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Max-Pressure Traffic Controller Based on Travel Times:
An Experimental Analysis
Pedro Mercadera, Wasim Uwayida, Jack Haddad*a
aTechnion–Israel Institute of Technology
Faculty of Civil and Environmental Engineering
Technion Sustainable Mobility and Robust Transportation (T-SMART) Laboratory
*Corresponding author:
Postal address: Technion City, Rabin building, Room 726, Israel.
Phone: +972 77 8871742, jh@technion.ac.il
Abstract
The traffic control of an arbitrary network of signalized intersections is con-
sidered. This work presents a new version of the recently proposed max-
pressure controller, also known as back-pressure. The most remarkable fea-
tures of the max-pressure algorithm for traffic signal control are: scalability,
stability, and distribution. The modified version presented in this paper
improves the practical applicability of the max-pressure controller by consid-
ering as input travel times instead of queue lengths. The two main practical
advantages of this new version are: (i) travel times are easier to estimate than
queue lengths, and (ii) max-pressure controller based on travel times has an
inherent capacity-aware property, i.e., it takes into account the finite capac-
ity of each link. Travel time tends to diverge when the queue length is close
to its capacity. It should be noted that previous max-pressure algorithms
rely exclusively on queue length measurements, which may be difficult to ac-
complish in practice. Moreover, these previous algorithms generally assume
queues with unbounded capacity. This may be problematic because a model
with unbounded capacity links is not able to reproduce spillbacks, which are
one of the most critical phenomena that a traffic signal controller should
avoid. After presenting the new version of the max-pressure controller, it is
analyzed and compared with existing control policies in a microscopic traffic
simulator. Moreover, results of a real implementation of the developed algo-
rithm to a signalized intersection, located at an urban arterial in Jerusalem,
are shown and analyzed. To the best of the authors’ knowledge, this ex-
periment is the first real implementation of a max-pressure controller at a
Preprint submitted to Transportation Research: Part C. December 27, 2020
signalized intersection.
Keywords: Max-pressure controller, Traffic signal control, Travel time
measurement, Bluetooth data
1. Introduction
Traffic congestions have become a major problem in urban areas due to
the increasing number of vehicles and transportation demand (Dargay et al.,
2007). Moreover, it is expected that this situation will continue to worsen in
the coming decades (Jia et al., 2016). Traffic congestions are originated when
there is a high demand for a common infrastructure with limited capacity.
This phenomenon results in the formation of queues while the infrastructure
is fully utilized. If the demand is too high, the situation may lead to a de-
graded use of the infrastructure, therefore increasing the congestion. The
spillback phenomena, i.e., upstream propagation of congestions, is an exam-
ple that leads to a degraded use of the infrastructure. A proper intersection
coordination by traffic lights may potentially alleviate this problem.
Several control policies have been proposed for traffic lights, see the review
paper Papageorgiou et al. (2003). Many approaches have been applied to
this problem, for example, optimal control and optimization (Haddad et al.,
2010; Ioslovich et al., 2011; Mercader et al., 2018), multivariable control
theory (Diakaki et al., 2002) and methods based on macroscopic fundamental
diagram (Keyvan-Ekbatani et al., 2012; Geroliminis et al., 2013; Haddad and
Shraiber, 2014; Leclercq et al., 2014).
Most of the actuated or responsive traffic control policies operate in a
centralized structure when applied to a large traffic urban network, i.e., there
is a control center which requires communication with all the sensors deployed
along the network and all the traffic lights that actuate in the network. This
requires a communication infrastructure that usually has high installation
and maintenance costs.
The potential problems with infrastructure and cost of centralized ap-
proaches motivate the emergence of decentralized control approaches, i.e., a
local traffic controller at a given intersection that only requires information
from adjacent links; therefore, the required communication infrastructure is
minimal. A decentralized self-organized traffic signal control is presented
in L¨ammer and Helbing (2008), which relies on an optimization strategy
along with a stabilizing strategy. In Cools et al. (2013), several decentralized
2
methods relying on simple rules but somewhat heuristic are presented, their
adaptation and self-organizing properties are analyzed by means of simula-
tions.
Another decentralized algorithm for traffic signal control is max-pressure
(MP), which was initially developed by Tassiulas and Ephremides (1992) for
scheduling packets in wireless communication networks. The MP controller
was adapted to traffic urban networks by Varaiya (2013a,b). The MP traffic
controller requires the measurement of queue lengths, and its main advan-
tages are: (i) computation is simple and distributed over intersections, (ii)
knowledge of the traffic demand is not needed, and (iii) it is provable stabi-
lizing whenever the network can be stabilized by any control under certain
assumptions. A remarkable fact about the MP algorithm is the strong as-
sumptions needed to prove stability, i.e., the need for separate queues for each
turn movement, the unboundedness of queue capacities, and the actuating
method that decides periodically which phase is active during a fixed time
step. Policies using this actuating method are referred as time-step based
policies. Unlike usual fixed cycle time policies that allocate periodically the
effective green time between an ordered set of phases, time-step based policies
may lead to erratic sequences of active phases.
In Le et al. (2015), the authors prove the stability of the MP controller
when using an actuating method that allocates the effective green time among
an ordered set of phases, leading to a control plan with a fixed cycle time and
cyclic phases, as it is usual in traffic control scenarios. As it was commented
before, previous MP traffic controllers adopt an actuating method that may
lead to erratic sequences of active phases. Therefore, the algorithm presented
in Le et al. (2015) was the first MP traffic algorithm adopting a fixed cycle
time policy with provable stability. Previously, in Kouvelas et al. (2014), an
MP traffic algorithm, which adopts a fixed cycle time policy but without sta-
bility guarantees, had been presented. The issue of assuming infinite queue
capacities, i.e., point queue model, in the MP controller was discussed in
Varaiya (2013a). In Varaiya (2013a), it was discussed that the MP controller
may lead to non-work conservation and congestion propagation situations.
