mathematics

Article
Understanding Complex Traffic Dynamics with the
Nondimensionalisation Technique
Juan Francisco Sánchez-Pérez 1, * , Santiago Oviedo-Casado 1 , Gonzalo García-Ros 2 , Manuel Conesa 1
and Enrique Castro 1

1 Department of Applied Physics and Naval Technology, Universidad Politécnica de Cartagena (UPCT),
30202 Cartagena, Spain; santiago.oviedo@upct.es (S.O.-C.); manuel.conesa@upct.es (M.C.);
enrique.castro@upct.es (E.C.)
2 Department of Mining and Civil Engineering, Universidad Politécnica de Cartagena (UPCT),
30202 Cartagena, Spain; gonzalo.garcia@upct.es
* Correspondence: juanf.sanchez@upct.es

Abstract: Hydrodynamic traffic models are crucial to optimizing transportation efficiency and
urban planning. They usually comprise a set of coupled partial differential equations featuring an
arbitrary number of terms that aim to describe the different nuances of traffic flow. Consequently,
traffic models quickly become complicated to solve and difficult to interpret. In this article, we
present a general traffic model that includes a relaxation term and an inflow of vehicles term and
utilize the mathematical technique of nondimensionalisation to obtain universal solutions to the
model. Thus, we are able to show extreme sensitivity to initial conditions and parameter changes, a
classical signature of deterministic chaos. Moreover, we obtain simple relations among the different
variables governing traffic, thus managing to efficiently describe the onset of traffic jams. We
validate our model by comparing different scenarios and highlighting the model’s applicability
regimes in traffic equations. We show that extreme speed values, or heavy traffic inflow, lead to
divergences in the model, showing its limitations but also demonstrating how the problem of traffic
jams can be alleviated. Our results pave the way to simulating and predicting traffic accurately on
a real-time basis.

Citation: Sánchez-Pérez, J.F.;

Keywords: traffic dynamics; nondimensionalisation; mathematical modelling; differential equations;
Oviedo-Casado, S.; García-Ros, G.;
traffic jams; sensitivity; engineering science; numerical simulations; urban planning
Conesa, M.; Castro, E. Understanding
Complex Traffic Dynamics with the
Nondimensionalisation Technique.
MSC: 00A73; 00A69; 00A79
Mathematics 2024, 12, 532. https://
doi.org/10.3390/math12040532

Academic Editor: Ripon Kumar

1. Introduction
Chakrabortty
Having an efficient transportation system is a necessary trait of any modern city; for
Received: 29 December 2023 this reason, great efforts are made in order to manage traffic and minimise the impact of
Revised: 5 February 2024 traffic jams [1]. Then, it follows that the ability to predict traffic in situ and in real time
Accepted: 6 February 2024 is crucial for a wide variety of reasons, ranging from economics to sustainability. In this
Published: 8 February 2024 scenario, the development of flexible, adaptive, and robust models can represent a great
advantage to administrations and users alike. Consequently, since the early research carried
out in the 1930s and 1950s, the field of traffic dynamics has received significant input from
both physicist and mathematicians [2,3], the reason being that traffic flow can be modelled
Copyright: © 2024 by the authors.
as if it were a continuous fluid.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
Early traffic models assumed a linear relation between the two fundamental variables
distributed under the terms and
that describe vehicles dynamics, namely, the traffic density at a point in space and time
conditions of the Creative Commons
(ρ( x, t) = ρ henceforth) and the mean velocity at said point (v( x, t) = v henceforth) [4]. The
Attribution (CC BY) license (https:// actual relation is based on a vehicle conservation law, which is formulated as
creativecommons.org/licenses/by/
∂ρ ∂Q(ρ)
4.0/). + = 0, (1)
∂t ∂x

Mathematics 2024, 12, 532. https://doi.org/10.3390/math12040532 https://www.mdpi.com/journal/mathematics

