1. The document reviews the optimal speed traffic flow model (OSM), which models traffic flow based on vehicles adapting their speed to an optimal value rather than the leader's speed. 2. In the OSM, a vehicle's acceleration is proportional to the difference between its optimal speed, determined by inter-vehicle distance, and its actual speed. The model can reproduce phenomena like traffic jams. 3. The OSM has advantages like realistically modeling traffic instabilities and congestion. However, it has difficulties avoiding collisions in emergency braking and accurately modeling driver behavior in jams. Further research is needed to validate and enhance the model.

1. A REVIEW OF OPTIMAL SPEED TRAFFIC FLOW MODEL
Master of Engineering Class Assignment
BY
AHMAD MUHAMMAD
(SPS/16/MCE/00079)
B. Eng. Civil Engineering
M. Eng. Transportation Engineering (IN progress)
Bayero University, Kano, Nigeria
COURSE LECTURER:
Prof. H. M. Alhassan, mnse, coren
May 10th, 2017

2. OPTIMAL SPEED TRAFFIC FLOW MODEL
Introduction
Traffic flow models are mathematical models of real world traffic (road traffic). They educate and
teach researchers and traffic engineers how to ensure an optimal flow with minimum traffic jams.
Broadly, there are three types of traffic flow models: Microscopic, Mesoscopic and Macroscopic traffic
flow models.
At microscopic level, each vehicle is considered as an individual; Ordinary Differential Equations can
be written for each vehicle. The models can model selective phenomena such as traffic jams. A
common and popular type of microscopic model is the car following model which has been in
existence for over sixty decades. They are the most suitable microscopic models to describe the
movement of each vehicle. There are 2 main objectives in car following process:
1. Reducing the speed difference
2. Maintain an appropriate spacing between the following vehicle and the leading vehicle.
A classical type of car following model is the optimal speed/velocity model (OSM). According to the
OSM, the vehicle/driver adapts its/his/her speed to a certain optimal value rather than to the speed of
the leader. In the OSM, the driver’s response is proportional to the difference between his/her optimum
and his/her actual speed. Features of the OSM are:
1. It is known to capture real world phenomena well
2. It incorporates driver’s delay
3. It provides modeling flexibility via the Optimal Speed Function (OSF).

3. Genesis and Basis Of OSM
The Optimal Speed Model (OSM) is a microscopic “car following type” model first proposed by
Bando et al (1994) .
The OSM describes the following features:
1. A car will keep the maximum speed with enough the distance to the next car
2. A car tries to run with an OS determined by the distance to the next car (Bando, et al., 1994).
OPTIMAL SPEED TRAFFIC FLOW MODEL

4. The OSM is based on the principle that for each situation, there is an Optimal Speed (OS) to
adopt by the drivers. Any deviation from this OS causes the vehicle to adopt an acceleration
proportional to the difference between the optimum and actual speed (Six, et al.,2012).
This acceleration is given by the following equation:
𝑥 = 𝑎 𝑛 𝑉 ∆𝑥 𝑛 − 𝑥 𝑛
Where
𝑥 is the acceleration of the nth vehicle
𝑎 𝑛 is the parameter of the model influencing how strong the reaction is. It is also known as the
sensitivity.
𝑉 ∆𝑥 𝑛 is the OS function for a given situation represented by the available gap or inter-vehicle
distance, and
𝑥 𝑛 is the speed of the nth vehicle
OPTIMAL SPEED TRAFFIC FLOW MODEL

5. The OS Function 𝑉 ∆𝑥 𝑛 is given by:
𝑉∆𝑥 𝑛 = tanh( ∆𝑥 𝑛 − 2) + tanh(2)
The OS Function has the following most observed noticeable properties: (1) There is a safety
distance h0 between car stops (2) The speed of the car increases with increased spacing (3)
There is a free flow speed; i.e., the speed of a car travelling alone (Del Castillo and Benitez, 1995).
The sensitivity and the OS function are assumed to be common to all cars.
According to Akihiro et al. (2016), the OS model predicts that a homogenous flow becomes
unstable and transits to a jammed flow if the density exceeds a critical value:
𝑑𝑉(ℎ)
𝑑ℎ
>
1
2
𝑎
Where h is the mean headway.
OPTIMAL SPEED TRAFFIC FLOW MODEL

6. Applications of OS Model
1. OS Model is used in the detection of vehicular reactions when the OS of the leading vehicle is
changed, otherwise known as the hysteresis phenomenon (Six et al, 2012)
2. A modified OS Model proposed by Salim et al. (2005) utilizes the time to collision (TTC) to
reduce the reference speed when the relative speed against the leading vehicle is negative. This
is achieved with the introduction of a weighting factor which reduces the OS by a factor of 0.45 –
0.50. The modified model also reduces the risk of rear-end collision
3. The OS function { 𝑉∆𝑥 𝑛 = tanh( ∆𝑥 𝑛 − 2) + tanh(2)} could be used to determine the critical
density
OPTIMAL SPEED TRAFFIC FLOW MODEL

7. Advantages of OS Model
1. The OS model is realistic enough to be able to reproduce spontaneous traffic jam as in real
traffic. The uniform solution of the OS model can be stable against a local perturbation
(Namiko and Hiizu, 1999)
2. The OS Model gives the expected behaviour of traffic flow, and the congestion phenomena
appears instead of accidents (Bando, et al., 1995)
3. OS Function could be determined from experiment (Akihiro, et al., 2016)
4. The OS model could reproduce many properties of real traffic flow, such as the instability of
traffic flow, the evolution of traffic congestion, and the formation of stop-and-go waves (Tian
et al., 2009)
5. The model could be used to develop a new continuous OS model that is intrinsically
asymmetric and collision-free (Antoine and Armix, 2014)
OPTIMAL SPEED TRAFFIC FLOW MODEL

