Appendix). Instead, the dynamic coupling
may now trigger coordinated service patterns.
In contrast to a pre-optimized coordination
based on periodic green waves, the demand-
driven service concept favors a self-organized
coordination based on local responses to travel-
ing vehicle platoons. Larger platoons are mostly
served without stopping, simply because giv-
ing them a green light immediately before
they arrive at an intersection helps minimiz-
ing delays. An illustrative example of platoon
propagation along an arterial road is depicted
in Fig. 8.
4. Real-world Network
4.1. Self-Control
The proposed supervisory mechanism (as pre-
sented in Sec. 2) is capable of stabilizing locally
optimizing traffic light controllers and to sup-
port them by an efficient throughput man-

10 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
Figure 8 Green-bands along an arterial road with 10 irregularly spaced intersections, mutually conflicting traf-
fic flows, and stochastic arrivals at the network boundaries. Left: The fixed-time control implements
regular green waves traveling from West to East with a cycle time of 120 seconds. The opposite flow
direction (for which the trajectory of a single vehicle is indicated by a black solid line) experiences large
delays compared to free traffic conditions (dashed line). Right: The flexible self-control, in contrast,
assigns shorter green times at a higher frequency and, thereby, serves the same amount of traffic with
significantly reduced average travel times (L¨ammer (2007)). The intersection controllers are loosely
coupled by vehicle platoons traveling between them. This creates the emergence of coordinated service
patterns.
agement, stabilizing queue lengths and red
times. In the following simulation study, we
have added the proposed stabilization mecha-
nism to the potentially unstable local optimiza-
tion strategy proposed by L¨ammer and Helbing
(2008). This strategy was based on travel time
minimization, utilizing a short-term anticipa-
tion of arrivals (see L¨ammer et al. (2007)). Since
the combination of anticipation, optimization,
and stabilization leads to a purely autonomous,
demand-responsive behavior of the traffic lights
at each intersection, we refer to this control
principle as “flexible self-control”. This new
control principle has furthermore been general-
ized such that complicated and combined ser-
vices of traffic flows, pedestrian flows, and the
prioritization of public transport can be consid-
ered (Helbing and L¨ammer (2005)).
The traffic and queueing dynamics in this
study have been simulated using the software
package PTV Vissim. The algorithms of the
self-control mechanism have been implemented
in Java. The Java application periodically reads
in the vehicle positions and writes back the
updated states of the traffic lights using a COM
interface. The following comparative analysis is
based on simulation runs for a typical afternoon
rush-hour. Related simulation videos are avail-
able at www.stefanlaemmer.de/selfcontrol.
4.2. Area of Investigation
The road network considered in the following
consists of 13 traffic-light-controlled intersec-
tions (see Fig. 9), which are part of Dresden’s
city center. This area was chosen as it experi-
ences unsatisfactory delays and notorious coor-
dination problems despite the use of an adap-
tive state-of-the-art-control. Two parallel main
roads pass the train station “Dresden Mitte” on
each side, which is used by more than 13.000
passengers on an average day. No less than 7
bus and tram lines cross the network every 10
minutes in opposite directions (Fig. 10), and 68
pedestrian crossings lie within that area. The
intersections are spaced very irregularly. Some
of them are closer than 100m. Due to the het-
erogeneity of transport, the irregularity of the

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 11
Figure 9 Illustration of 13 traffic-light-controlled intersections (see circles) in the city center of Dresden, Ger-
many, for which the proposed traffic light control is simulated. The particular challenge is to coordinate
heterogeneous traffic streams in an irregular road network topology and to prioritize several public
transport lines at the same time (see Fig. 10).
Figure 10 Several public transportation lines (trams
and buses) cross the network every 10
minutes in each direction. An exception
is the “CarGoTram”, which was simulated
to run about once per hour.
network topology, and the relatively high traffic
load, local traffic authorities agree that this part
of Dresden is most challenging to control (see
the corresponding statements in Ref. L¨ammer
et al. (2009)).