Mathematics 2024, 12, 532 2 of 14

where Q(ρ) denotes the flow of vehicles and corresponds to Q(ρ) = ρV (ρ), where V (ρ)
is some non-increasing function of the vehicle density, which is usually referred to as the
fundamental diagram of the model [5,6]. Linear models of this type, however, do not
capture well the complex dynamics of traffic, as the steady states to which they lead are one-
dimensional [7]. Thus, a whole set of non-linear models have been developed throughout
the years to account for the anisotropies that can develop in traffic flow [8–11], in a process
much like having different diffusion terms in the classical Navier–Stokes equations [12,13].
Among the new types of high-precision models, we find those that modify the funda-
mental diagram in such a way that the dependence of vehicle speed is no longer a unique
function of vehicle density. Rather, it becomes another differential equation whose solution
is, at the same time, part of and dependent on the model considered, thus compounding a
set of non-linear models with which anisotropies in traffic due to, e.g., congestion, changes
in the topology of the road considered, weather variability, or relaxation of the system, can
be simulated [14–18]. These models usually result in complicated coupled second-order
non-linear partial differential equations of the hyperbolic kind [19–21], which can be quite
involved to interpret and whose solutions must normally be obtained with cumbersome
numerical analysis that impedes their real-time usage [22]. Here, we take advantage of the
nondimensionalisation technique to solve and interpret a general non-linear second-order
partial differential equation model of traffic that includes the possibility of congestion,
changes in the number of vehicles due to entries and/or exits, and the relaxation of the
system, offering an alternative approach to understanding traffic models.
The nondimensionalisation technique requires normalising all variables in the problem,
reducing them to dimensionless independent and dependent variables and, consequently,
resulting in dimensionless equations and monomials, which are the ratios between the
different coefficients appearing in the problem. These monomials provide valuable in-
formation about the relative weight that each of the variables contributes to the problem
and permit analysing the influence that each of the variables exerts on the others. With
this information, it is possible to group the different variables by applying the π-theorem,
ensuring that each unknown variable appears only once within a group. Then, it is a simple
task to obtain universal solutions for each of the variables, with which the original problem
can be solved without having to resort to complicated numerical techniques.
The procedure for the nondimensionalisation of differential equations is well known
and thoroughly explained in the literature [23,24]. Moreover, it has been successfully
applied to a wide range of non-linear problems, such as diffusion of chlorides in reinforced
concrete structures or soil consolidation [25–28], where the universal curves obtained offer
invaluable information about the problems considered while offering a significant reduction
in computational time with a minimal increase in error. In this article, we tackle the problem
of understanding the involved behaviour of traffic by searching for the universal curves
in a complex traffic model. We do so by using nondimensionalisation in the general
traffic model. We show how the nondimensionalised model can be used to obtain general
solutions for the behaviour of traffic in an inexpensive computational time and use it to
demonstrate extreme sensitivity to initial conditions and parameter changes, which is a
typical signature of deterministic chaos that explains certain odd traffic behaviours [29].
The approach that we demonstrate here opens up new possibilities to explore complex
traffic models and paves the way for real-time studies and on-site applications [30].
The article is organised as follows: We first present the mathematical model of traffic,
and describe its main characteristics. Then, we proceed with the nondimensionalisation of
the model and the monomials that it leads to. Next, we show how for certain parameters,
these monomials become insensitive, which means that the universal curves tend to infinity
and cannot offer a proper solution due to the extreme sensitivity to the values of the
parameters. Finally, we demonstrate, for the rest of the monomials, the universal solutions
and their validation and draw some conclusions.
Mathematics 2024, 12, 532 3 of 14

2. Mathematical Model
We consider a traffic model that allows for inbound or outbound vehicles by breaking
the homogeneity of the continuity equation, such that for us, the number of vehicles is not
conserved. The system of equations that models the traffic flow is, then [31],
( ∂ρ ∂ρ ∂v
∂t + v ∂x + ρ ∂x = f,
∂v ∂v ∂v ∂v ∂v (2)
∂t + v ∂x + ρ ∂ρ ∂x = ∂ρ f,

where f is the relative change in the number of vehicles, a term that can be used to simulate
an inflow of traffic, accounting for possible lane changes, sudden speed variations, or
worsening traffic conditions. v here is the vehicle speed, and x is the distance. Thus,
Equation (2) accounts for the density of vehicles (ρ), which is spatially dependent as well as
time-dependent, containing, as a consequence, the possibility of local density variations, i.e.,
traffic jams. Additionally, we consider the relative velocity of the congestion propagation,
analogous to the sound speed of waves in a fluid medium, to be of the form

