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SIMULATION ANALYSIS OF TRAFFIC CONGESTION AT UNSIGNALIZED

CIRCULAR ROUNDABOUT USING MICROSCOPIC TRAFFIC FLOW MODEL

Article in Journal of Computational Analysis and Applications · April 2025

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Journal of Computational Analysis and Applications VOL. 34, NO. 4, 2025

SIMULATION ANALYSIS OF TRAFFIC CONGESTION AT UNSIGNALIZED CIRCULAR

ROUNDABOUT USING MICROSCOPIC TRAFFIC FLOW MODEL

Déo KABANGA 𝟏
School of Pure and Applied Mathematics(SPAS)
Department of Mathematics and Actuarial Sciences
Kenyatta University
Nairobi, Kenya.
kabangado@gmail.com

Kennedy Otieno Awuor 𝟐

School of Pure and Applied Mathematics(SPAS)
Department of Mathematics and Actuarial Sciences
Kenyatta University
Nairobi, Kenya.
awuor.kennedy@ku.ac.ke

Abstract
The Effective management of roundabout is very important as a part of the current traffic
infrastructures in developing countries. A comprehensive understanding of its dynamics especially
at unsignalized roundabouts is imperative. In this paper, we study and present a Microscopic
Simulation of traffic in an unsignalised roundabout using a Microscopic Car Following model
based on the Intelligent Driver Model (IDM). At an unsignalized roundabout, traffic flow relies
heavily on the principles of yielding and right-of-way. Vehicles entering the roundabout must yield
to circulating traffic already inside. This dynamic often leads to a smoother flow compared to
traditional intersections with traffic lights. However, during peak hours or when drivers fail to
yield properly, congestion can occur, causing delays. Proper understanding and adherence to
roundabout rules are essential for maintaining efficient traffic flow and preventing accidents.
Hence, this paper delves into the mathematical modeling of traffic dynamics specifically tailored
for unsignalized circular roundabouts. we meticulously examine mathematical models governing
traffic flow, treating it as a continuous phenomenon, while delving into their inherent limitations.
The impacts of the additional straight and turning movements in the Intelligent Driver Model
(IDM) on traffic dynamics inside the circular roundabout are numerically analyzed using
MATLAB, employing a fourth-order Runge-Kutta method after Fast Fourier Transform(FFT).
Four Vehicles are put in a simulation environment over a specified period of time over which they
are considered for analysis and the simulated results culminate in the determination of the range
of velocities for three different types of movement considered in the analysis and different types
of manoeuvers serving as a gauge for roundabout capacity and management. This study furnishes
invaluable insights for scientist and road engineers involved in roundabout design in developing
countries.

Key Words: Microscopic model,Intelligent Driver Model (IDM), Simulation analysis, Equation of
traffic flow, Unsignalized Circular Roundabout.
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1 Introduction

Roundabouts are essential in developing countries for several reasons. Firstly, they help improve
traffic flow by minimizing congestion and reducing the need for traffic signals, thus
enhancingoverall efficiency. Secondly, they are cost-effective solutions compared to traditional
intersectionswith traffic lights, making them more feasible for countries with limited resources.
Thirdly,roundabouts enhance road safety by reducing the risk of severe accidents, as they
encourageslower speeds and provide clearer traffic patterns. Lastly, they contribute to urban
planning and development by accommodating increased traffic volumes as cities expand, offering
sustainable solutions for transportation infrastructure.

Developing countries have been relaying on roundabout as a major intersection of many

roads[21, 1], However the auto-control of this infrastructure is a big issue due to the cost of energy
and technological capacity in many developing countries[8, 4]. One of the major indicators of the
development of any nation is the effectiveness,safety and the ability to move people and goods[18,
22]. Unsignalized roundabout are examples of bottlenecks that are interesting in the study of traffic
congestion since they are not directly controlled in many developing countries. More recently,
many researchers suggested models on networks that are able to describe the dynamics at
junctions.[7, 11] and the roundabout were not left behind.[2, 9]. Circular roundabouts have become
very popular in recent years due to its management flexibility, safety and mobility [16]. Traffic
congestion is a major problem of transport in many countries and it’s not expected to end soon [6,
13], Recent researchers in civil engineering and mathematical modeling such as [15, 19] have
formulated different microscopic and macroscopic models which described the source of traffic
congestion and have in common findings the poor management of different types of intersections
such as T-Junctions and Roundabouts[12, 17], The cities have been developed in a disintegrated
urban form spreading along major traffic corridors [20]. At the unsignalized Roundabout, we have
three different type of manoeuvers, two type of straight movements and a lane changing movement
[2, 24].