Today, the traffic lights in this area are oper-
ated with a centralized traffic-adaptive state-
of-the-art control system of type VS-PLUS
(see www.vs-plus.com). The intersections are
coordinated such that green waves propagate
on both main arterial roads in both direc-
tions, and also on one of the orthogonal roads
(Sch¨aferstraße/Schweriner Straße). The com-
mon cycle time is 100 s most of the day. For
Figure 11 In comparison with the currently used
state-of-the-art control (left columns),
the flexible self-control (right columns)
reduces the average delay for all modes of
transport.
trams and buses, there are two time slots
reserved per cycle, in which they can be sched-
uled upon request. Pedestrians get a green light
with a delay and only on demand by pressing a
button.
4.3. Results
Note that we are using the same simulation
suite and the same empirically measured traffic
flow data, for which the currently used state-
of-the-art control has been optimized. A com-
parison with our proposed flexible self-control
strategy is particularly interesting, as both
traffic light control approaches follow different
“philosophies”: The first one tries to achieve
globally coordinated network flows, while the
second one aims at locally minimizing the wait-
ing times of all modes of transport.

12 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
Figure 12 The reduction of average delays that
we observed in the reference scenario (a
typical afternoon rush-hour, see marked
points), is also found, if the overall traffic
demand is varied.
As Fig. 11 shows, the traffic-light-induced
delay of trams and buses could be reduced by
more than 50 percent. Even cars and trucks,
which are currently served by green waves, can
travel faster through the network, when the
flexible self-control is applied. Pedestrians wait
36 percent less for the next green light.
The significant reductions in waiting times
for all modes of transport are also found for
different overall traffic demands, as Fig. 12 indi-
cates. This becomes possible due to the more
flexible, demand-oriented service concept, as we
will explain in the following.
For example, the flexible self-control typically
does not stop cars and trucks at the entrance
to the arterial roads, while the current system
builds up large queues that wait for the next
green wave (see Fig. 13). Instead, each inter-
section serves incoming vehicles as soon and as
often as possible, which favors the formation
of smaller vehicle platoons. Sometimes, these
small platoons are stopped at some downstream
intersection, for example, in favor of prioritizing
a tram or bus while large platoons are hardly
ever stopped. Buses and trams are weighted like
15 vehicles to reach a harmonic prioritization
of public transport. Gaps between the platoons
are utilized by the locally optimizing controller
to serve vehicle flows of side streets, left turn-
ers, or pedestrians. This principle may be called
“gap” or “chance management”.
Figure 14 Frequency distribution of red time dura-
tions at pedestrian crossings. The red
time periods of the self-controlled traf-
fic lights (bottom) are significantly shorter
and more naturally distributed, as the traf-
fic lights utilize randomly occurring gaps
in the traffic flows.
Pedestrians get a green light whenever the
time gap between the crossing vehicles is large
enough. In our model, pedestrian streams are
treated as virtual flows, e.g. assuming a pedes-
trian arrival every 10 seconds. As a conse-
quence, the “pressure” on the traffic lights is
permanently increasing during the red time.
The distribution of red times for pedestrians is
shown in Fig. 14. For the state-of-the-art con-
troller, the distribution shows several promi-
nent peaks, which reflect that the program
underlay certain temporal restrictions. In con-
trast, the proposed flexible self-control can fully
adopt to fluctuations in the traffic flows. The
resulting red time distribution appears more
natural and has, in addition, a significantly
smaller mean value.
In order to evaluate how the two control
strategies can deal with randomness, we varied
the random seed of the simulation. The result
of 24 independent simulation runs are plotted
in Fig. 15. The variance in horizontal and ver-
tical direction indicates how sensitive each of
the strategies is to random fluctuations. The
variance of delays is much higher for the state-
of-the-art controller (horizontal axis). This can
be explained by the dominance of green waves,
which leave buses and trams only two time slots
per cycle to cross an intersection. If they missed
a time slot, they need to wait half a minute or

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 13
Figure 13 Vehicle trajectories along K¨onneritzstraße southbound. With the state-of-the-art control (top), large
vehicle platoons are propagated through the network by regular green waves. The price for this is the
formation of long vehicle queues at the entrance to the coordinated arterial. Self-controlled traffic
lights (bottom), in contrast, can serve the incoming vehicles as soon as possible. This concept favors
smaller vehicle platoons, which allows one to solve conflicts with crossing trams or buses in a much
more flexible way.