∂v
c(ρ) = ρ = αρ, (3)
∂ρ

which includes parameter α to account for the possible relaxation of the resultant wave
characterising traffic dynamics and which describes, e.g., alleviating measures introduced
at a given time to reduce heavy traffic. Thus, α permits traffic dissipation in the model
to occur. Our task is to solve this model by extracting the principal monomials using the
method of nondimensionalisation [23,32].
First, we choose the dependent variables that will be related to one another through the
independent variable; then, we define the dimensionless variables that span the monomials.
To do so, we first bound density and speed, such that for a given road, 0 ≤ v ≤ v M with a
certain fixed maximum speed (v M ), and 0 ≤ ρ ≤ ρ M . While v M depends on the kind of road
considered, typically, ρ M , the maximum density of vehicles allowed on a given road, is fixed.
The literature usually considers numbers in the range 0.12 ≤ ρ M (vehicles/km) ≤ 0.14 [33],
which assume a typical average length of 4.5 metres per vehicle, plus a minimum of 1 metre
in front and 1 in the back of each vehicle at any given time, and the possibility of a few
longer vehicles, such as trucks. Then, the dimensionless variables are

ρ − ρin v − vin t x
ρ′ = v′ = t′ = x′ = , (4)
ρ M − ρin v M − vin τ xM

where ρin and vin are the initial vehicle density and speed, respectively; x M is the maximum
road distance considered; and τ is the critical time at which the speed of vehicles goes to
zero, i.e., the extreme value for the traffic jam formation time. In these expressions, we can
see that ρ′ is the change in traffic density with respect to its value of maximum variation,
ρ M − ρin , which also ensures that this variable is contained in the interval ρ′ ∈ [0, 1]. The
great advantage of introducing this new variable is that in addition to the subsequent
dimensionless treatment and the obtaining of dimensionless monomials, when its value
is close to 0, it tells us that traffic flows easily and, when it is close to 1, that traffic tends
to get stuck. In the case of v′ , the dimensionless speed, the meaning is the variation in
the speed of vehicles with respect to the initial speed and also relative to its maximum
variation value, this variable being defined between a negative value and 1. In this case,
the more negative the value is, the more often traffic jams will occur, and the closer it is to
1, the more easily traffic will flow. Finally, for both t′ and x ′ , we also obtain dimensionless
variables by dividing them by τ and x M , respectively.
Mathematics 2024, 12, 532 4 of 14

We can use the variables defined in Equation (4) to find the dimensionless form of
the traffic model in Equation (2). By also using the velocity of congestion propagation,
Equation (3), we obtain

∂[(ρ M − ρin )ρ′ + ρin ]   ∂[(ρ M − ρin )ρ′ + ρin ]

′
+ (v M − vin )v′ + vin
∂(τt ) ∂( x′ X M )
(5)
 ∂[(v M − vin )v′ + vin ]
+ (ρ M − ρin )ρ′ + ρin

= f,
∂( x′ X M )

∂[(v M − vin )v′ + vin ]   ∂[(v M − vin )v′ + vin ]

′
+ (v M − vin )v′ + vin
∂(τt ) ∂( x′ X M )
(6)
 ∂[(v M − vin )v′ + vin ]
+ α (ρ M − ρin )ρ′ + ρin

= αf.
∂( x′ X M )
In cases where the problem exhibits linearity or pseudo-linearity, we can assume that
each term has an order of magnitude around 1 [32,34,35]. Then, the factors can be extracted
from the equations to obtain the relevant coefficients of the model, which are

ρ M − ρin vM ρM
(ρ M − ρin ) (v M − vin ) f (7)
τ xM xM

and
v M − vin vM ρM
(v M − vin ) α(v M − vin ) αf, (8)
τ xM xM
respectively. Dimensionless groups are then derived from the ratios of coefficients, with the
maximum number of groups being equal to the number of addends in the dimensionless
equation minus one. In systems of coupled equations, certain groups may be shared across
multiple equations. These groups can be further manipulated with basic mathematical
operations to be presented in the most advantageous form. For instance, one optimization
strategy involves ensuring that each unknown is associated with only one group [36]. We
now combine these coefficients to obtain dimensionless monomials Πi,j for each of the (i )
equations in our traffic model:

τv M ρ M (v M − vin ) f xM
Π1,1 = Π1,2 = Π1,3 = , (9)
xM v M (ρ M − ρin ) v M (ρ M − ρin )