This problematic of finite buffers was addressed in the communication field
by Giaccone et al. (2005), leading to a very complex policy. In the trans-
portation field, this issue was recently tackled in Gregoire et al. (2015). The
proposed solution in Gregoire et al. (2015) is to use generalized weights, in
particular, normalized convex functions of the queue lengths, in order to
guarantee work conservation under queue capacities consideration. In the
3
context of Gregoire et al. (2015), work-conservation situations in a intersec-
tion occur when there is a transfer of vehicles through the intersection during
a time interval. However, there is no proof of stability for the algorithm in
Gregoire et al. (2015). It should be stressed that the issue of stability of
variations of MP traffic controllers under finite queue capacities is still an
open research problem.
The combined problem of traffic signal and routing can also be addressed
in an integrated way using a modification of the MP controller, as it was
proposed in Zaidi et al. (2016). An early consideration of travel times in MP
algorithm was done in Taale et al. (2015) in a combined signal control and
route guidance problem. In Taale et al. (2015), the travel time is used as a
factor to account for the user preference in the route guidance problem. In the
context of MP algorithms, an upgraded algorithm has been recently presented
in Nilsson and Como (2017, 2018). This algorithm is provable stable and
does not require knowledge about the turning ratios unlike the classic MP in
Varaiya (2013a). In addition, the controller presented in Nilsson and Como
(2017) also determines the cycle length, and allows to increase the cycle
length in situations of high demand. Recently, in Levin and Boyles (2017),
the MP algorithm has been adapted and applied to the problem of controlling
reservation-based intersections for autonomous vehicles. A common caveat of
the aforementioned algorithms, with the notable exception of Kouvelas et al.
(2014); Gregoire et al. (2015), is the assumption of queues with unbounded
capacities. Under this assumption, the algorithms are not able to take into
consideration the spillback phenomena, therefore, one of the most crucial
issues that a traffic signal control has to tackle is completely ignored.
The developed traffic controller in this paper is based on the MP controller
presented in Varaiya (2013a); Kouvelas et al. (2014). The main difference
of the presented controller with respect to all previous versions of the MP
controller is that it uses travel times as input instead of queue lengths. The
motivation for using travel times as input to the algorithm is twofold: (i)
travel times are easier to measure and to estimate than queue lengths, and
(ii) MP traffic controller based on travel times provides an inherent capacity-
aware property. Regarding (i), it is important to note that queue lengths
are extremely challenging to measure or estimate using current detection
technology (Amini et al., 2016). Several studies have been carried out in this
topic, see, e.g., Geroliminis and Skabardonis (2005); Ban et al. (2011); Zhan
et al. (2015).
The problem of measuring or estimating queue lengths is somewhat cir-
4
cumvented here by measuring and using as input for the control algorithm
travel times instead of queue lengths. Many technologies make it possible to
measure travel times, e.g., Bluetooth, GPS-equipped vehicles, license plate
matching, or probe vehicle (Turner et al., 1998). In recent years, Blue-
tooth technology has emerged as an attractive sensing technology in traffic
engineering (Barcel´o et al., 2010; Haghani et al., 2010), that is partly due
to its low installation and maintenance costs. Another benefit of utilizing
Bluetooth sensors is that they also provide measurements about the turning
ratios, which are also required by the MP traffic controller. Regarding (ii),
the MP traffic controller based on travel times bears some relationship with
the framework of capacity-aware MP controllers presented in Gregoire et al.
(2015). Travel times should provide a better characterization of the degree
of congestion in an urban link than queue lengths. Note that the travel time
tends to diverge when the queue length is close to its capacity. This corre-
lation can be utilized to prevent queue spillback. The developed algorithm
in this paper uses a normalized travel time variable in the spirit of other
works that also consider normalized variable like normalized queue lengths
(Kouvelas et al., 2014) or generalized (normalized) weights (Gregoire et al.,
2015). This makes it possible to prove the work-conservation property of the
proposed traffic controller.
The rest of this work is organized as follows. The traditional MP traffic
controller based on queue lengths (Varaiya, 2013a) is introduced in Section 2.
The proposed MP traffic controller based on travel times is presented and
analyzed in Section 3. Application results under simulated and fields scenar-
ios are given in Sections 4 and 5, respectively. Finally, some conclusions and
future research are drawn in Section 6.
2. Max-pressure traffic controller based on queue lengths
The MP traffic controller presented in Varaiya (2013a) is an algorithm for
traffic signal control with the properties of being scalable, distributed over in-
tersections, and provable stable. The stability is proved in terms of expected
long-term average of total queues, by making simplifying assumptions like
the need for separate queues for each turn movement, the unboundedness of
queue capacities, and an actuating method which follows a time-step based
policy. Its implementation requires real-time measurements or estimations
about the queue lengths, turning ratios, and saturation flows. It is remark-
able that knowledge of the traffic demand is not needed. The framework in
5
which the algorithm was presented and analyzed assumes a store-and-forward
queuing network model, where the queue length at link lwith destination m
at the beginning of the time step tis represented by xl,m(t) [veh], see Fig. 1,
and is modeled as:
xl,m(t+ 1) =xl,m(t)−min{cl,m(t+ 1) ·sl,m (t), xl,m(t)}
+X
k
min{ck,l(t+ 1) ·sk,l(t), xk ,l(t)} · rl,m(t+ 1)
+dl,m(t+ 1),
(1)
where cl,m(t+ 1) [veh] is the saturation flow, which represents the maximum
number of vehicles that can potentially depart from link lto mduring the
time step t+ 1 when this movement has right of way, sl,m(t) is a variable
that takes the value 1 when the movement from link lto mhas right of
way during the time step tand 0 otherwise, rl,m(t+ 1) is the turning ratio,
i.e., the proportion of vehicles that are queuing at lwith destination m, and
dl,m(t+ 1) [veh] represents the external demand during the time step t+ 1.
Note that in addition to the assumption of queues with infinite capacities,
the dynamic equation (1) assumes that there exists a separate queue for
each turn movement. This assumption does not generally hold at real urban
networks and it is not considered in many work about controlling signalized
traffic intersections, see, e.g., Kouvelas et al. (2014).
The MP traffic controller decides which phase is active at each intersection
during the time step tby activating the phase with the maximum pressure.