This research analyses the traffic congestion evolution process using simulation analysis and
proposes a theory to analyze roundabout traffic flows and a strategy to determine the effective use
of a unsignalized roundabout focusing on three different types of movement inside the roundabout.
The generated microscopic model adding lane change in Intelligent Driver Model (IDM) is
developed to address the problem of congestion in the roundabout which is the source of accident
and source of traffic congestion and delay in most roads in Africa [10, 3]. The objective is to
suggest safety measures and Planning improvements in transport infrastructure in many
developping countries. Therefore, it is very important to redevelop procedures for integrating
various local traffic characteristics for a thorough analysis as planning expanding the roads and
resources to achieve it are quite expensive and unsustainable hence, cannot be relied upon in
African Countries. we assure that All the parameters and variables chosen for the study are well
evaluated, and Our research consists of five steps consecutively,namely: mathematical modeling,
model discretisation, computer programming,numerical simulation, and observation of simulation
results.
In a real world road network, there are a few specific events that occur as vehicles move around
the system: Arrivals In a simulation, only a restricted area of the road system is included and there
must be a model that controls the arrival of vehicles to the network. Traffic Flow Once vehicles
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Journal of Computational Analysis and Applications VOL. 34, NO. 4, 2025

are in the network, their behavior as they move along roads should be as realistic as possible.
Turning When vehicles arrive at intersections, they must choose a direction to continue their travel.
In our system we have formulated several essential models that govern the movement, arrival, and
decisions of vehicles in the network. A model for the flow of traffic and the behavior of vehicles
as they either drive freely or follow other vehicles is given. Also, models for vehicle arrivals and
turning behavior,both of which are based on real-world vehicle count data, are presented.

2 MODEL PROBLEM

The Model developed in this research is generated based on Intelligent Driver Model (IDM) one
of the types of car-following models. To determine the simulation analysis of traffic congestion
by adding lane change to the straight movement inside the roundabout is the major work of this
research. In this research we considered the common roundabout of four entering and four exiting
roads as one that allows the best potential vehicles throughput while reducing vehicles delay and
that has the best use in many developing countries. The major goal is to discover the optimum
range of velocities allowed in order to simulate the traffic congestion analysis,time needed for
action to clear the roundabout for straight and turning movements. We considered a roundabout in
the following geometric forms for mathematical analysis, the two unsignalized T-junctions
Combined formed our unsignalized circular roundabout for analysis.

Figure 1: Turning Movement along circular roundabout Image from [14]

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3 EQUATIONS OF THE MODEL

A microscopic model of a traffic describes the vehicle (car) following behavior as well as the lane-
changing behavior of every vehicle in the traffic. A modern roundabout gives priority to a
circulating flow; however, a conventional roundabout gives priority to traffic that enters the
roundabout [5, 23], this theory is used to describe all my following equations.The initial condition
for motion through the path A,B or C Paths (5) is given by 0 < 𝜃 < 2𝜋. The initial condition
will keep changing depending on the direction of the turn. For the motion described by AB, a
vehicle moves from the straight path and to the connecting path. This motion makes an arc like
path. A parametrization of the formed arc is done to help us in describing the motion. If we
continue this path, the vehicle(s) moves in such a way that it forms part of a circle with a center
somewhere and with radius 𝑟 which is always constant throughout the motion while 𝜃 as the
vehicle moves along path AB. The movement is occurring on a two-dimensional plane on the 𝑥
and 𝑦 -axis where considering 𝑥 and 𝑦 in terms of the polar coordinates; 𝑥 = 2𝜋𝑟cos𝜃 and
𝑦 = 2𝜋𝑟sin𝜃. Looked this way, 𝑟 is a vector and is given as 𝑟⃗(𝜃) = 2𝜋𝑟(cos𝜃𝑖 𝑒1 + sin𝜃𝑗 𝑒2 ).
A velocity 𝑣𝑖 is applied on the vehicle as it travels through the desired distance 𝑥𝑦 or the 𝑟
path., The effects of the radius,the number of different paths and the gradient used to reach every
path are the three parameters to be incorporated in the equation of the traffic motion in order to
analyze their effects on the flow in and out of the roundabout.
According to the two parts of the roundabout A,B and C (5) entering and exiting which
form the entering and getting out parties of the roundabout the equation of continuity changes to
the following equation as they are cars entering and exiting the roundabout,𝛼 is characterizing the
cars entering the roundabout when 𝛽 is for vehicles leaving the roundabout over a specified
period of time
𝜕𝜌(𝑥,𝑡) 𝜕𝑞(𝑥,𝑡)
+ 𝜕𝑥 = 𝛼𝑗 (𝑥, 𝑡) − 𝛽𝑖 (𝑥, 𝑡) (1)
𝜕𝑡
. 𝛼𝑗 : The leading car 𝛽𝑖 : The leading car The right part of equation (1) is the difference of
distance between the leading cars and the following cars exiting in one of the three considered
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types of movement inside the roundabout which we considered to be constant due to time and
position 𝛾(𝑥, 𝑡) ,hence we have the final equation of the movement.