τv M αρ M f xM
Π2,1 = Π2,2 = Π2,3 = . (10)
xM vM ρ M (v M − vin )
By combining the monomials appropriately and eliminating those which appear
repeated or are simply linear combinations of the others, we find the final four monomials
that completely characterise Equation (2):

τf ρ M (v M − vin ) f xM αρ M
Π1 = Π2 = Π3 = Π4 = . (11)
ρM v M (ρ M − ρin ) v M (ρ M − ρin ) vM

We can now apply the π-theorem, which states that we can express the monomial
containing the unknown as a function of all the other monomials that do not contain
unknowns, such that, e.g., Π1 = Ψ(Π2 , Π3 , Π4 ), from which the characteristic time of the
problem can be obtained as

ρ v − vin
 
ρ f xM αρ
τ = MΨ M M , , M , (12)
f v M ρ M − ρin v M (ρ M − ρin ) v M

where Ψ is some unknown function of the coefficients, which we will determine in the
ensuing sections for each particular case.
Mathematics 2024, 12, 532 5 of 14

3. Results
We now apply the expressions derived above to obtain specific functional forms for
Equation (11), considering certain initial conditions and particular road circumstances. Our
focus here is to understand traffic jam formation as a function of vehicle speed, density, and
change over time. To achieve this, we simulate the traffic model described by Equation (2)
using the Network Simulation Method [34,37], searching for the time (τ) at which the speed
goes to zero. As our starting point, we consider a given stretch of road in which, initially,
conditions are those of fluid traffic, such that vehicles circulate at an initial speed (vin )
which is the maximum speed permitted in said road.
The Network Simulation Method is a mathematical modelling approach employed for
simulating and analysing complex systems [34,37], particularly those characterised by mul-
tiple variables and non-linear relationships. This method revolves around the creation of an
electrical network comprising interconnected nodes that serves as a symbolic representation
of the physical medium or system under consideration. Within this network, the variables
of interest are defined, along with their interrelationships, enabling a comprehensive ex-
ploration of the system’s behaviour and dynamics. The Network Simulation Method has
been used in a wide variety of applications, including the simulation of economic [38–40],
social [41], and biological systems [42], among others [43–47].
To comprehend the distinct effects of each variable on traffic jam formation, we fix
two of the monomials at constant values throughout the time evolution of the problem.
This involves selecting appropriate values for the specific variables in these monomials.
Subsequently, we vary the other two monomials along a given dimension (variable) to
study their influence on traffic jam formation. Since the monomials in Equation (10) fully
characterise the differential equations of the traffic model, this approach allows us to
understand the individual contributions of different variables.
In the following analysis, we first explore the relationship between distance x M and
traffic jam formation time by keeping the speed constant, representing Π1 vs. Π3 . Sub-
sequently, we examine the influence of both speed and relaxation time α on traffic jam
formation time by keeping Π3 constant, representing Π1 vs. Π4 . In all cases, we choose
vin = v M and ρin = 0, resulting in Π2 = 0 [see Equation (12)] for the remainder of
the article.

3.1. Π1 vs. Π3 : Sensitivity to Initial Conditions

We begin by fixing Π4 and investigating the relationship between the traffic jam
formation time and distance with Π3 . To conduct this analysis, we maintain a constant
maximum density of ρ M = 0.14 (vehicles/km); an initial density of ρin = 0; an inflow
of vehicles of f = 10−3 (vehicles/m·s)—meaning a new vehicle enters the road per km
and per second; and a relaxation time of α = −300 (m2 /vehicles·s). Further, we select a
fixed initial and maximum speed, vin = v M = 20 km/h. Figure 1 illustrates the results of
simulating the traffic model using Equation (2) with the Network Simulation Method. The
simulation is conducted for various values of distance x M , determining traffic jam time
τ when the speed reaches zero. We calculate Π1 = Ψ(Π3 ) with all other monomials held
constant as per definition. Additionally, we present the best functional fit that provides the
specific Ψ(Π3 ), achieving an R2 close to unity in all cases.
There are two noteworthy observations about the behaviour of all Π1 = Ψ(Π3 )
calculated in Figure 1, providing valuable insights into relevant traffic dynamics. The
first is that as the distance approaches zero, the traffic jam formation time diverges to
infinity. This outcome is expected; if the stretch of road considered is sufficiently small, any
vehicle present can be considered a traffic jam. This implies that the model is only valid for
distances spanning several vehicle lengths, as defined by the maximum allowed density of
vehicles, ρ M . This density depends on the specific speed considered, changing from ∼50 in
Figure 1a,b, featuring initial and maximal speeds of 20 km/h and 50 km/h, respectively, to
∼20 in Figure 1c, where the speed considered is 100 km/h.
Mathematics 2024, 12, 532 6 of 14