The pressure of phase jat intersection nis calculated as
pj,n(t) = X
(l,m)∈Mj,n
wl,m(t)·cl,m (t),∀j∈Fn,(2)
where Mj,n is a set that contains all link pairs which define movements that
have right of way during the phase jat the intersection n,wl,m(t) represents
the weight of the movement from link lto m, and Fnis the set of all phases
of the intersection n. The weight of the movement from link lto mis defined
as
wl,m(t) = xl,m (t)−X
p∈On
rm,p(t)·xm,p (t),∀l∈In,(3)
where Inis the set of incoming links of intersection n, and Onis the set of
outcoming links of intersection n. Note that this scheme does not generate a
6
Link 1
Link 2
Link 4
Link 3
x1,3(t)
x1,4(t)
x2,4(t)
x2,3(t)
I1={1,2}
O1={3,4}
F1={1,2}
M1,1={(1,4),(1,3)}
M2,1={(2,4),(2,3)}
Figure 1: Example of two one-way roads intersection with the notations used in the
dynamic equation (1) and in the MP traffic controller (2), (3).
control plan with fixed cycle time and cyclic phases. Traffic control scenarios
generally consider plans with fixed cycle time and cyclic phases.
In Varaiya (2013a), it is proved that the MP controller is stable in terms of
expected long-term average of total queues, which means that this algorithm
prevents any queue from growing indefinitely. Indeed, this is a very desirable
property, however, it is important to note that under this definition it is
possible to have a stable network where some of the queues take arbitrary
large values but at some point the queues stop increasing. The stability of
the MP traffic controller is guaranteed for any demand for which a stabilizing
control plan exists. However, in addition to the use of queue lengths as input
of the algorithm, there are other issues which limit its applicability to real
urban networks. These issues are: (i) the need for separate queues for each
turn movement, (ii) the assumption of queues with unbounded capacities,
i.e., point queues model, and (iii) an actuating method without a fixed cycle
time and cyclic phases. The main limitation is the infinite storage capacity in
each link. This may lead to situation in which a movement is blocked when
it has right of way, although this issue is not always leading to a permanent
throughput reduction.
7
The facts presented before are probably the reasons why practical-oriented
papers, which implement the MP controller on a microsimulation environ-
ment, see e.g. Kouvelas et al. (2014); Manolis et al. (2018), modify the
MP controller presented in Varaiya (2013a) to alleviate some of these issues.
These are also the reasons why it is considered here the variation of the MP
controller proposed in Kouvelas et al. (2014) instead of the one presented
in Varaiya (2013a). Unfortunately, these modifications make the MP traffic
controller to lose its stability property, since some of the simplifying assump-
tions that were made in Varaiya (2013a) are removed.
The modified MP controller presented in Kouvelas et al. (2014) is based
on the following equations for the computation of pressures and weights:
pj,n(t) = max
0,X
(l,k)∈Mj,n
wl(t)·cl(t)
,∀j∈Fn,(4)
wl(t) = xl(t)
xmax,l
−X
m∈On
rl,m(t)·xm(t)
xmax,m
,∀l∈In,(5)
where wl(t) is the weight associated to the link l,xl(t) [veh] is the maximum
queue length at the link lduring the cycle t, and xmax,l [veh] is the maxi-
mum capacity of queuing vehicles in the link l. It should be stressed that t
represents here the index for the cycle of the traffic light, the same notation
was used in the time-step based policy (2), (3) to represent the index for the
time step. The rest of the variables were already defined previously in this
section, and Fig. 2 illustrates the notations used in the MP traffic controller
(4), (5). Note that unlike the previous algorithm defined by (2), (3), all the
variables considered in (4), (5) are referred to the link where the vehicles are
queuing independently on the destination links, i.e., the link queue is not
decomposed into separate queues based on destinations.
There are several alternatives to measure the state of a queue during
a cycle, here we choose the maximum queue length in the link, i.e., the
maximum number of queuing vehicles in a link during a cycle, but other
options are the queue length at the beginning or at the end of the cycle, or
the average queue length. Both maximum and average queue lengths take
into account the dynamic of the queue length during the cycle, unlike queue
length at the beginning or at the end of the cycle that only capture the
status of the queue at a specific instant. We recall that here the state xl(t)
8
represents the maximum queue length at the link lduring the cycle tand it
should not be confused with maximum capacity of queuing vehicles in the
link lthat is represented by xmax,l.
Link 1
Link 2
Link 4
Link 3
x1(t)
x2(t)
I1={1,2}
O1={3,4}
F1={1,2}
M1,1={(1,4),(1,3)}
M2,1={(2,4),(2,3)}
Figure 2: Example of two one-way roads intersection with the notations used in the MP
traffic controller (4), (5).
Once the pressure for each phase has been computed, the allocation of
the effective green time for a given intersection n, denoted as Gn[s], among
the different phases is performed according to
˜gj,n(t) = pj,n(t)
Pi∈Fnpi,n(t)·Gn,∀j∈Fn,(6)
where ˜gj,n(t) [s] represents the portion of effective green time allocated to the
phase jat the intersection nduring the time interval t. When all the pres-
sures are equal to zero, equation (6) is not defined and effective green time is
equally distributed among the different phases. The motivation of allocating
the effective green time according to equation (6) can be interpreted in terms
of adopting a proportional fairness criterion, see Bertsimas et al. (2011) for
more details about fairness. The effective green time for a given intersection
9
nis obtained by
Gn=Cn−Llost,n −X
j∈Fn
gmin,j,n,∀n∈N , (7)
where Nrepresents the set of all intersections in the network, Llost,n [s]
represents the total fixed lost time in a cycle, Cn[s] represents the total
cycle time, and gmin,j,n [s] represents the minimum green time for the phase
j. Recall that the subscript nis used to refer to the intersection n. Finally,
the total green time which is allocated to each phase for a given intersection
nis given by
gj,n(t) = ˜gj,n(t) + gmin,j,n ,∀j∈Fn.(8)
This algorithm will serve as the basis for developing the proposed MP con-
troller based on travel times that will be introduced in the next section. The
algorithm is invoked at the end of each cycle generating the green times to
be applied in each phase during the following cycle. In this way, the phases
can be actuated in a predefined order, see, Kouvelas et al. (2014) for further
details.
3. Max-pressure traffic controller based on travel times
Previous research on MP algorithms applied to traffic network relies ex-
clusively on queue lengths as input variables, see, e.g., Wongpiromsarn et al.