8. 6. The OS Model does not have a time delay in its model expression, thereby making it
convenient for its model expression (Hao, 2014).
7. The OS Model successfully reproduce the instability of high-flux traffic, in which every vehicle
maintains the same distance from the leading vehicle (Tetsuji, 2014)
Disadvantages of OS Model
1. Classical OS model can locally oscillate, leading to collisions or negative speeds (Antoine and
Armin, 2014)
2. Linear OS Functions do not describe realistic microscopic behaviours (Antoine and Armin,
2014).
3. OS Model has difficulty to avoid collision in urgent braking cases (Doudou et al., 2016).
OPTIMAL SPEED TRAFFIC FLOW MODEL

9. Current State of Research
1. Hao (2014) has proposed a simple car following model called the Dual-Boundary-Optimal-
Speed-Model, which is based on the original OS Model. The proposed model improves the
stability of traffic.
2. Doudou et al. (2016) have proposed an approach which permits to take into account the
phenomenon of anticipation of driver behaviour. The proposal was validated by simulation.
Future Research
1. Hao (2014) have proposed that the dual boundary steady region can also be introduced into
other well known car following models. Furthermore, the parameters of his proposed Dual-
Boundary-Optimal-Speed-Model are required to be calibrated by real traffic data and this calls
for further research in the area for determining its viable applications.
2. Krishna et al. (2016) have recommended a further research to be conducted that considers
the finite length of road segments – lengths small enough to prevent systems from reaching their
respective equilibria.
OPTIMAL SPEED TRAFFIC FLOW MODEL

10. Conclusion
In OSM, it has been explained that small disturbances to traffic flow can lead to
spontaneous jam formation. The model is used in the analysis of backward
velocity jam clusters, stop-and-go flows and rear end collisions. It has been found
also that the uniform solution of the OS model can be stable against local
perturbation even in a linearly region if OBC is employed.
However, it is difficult to accurately predict the behaviour of drivers during jam
formation and congestion. And consequently it is difficult to predict the OS
function in such cases. For the OSM to receive global and uniform acceptance
and approval, researchers and traffic engineers around the world should embark
on validating previous works that match their geographical settings; improve and
enhance the model to solve current and future traffic problems, and; upgrade
and correlate the model with other microscopic models to generate new models
that enhance traffic flow safely and conveniently.
Recommendations
OPTIMAL SPEED TRAFFIC FLOW MODEL

11. OPTIMAL SPEED TRAFFIC FLOW MODEL
Recommendations
Based on the review conducted, it is recommended that further traffic research studies be carried
out on:
1. Physical and mathematical characteristics of OS model
2. Chaotic structure of non-linear OS model equations
3. Properties of clusters of congestions, etc.
It is also recommended to Nigerian researchers to key-in into the world of OS model research so as
to solve the numerous problems of our highways and streets. This will bring about positive traffic
engineering development and an international recognition and respect for our indigenous traffic
engineers.

12. References
1. Akihiro, N., Macoto, K., Akihiro, S., Yuki, S., Shin-ichi, T. and Satoshi, Y. (2016) ‘Quantitative Explanation of Circuit Experiments and Real Traffic Using the
Optimal Velocity Model’. New Journal of Physics. 18(2016)043040.
2. Antoine, T. and Armin, S. (2014) ‘Collision-Free Non-Uniform Dynamics within Continuous Optimal Velocity Models’. Physical Review. E. 90, 042812.
3. Bando, M., Hasebe, K., Nakayama, A., Shibata, A. and Sugiyama, Y. (1994) ‘Structure Stability of Congestion in Traffic Dynamics’. Japan Journal of Industrial and
Applied Mathematics. 11(2). p. 203 – 223.
4. Bando, M., Hasebe, K., Nakayama, A., Shibata A. and Sugiyama, Y. (1995) ‘Dynamical Model of Traffic Congestion and Numerical Simulation’. The American
Physical Society Journal. 51. p. 1035 – 1042.
5. Doudou, G., Roger, M. F. and Mampassi, B. (2016) ‘A Car Following Model for Traffic Simulation’. International Journal of Applied Mathematical Sciences. ISSN
0973-0176 Vol. 9; No. 1; p. 1 – 9.
6. Hao, W. (2014) ‘Optimal Velocity Model with Dual Boundary Optimal Velocity Function’. Symposium Celebrating 50 Years of Traffic Flow Theory.
Krishna, G. K., Krishna, J. and Gaurav, R. (2016) ‘A Computational Study of a Variant of the Optimal Velocity Model with no Collisions’. Indian Institute of
Technology. Madras, Chennai, India.
7. Namiko, M. and Hiizu, Nakanishi (2009) ‘Stability Analysis of Optimal Velocity Model for Traffic and Granular Flow under Open Boundary Condition’.
Department of Physics, Kyushu University 33, Fukuoka 812-8581.
8. Six, L., Leng, S., Saunier, J. and Guessoum, Z. (2012) ‘Understanding simulated driver behaviour using hysteresis loops’. International Federation of Automatic
Control. 13. p. 334 – 340.
9. Tetsuji, T. (2009) ‘Ultradiscrete Optimal Velocity Model: A Cellular Automaton Model for Traffic Flow and Linear Instability of High Flux Traffic’. Physical Review.
E 79. 056108.
10. Tian, J. F., Jia, B., Li, X. G. and Gao, Z. Y (2010) ‘A New Car-Following Model Considering Velocity Anticipation’. Chin. Phys. B. Vol. 19, No. 1, 010511.
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