0.1350
0.1345
0.1340
0.1335
Π1
0.1330
0.1325
0.1320
0.1315
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Π3

0.33095
0.33090
0.33085
0.33080
Π1

0.33075
0.33070
0.33065
0.00 0.05 0.10 0.15 0.20 0.25

Π3

0.661400

0.661395

0.661390
Π1

0.661385

0.661380

0.661375
0.00 0.02 0.04 0.06 0.08 0.10 0.12

Π3
Figure 1. Π1 as a function of Π3 for different values of distance x M (dots). Fitting of functional Ψ
describing Π1 = Ψ(Π3 ) (full line) with the particular functional form specified above each figure.
Throughout, Π2 = 0 and Π4 = −42/v M are kept constant, with v M = vin ranging from 20 km/h in
(a) (Π4 = −7.56) to 50 km/h in (b) (Π4 = −3.024) and 100 km/h in (c) (Π4 = −1.512). In all cases,
initial density of vehicles ρin = 0.
Mathematics 2024, 12, 532 7 of 14

On the opposite end, we observe that the traffic jam formation time is essentially
constant, indicating that given certain speed and density of vehicles, traffic jam formation
is inevitable. This also suggests that higher speeds lead to slower traffic jam formation, but
the latter can also occur in shorter distances.
The asymptotic behaviour with constant τ independent of Π3 and distance x M and the
smallness of the fitting parameters that control the influence of Π3 on Π1 indicate extreme
sensitivity to the values of the variables in the problem. Minute differences can result in
significant changes in the predicted traffic jam time. Neither the exact equations and their
numerical solution nor the fitted models can capture such behaviour, leading to substantial
differences in the predicted traffic jam time for apparently similar initial conditions.
This sensitivity explains a phenomenon experienced by any driver on the road—the
apparent spurious formation of traffic jams following smooth conditions in what seemed to
be dense but fluid traffic [48–50]. It also suggests that traffic jam alleviation measures should
focus on reducing the number of incoming vehicles on the road (namely, f ) and accelerating
relaxation times. Any other measure is likely to have a random or negligible impact.

3.2. Π1 vs. Π4 : Speed Dependence

We now examine the relationship between Π1 and Π4 while keeping Π3 constant
(with Π2 = 0 throughout, as done previously). We seek to understand the impact of speed
and relaxation measures on traffic jam formation and to illustrate how the nondimension-
alisation of the differential equation system facilitates the accurate, simple, and efficient
modelling of traffic dynamics.
Initially, we determine the traffic jam formation time (τ) by solving the exact model
for two instances of the relaxation parameter (α) while varying the speed (v M = vin )
and keeping the ratio x M /v M = 1/10 constant to maintain Π3 unchanged. By selecting
ρ M = 0.14 and f = 10−3 , we obtain the results depicted in Figure 2a (with α = −100) and b
−c/Π
(with α = −200). In both cases, we fit the data points to a functional form Π1 = a + b e Π 4 ,
4
as shown above each figure with the best-fitting parameters. This functional form effectively
reproduces the traffic behaviour with a near-unity R2 . Subsequently, we use these fits to
validate the model.
Observing the results in Figure 2, we note that the traffic jam formation time is highly
dependent on speed, favouring fast traffic jam formation at slower speeds. This is due to the
fact that for slow-moving vehicles, the addition of more traffic through the f term increases
the density faster than it can dissipate it. This suggests that alleviating measures, once the
speed is limited to a certain high value, should focus on introducing relaxation mechanisms,
such as facilitating road exits. This is demonstrated in Figure 3, where instead of changing
Π4 through speed, we fix v M at three different values and change α using the Network
Simulation Method. We observe a greater sensitivity to the relaxation parameter, indicating
that a small relaxation parameter can significantly extend the traffic jam formation time.
Finally, note that all panels in Figures 2 and 3 present similar results with different
ranges of validity. This suggests the general applicability of the traffic model we are
analysing and demonstrates that the nondimensionalisation technique accurately captures
the correct dynamics over a wide range of parameter regions. This means that it can be
effectively used to simplify the resolution of traffic differential equations, reducing the
required time and enhancing their applicability under real-time conditions. To demonstrate
the predictive capabilities of the obtained functionals, Ψ, we validate the model using the
data from Figure 2b and interpolate for different cases.
Mathematics 2024, 12, 532 8 of 14