(2012); Varaiya (2013a); Gregoire et al. (2015); Le et al. (2015); Zaidi et al.
(2016). As it was already commented in Section 1, this issue poses serious
impediments to practical implementations in real signalized urban networks.
This is mainly due to the difficulties in measuring or even estimating in real-
time queue lengths with current sensor technologies (Amini et al., 2016).
The proposed algorithm tries to overcome this difficulty by resorting to mea-
surements of travel time of each link connected to the intersection. The
underlying principle of the proposed modification is that the travel time in
an urban link can be used as a proxy for the queue length. In particular,
the normalized travel time is used to compute the weights instead of using
queue lengths. In addition, normalized travel time provides capacity-aware
property. Recall that Gregoire et al. (2015) shows that the use of convex
normalized weights avoids non-work-conservation situations and prevents the
queue spillback phenomena.
10
The proposed MP controller is defined by the equations (4)–(8), as it
is usual in MP algorithms that consider an actuating method with a fixed
cycle time, cyclic phases, and minimum green time per each phase, i.e., fixed
cycle time policies. The difference with respect to previous MP algorithms
is the computation of the weights that define the pressures. In this case, the
expression used to compute the weight for the link lis given as follows:
wl(t) = T Tl(t)
T Tff,l
−X
m∈On
rl,m(t)·T Tm(t)
T Tff,m
,(9)
where T Tl(t) [s] stands for the average travel time during the cycle tat the
link land T Tff,l [s] stands for the free-flow travel time at the link l. Note
that T Tl(t)/T Tff,l represents the normalized travel time1during the cycle t
at the link l.
There is a relationship between the use of the normalized travel time
for the computation of weights in the MP algorithm and the framework of
convex normalized weights for capacity-aware MP controllers presented in
Gregoire et al. (2015). The latter framework uses convex normalized func-
tions of the queue lengths to compute the weights, unlike previous methods
that use linear functions of the queue lengths. One of the reasons of using
convex normalized functions of the queue lengths to compute the weights
is to produce a marginal function (derivative) that increases as the queue
grows. Therefore, it takes into consideration the fact that adding a vehicle
to a queue that is near to its maximum capacity is more problematic than
adding it to a queue that is far from its maximum capacity. In this way, the
finite capacity of the queues is taken into account and it is expected to reduce
the risk of occurrence of spillback phenomena. The connection between the
current proposed algorithm and the framework presented in Gregoire et al.
(2015) is that the normalized travel time can be modeled as a convex function
of the queue length.
Many formulas have been presented in the literature attempting to model
the travel time as a function of the arrival rate. Take into account that the
maximum queue length and the arrival rate are approximately proportional,
1The distance-normalized travel time T Tl(t)/Llis normalized using the free-flow
distance-normalized travel time T Tff,l (t)/Ll, this leads to T Tl(t)/T Tf f,l (t), i.e., normal-
ized travel time. In situations where are the involved links have the same free-flow velocity,
it is equivalent to use distance-normalized travel time or the normalized travel time.
11
see, e.g., Treiber and Kesting (2013). One of the most popular formulas for
estimating the travel time T T [s] is the standard BRP formula published in
Bureau of Public Roads (1964):
T T =T Tff·"1 + α·A
Qβ#,(10)
where T Tff[s] denotes the free-flow travel time, A[veh/s] is the arrival rate,
Q[veh/s] is the capacity, and αand βare fitting parameters. Common
values for these fitting parameters are α≈0.5 and β≈4, see, e.g., Helbing
(2009). In this way, the (average) travel time can be interpreted as a convex
function of the maximum queue length during a cycle, whenever β > 1,
which is the common case. We use this equation to support the convex
relationship between the normalized travel time per cycle, T Tl/T Tmax,l , and
the normalized queue length during a cycle, xl/xmax,l.
To conclude this section, we present a relationship based on microscopic
simulations between the normalized travel time and the normalized queue
length. The data used to obtain this relationship are taken from the sim-
ulations presented in Section 4 that contain scenarios with different traffic
controllers and demand profiles. This relationship is based on data obtained
by microsimulation over a wide range of conditions. Detailed description of
the microsimulations are given in Section 4. Fig. 3 shows the average normal-
ized travel time per each value of normalized queue length. The data used
in this figure correspond to the link level. Error bars represent the standard
deviation. A curve fitting is performed on the data using a power function
with the structure
T Tl
T Tff,l
=a·xl
xmax,l b
+c, (11)
where a,band care fitting parameters. A regression analysis using these data
is also displayed in Fig. 3 and the following values for the fitting parameters
are obtained: a= 58.93, b= 3.59, and c= 6.17. Note that whenever b > 1
this function is strictly convex. The issue of scattered data in the congested
regime has been discussed in previous research. There are some alternatives
that propose different relationships per each regime of the road, leading to
more complex models, see, e.g., Helbing (2009).
12
0 0.2 0.4 0.6 0.8 1
0
50
100
150
xl
xmax,l [−]
T Tl
T Tff,l [−]
Figure 3: Convex relationship between normalized travel time and the normalized queue
length. Error bars in black and power fitting curve in blue.
3.1. Work conservation property of max-pressure traffic controller based on
travel times
This subsection presents a study of the work-conservation property of the
proposed MP traffic controller based on travel times, defined by (4), (6)–(9).
The work-conservation property has been previously studied in the MP traffic
controller based on queue lengths proposed in Gregoire et al. (2015). It is said
that an intersection works during a cycle tif there is a transfer of queuing
vehicles through links served by the phases of the intersection. In the context
of a time-step based policy, as the one proposed in Gregoire et al. (2015), a
traffic controller is said to be work conservative if every intersection, at any
time step, is not idle when there is a non-empty upstream queue with a non-
full downstream queue. In contrast to the time-step based policy presented
in Gregoire et al. (2015), the proposed MP traffic controller based on travel
times distributes the effective green time among an ordered set of phases,
leading to a control plan with fixed cycle time and cyclic phases. Therefore,
the definition of work-conservative traffic controller should be adapted to
traffic controllers with fixed cycle time and cyclic phases.
Definition 1. The traffic controller defined by equations by (4), (6)–(9) is
said to be work conservative if a phase that is not serving any queuing vehicle
13
has a portion of effective green time equal to the minimum portion of effective
green time among the all the phases in its intersection.
