This document reviews several extensions and applications of the optimal speed traffic model. The original optimal speed model introduced in 1995 assumes that each vehicle has a legal velocity that depends on the following distance. Later models like the generalized force model and full velocity difference model address issues like unrealistic acceleration and deceleration in the original model. Other extensions examine the effects of next-nearest neighbor interaction and backward-looking behavior. Applications of optimal speed models include autonomous vehicle control, evaluating ITS strategies, and gaining insights into traffic congestion formation and flow stability. The conclusion recommends developing a more systematic "almighty model" that incorporates the various extensions and applications.

1. Review of Optimal Speed
Traffic Models
By:
Amir Idris Imam
(SPS/16/MCE/00026)
Submitted to:
Engr. Prof. H.M. Alhassan
Masters class Assignment (Highway and Transportation option, 2016)
Department of Civil Engineering, Bayero
University Kano

2. Background
In recent decades, traffic flow problems have been
given much attention by several researchers from
different outlook to consider the various aspect of
traffic phenomena. One of the most notable
problems of traffic dynamics is traffic congestion.
The Optimal Speed Traffic Model is a car following
model that can be used to describe many properties
of traffic flow such as the instability of traffic flow,
the evolution of traffic congestion, and the
formation of stop and go waves.

3. The Optimal Speed Model
This simple and realistic model was introduced by
some researchers (Bando et. al, 1995). The model
induces traffic congestion spontaneously, they
noticed that congestion really exists and is caused
by a small perturbation in the traffic flow without
any origin such as traffic accident or congestion.
These researchers adapted a theory for regulation
thus:
“Each vehicle has the legal velocity, which depends
on the following distance of the preceding vehicle”
In this approach, the stimulus is a function of a
following distance and the sensitivity is a constant.

4. The dynamical equation…
These researchers assumed that each vehicle has a legal
velocity V and that each driver responds to a stimulus from
the vehicle ahead of him. He must control the acceleration in
such a way that he can maintain the safe legal velocity with
regard to the vehicle ahead of him.
Thus, the equation is given as:
(1)
For each car number n (n = 1, 2,...), is the position of the nth
car and is the headway of this car, a is a constant
called “Sensitivity”, which was set at the same value for all
drivers. V(x) is called the optimal velocity function (OV
function), which expresses the relation between headway
and the optimal velocity of each car.

5. According to (Bando et. al, 1995), this model
assumes that the sensitivity of drivers is identical
and has no dependence of velocity, headway or the
relative velocity of the preceding vehicle.
A model could be adopted in which a depends on
each driver; the legal velocity function can also be
dependent on the drivers with different maximum
speed or the slope of the OV model curve.
They also recommend other aspects of further
studies of their model such as physical and
mathematical characteristics, chaotic structure of
non-linear equations, properties of clusters of
congestion etc.

6. Review
As it was stated in earlier slides, the optimal speed
traffic model has been reviewed by several
researchers from different approaches in order to
determine how this model can be applied to explain
various traffic flow problems.
Some of these works are been reviewed here
independently to enable proper understanding of
this model and how it can be applied in the future
to come up with traffic flow explanations that will
enable the tendering of proper solutions to the
various traffic problems on our freeways and
highways.

7. Generalized Force Model (GFM) and
Full Velocity Difference Model (FVDM)
(Helbing and Tilch, 1998) carried out a calibration of the OVM
with respect to the empirical data. They adopted the optimal
velocity function as:
(2)
After getting the results of the simulations, they realized that the
comparison of this empirical data with field data shows OVM
encountering the problems of too high acceleration and
unrealistic deceleration.
In this regard, they proposed a Generalized Force Model (GFM) in
which one term is increased on the right-hand side (RHS) of the
OV model equation to solve these problems of the OVM:
(3)
Where Θ is the Heaviside function.

8. The GFM (Eq. 3) can be re-written as :
(4)
They also applied calibration in GFM, and the results show that
GFM reaches better agreement with the field data than OVM
by having a smaller value of sensitivity (κ).
Consequently, Comparing GFM with OVM, (Wu et. al, 2001)
find out that when ΔV≥0, GFM has the same form as OVM, the
difference lies in that they have different values of sensitivity .
They also noticed that GFM is poor in anticipating the delay
time of a car motion and the kinematic wave speed at jam
density which they argued is as a result of not including
positive ΔV in the GFM, they therefore come up with a more
systematic model which they called the Full Velocity
Difference Model (FVDM) taking positive ΔV into account.
(5)

9. The FVDM (Eq. 5) can also be re-written as:
(6)
Where vm is the maximum speed. The first term on the RHS is
the acceleration force, and the last two terms represent the
interaction force.
Lastly, (Wu et. al, 2001) also apply FVDM to several
simulations just as in the case of GFM. The results reveal that
FVDM predicts correct delay time of car motion and kinematic
wave speed at jam density and moreover, unrealistically high
acceleration will not appear.