3.0

2.5

2.0

Π1
1.5

1.0

0.5

- 0.8 - 0.7 - 0.6 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1

Π4

1.4
1.2
1.0
Π1

0.8
0.6
0.4
0.2
- 1.6 - 1.4 - 1.2 - 1.0 - 0.8 - 0.6 - 0.4 - 0.2

Π4
Figure 2. Π1 as a function of Π4 for two instances of the relaxation parameter, with α = −100 in (a)
and α = −200 in (b). The dots show the result of numerically solving the traffic model differential
equations with the Network Simulation Method for different instances of the speed (v M ) while
keeping the ratio x M /v M = 10 (and thus Π3 ) constant. The full line is the best fit, Π1 = Ψ(Π4 ),
whose functional form is specified above each figure. In both cases vin = v M , ρin = 0, ρ M = 0.14 and
f = 10−3 . In both cases, Π3 = 0.257.

3
Π1

1
- 1.6 - 1.4 - 1.2 - 1.0 - 0.8 - 0.6 - 0.4 - 0.2

Π4
Figure 3. Cont.
Mathematics 2024, 12, 532 9 of 14

3.0

2.5

2.0
Π1 1.5

1.0

0.5
- 3.0 - 2.5 - 2.0 - 1.5 - 1.0 - 0.5

Π4

1.2

1.0

0.8
Π1

0.6

0.4

0.2

-8 -7 -6 -5 -4 -3 -2 -1

Π4
Figure 3. Π1 as a function of Π4 for three different values of v M , i.e., 100 km/h in (a), 50 km/h in
(b) and 20 km/h in (c), changing the relaxation parameter (α) while keeping the distance (x M = 500 m)
constant. The dots show the result of numerically solving the traffic model differential equations
with the Network Simulation Method. The full line is the best fit, Π1 = Ψ(Π4 ), whose functional
form is specified above each figure. In all cases, vin = v M , ρin = 0, ρ M = 0.14, and f = 10−3 . In this
case, Π3 = 0.129 in (a), Π3 = 0.257 in (b), and Π3 = 0.643 in (c).

Model Validation
We now validate the methodology employed and the results obtained for the func-
tionals Π1 = Ψ(Π4 ) by choosing various study cases, displayed with their corresponding
parameters of choice in Table 1. For these, we calculate both Π3 and Π4 , which shows that
there is a slight remaining dependence on Π3 , which is nonetheless inconsequential for
the results.

Table 1. Four different scenarios with which to compare the results of the exact traffic model
simulation and the approximate solution provided by the nondimensionalisation technique. We
benchmark both pathways through the traffic jam formation time (τ) shown in the last two columns.
We find great agreement between the model and the exact result.

2
Case veh
f (m ·s ) vM ( km
h )
x M (m) m
α ( veh ·s )
ρ M ( veh
m )
Π3 Π4 Π1 τUS (s) τsim (s)
1 10−3 50 500 −100 0.14 0.257 −1.008 0.276 138.949 138.890
2 2 · 10−3 100 400 −150 0.14 0.201 −0.756 0.368 92.613 92.593
3 3 · 10−3 120 300 −300 0.14 0.193 −1.260 0.221 37.061 37.036
4 4 · 10−3 110 200 −293 0.14 0.187 −1.343 0.207 26.080 26.072
Mathematics 2024, 12, 532 10 of 14