A phase may be not able to serve any queuing vehicle due to two reasons,
these are the absence of queuing vehicles in the upstream links and the pres-
ence of queues reaching the maximum queue capacity in the downstream
links.
Theorem 3.1. The MP traffic controller based on travel times defined in
(4),(6)–(9) is work conservative under the assumption that travel times are
modeled by (11).
Proof. Assume that there is a phase jat an intersection nthat is not serving
any queuing vehicles while having higher portion of effective green time than
other phase that is able to serve at least one queuing vehicle. Therefore, for
all pairs of links (l, m)∈Mj,n with rl,m >0 it holds that there are no vehicles
waiting at l, i.e.,
T Tl(t)
T Tff,l
=c, (12)
or that the outcoming links are at their maximum queue capacity, i.e.,
X
m∈On
rl,m(t)·T Tm(t)
T Tff,m =c+a, (13)
according to the assumed model for the travel time given in (11). In both
cases, the resulting weight satisfies wl(t)≤0. Therefore, the pressure of this
phase is pj,n(t) = 0 and it is not possible to have higher portion of effective
green than any other phase. Since, according to the proportional allocation
of effective green time done by (6), the portion of effective green time is zero
for a pressure equals to zero.
In the previous proof, it is implicitly assumed that a queuing vehicle
could move to other downstream links, if the destination link operates at
its maximum. This prevent the occurrence of blocking because one of the
downstream link is at its maximum capacity. It is important to note that this
assumption is only required to prove the work conservation property, but it
is not required by the proposed controller to be applied in real scenarios. If
this assumption is considered to be restrictive, it could be directly replaced
by assuming queues with unbounded capacities in the same way as it is done
14
in Varaiya (2013a). Another important point to note is that normalization
of travel times is what makes it possible to prove the work-conservation
property, as it can be seen in the proof of Theorem 3.1.
4. Simulation-based results
This section is devoted to present simulation-based results for the pro-
posed MP traffic controller based on travel times, i.e. (4), (6)–(9). The per-
formance of the MP controller based on travel times (MP-TT) is compared
to the MP controller based on queue lengths (MP-QL) that was presented
in Kouvelas et al. (2014), and a fixed-time controller (fixed plan) that dis-
tributes equally the effective green time between the existing phases. The
comparison is performed using microscopic simulations in AIMSUN (Casas
et al., 2010), where the urban signalized network shown in Fig. 4 is mod-
eled and controlled. The MP-TT, MP-QL, and fixed plan controllers have
been implemented using Aimsun API extensions in Python programming
language (Transport Simulation Systems, 2013). In this network, each link
has one lane, the length of each link is approximately 100 m, and the max-
imum speed is 50 km/h. The network includes 12 signalized intersections
with a common cycle length of 72 seconds for each intersection and a com-
mon minimum green time of 5 seconds for each phase. The control plans
have three phases for the intersections located along the network boundary
and four phases for the rest of intersections.
Three demand profiles are considered in these simulations. These demand
profiles define the traffic demand for 2 hours and all of them have an increase
of the demand starting at 0.5 hours and ending at 1 hour, the intensity of
this peak is different for each demand. The demand profiles are shown in
Fig. 5. The vehicles enter and exit the network through four origins and
destinations that are placed at the network boundary, as indicated in Fig. 4.
We consider nine scenarios that result from all the possible combinations
between the three demands and the three traffic controllers. These simula-
tions include a warm-up period of 20 minutes in which the fixed plan con-
troller is active; after this period, the traffic controller under consideration in
the current scenario is activated. For each scenario, 100 replications of two
hours duration are simulated. The only difference between these replications
is the random seed. The most revealing outcome of this comparison is shown
in Fig. 6, where the percentage of unstable replications for each scenario is
shown. A replication in which the number of vehicles queuing, including
15
Figure 4: Arterial network with 12 four-way signalized intersection with 4 origins and 4
destinations modeled in AIMSUN.
0 0.511.5 2
1,000
1,500
2,000
2,500
Time [h]
Demand [veh/h]
Demand 1
Demand 2
Demand 3
Figure 5: Demand profiles considered for the rectangular network.
16
vehicles queuing outside the network (virtual queues), becomes steadily in-
creasing with time is refereed here as an unstable replication. Note that for
the same demand and control plan, it is possible to have both stable and
unstable replications due to the difference between the initial seeds.
1 2 3
0
20
40
60
80
100
Demand
Unstable replications [%]
Fixed plan
MP-QL
MP-TT
Figure 6: Percentage of unstable replications for each demand and traffic controller.
As it is natural to expect, the results shown in Fig. 6 show that the
percentage of unstable scenarios increases as the demand increases, which
hold for all traffic controllers. These results also suggest that the MP con-
troller based on travel times is more effective for stabilizing queues than the
MP controller based on queue lengths and the considered fixed plan. It is re-
markable that the superiority of the MP controller based on travel times over
the other traffic controllers holds for the three considered demands. There-
fore, these simulation-based results demonstrate the ability of the proposed
MP controller based on travel times to avoid spillbacks, which cause drop
in capacity of the network to serve vehicles and unstable behavior of the
queue lengths. This result is expected because the functions used to com-
pute the weights in MP-TT are convex functions of the queue lengths, see
Fig. 3. Therefore, the marginal function increases as the queue grows (Gre-
goire et al., 2015), limiting vehicles from entering to highly congested links,
and also giving more priority to vehicles queuing in highly congested links.
Further results of these simulations are discussed. Table 1 shows the
average exit flow, density, velocity, travel time, and delay among the replica-
tions for each of the considered scenario. Only common stable replications
17
for each controller have been taken into account in the computation of these
values shown in Table 1. Since only stable replication are considered, all the
control plans exhibit very similar values of exit flow for each demand. The
main difference between the control plans occurs in the values of density,
velocity, travel time, and delay. Both MP traffic controllers result in higher
values of density than the values obtained by the fixed plan. Therefore,
they deliver lower values for velocity, travel time, and delay. In particular,
the MP based on travel times achieves the highest value of density. These
findings are illustrated in Fig. 7 that shows flow-density diagrams for each
control plan and demand. Note that this data corresponds to the network
level. Again, only data from common stable replications for each controller
have been considered. These graphs reveal that a side effect of the ability
to decrease the probability of spillback phenomena is the possible decrease
on velocity, and hence, increase on travel time, experimented under high de-
mands. This behavior is observed in the two MP traffic controllers. The
underlying principle of MP controllers is not only to prioritize movements of
vehicles towards less congested links, but also to limit movements of vehicles
towards congested links. This principle resembles other techniques to avoid
or retard traffic congestions like perimeter control in urban networks, variable
speed limit (VSL), and ramp metering in freeways, see e.g., Carlson et al.