𝜕𝜌(𝑥,𝑡) 𝜕𝑞(𝑥,𝑡)
+ = 𝛾𝑘 (𝑥, 𝑡) (2)
𝜕𝑡 𝜕𝑥

The equation of a fluid flow is described by 𝜌(𝑥, 𝑡) the density of the traffic stream at a
given point in the roundabout 𝑥 at a given time 𝑡 and 𝑞 be the flux of traffic through the point
(𝑥, 𝑡).
Now we consider a dynamical system of 𝑛 vehicles, translated as follows;

𝑥1 (𝑡)
𝑥2 (𝑡)
𝑥(𝑡) = . (3)
.
(𝑥𝑛 (𝑛))

The 𝑖 𝑡ℎ component of the 𝑥(𝑡) is 𝑥𝑖 (𝑡) 𝑖 ∈ {1, … . , 𝑛} and 𝑥(𝑡) is again the vector position.
By differentiation, we obtain;

𝑑
𝑥 (𝑡)
𝑑𝑡 1
𝑑
𝑥 (𝑡)
𝑑𝑥 𝑑𝑡 2
𝑣(𝑡, 𝑥) = = ⋅ (4)
𝑑𝑡
⋅
𝑑
(𝑑𝑡 𝑥𝑛 (𝑡))
𝑑𝑥
It can be seen that 𝑑𝑡 = 𝑓(𝑡, 𝑥) where 𝑓(𝑡, 𝑥) is the vector field of the system. In our
case, 𝑓(𝑡, 𝑥) = 𝑣(𝑡, 𝑥). By integration;
𝑑𝑥
∫ 𝑑𝑡 𝑑𝑡 = ∫ 𝑣(𝑡, 𝑥)𝑑𝑡 For the turning movement, we now use approximation for 𝑥
which is given by; 𝑥(𝑡) = ∫ 𝑣(𝑡, 𝑥)𝑑𝑡. This approximation will give us a formula for the turning
angles 𝜃. For each vehicle 𝑖, its position 𝑥𝑖 (𝑡) is given by;

𝑥𝑖 (𝑡) = ∫ 𝑣𝑖 (𝑡, 𝑥)𝑑𝑡; where𝑖 = 1,2, … … … , 𝑛 (5)

Looking at the Fig (2) (3) (4) bellow, there will be three types of movements to be analyzed.
We will make our equation using the idea that no cars appear or vanish. So, any car that enters a
part of the roundabout at A will eventually leave at B. The way they go depends on which way
they turn and where they exit the roundabout in the three considered movement.

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𝑥̇ 𝑖 = 𝑣𝑖 (𝑡)
𝑥̇ 𝑖 (𝑡) = 2𝜋𝑟1 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]
𝑥̇ 𝑖 (𝑡) = 2𝜋𝑟2 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]
𝑥̇ 𝑖 (𝑡) = 2𝜋𝑟3 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]

Where; 𝜃𝑖 (𝑡) ≅ ∫ 𝑣𝑖 (𝑡)𝑑𝑡 and 𝑒1 and 𝑒2 are unit orthogonal vectors,the final function of the
motion 𝐹(𝑀) in the whole roundabout is finally given by a microscopic model of traffic flow
that describes the behavior of individual vehicles within the traffic stream. One common
microscopic model is the car-following model the Intelligent Driver Model (IDM) which
represents the interactions between a vehicle and the vehicle immediately in front of it .