All the models presented in Table 1 were deliberately selected to encompass a broad
spectrum of parameters. This intentional diversity aims to underscore the comprehensive
applicability of the nondimensionalisation methodology and the universality of the results.
The chosen models span a wide range of speeds, allowing us to calculate traffic jam times
at various distances. Simultaneously, we explore changes in the relaxation parameter
and an increase in the number of allowed incoming vehicles. Intriguingly, despite the
expected delay in traffic jam formation due to speed increase, we observe a counterintuitive
trend: the traffic jam formation time decreases across the studied cases. This is due to
the role played by the term inflow of vehicles, f , which highlights its importance as the
key factor requiring attention for preventing traffic jam formation, advocating for targeted
interventions on this parameter to mitigate traffic congestion.
It is worth noting that in all cases, we set the maximum allowed density (ρ M ) to the
highest conceivable value [33], essentially testing the model’s validity at the extremes. This
approach ensures the robustness of the model across a range of scenarios, validating its
efficacy even under conditions where traffic jam formation might be delayed due to lower
density limits.
With the value of Π4 , one either chooses to calculate Π1 using one of the follow-
ing equations:

e−0000124232/Π4
Π1 = 0.000282782 + 0.277576 Π3 = 2.57 · 10−3 (13)
Π4

e−0.0996657/Π4
Π1 = 0.352325 + 0.728287 Π3 = 0.129 (14)
Π4
or, provided that the values of Π3 and Π4 do not coincide with those chosen in Figure 2b,
interpolates between them to obtain Π1 , from which we can readily obtain the traffic jam
formation time, τUS . We compare this with the exact result τsim for those same parameters
obtained by solving the exact model with the Network Simulation Method [34,37]. An
extraordinarily good agreement between exact result and model is found. Figure 4 displays
the results for the speed evolution obtained by solving the exact traffic model differential
equations numerically, up to the point at which traffic jam formation occurs, signalled by
the speed reaching zero.

30
14
12 25

10 20
v (m/s)
v (m/s)

8 15
6
10
4
2 5

0 0
0 20 40 60 80 100 120 140 0 20 40 60 80 100
t (s) t (s)
35 35
30 30
25 25
v (m/s)
v (m/s)

20 20
15 15
10 10
5 5
0 0
0 10 20 30 40 0 5 10 15 20 25 30
t (s) t (s)
Figure 4. Numerical time evolution of traffic Equation (2) simulated with the Network Simulation
Method [34,37]. Each figure shows the vehicle speed as a function of time up to the point at which
traffic jam forms, labelled τsim , at which the speed goes to zero. (a–d) correspond to cases 1 to 4
shown in Table 1.
Mathematics 2024, 12, 532 11 of 14

4. Discussion
In this article, we introduce a generalised traffic model that incorporates discontinuities
and anisotropies through a relaxation parameter, along with the ability to dynamically
adjust the number of vehicles over time. Using a numerical approach, we determine the
traffic jam formation time as the time at which the speed of vehicles drops to zero. This
is a necessary approximation taken in order to have a definite rule that can be used for
any parameter set. Note that actually, traffic jam formation starts earlier, as the density
of vehicles overcomes a critical value. However, said critical value is dependent on the
choice of parameters f and α and the initial conditions and thus cannot be used as a
universal measure to validate the model, hence our choice. We understand that this
represents an advantage, as it allows us to demonstrate our approach and perform a
universal analysis, but it is also a drawback, as it somewhat limits the accuracy of the
prediction. Nonetheless, a precise traffic jam formation time prediction can be obtained
by simply shifting the definition for a particular set of parameters; it is, therefore, not a
limitation of our approach. Future work will explore the different particulars of the traffic
model that we generally present here, as the traffic jam formation time offers a unique
benchmark, crucial to evaluating the model’s ability to reflect real-world road conditions.
By using nondimensionalisation, we decompose the traffic differential equations
into their principal monomials, establishing unique relationships among dimensionless
variables. This technique enables a more efficient determination of the traffic jam formation
time, or any other parameter, compared with solving the entire model numerically, without
a noticeable loss of precision, and shows the stability regions of the model by easily
demonstrating where divergences appear. The insights that we gain in this way can,
therefore, be used to determine efficient ways to manage traffic and show that the focus
should be placed not on long-term prediction but rather on rapid adaptable measures.
Our initial study reveals a high sensitivity of the model to certain parameters, a char-
acteristic feature of deterministic chaos. This sensitivity explains the seemingly irrational
occurrence of traffic jams under fluid driving conditions. Such instabilities, common in
complex models with coupled partial differential equations, result in significant variations
over time due to slight changes in certain parameters. This leads to exponential numerical
complexity, making long-term predictions challenging, as the system effectively behaves
in a random manner after a critical time. This behaviour is evident in the divergence to
infinity or constant values of the functionals of the monomials. Such behaviour is critical,
as it inherently limits our ability to predict traffic beyond a certain point, regardless of the
complexity of the modelling that we use, and means that traffic prediction models should
rather focus on short-term prediction and traffic-alleviating measures in real time, a goal
for which our analysis is specially suited, given the speed boost provided by simplifying
the model while maintaining accuracy through nondimensionalisation.
Expanding our study, we analyse different relationships between the monomials and
validate the results by comparing them to exact simulations. The analysis demonstrates a
remarkable ability to predict the traffic jam formation time based on the obtained function-
als, eliminating the need for complex simulations each time. This simplification reduces the
prediction problem to interpolating the correct functional. While the nondimensionalisation
technique is limited to a specific range of values, its solutions are universal enough to be
applied to a broad range of traffic situations, offering a simple and straightforward method
to obtain key parameters. It is true and a limitation of our and any traffic modelling ap-
proach that not all variables affecting real-life traffic can be taken into account, but it is also
true that by introducing a traffic inflow and relaxation parameters, we provide our model
with enough flexibility to accommodate a wide range of nuances, such as weather condi-
tions or certain human behaviours, which can be thoroughly explored with the technique
presented here.
Mathematics 2024, 12, 532 12 of 14

