This document discusses various mathematical models for traffic flow analysis. It begins by introducing Greenshield's linear speed-density model from 1935 as the first macroscopic traffic flow model. It then discusses limitations of this model and introduces other models like Greenberg's logarithmic model and Underwood's exponential model. The document also discusses Pipe's generalized model that can produce different models by varying a parameter. Finally, it introduces the concept of multi-regime models that use different speed-density relationships for congested and uncongested traffic conditions.

1. MODUL KE 6
PERTEMUAN KE 9
RENI KARNO KINASIH, S.T.,M.T.
UNIVERSITAS MERCU BUANA
Model Matematik Lalu
Lintas

2.  Penggunaan sistem kontrol traffic light pada lalu lintas
belum memberikan prioritas berupa nyala lampu hijau
lebih lama pada jalur-jalur yang lebih padat
penggunanya.
 Hal tersebut dapat menyebabkan antrian panjang pada
sebuah ruas.
 Sehingga bila dilihat secara gambaran besar, sistem
kontrol traffic light yang ada ternyata belum maksimal.
 Belum adanya informasi yang bisa kita andalkan untuk
memberikan prioritas atau mengambil keputusan.
 Bahkan sering kali justru sistem kontrol traffic light lah
yang membuat kemacetan pada sebuah ruas.

3.  Dalam simulasi traffic light, untuk menganalisa arus
kendaraan pada sebuah ruas, terdapat beberapa
metode yang bisa diterapkan. Diantaranya metode
Greenshield, teori antrian dan metode
Computational Fluid Dynamic

4. Model Greenshield
 Berguna untuk membantu peneliti dibidang
transportasi dalam memahami arus tanpa
hambatan.
 Model ini memberikan rumusan matematika arus
kendaraan sebagai fungsi dari kepadatan lalu-lintas,
serta arus kendaraan sebagai fungsi dari kecepatan
kendaraan.
 Akan tetapi, model greenshields tidak dapat
mengatasi kerumitan yang dihasilkan oleh kondisi
arus yang memiliki hambatan.

5. Interrupted Flow
 Arus dikatakan memiliki hambatan, jika arus lalu-
lintas terhenti secara periodik yang disebabkan oleh
rambu-rambu lalulintas.
 Arus yang berhambatan itu memerlukan
pemahaman teori antrian, yang sepenuhnya
merupakan model terpisah dari model arus lalu
lintas

6. Teori Antrian
 Teori antrian dapat digunakan untuk menganalisis arus
lalu-lintas melalui pendekatan sebuah persimpangan
jalan yang dikontrol oleh rambu lalu-lintas.
 Namun teori ini harus memiliki asumsi bahwa arus
kendaraan harus dalam keadaan rapi dan tidak fleksibel.
 Sehingga diperlukan metode yang dapat digunakan
untuk mengatasi permasalahan ini . Salah satunya
adalah metode Computational Fluid Dynamic.
Computational Fluid Dynamic (CFD) adalah metode
yang digunakan untuk menganalisis aliran fluida atau air

7. Computational Fluid Dynamic (CFD)
 Adalah metode yang digunakan untuk menganalisis
aliran fluida atau air, namun seiring perkembangan
zaman, metode ini mulai diterapkan di bidang
engineering, salah satunya adalah transportasi
 CFD juga dapat memainkan peran penting dalam
mengatur waktu sinyal, sesuai dengan kondisi lalu
lintas, sehingga akan menjamin arus lalu lintas
seragam bahkan ketika tingkat aliran tinggi

8. How CFD Works?
 CFD menganalisa aliran fluida dengan cara pemodelan
matematika (persamaan diferensial parsial), metode
numerik (diskritisasi dan solusi teknik) dan perangkat
lunak (pemecah, pra-dan utilitas postprocessing)
 CFD menggunakan sudut pandang Eulerian, yakni bukan
memandang kendaraan secara individual dalam aliran,
tetapi memandang arus lalu lintas sebagai aliran
sederhana yang didistribusikan secara terus menerus,
dengan melihat kesenjangan atau selang yang konsisten
antara jumlah mobil dengan panjang jalan yang dikenal
sebagai density.
 Penekanan metode CFD adalah pada aliran secara
keseluruhan atau sistem dan bukan pada individu
kendaraan.

9.  CFD memungkinkan untuk melakukan eksperimen
berupa perhitungan numerik dan simulasi komputer.
Sedangkan metode yang digunakan untuk menghitung
waktu nyala lampu lalu lintas adalahaturan Manual
Kapasitas Jalan Indonesia (MKJI)
 Computational Fluid Dynamic merupakan metode yang
digunakanuntuk dapat menganalisis keadaan arus
antrian pada ruas jalan. Sedangkan aturan MKJI
digunakan untuk mendapatkan nilai kapasitas jalan,
waktu nyala lampu lalu lintas dan derajat kejenuhan

10. Model – Model Makroskopis
Arus Lalu Lintas

11. Greenshiled’s Linear Model (1935)
 Macroscopic stream models represent how the
behaviour of one parameter of traffic flow changes
with respect to another.
 Most important among them is the relation between
speed and density. The first and most simple relation
between them is proposed by Greenshield.
 Greenshield assumed a linear speed-density
relationship as illustrated in figure 1 to derive the
model.

13.  The equation for this relationship is shown below.
………………………. (1)
 Where v is the mean speed at density , vf is the free
speed and kj is the jam density. T
 This equation (1) is often referred to as the Greenshields'
model. It indicates that when density becomes zero,
speed approaches free flow speed (ie. V  vf when k 
0).