Figure 15 Average delay of public transport vehicles
in 24 simulation runs with different ran-
dom seeds. Public transport delays have a
very low variance, if traffic lights are self-
controlled (vertical axis), in contrast to
the results for the state-of-the-art control
obtained for the same random seeds (hor-
izontal axis). That is, buses and trams are
not only faster, but they are also on time
more often.
longer for the next one. Our self-control strat-
egy, in contrast, tries to give buses and trams a
green light as soon as they approach an intersec-
tion. This means that public transport vehicles
are not only faster, but also keep their schedule
more reliably. Note that we do not require any
sophisticated decision logic to handle all differ-
ent modes of transport and to reduce the aver-
age of the travel times as well as their variance.
5. Summary and Discussion
Due to limited prognosis horizons and dynamic
feedback loops in the network, locally optimiz-
ing control strategies are facing problems to find
optimal solutions for a traffic network. There-
fore, we have proposed a decentralized stabiliza-
tion mechanism which ensures that all traffic
flows get at least the same green time as a stable
fixed-time schedule would attribute to them. In
contrast to a fixed-time controller, however, the
green times are requested only when there is
definite demand for them, the cycle time is not
fixed, and the service is not necessarily periodic.
The “cycle time” rather becomes a random vari-
able, which is distributed around a desired ser-
vice interval Z and limited by a maximum red
time period Z
max
. An implementation of the
proposed self-control concept is based on the
following elements:
(i) Short-term anticipation of vehicle arrivals
by measurements of arrival and departure flows

14 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
of the road sections (see L¨ammer et al. (2007)).
Where measurements are unavailable, typical
flows can be assumed instead.
(ii) Local optimization of vehicle delays
based on a dynamic prioritization of traffic flows
(see L¨ammer and Helbing (2008)).
(iii) Local stabilization based on the super-
visory principle presented in Sec. 2.
The method can be augmented by real-time
simulations of network traffic (see Helbing et al.
(2007)), and it can also operate together with
existing traffic lights controlled by fixed-time or
other control schemes.
The acyclic operation of the traffic lights is an
essential feature of the proposed concept, which
compensates for random fluctuations in the
traffic flows, and which consequently reduces
the mean value as well as the variance of vehicle
and pedestrian queues and delay times. Other
interesting features are:
(i) The traffic light switching sequences are
not centrally imposed on the traffic flows, but
it is the traffic flows, which determine the pri-
oritization scheme. This is done “on the fly”.
(ii) Instead of implementing a particular
event-logic to solve conflicts between different
transportation modes, we just attribute higher
weights to trams and busses in the goal func-
tion, and thereby reach a public transport pri-
oritization, which harmonizes with other flows.
(iii) The service concept is demand-driven:
Order, beginning, and duration of green time
periods are flexibly adjusted to local traffic con-
ditions.
(iv) Instead of serving each flow exactly once
in a cycle time, self-controlled traffic lights tend
to give green lights to heavier traffic flows as
long as possible and more often.
(v) Growing queues are stabilized to avoid
spill-overs (see Mazloumian et al. (2010)).
(vi) The short-term anticipation of gaps and
platoons enables a flexible and efficient oppor-
tunity management. Pedestrians, for example,
get a green light whenever there is a larger gap
in the crossing vehicle stream.
(vii) Travel times are reduced on average and
become more predictable.
As we could demonstrate in the simulation of
a realistic, very irregular road network with 13
traffic-light-controlled intersections and many
crossing public transport lines, the proposed
flexible self-control mechanism provides a supe-
rior performance for all modes of transport as
compared to the currently implemented state-
of-the-art control.