5. Conclusions
Having reliable traffic models that can be utilised in real time is a crucial asset for
any society. However, traffic models often prove to be cumbersome, complex, and diffi-
cult to solve, limiting their interpretability and applicability. In this article, we consider
one such complex model and demonstrate that it is possible to simplify its analysis with-
out compromising the accuracy of the short-term predictions that can be drawn. We do
so by nondimensionalising the model and thoroughly analysing the resulting monomi-
als, with which we obtain universal solutions. Focusing on traffic jam formation as a
benchmark, we show extreme sensitivity to initial conditions and parameter changes and
demonstrate the validity of our approach, which shows that the simplified model obtained
through nondimensionalisation has essentially the same accuracy as the solutions of the
full-model equations.
Nondimensionalisation techniques can become an essential tool for simplifying com-
plex traffic models in transportation engineering. By normalizing variables and removing
physical units, the mathematical expressions are greatly simplified for analysis. This
approach is particularly useful in traffic simulations, where various interacting factors
contribute to model complexity. Identifying dimensionless parameters streamlines the
system’s representation. Simultaneously, integrating network simulations is crucial to
solving these models. These simulations allow for a detailed exploration of traffic dynamics
and intervention impacts, offering valuable insights for optimizing urban mobility and
transportation systems through informed decision making.
Following the first demonstration of the usefulness of nondimensionalisation for traffic
models, future dedicated work will explore in detail the onset of chaotic behaviour and the
extreme dependence of minor variations on input parameters, which have a huge impact
on the ability to make long-term predictions, which suggests that these models should
target quick response to changing traffic conditions rather than extending the predictions
in time. Furthermore, given the simplicity that nondimensionalising the model brings,
we will analyse the parameter range in full detail and include different parameters that
account for various traffic effects. Additionally, future work will also test the validity of the
model beyond the mere numerical analysis by using real traffic data.

Author Contributions: Conceptualization, J.F.S.-P., S.O.-C. and G.G.-R.; methodology, J.F.S.-P. and
S.O.-C.; software, J.F.S.-P. and S.O.-C.; validation, J.F.S.-P., S.O.-C., G.G.-R., M.C. and E.C.; formal
analysis, J.F.S.-P., S.O.-C. and G.G.-R.; writing—original draft preparation, J.F.S.-P., S.O.-C., G.G.-R.,
M.C. and E.C.; writing—review and editing, J.F.S.-P., S.O.-C., G.G.-R., M.C. and E.C. All authors have
read and agreed to the published version of the manuscript.
Funding: S.O.-C. is grateful for the support of the María Zambrano program of postdoctoral stays
(Ministerio de Ciencia, Innovación, y Universidades).
Data Availability Statement: Any data not directly available on the article itself will be provided by
the corresponding author upon reasonable request.
Conflicts of Interest: The authors declare no conflicts of interest.

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