15.  Once the relation between speed and flow is
established, the relation with flow can be derived.
This relation between flow and density is parabolic
in shape and is shown in figure 3. Also, we know that

17.  Now substituting equation 1 in equation 2, we get
………………….. (3)
 Similarly we can find the relation between speed and
flow. For this, put in equation 1 and solving, we get
………………….. (4)

18.  This relationship is again parabolic and is shown in
figure 2.
 Once the relationship between the fundamental
variables of traffic flow is established, the boundary
conditions can be derived.
 The boundary conditions that are of interest are jam
density, freeflow speed, and maximum flow.
 To find density at maximum flow, differentiate
equation 3 with respect to and equate it to zero. ie.,


20.  Denoting the density corresponding to maximum flow as k0,
 Therefore, density corresponding to maximum flow is half the
jam density Once we get , we can derive for maximum
flow, qmax. Substituting equation 5 in equation 3

21.  Thus the maximum flow is one fourth the product of
free flow and jam density. Finally to get the speed at
maximum flow, v0, substitute equation 5 in equation 1
and solving we get,
……………………………………… (6)
 Therefore, speed at maximum flow is half of the free
speed.

22. Calibration of Greenshield's model
 Inorder to use this model for any traffic stream, one
should get the boundary values, especially free flow
speed (vf) and jam density (kj).
 This has to be obtained by field survey and this is
called calibration process.
 Although it is difficult to determine exact free flow
speed and jam density directly from the field,
approximate values can be obtained from a number
of speed and density observations and then fitting a
linear equation between them.

23.  Let the linear equation be y = ax = b such that y is
density, k and x denotes the speed v. Using linear
regression method, coefficients a and b can be
solved as,
 (7)
 (8)

24.  Alternate method of solving for b is,
……(9)
 where xi and yi are the samples, n is the number of
samples, and are the mean of xi
and yi respectively.

25. Problem
 For the following data on speed and density,
determine the parameters of the Greenshields'
model. Also find the maximum flow and density
corresponding to a speed of 30 km/hr.

26. Solution
 Denoting y = v and x = k, solve for a and b using
equation 8 and equation 9. The solution is tabulated
as shown below.
 Step 1

27.  Step 2: From equation 9, define b = ….. And a =……
 Step 3: Define the linear regression from Step 2
above
v =……. (10)
 Here vf = …….. and vf/kj= …….. This implies, =
……/…….. = …….. veh/km
 The basic parameters of Greenshield's model are free
flow speed and jam density and they are obtained as
….. kmph and ……… veh/km respectively.

28.  To find maximum flow, use equation 6
q max = …… veh/hr
 Density corresponding to the speed 30 km/hr can be
found out by substituting in equation 10. i.e,
30 = 40.8 - 0.2 k
Therefore, k = ………. veh/km

29. G R E E N B E R G ’ S A N D U N D E R W O O D ’ S
Other macroscopic
stream models

30.  In Greenshield's model, linear relationship
between speed and density was assumed. But in
field we can hardly find such a relationship
between speed and density.
 Therefore, the validity of Greenshields' model was
questioned and many other models came up.
Prominent among them are Greenberg's
logarithmic model, Underwood's exponential
model, Pipe's generalized model, and multiregime
models. These are briefly discussed below.

31. Greenberg's logarithmic model
(1959)
 Greenberg assumed a logarithmic relation between
speed and density. He proposed,

32.  This model has gained very good popularity because
this model can be derived analytically. (This
derivation is beyond the scope of this notes).
 However, main drawbacks of this model is that as
density tends to zero, speed tends to infinity. This
shows the inability of the model to predict the speeds
at lower densities.

33. Underwood's exponential model
 Trying to overcome the limitation of Greenberg's
model, Underwood put forward an exponential
model as shown below.
…………………………………… (12)
 Where vf The model can be graphically expressed as
in figure 5 is the free flow speed and k0 is the
optimum density, i.e. the densty corresponding to
the maximum flow.

34.  In this model, speed becomes zero only when density
reaches infinity which is the drawback of this model.
Hence this cannot be used for predicting speeds at
high densities.

35. Pipes' generalized model
 Further developments were made with the
introduction of a new parameter (n) to provide for a
more generalised modelling approach. Pipes
proposed a model shown by the following equation.
……………………………………(13)
 When is set to one, Pipe's model resembles
Greenshields' model. Thus by varying the values of ,
a family of models can be developed.

36. Multiregime models
 All the above models are based on the assumption that
the same speed-density relation is valid for the entire
range of densities seen in traffic streams.
 Therefore, these models are called single-regime models.
However, human behaviour will be different at different
densities. This is corraborated with field observations
which shows different relations at different range of
densities.
 Therefore, the speed-density relation will also be
different in different zones of densities.
 Based on this concept, many models were proposed
generally called multi-regime models. The most simple
one is called a two-regime model, where separate
equations are used to represent the speed-density
relation at congested and uncongested traffic.

37. Resources
 https://www.civil.iitb.ac.in/tvm/1100_LnTse/503_l
nTse/plain/plain.html
 Khisty, J & Lall, K. 2003. Dasar-Dasar Rekayasa
Transportasi, 3rd Edition. Prentice Hall.

38. S E E Y O U I N T H E N E X T C H A P T E R !
Thank You