Figure 3: Illustration of Car following Model in straight Movement

The IDM is expressed by the Four following equations:

𝜕𝜌(𝑥,𝑡) 𝜕𝑞(𝑥,𝑡)
+ 𝜕𝑥 = 0 (6)
𝜕𝑡
𝑑𝑣
𝑣(𝑡 + Δ𝑡) = 𝑣(𝑡) + 𝑑𝑡 Δ𝑡 (7)
Δ𝑣
𝑎(𝑡 + 𝑇) = 𝑐𝑣 𝑚 (Δ𝑥)𝑙 (8)
𝑑𝑣 𝑣 𝛿 𝑠∗ 2
= 𝑎 (1 − (𝑣 ) − ( 𝑠 ) ) (9)
𝑑𝑡 0

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Where: 𝑣 is the velocity of the vehicle, 𝑎 is the maximum acceleration, 𝑣0 is the desired
velocity, 𝑠 is the spacing between the current vehicle and the vehicle in front, 𝑠 ∗ is the desired
minimum spacing, 𝛿 is the exponent determining the sensitivity to velocity differences.
Update Equation of Velocity adding the change of position:
1 𝑑𝑣
𝑥(𝑡 + Δ𝑡) = 𝑥(𝑡) + 𝑣(𝑡)Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2 (10)

Final equation for Position in circular roundabout:

1 𝑑𝑣
𝑥(𝑡 + Δ𝑡) = 𝑥(𝑡) + 𝑣(𝑡)Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2 (11)

For circular Roundabout the turning angle is given by the following equations:
𝑣(𝑡)
𝜃(𝑡 + Δ𝑡) = 𝜃(𝑡) + 𝑅 Δ𝑡
Where: - 𝜃 is the angular position of the vehicle on the roundabout. - 𝑅 is the radius of the
roundabout/circular path.
adding the turning movement to the above equations we have the following equations:
1 𝑑𝑣
𝑥(𝑡 + Δ𝑡) = 𝑥(𝑡) + 𝑣(𝑡) ∗ 𝑟𝑖 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2 (12)

Where; 𝜃𝑖 (𝑡) ≅ ∫ 𝑣𝑖 (𝑡)𝑑𝑡 and 𝑒1 and 𝑒2 are unit orthogonal vectors, the final function
of the motion 𝐹(𝑀) in the whole roundabout. Which helps to get the final update equation for
the position of vehicle i in Microscopic Model inside the circular roundabout motion.
For straight Movement:
1 𝑑𝑣
𝑣1 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑥̇ (𝑡) + 𝑣(𝑡)Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2
1 𝑑𝑣
𝑣2 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑥̇ (𝑡) + 𝑣(𝑡)Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2
1 𝑑𝑣
𝑣3 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑥̇ (𝑡) + 𝑣(𝑡)Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2
𝐹(𝑆𝑡𝑟𝑀𝑜𝑣 ) = 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝑣 𝜕𝜌2 (𝑥,𝑡+Δ𝑡) (13)
+ 𝑣max − 𝜌max = 𝛾1 (𝑥, 𝑡 + Δ𝑡)
𝜕𝑡 𝜕𝑥 max 𝜕𝑥
𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝑣max 𝜕𝜌2 (𝑥,𝑡+Δ𝑡)
+ 𝑣max −𝜌 = 𝛾2 (𝑥, 𝑡 + Δ𝑡)
𝜕𝑡 𝜕𝑥 max 𝜕𝑥
𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝑣 𝜕𝜌2 (𝑥,𝑡+Δ𝑡)
( + 𝑣max − 𝜌max = 𝛾3 (𝑥, 𝑡 + Δ𝑡))
𝜕𝑡 𝜕𝑥 max 𝜕𝑥

For two types of turning movement we have the following system:

𝐹(𝑇𝑢𝑟𝑛𝑀𝑜𝑣 ) =

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1 𝑑𝑣
𝑣1 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑥̇ (𝑡) + 𝑣(𝑡) ∗ 𝑟1 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2
1 𝑑𝑣
𝑣2 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑥̇ (𝑡) + 𝑣(𝑡) ∗ 𝑟2 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2
1 𝑑𝑣
𝑣3 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑥̇ (𝑡) + 𝑣(𝑡) ∗ 𝑟3 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ]Δ𝑡 + 2 𝑑𝑡 (Δ𝑡)2
𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝑣 𝜕𝜌2 (𝑥,𝑡+Δ𝑡) (14)
+ 𝑣max − 𝜌max = 𝛾1 (𝑟1 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ], 𝑡 + Δ𝑡)
𝜕𝑡 𝜕𝑥 max 𝜕𝑥
𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝑣 𝜕𝜌2 (𝑥,𝑡+Δ𝑡)
+ 𝑣max − 𝜌max = 𝛾2 (𝑟2 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ], 𝑡 + Δ𝑡)
𝜕𝑡 𝜕𝑥 max 𝜕𝑥
𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝜕𝜌(𝑥,𝑡+Δ𝑡) 𝑣 𝜕𝜌2 (𝑥,𝑡+Δ𝑡)
( + 𝑣max − 𝜌max = 𝛾3 (𝑟3 [cos𝜃𝑖 (𝑡)𝑒1 + sin𝜃𝑖 (𝑡)𝑒2 ], 𝑡 + Δ𝑡))
𝜕𝑡 𝜕𝑥 max 𝜕𝑥

These equations describe how the velocity and position of each vehicle change over time
based on their interactions with neighboring vehicles. The IDM captures the tendency of drivers
to maintain a desired velocity while keeping a safe distance from the vehicle ahead, while also
considering the ability to accelerate and decelerate.
In practice, various modifications and extensions to the IDM exist to account for additional
factors such as driver heterogeneity, traffic conditions, and road geometry. Microscopic traffic
flow models like the IDM are fundamental tools in understanding and simulating traffic dynamics
at the individual vehicle level.

4 Method of Solution

The obtained non-linear first order partial differential equation in the model will be transformed
to first order ordinary differential equation using fast fourier transform(FFT). For the next step I
fixed the initial parameters for initial conditions for the specific roundabout and solve for 𝜌 in
equation numerically in MATLAB using a fourth order Runge-kutta by assuming periodic
boundary conditions of maximum velocity and maximum density, in order to determine the
movement of each considered vehicles.

5 NUMERICAL ANALYSIS

5.1 Discretisation using Fast Fourier Transform(FFT)

By Fast Fourier Transform (𝐹𝐹𝑇) we transform our partial differential equation(PDE) to

ordinary differential equation(ODE)
(𝑡 + Δ𝑡) ≠ 0:

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𝜕𝜌 1 ∞ 𝜕𝜌 ̂ (𝑘,𝑡+Δ𝑡)
𝜕𝜌
𝐹 ( 𝜕𝑡 ) = 2𝜋 ∫−∞ 𝜕𝑡 𝑒 𝑖𝜔𝑥 𝑑𝑥 = 𝜕𝑡
𝜕𝜌 ∞ 𝜕𝜌 𝑖𝜔𝑥
1
𝐹 (𝜕𝑥 ) = 2𝜋 ∫−∞ 𝜕𝑥 𝑒 𝑑𝑥 = 𝑖𝜔𝜌̂(𝑥, 𝑡 + Δ𝑡)
𝜕𝜌2 1 ∞ 𝜕𝜌2
𝐹 ( 𝜕𝑥 ) = 2𝜋 ∫−∞ 𝜕𝑥 𝑒 𝑖𝜔𝜋 𝑑𝑥 = 𝑖𝜔𝜌̂2 (𝑥, 𝑡 + Δ𝑡)
1 ∞ 1
(15)
𝐹(𝛾(𝑥, 𝑡 + Δ𝑡)) = 2𝜋 ∫−∞ 𝛾(𝑥, 𝑡 + Δ𝑡)𝑒 𝑖𝜔𝑥 𝑑𝑥 = 𝛾(𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝑡 + Δ𝑡)
1 ∞
𝐹(𝑣max ) = 2𝜋 ∫−∞ 𝑣max 𝑒 𝑖𝜔𝑥 𝑑𝑥 = 𝑣max × 2𝜋𝛿(𝑡 + Δ𝑡)
𝑣max 1 ∞ 𝑣max 𝑖𝜔𝑥 𝑣max
{𝐹 (𝜌max ) = 2𝜋 ∫−∞ 𝜌max 𝑒 𝑑𝑥 = 𝜌max × 2𝜋𝛿(𝑡 + Δ𝑡)