(2010); Geroliminis et al. (2013); Ferrara et al. (2018). These techniques,
as MP traffic controllers, limit the flow of vehicles towards regions that are
almost congested. Although these actions may lead to a deterioration of the
traffic in terms of velocity decrease, and hence, travel time increases, they
are effective in delaying the occurrence of congestions. Regarding the differ-
ences between MP based on queue lengths and MP based on travel times, the
convex property of the normalized travel time as function of the maximum
queue length highly limits movements of vehicles towards congested links,
which implies that the decrease of velocity during the increase of congestion
is more noticeable. Therefore, the ability of MP based on travel times to de-
crease the probability of instability is also superior, as it was shown in Fig. 6.
In our opinion, this is the most desirable property of the proposed MP traffic
controller. Finally, Fig. 8 shows some examples of the evolution of vehicles
queuing for some particular replications corresponding to different scenarios.
We can appreciate in these graphs that the MP controller based on travel
time is less effective than the other controllers at maintaining low values of
queue vehicles in a non-congested regime; but, on the other hand, its perfor-
mance reducing the probability of instability is clearly superior. This effect
18
may be originated by the flat behavior of the distance-normalized travel time
in the region of low values for the maximum queue length, see Fig. 3.
4.1. Dependence of performance with respect to penetration rate
The simulation study presented in the previous subsection assumes that
the travel time measurements are available from all vehicles in the network.
Unfortunately, this assumption is not always valid with the available measur-
ing systems for travel time, e.g., probe vehicles, cellular devices, Bluetooth
devices, and radio frequency identification. The technologies for measuring
travel time usually collect measurements from a proportion of the vehicles
traveling in a link. If this proportion is too small, it might not be statistically
representative of the entire population and the performance of the controller
may be degraded with respect to the ideal case, where measurements are
available from all vehicles. We refer as a penetration rate of equipped vehi-
cles to the proportion of equipped vehicles in the total flow. Previous research
has studied the relationship between the penetration rate and the accuracy
of travel time measured by Bluetooth devices, see, e.g., Rim et al. (2011);
Bhaskar and Chung (2013a).
Simulations are carried out with the purpose of showing the dependence
of percentage of unstable replications with respect to the penetration rate.
This simulation considers the same network presented in the previous sub-
section, the MP controller based on travel times and several values of the
penetration rate. The results are displayed in Fig. 9, where the percentage of
unstable replications is displayed for different values of the penetration rate.
The results presented in Fig. 9 show that there is a decrease in the percent-
age of unstable replications as the value of the penetration rate increases.
This degradation is accentuated for values of the penetration rate less than
20 % and low demands. In the case of 0 % penetration rate, the MP traffic
controller based on travel times is not well defined and the available green
time is equally distributed among the phases as in the considered fixed time
control plan. Fig. 9 shows that the higher percentage of unstable scenarios
is attained at a percentage rate of 0 %; therefore, even in scenarios with low
values of penetration rate, the percentage of unstable scenarios is lower than
the obtained by the considered fixed time control plan, see Fig. 6.
4.2. Alternative MP traffic controller based on velocities
An alternative method to compute the weights in the MP algorithm is to
use velocities instead of queue lengths or travel times. At a first glance, it
19
Table 1: Summary of the simulation-based results considering common stable replications.
Flow Density Velocity Travel time Delay Unstable replications
[veh/h] [veh/km] [km/h] [s/km] [s/km] [%]
Demand 1
Fixed plan 1357.0 10.86 14.15 349.1 280.8 23
MP-QL 1355.0 12.20 12.44 391.1 322.9 18
MP-TT 1354.3 13.44 11.68 432.0 363.8 10
Demand 2
Fixed plan 1381.6 11.71 13.86 369.8 301.5 53
MP-QL 1379.9 13.18 12.15 415.1 346.8 44
MP-TT 1378.7 14.20 11.45 448.4 380.2 33
Demand 3
Fixed plan 1406.7 13.07 13.46 406.0 337.8 81
MP-QL 1404.4 14.11 11.85 437.6 369.3 78
MP-TT 1403.8 15.80 10.99 491.8 423.6 64
20
1,000
1,500
2,000
2,500
Flow [veh/h]
Fixed plan
MP-QL
MP-TT
1,000
1,500
2,000
2,500
Flow [veh/h]
Fixed plan
MP-QL
MP-TT
5 10 15 20 25 30 35 40 45
1,000
1,500
2,000
2,500
Density [veh/km]
Flow [veh/h]
Fixed plan
MP-QL
MP-TT
Figure 7: Flow-density diagrams for demands 1 (top), 2 (middle), and 3 (bottom). The
beginning and the end of the high-demand period is denoted with a square and a circle,
respectively.
21
0
100
200
300
400
Vehicles queuing [veh]
Fixed plan
MP-QL
MP-TT
0
100
200
300
400
500
Vehicles queuing [veh]
Fixed plan
MP-QL
MP-TT
0 0.5 1 1.5 2
0
200
400
600
Time [h]
Vehicles queuing [veh]
Fixed plan
MP-QL
MP-TT
Figure 8: Examples of evolution of vehicles queuing for demand 1 (top), 2 (middle), and
3 (bottom).
22
0 20 40 60 80 100
0
20
40
60
80
100
Penetration rate [%]
Unstable replications [%]
Demand 1
Demand 2
Demand 3
Figure 9: Percentage of unstable replications as function of the penetration rate for MP-TT
under the demands 1, 2, and 3.
may seem to be equivalent to the MP traffic controller based on travel time,
but there are some fundamental differences. The velocities can be employed
for computing the weights according to the following equation
wl(t) = 1−vl(t)
vff,l −X
m∈On
rl,m(t)1−vm(t)
vff,m ,(14)
where vl(t) [m/s] stands for the average velocity during the cycle tat the link
land vff,l [m/s] for free-flow velocity of the link l.