Our results show that it is crucial to admit a
heterogeneity in the service intervals, which is
in sharp contrast to the classical synchroniza-
tion approach, requiring homogeneous (identi-
cal) cycle times. This heterogeneity allows to
resolve the conflict between two incompatible
optimization goals resulting from travel time
minimization in the network, namely through-
put maximization and avoidance of spill-over
effects. These conflicting goals are due to the
fact that traffic optimization involves two dif-
ferent kinds of capacities, the maximum depar-
ture flow from road sections and their stor-
age capacity for vehicles. Putting it differently,
travel time minimization in the network implies
a multi-goal optimization problem at the nodes,
which cannot be solved in a satisfactory way
by a prioritization, superposition, or weight-
ing of several goal functions for a node (or by
other classical approaches in multi-goal opti-
mization). It rather requires a suitable switch-
ing strategy between both optimization goals,
considering that the finite setup time τ discour-
ages frequent switching. It is interesting that
classical optimization approaches fail, when
applied to the nodes of a flow network with vari-
able arrival flows, since dynamical interaction
effects between nodes are highly significant. We
believe that this also applies to logistic and pro-
duction networks (Helbing (2003), Sipahi et al.
(2009)), and even to administrative processes.
The highest possible system performance can
only be reached, when each system element does
not behave strictly optimally, but if there is a
local variability that allows for a flexible coordi-
nation of the interacting elements (Helbing and
L¨ammer (2005)).
Acknowledgments
The authors are grateful for partial financial support
by the German Research Foundation (DFG projects
He 2789/5-1,2 and 8-1), the Volkswagen Foundation
(project I/82 697), and the Gottlieb Daimler and
Karl Benz Foundation (BioLogistics project).

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 15
1
2
Figure 16 Network of two intersections with two
symmetrically entangled traffic flows. The
same controller, which controlled each of
the intersections optimally in isolation,
will fail as soon as the intersections are
dynamically coupled by network flows.
Appendix. Analytical Example
An isolated intersection with two traffic flows and
no setup times can be optimally controlled by a
clearing policy, i.e. by clearing the queues in an
alternating way one after another without green
time extensions (see Hofri and Ross (1987), van
Oyen et al. (1992), Chase and Ramadge (1992)).
This very simple, purely traffic-responsive control
strategy provides the intersection with maximum
throughput, while is able to dynamically counterbal-
ance inflow variations by variations of green times,
i.e. longer queues are given longer green times auto-
matically. These favorable dynamic properties, how-
ever, get lost as soon as the intersection is coupled
with others in a network.
The following example refers to the simple net-
work of two intersections and two traffic flows shown
in Fig. 16. Similar network topologies have been
studied by Kumar and Seidman (1990), Rybko and
Stolyar (1992) in the context of manufacturing sys-
tems. In order to keep the analysis simple, we assume
constant arrival rates q
1
and q
2
and also neglect the
travel times between the intersections. The satura-
tion flow rates per lane are denoted by S. The net-
work flows can be served in a stable way with a fixed-
time program as long as none of the intersections is
saturated, i.e. as long as both conditions
q
1
2S
+
q
2
S
< 1 and
q
1
S
+
q
2
2S
< 1 (5)
hold. This is the case, for example, if we choose
q
1
= q
2
= 0.6 S. In order to show that the alternating
clearing of queues may produce dynamic instabili-
ties, we now analyze the evolution of the queues in
the network over one switching period. The analy-
sis is similar to the second example in Ref. Kumar
and Seidman (1990) and can be followed in Fig. 17.
Corresponding simulation videos are provided at
www.stefanlaemmer.de/stability.
Initially, there are n
1
(t
0
) vehicles waiting on the
main street at the left intersection, but nowhere else.
According to the switching rule under considera-
tion, the left intersection switches to green for the
queue on the two-lane main street. While the queue
is resolved within the next n
1
(t
0
)/(2S − q
1
) seconds,
vehicles proceed to the second intersection, where
they have to merge on the single, left-turning lane.