By replacing in the equation of continuity I have the following equation,

̂ (𝐾,𝑡+Δ𝑡)
𝜕𝜌 1 𝑣 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂(𝑥, 𝑡 + Δ𝑡) − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 (𝐾, 𝑡 + Δ𝑡) = 𝛾(𝑥, 𝑡 +
𝜕𝑡 max
1
Δ𝑡) × 2𝜋 𝛿(𝐾) (16)

I have the final equation of continuity in the ordinary differential form:

For (t+Δ t) ≠ 0 :

̂
𝑑𝜌 1 𝑣 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾(𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾) (17)
𝑑𝑡 max

we have the final model which will be analyzed using simulated data for one straight
movement and two turning movement:

𝐹(𝑆𝑡𝑟𝑀𝑜𝑣 ) =

𝑣1 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡)
𝑣2 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡)
𝑣3 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡)
̂
𝑑𝜌 1 𝑣 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾1 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾) (18)
𝑑𝑡 max
̂
𝑑𝜌 1 𝑣 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾2 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾)
𝑑𝑡 max
̂
𝑑𝜌 1 max 𝑣 2 1 1
( 𝑑𝑡 + 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ = 𝛾3 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾))

For first turning Movement we incorporate the gradient of turning in different directions inside the
roundabout:

𝐹(𝐹𝑖𝑟𝑇 𝑢𝑟𝑛𝑀𝑜𝑣 ) =

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Journal of Computational Analysis and Applications VOL. 34, NO. 4, 2025

𝑣1 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑟1 [cos𝜃𝑖 (𝑡 + Δ𝑡)𝑒1 + sin𝜃𝑖 (𝑡 + Δ𝑡)𝑒2 ]

𝑣2 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑟2 [cos𝜃𝑖 (𝑡 + Δ𝑡)𝑒1 + sin𝜃𝑖 (𝑡 + Δ𝑡)𝑒2 ]
𝑣3 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 𝑟3 [cos𝜃𝑖 (𝑡 + Δ𝑡)𝑒1 + sin𝜃𝑖 (𝑡 + Δ𝑡)𝑒2 ]
̂
𝑑𝜌 1 𝑣 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾1 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾)
𝑑𝑡 max
̂
𝑑𝜌 1 𝑣max 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌 × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾2 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾)
𝑑𝑡 max
̂
𝑑𝜌 1 𝑣max 1
2 1
( 𝑑𝑡 + 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ = 𝛾3 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾))

(19)

For second type of turning Movement we incorporate the gradient of turning in different directions
inside the roundabout and the radius becomes a conference of the circle around the radius of the
turning movement:

𝐹(𝑆𝑒𝑐𝑇 𝑢𝑟𝑛𝑀𝑜𝑣 ) =

𝑣1 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 2𝜋𝑟1 [cos𝜃𝑖 (𝑡 + Δ𝑡)𝑒1 + sin𝜃𝑖 (𝑡 + Δ𝑡)𝑒2 ]

𝑣2 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 2𝜋𝑟2 [cos𝜃𝑖 (𝑡 + Δ𝑡)𝑒1 + sin𝜃𝑖 (𝑡 + Δ𝑡)𝑒2 ]
𝑣3 = 𝑥̇ 𝑖 (𝑡 + Δ𝑡) = 2𝜋𝑟3 [cos𝜃𝑖 (𝑡 + Δ𝑡)𝑒1 + sin𝜃𝑖 (𝑡 + Δ𝑡)𝑒2 ]
̂
𝑑𝜌 1 𝑣 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾1 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾)
𝑑𝑡 max
̂
𝑑𝜌 1 𝑣max 1 1
+ 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌 × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂2 = 𝛾2 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾)
𝑑𝑡 max
̂
𝑑𝜌 1 𝑣max 1
2 1
( 𝑑𝑡 + 𝑣max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ − 𝜌max × 2𝜋 𝛿(𝐾)𝑖𝐾𝜌̂ = 𝛾3 (𝑥, 𝑡 + Δ𝑡) × 2𝜋 𝛿(𝐾))

(20)

6 Results and Discusions

In this section, I will present the results obtained from a simulation analysis involving four cars
engaged in various types of movements within a roundabout. The study considers three distinct
scenarios: straight movement inside the roundabout (case A), and two turning movements inside
the roundabout (case B and Case C). Furthermore, we extend our analysis to include simulations
involving multiple vehicles executing the same movement, and the outcomes of these simulations
will also be discussed. The simulation employs the Fast Fourier Transform (FFT) method in
conjunction with the fourth-order Runge-Kutta scheme, implemented using the MATLAB
programming language. This approach allows for the generation of code and graphical
representations within the MATLAB environment. The simulation setup involves the random
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generation of cars, each assigned specific movement patterns within the roundabout located in
Bujumbura [14].