This is closely related to the use of normalized travel time since the inverse
of the normalized travel time is the normalized velocity, vl(t)/vff,l. However,
there is an important difference in favour of the use of normalized travel
time. While the normalized travel time can be modeled as a convex function
of the normalized queue length, the variable 1 −vl/vff,l used to compute the
weights in the case of MP based on velocities can be modeled as a concave
function of the normalized queue length. Therefore, the marginal function
decreases as the normalized queue length increases. That is the reason why
this controller is expected to have a lower performance avoiding spillback
than the proposed MP based on travel times.
The performance of the MP controller based on velocities (MP-V) was
measured by implementing it for the same network of the previous study
under the three considered demands. The results of these simulations show
23
that the performance of the MP controller based on velocities is comparable
to the MP controller based on queue lengths, while it has a lower performance
compared to the MP controller based on travel times. In particular, we obtain
the following values for the percentage of unstable replications 24 %, 44 %,
and 72 %, which correspond to the demands 1, 2, and 3, respectively.
5. Experimental results
This section presents experimental results of the proposed MP traffic con-
troller based on travel times. The proposed MP controller was implemented
and tested at a real intersection in Jerusalem. In particular, it was imple-
mented and evaluated at the major intersection between the streets Kanfei
Nesharim, Yemin Avot, and Yehoshu’a Farbstein in Jerusalem. Prior to the
experiment, Bluetooth detectors were installed at the positions marked in
Fig. 10, while Fig. 11 shows photographies of the Bluetooth and loop detec-
tors.
A Bluetooth detector consists of a Bluetooth device that scans for other
Bluetooth devices and captures an unique electronic identifier, denoted as
Machine Access Control (MAC) address, and the time stamp (Haghani et al.,
2010; Bhaskar and Chung, 2013b). The list of detectable Bluetooth devices
includes smartphones, laptops, hands-free kits, and mainly in-vehicle audio
systems. A local monitoring system, that receives the data from all the
Bluetooth detectors, calculates travel times by matching these identifiers at
successive sensors. The data are processed to eliminate multiple detections on
the same vehicle, e.g., several devices in a bus, and outliers, e.g., pedestrian.
Therefore, this system is able to measure the travel times of a subset of
the vehicles traveling thought the network, i.e., vehicles which contain a
Bluetooth device.
It is remarkable that the Bluetooth sensing technology overcomes most of
the drawbacks of magnetic loop detectors, that are the traditional sensors for
monitoring traffic flow. These drawbacks are: the magnetic loop detectors
installation is complicated and expensive, and their failure rates are quite
high, see e.g. Leduc (2008). In addition, the same reference indicates that
loop detectors have a lifetime of only 5 years, since wire loops are subjected
to stresses of traffic and changes of temperature. It is also important to
note that the installation of this kind of sensor decreases the lifetime of the
pavement. It should be emphasized that loop detectors provide information
that Bluetooth sensors cannot provide (Day et al., 2014). However, the
24
practical advantages of the Bluetooth sensors are what motivate us to develop
a variant of the MP traffic controller based on travel times, which can be
measured using Bluetooth sensors. On the other side, the main drawback of
the Bluetooth technology is that it might be not valid in scenarios with very
low penetration rate, i.e., low fraction of vehicles with an active Bluetooth
device. Another issue is that it may be a concern with respect to privacy
but it should be noted that the MAC address of Bluetooth devices generally
is not associated with any personal data. Moreover, the MAC address can
be encrypted before storing the data in the database of the local monitoring
system.
Figure 10: Case study intersection in Jerusalem with the location of the Bluetooth detec-
tors (BD).
The MP controller based on travel times allocates the effective green
time among the existing designed phases in the traffic control plan, which is
currently running by the Jerusalem traffic control center. The value of the
cycle length used in the MP controller is also the same as the one used in
this traffic control plan. There are three phases, and they are defined, by
using as a reference the position of the Bluetooth detectors, as follows:
•Phase 1 serves movements of vehicles BD3-BD2-BD1 and BD1-BD2-
BD3.
•Phase 2 serves movement of vehicles BD1-BD2-BD4.
•Phase 3 serves movements of vehicles BD4-BD2-BD1 and BD4-BD2-
BD3.
25
a) Bluetooth detectors.
b) Loop detectors.
Figure 11: Details of the Bluetooth (top) and loop (bottom) detectors.
26
The rest of the control plan parameters are given as follows: a fixed cycle
length C= 120 seconds, minimum green times are G1= 12 seconds, G2= 9
seconds and G3= 11 seconds, and lost time is Llost = 11 seconds. As a
result, the effective green time is G= 74 seconds.
Before showing the results of the experiment, we present an experimental
analysis of the penetration rate of the Bluetooth technology during the day
of the experiment. The penetration rate of the Bluetooth technology is an
important non-controlled variable in this experiment. During the morning
that the experiment was performed, a penetration rate for the Bluetooth
technology of about 20 % was measured using as reference data from loop
detectors. Fig. 12 shows the flow of vehicles measured by loop detectors and
Bluetooth detectors (top) and penetration rate of the Bluetooth technology
(bottom) from 12:00 am (midnight) to 13:00 pm of the day of May 30, 2018.
Since not all lanes are equipped with loop detectors, the traffic flow shown
in Fig. 12 is not the total flow at the intersection during the experiment.