10. Stabilization and Enhancement of Traffic Flow
by the Next-Nearest-Neighbor Interaction
Another extension of the OV model was presented by (Nagatani, 1999) to
take into account the next-nearest-neighbor interaction. He investigated
the effect of the next nearest- neighbor interaction on the traffic current
and the jamming transition by the use of the numerical and analytical
methods to address whether or not the next nearest- neighbor
interaction enhances the traffic current and stabilizes the traffic flow.
He extends the difference equation model described by Newell and
Whitham that include only the nearest-neighbor interaction to take into
account the next-nearest-neighbor interaction.
He assumes that a driver can obtain the information of the car position
before the next car ahead and the extended difference model is given by
the following:
(7)
The last term on the left side of Eq. 14 represents the additional term of
the next-nearest-neighbor interaction.

11. The parameter γ represents the strength of the next-nearest
neighbor interaction and 0<γ<1.
When γ=0, the equation of the extended difference equation
reduces to the original Newell and Whitham equation of
difference model.

12. The Backward Looking Optimal Velocity
Model (BL-OV)
Another group of researchers (Nakayama et. al, 2001) produced
an extended OV models to suppress the formation of congestion.
In the OV model, the appearance of congestion can be suppressed
by choosing high sensitivity. In their extended model where a
driver looks at the following car as well as the preceding car, they
showed that it was not the best way and presented another
possibility.
The equation of the model is given as follows:
(8)
Where VF(x) is the OV function for forward looking and VB(x) is
the OV function for backward looking, which is a function of the
headway of the following car.
In this model, each car is controlled so as to be positioned at the
middle point between the preceding car and the following car,
and the function has the effect of increasing the velocity of the
car, if the headway of the following car becomes small.

13. The free flow becomes stable in the BL-OV model, even if low sensitivity is
taken where the flow is unstable in the original OV model.
From the diagram, it can be seen that in the upper region, the homogeneous
flow is stable for both models. In the middle region it is unstable only for the
OV model, and in the lower region it is unstable for both models.
Therefore, the idea of backward looking is beyond the usual control of
drivers, but can be realized by some engineering techniques of the ITS. Such
a realization seems easier than the development of a sensitive control system.

14. Applications Of The Optimal Speed
Traffic Models
 They are of great importance with regard to an autonomous
cruise control system.
 They are important evaluation tools for Intelligent
Transportation System strategies since the early 1990s.
 They give more efficient methods for tracing out the key
boundaries in parameter space (e.g., the characterization of
bistability regions).
 Current research shows that the chaos which has been found
in these models with delay by using numerical simulations can
be applied in the future to gain concise information about the
routes to chaos.
 Since the formation mechanism of traffic congestion is
naturally described by the Optimal Speed models, the stability
of the traffic flow can be achieved using these models.

15. Conclusion and Recommendation
As we have seen from the earlier discussions, the optimal speed
(otherwise, velocity) traffic flow models which are integral part of
the microscopic car – following models have received considerable
attention for the development of theories concerning traffic
phenomena over the last few decades. Many researchers with
different background and ideas have considered various aspects of
traffic phenomena with very satisfying results after applying the
models to several simulations and tests.
It is recommended that the various results obtained by these
researchers be used to develop a more systematic optimal speed
traffic model (The Almighty Model). This model will covers all the
theories and individual recommendations made thorough the
researches and as such be effectively applied to solve the various
problems of traffic flow especially congestion.

16. References
 Bando, M.; Hasebe, K.; Nakayama, A.; Shibata, A.; Sugiyama, Y. (1995).
Dynamical Model of Traffic Congestion and Numerical Simulation. Phys.
Rev. E Vol. 51, No 2: pp. 1035 -1042.
 Helbing, D.; Tilch, B (1998). Calibration of Optimal Velocity Traffic Model
with Respect to Empirical Data. Phys. Rev. E Vol. 58, No 133: pp. 1 – 4.
 Nagatani, T. (1999). Stabilization and Enhancement of Traffic Flow by the
Next Nearest – Neighbour Interaction. Phys. Rev. E. Vol. 60, No. 2: pp. 6395
– 6401.
 Nakayama, A.; Sugiyama, Y.; Hasebe, K. (2001). Effect of Looking at the Car
that Follows in an Optimal Velocity Model of Traffic Flow. Phys. Rev. E Vol.
65, No 016112: pp. 1 -6
 Orosz, G.; Wilson, R. E.; Krauskopf, B. (2004). Global Bifurcation
Investigation of an Optimal Velocity Traffic Model with Driver Reaction
Time. Phys. Rev. E 70, No 026207: pp. 1 -10.
 Wu, Q.; Jiang, R.; Zhu, Z. (2001). Full Velocity Difference Model for a Car
Following Theory. Phys. Rev. E Vol. 64, No 017101: pp. 1 – 4.

17. Thank you