The study investigates the dynamics of vehicle movements within the roundabout under different
scenarios. For instance, when two cars approach the roundabout, with one intending to turn right
and the other left, the car making the left turn is given priority, while the right-turning vehicle
waits until the intersection clears. Similarly, when multiple movements are occurring
simultaneously, cars must yield to others until the roundabout is free for passage.
The analysis yields insights into the efficiency and safety of vehicle movements within the
roundabout. Graphical representations, such as the Red, Blue, and Green lines, depict the
trajectories of turning vehicles, highlighting their paths and interactions with other vehicles. The
simulation results shed light on various factors influencing traffic flow, including the number of
vehicles, their respective movements, and the sequencing of their actions within the roundabout.
Understanding the dynamics of vehicle movements within roundabouts is crucial for optimizing
traffic flow and ensuring road safety. The insights gained from this simulation study can inform
urban planning and traffic management strategies aimed at improving transportation systems in
cities. By identifying potential bottlenecks and optimizing traffic flow patterns, policymakers can
enhance the overall efficiency and safety of road networks, benefiting both motorists and
pedestrians alike.

6.1 Simulation Analysis for the straight Movement(Case A (2))

In our analysis, we focused on roundabouts with radii ranging from 15 meters to 45 meters. Over
a period of 5 minutes, we meticulously assessed the velocities of vehicles along their designated
paths, particularly in segments devoid of conflicts within the roundabout. Our evaluation involved
adjusting velocities based on several factors, including the distance to the roundabout, the time
required to navigate it, and the velocities needed upon exiting the roundabout. To accomplish this,
we integrated an Intelligent Drive Model (IDM), which incorporates algorithms for adjusting
velocities according to changing conditions. Specifically, we concentrated on the straight
movement path within the roundabout’s conflict-free zone.
Our simulation results revealed intriguing insights. For instance, we observed that drivers typically
begin to decelerate approximately 25 meters before approaching the roundabout. Furthermore, it
takes an average of 2 to 3 minutes for vehicles to complete their passage through the roundabout,
maintaining a minimum velocity of 45 kilometers per hour upon exit. These findings underscore
the nuanced dynamics of vehicle movement within roundabouts and highlight the importance of
incorporating intelligent driving models to optimize traffic flow and ensure safe navigation.

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In the context of an unsignalized roundabout, a V-shaped pattern in velocities and accelerations

typically indicates a rapid increase followed by a sharp decrease in both parameters as vehicles
navigate the roundabout. As vehicles approach the entry point, velocities and accelerations rise as
drivers accelerate to merge into circulating traffic. Once within the roundabout, velocities and
accelerations decline as vehicles negotiate the curvature and adjust speed to safely exit. This V-
shaped pattern reflects the dynamic nature of roundabout traffic flow, with acceleration and
deceleration occurring in response to changing geometric and traffic conditions, promoting
efficient and safe movement through the intersection. As vehicles approach the roundabout entry,
velocities and accelerations typically increase as drivers accelerate to merge into the circulating
flow of traffic. This acceleration phase forms the upward slope of the V-shape. Once within the
roundabout, velocities remain relatively high but begin to decrease gradually as vehicles negotiate
the curvature of the roadway. Simultaneously, accelerations decrease as drivers maintain a steady
speed or slightly decelerate to navigate the roundabout safely. This deceleration phase constitutes
the downward slope of the V-shape. The V-shaped profile in velocities and accelerations thus
reflects the transition from entry acceleration to circulating speed maintenance and exit
deceleration. Understanding this pattern aids in designing efficient roundabouts and implementing
traffic management strategies to ensure smooth and safe traffic flow through these intersections.