The MP traffic controller based on travel times was implemented in the
considered intersection from 8:30 am to 10:15 am of May 30, 2018. This
period of time corresponds to the rush hour in the morning for this intersec-
tion; observe that in Fig. 13 historic data of distance-normalized travel times
(dashed blue lines) take larger values during this period of time. The aver-
age of the distance-normalized travel times among all incoming links to the
intersection is considered here as the performance index in this experiment,
see Fig. 13. The solid black line data corresponds to the day of the exper-
iment and the dashed blue lines data correspond to previous days. These
historic data are shown in order to compare the performance of the proposed
controller with a reference traffic plan. The previous days were chosen on
the same day of the week and days with unusual activities were excluded. It
is possible to appreciate in Fig. 13 that during most of the experiment, indi-
cated by the shadow region in Fig. 13, the proposed MP controller based on
travel times has achieved better performance than the historic performance
delivered by the optimized fixed control plan employed by the traffic con-
trol center. Note that a broken vehicle blocked one of the lanes of the link
between the Bluetooth detectors BD2 and BD3 at 8:45. This vehicle was
stuck until the end of the experiment. This event corresponds to the peak of
the travel time, see Fig. 13. The results shown in Fig. 13 indicate the abil-
ity of the proposed MP controller to deal with disturbances, e.g., the traffic
flow disturbance occurred from the broken vehicle during the experiment. In
order to better measure the performance of the proposed controller, we pro-
27
00:00 02:00 04:00 06:00 08:00 10:00 12:00
0
1,000
2,000
3,000
Time
Flow [veh/h]
Bluetooth detector
Loop detector
00:00 02:00 04:00 06:00 08:00 10:00 12:00
0
10
20
30
Time
Penetration rate [%]
Figure 12: Flow of vehicles measured by loop detectors and Bluetooth detectors (top),
and penetration rate of the Bluetooth technology (bottom), from 12:00 am to 13:00 pm
of the day of the experiment. The dashed line in the bottom figure indicates the average
of the penetration rate during the considered period of time.
28
vided the average distance-normalized travel times during the experiment,
0.291 [s/m], and during previous days, 0.303 [s/m]. This implies a reduction
of about 4 % in spite of the disturbance due to the stuck vehicle.
07:00 08:00 09:00 10:00 11:00 12:00
0.1
0.2
0.3
0.4
0.5
Time
Distance-normalized travel time [s/m]
Figure 13: Comparison of the distance-normalized travel times between the day of the
experiment (solid black line) and previous days (dashed blue lines). The time interval
during, in which the proposed MP controller based on travel times was active the day of
the experiment, is delimited by the shadow region.
6. Conclusions
This paper presented a modification of the max-pressure (MP) controller
for signal timing control. Unlike all previous versions of the MP controller
for signal timing control, travel times are used as input data instead of queue
lengths. This modification substantially improves the applicability of the
presented version of MP with respect to previous versions of MP. This im-
provement in the applicability does not come at the cost of lower perfor-
mance. Simulation-based results have shown that MP based on travel times
is more effective for avoiding spillback phenomena, and hence avoiding unsta-
ble behavior of the queue lengths. The same behavior is experimented with
other traffic control techniques aimed at avoiding or retarding traffic conges-
tions like perimeter control in urban networks and, variable speed limit and
ramp metering in freeways. Note that the underlying principle in all these
29
techniques is to limit the flow of vehicle towards regions that are almost con-
gested. Nevertheless, it was shown in this paper that MP based on travel
times may have a negative impact on the traffic in terms of velocity decrease,
and hence, travel time increase.
There are several theoretical issues that deserve more attention in future
research. Theoretical guarantees for stability are lacking and more efforts
should be done in this direction. Note that the problem of stability for any
variant of MP controller, when the queues are assumed to have finite capac-
ities, is still an open problem. Another aspect that deserves more attention
and was not addressed in this work is the possible presence of communication
time delays (time lags) in the measuring systems. Measurement delays are
also present in the system, since the MP controller uses the measurement
corresponding to the previous cycle. The issue of the presence of time de-
lays has been studied before in the framework of MP controller for wireless
networks, but it leads to more complicated controllers whose application in
large scale transportation networks may be very challenging, see, Neely et al.
(2005). Finally, the average values of normalized travel times over several
cycles could be used in the control algorithm to improve the scattering of
normalized travel times in the oversaturated case (see Fig. 3).
The most debatable issue in our work is the assumption that a queuing
vehicle whose destination link is at their maximum capacity will reroute to
another link if this is possible. This assumption (that may be realistic or
not) is only taken into account to prove the work conservation property, but
it is not required by the proposed controller to be applied in real scenarios.
The rationale behind this assumption about the behavior of the drivers is to
avoid blocking situations in the proof of Theorem 3.1, without requiring the
assumption of a point-queue model as it is done in Varaiya (2013a). Note that
the assumption of a point-queue model is recognized in Varaiya (2013a) as a
big limitation. In this work, we take into account the effect of spatial queues
and their relative to their capacity (“horizontal” queues) towards spillback,
although the effect of full spillback is attenuated by the assumed adaptation
of drivers.
On the experimental side, the infrastructure required by the proposed
control algorithm is simple and also relatively inexpensive to install and
maintain. A successful field implementation of the algorithm at an intersec-
tion in Jerusalem has been presented, which demonstrates the applicability
of the presented control algorithm. Recall that current technologies face
many difficulties at measuring queue lengths. In addition, the experimental
30
results indicate that the proposed controller improves the performance of the
optimized fixed control plan, used by the traffic control center. It should be
noted that this comparison is difficult to address due to many variables that
cannot be controlled. The performance of the controller depends on the pen-
etration rate for the Bluetooth technology, which is an exogenous variable
that cannot be controlled in our experiment. We measured a penetration
rate of approximately 20 % during the day of the experiment. However, it
is expected that the penetration rate will increase in the near future due to
many factors as for example the increase of vehicles with a detectable built-in
Bluetooth device and the widespread use of Bluetooth wireless audio devices.
Finally, from the experimental point of view, it could be interesting to
compare in future applied research whether using traditional MP based on
queue lengths measured by video-based systems is preferred to the proposed
MP based travel times collected by Bluetooth sensors. While the use of the
traditional MP based on queue lengths has provable stability properties under
idealized conditions with strong assumptions, video-based systems require
complex and expensive installation and data processing. Our belief is that
the implementation based on Bluetooth sensors is much easier and simpler
than on video-based systems, and it has work-conservative property.
7. Acknowledgements
This research was financially supported by the Technion and the Israeli
Ministry of Transport and Road Safety. The first author was also supported
at the Technion by a fellowship from the Lady Davis Foundation. The au-
thors would like to thank the Jerusalem urban traffic management center
(“MANTI”) for the support in the experimental case study.
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