On that lane, a queue of
n
2
0
(t
1
) = n
1
(t
0
)
S
2S − q
1
(6)
vehicles builds up until t
1
requiring another
n
1
(t
0
)/(S − q
1
) seconds to resolve. Note that in time
interval [t
1
, t
2
] a problematic situation occurs: The
left intersection has successfully cleared the queue on
the main street, when there are no vehicles arriving
from the right, while the intersection on the right is
still busy serving vehicles turning left on 2’. In the
words of Kumar and Seidman (1990), the left inter-
section “is being starved of input”. Due to the lack
of alternatives, the left intersection continues serv-
ing the arrivals at a rate lower than the saturation
flow rate. Doing so, the left intersection is continu-
ously sending new vehicles to the right intersection
and, thereby, delaying the time point at which it will
eventually switch to the main street. In the mean-
time, there are
n
1
0
(t
2
) = n
1
(t
0
)
q
2
S − q
1
(7)
vehicles queued up on the main street 2 and nowhere
else in the network. Since this is symmetrical to the
initial state, we can conclude that, after a full control
cycle, there are
n
1
(t
4
) = a n
1
(t
0
) where a =
q
1
S − q
1
q
2
S − q
2
(8)
vehicles waiting in queue 1. Obviously, the queues
are reduced from one cycle to the next, only if a < 1.
This condition for the stability of the local clearing
policy can be rewritten as
q
1
S
+
q
2
S
< 1. (9)
It clearly covers a smaller region in the (q
1
, q
2
)-plane
compared to the condition (5) for a stable fixed-time
program, see Fig. 18. The example demonstrates
that a controller, which is optimal for a single inter-
section in isolation, may fail to be stable, if the inter-
section is linked with others in a network.
The same clearing policy can be stabilized, how-
ever, if combined with the supervisor proposed in
Sec. 2. Fig. 19 shows the resulting solution, in which
each traffic stream is being served sufficiently long
at least once within the desired service period of Z =
90 s.
The above example should learn us that even if
the traffic light controllers at the intersections oper-
ate independently of each other, the switching in the

16 L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic
2'
1'
0
3 41 2
2
1
2
2
1
1
1
2
1
2
1
2
2'
1'
2'
1'
Figure 17 Temporal evolution of the unstable queues at the two intersections of Fig. 16. If the queues are cleared
in an alternating way one after another, oscillations start to grow. The right intersection has most
of the traffic load in the interval [t
1
, t
2
], the left one in the interval [t
3
, t
4
]. At the end of a control
cycle, the queues are longer than before. The chosen parameters are: n
1
(t
0
) = 10, S = 1/1.8s, and
q
1
= q
2
= 0.6 S.
2'
1'
2
1
1
2
1
2
1
2
1
2
2'
1'
2'
1'
2'
1'
2'
1'
Figure 19 Temporal evolution of the stabilized queues for the Kumar-Seidman-network depicted in Fig. 17. The
proposed supervisory stabilization guarantees that all traffic flows are assigned green times often and
long enough. Note that the stationary solution extends over a period that is larger than Z, during
which each traffic flow is being served twice. However, the service interval itself (sum of red and
subsequent green time) never exceeds Z. The parameters used are: Z = 90 s, Z
max
= 120 s, S = 1/1.8 s,
and q
1
= q
2
= 0.6 S.

L¨ammer and Helbing: Self-Stabilizing Decentralized Signal Control of Realistic, Saturated Network Traffic 17
1
2
Figure 18 Illustration of the stability regions for an
optimal fixed-time controller and a local
clearing policy of the setup depicted in
Fig. 16. The local clearing policy does not
manage to provide a stable control for all
circumstances in which a stable fixed-time
program exists.
network is correlated, as the intersections are cou-
pled by the network flows. The switching sequence
of one intersection directly influences the arrival pat-
terns of vehicles at other intersections, and as soon
as its controllers dynamically respond to it, feed-
back loops are created. Any street segment with bidi-
rectional traffic flows can act as a feedback loop.
Clearing policies are stable only in acyclic networks
(Kumar and Seidman (1990), Rybko and Stolyar
(1992)). Road networks, however, are by definition
never acyclic, because one can travel in the net-
work from any node to any other and back. That
means, whenever traffic lights are programmed never
to interrupt the clearing of queues, there can be sit-
uations in which they become unstable.
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