6.2 Simulation Analysis for the first turning Movement (Case B (3))

The considered roundabout should have the radius varying from 15 m to 45 m. within a period of
5 min, we evaluated the velocities along the movements for the movement without conflict part in
our simulation Analysis. Changing the velocities according to the distance approaching the
roundabout and the required time to clear the roundabout and the velocities of New path after
Leaving the roundabout while the lane changing is incorporated in intellignet drive model(IDM),
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we considered the straight movement that path in the roundabout without conflict zone. The
simulation analysis show that it takes around from 25m for a driver with a stimulus of straight
movement to start slowing down in approachinf the roundabout and from 2 to three minutes to
exist the roundabout with a minimum velocity of 45km/h to exist the roundabout.

In the context of turning movements within an unsignalized roundabout, a W-shaped pattern in

velocities and accelerations may occur due to the interaction between multiple vehicles navigating
the roundabout. Initially, as vehicles approach the entry point, velocities and accelerations increase
as they accelerate to merge into the circulating flow of traffic, forming the first peak of the W-
shape. Subsequently, as vehicles negotiate the curvature of the roundabout, velocities and
accelerations may decrease as drivers adjust their speed to safely navigate the turn, resulting in the
first trough of the W-shape. As vehicles continue through the roundabout, velocities and
accelerations rise again as they straighten their trajectory and exit the roundabout, forming the
second peak of the W-shape. Finally, as vehicles complete the turn and exit the roundabout,
velocities and accelerations decrease once more, returning to a lower level, forming the second
trough of the W-shape. This complex pattern illustrates the dynamic nature of turning movements
within unsignalized roundabouts, influenced by factors such as traffic volume, roundabout
geometry, and driver behavior.

6.3 Simulation Analysis for the Second turning Movement(Case C (4))

The considered roundabout should have the radius varying from 15 m to 45 m. within a period
of 5 min, we evaluated the velocities along the movements for the movement without conflict part
in our simulation Analysis. Changing the velocities according to the distance approaching the
roundabout and the required time to clear the roundabout and the velocities of New path after
Leaving the roundabout while the lane changing is incorporated in intelligent drive model(IDM),
we considered the straight movement that path in the roundabout without conflict zone. The
simulation analysis show that it takes around from 25m for a driver with a stimulus of straight
movement to start slowing down in approaching the roundabout and from 2 to three minutes to
exist the roundabout with a minimum velocity of 45km/h to exist the roundabout.

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In the context of turning movements within an unsignalized roundabout, a combined W and V-

shaped pattern in velocities and accelerations may illustrate the intricate dynamics of vehicle
interactions and navigation through the roundabout. Initially, as vehicles approach the entry point,
velocities and accelerations increase as drivers accelerate to merge into the circulating flow of
traffic, forming the upward slope of the V-shape. Once within the roundabout, velocities and
accelerations may fluctuate due to adjustments made by drivers to navigate curves and yield to
other vehicles, resulting in periodic decreases and increases, akin to the peaks and troughs of a W-
shape. As vehicles exit the roundabout, velocities and accelerations typically decrease as drivers
decelerate to transition back onto the main road, forming the downward slope of the V-shape. This
combined pattern reflects the complex interplay of acceleration, deceleration, and navigation
within unsignalized roundabouts, influenced by factors such as traffic flow, geometry, and driver
behavior. Understanding these patterns is crucial for optimizing roundabout design and enhancing
traffic flow efficiency and safety.

7 Conclusions and Future Work

The main objectif of this work was to examine how the radius (r), the gradient of the turning
movement (𝜃) and the number of lanes (n) affect throughput velocities in the roundabout. We build
the mathematical model for the specific type of the roundabout and test the output by considering
the effects of any considered parameter on velocities to all scenarios of all getting out movements
in the roundabout.The simulation analysis presented in this study offers valuable insights into the
behavior of vehicles within a roundabout environment. By employing advanced simulation
techniques and considering various scenarios, we gain a better understanding of traffic dynamics
and can identify opportunities for enhancing transportation systems. Moving forward, further
research and experimentation can build upon these findings to develop more comprehensive
models for traffic analysis and urban planning.

In order to determine the optimal roundabout design, he simulation results show that the
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proposed model and method were effective and feasible.The proposed method has been
implemented to this area and has a good effect.The followed research will be the collaborative
research on the road channelization and the traffic information optimization and the study of the
real time traffic simulation system using real data.To validate the model with real data on
roundabout would be the future work.

Acknowledgment

The author would like to thank Kenyatta University in Kenya for their support during this
research work. The authors acknowledge the close collaboration of all staff members of the
Department of Mathematics and Actuarial Sciences at Kenyatta University.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this
paper.

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