1 23

International Journal of Advances inEngineering Sciences and AppliedMathematics ISSN 0975-0770Volume 5Combined 2-3 Int J Adv Eng Sci Appl Math (2013)5:158-176DOI 10.1007/s12572-013-0087-1

State-of-the art of macroscopic traffic flowmodelling

Ranju Mohan & GitakrishnanRamadurai

1 23

Your article is protected by copyright and all

rights are held exclusively by Indian Institute

of Technology Madras. This e-offprint is

for personal use only and shall not be self-

archived in electronic repositories. If you wish

to self-archive your article, please use the

accepted manuscript version for posting on

your own website. You may further deposit

the accepted manuscript version in any

repository, provided it is only made publicly

available 12 months after official publication

or later and provided acknowledgement is

given to the original source of publication

and a link is inserted to the published article

on Springer's website. The link must be

accompanied by the following text: "The final

publication is available at link.springer.com”.

State-of-the art of macroscopic traffic flow modelling

Ranju Mohan • Gitakrishnan Ramadurai

Published online: 15 August 2013

� Indian Institute of Technology Madras 2013

Abstract Macroscopic traffic flow models are suited for

large scale, network wide applications where the macro-

characteristics of the flow are of prime interest. A clear

understanding of the existing macro-level traffic flow

models will help in modelling of varying traffic scenarios

more accurately. Existing state-of-the-art reports on traffic

flow models have not considered macro-level models

exclusively. This paper gives a review of macroscopic

modelling approaches used for traffic networks including

recent research in the past decade. The modelling of the

two main components of the network i.e. links and nodes

are reviewed separately in two sections and solution pro-

cedures are discussed followed by a synthesis on the

advantages and disadvantages of these models. This review

should encourage efficient research in this area towards

network level application of these models.

Keywords Macroscopic traffic flow models �Link modelling � Cell transmission model �Riemann problem � Porous flow approach � Node modelling

1 Introduction

Traffic phenomena are complex and nonlinear depending on

the interactions of a large number of vehicles. Network level

traffic flow model must include three features—capturing of

bottleneck effects, computing wave propagation, and

modelling of intersections. The transportation system

infrastructure, time-dependent traffic demand, level of traffic

control, and stochastic behavior of people make the traffic

flow modelling difficult. Traffic flow can be analyzed at

macroscopic, mesoscopic, and microscopic levels of aggre-

gation. Macroscopic models are the aggregation of individ-

ual vehicle dynamics and mainly focus on describing the

overall stream features such as congestion, delay, and queue

formation. These models are suited for large scale, network

wide applications where the macro-characteristics of traffic

(speed, density, and flow) are of prime interest. Microscopic

models focus on individual vehicles and their interaction

with neighboring vehicles and describes phenomena such as

vehicle following, overtaking, lane changing, and gap

acceptance. Though these models are often seen as more

‘realistic’ representation of traffic flow, computational

complexity limits use of these models for network wide

applications. Mesoscopic models model traffic as vehicular

packets or even individual vehicles, but still governed by

macroscopic flow laws. Thus, retaining the advantage of less

complexity, mesoscopic models can capture the level of

detail closer to that by microscopic models.

At network level macroscopic models will be the best

choice to balance the tradeoff between solution detail and

computational effort. A clear understanding of the existing

macro-level traffic flow models will help in modelling of

varying traffic scenarios more accurately. Existing state-of-

the-art reports on traffic flow models have not considered

macro-level models exclusively (the last comprehensive

review was by Hoogendoorn and Bovy [1]—over a decade

earlier). This paper gives a review of traffic flow models,

restricted to macroscopic modelling approaches, for links

and nodes.

The important modelling approaches for link are clas-

sified in this paper as follows.

R. Mohan (&) � G. Ramadurai

Department of Civil Engineering, Indian Institute of Technology,

Madras, Chennai 600032, India

e-mail: sasthamgmrm@gmail.com

G. Ramadurai

e-mail: gitakrishnan@iitm.ac.in

123

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

DOI 10.1007/s12572-013-0087-1 IIT, Madras

Author's personal copy

(1) Continuum models

(a) First-order model (LWR model)

(b) Higher order models

(2) Non-continuum model.

Node models are classified as follows.

(1) First generation models

(2) Models based on constraint flow maximization.

Section 2 discusses the important macroscopic link

models and in Sect. 3, several approaches for node mod-

elling are reviewed. Section 4 shows a synthesis on the

choice of the above models and Sect. 5 draws conclusions.

2 Link modelling

2.1 Continuum models

Continuum models assume traffic flows similar to a com-

pressible fluid flow. This family of models started with the

simple, well-known first-order (time and space derivatives

are the order of one) LWR model [2, 3] and later developed

into higher order models. The basic, common equations for

all these models are the laws of conservation, fundamental

equation of traffic flow, and the speed–density relationship.

These models are reviewed in the following subsections.

2.1.1 First-order model (LWR model)

This model gives the dynamic equations for density

through Eqs. (1)–(3) [4].

ok

otþ oq

ox¼ 0; ð1Þ

u ¼ ueðkÞ; ð2Þq ¼ ku: ð3Þ

Here, x, t, u, k and q represent the space, time, speed,

density, and flow, respectively, and ue is the equilibrium

speed from the fundamental diagram of traffic flow.

Equation (1) is the flow conservation equation which is a

nonlinear hyperbolic partial differential equation and can

be rewritten as Eq. (4).

oU

otþ A

oU

ox¼ 0; ð4Þ

where U = [k], A = qF/qU and F = [q]. An equation

(system of equations) is said to be hyperbolic if Eigen

value(s) of A is (are) real (real and distinct). The dynamic

density, k(x, t) can be obtained by using the property of

characteristic curves. In LWR model, characteristic curves

are straight lines emanating from the boundary t = 0 and

passing through the points of equal density. The slope of

the characteristic curve is the speed with which the small

disturbances (flow change due to small change in density)

propagate and is given by the Eigen value of the matrix A,

dq/dk. The speed of the large disturbances—shockwaves

formed by two different steady states—is given by the

Rankine-Hugoniot condition as below:

w12 ¼q1 � q2

k1 � k2

; ð5Þ

where w is the shockwave speed and the indices 1 and 2

represent the two different steady states.

Though the model can represent explicitly the formation

of shockwaves, it is not capable of explaining other phe-

nomena such as steady state speed–density relationship,

discontinuities in the density, regular start–stop waves,

traffic hysteresis, localized clusters and phantom jams.

2.1.1.1 Extension of LWR model LWR model assumes

single lane, homogeneous traffic flow. As an extension to

the model, several papers tried to describe traffic with two

approaches. The first approach was to make a distinction

between lanes. Munjal and Pipes [5], Munjal et al. [6],

Holland and Woods [7] and Greenberg et al. [8] used this

approach and a separate LWR model and fundamental

diagram is used for each lane. This leads to a parallel

coupling of several LWR model using exchange terms. A

general formulation of this simple extension of LWR

model to multilane traffic is given in [4] as below.

oki

otþ oqi

ox¼ gi þ Qi i ¼ 1; 2; . . .;N; ð6Þ

Qi ¼ ai;i�1 ki�1ðx; t � sÞ � kiðx; t � sÞð Þ � kði�1Þ0; �ki0

� �� �

þ ai;iþ1 kiþ1ðx; t � sÞ � kiðx; t � sÞð Þ½� kðiþ1Þ0; �ki0

� ��i ¼ 1; 2; . . .;N;

ð7Þ

where N is the number of lanes, Qi is the lane changing rate

for lane i, i ? 1 and i - 1 are the neighboring lanes to lane

i, ai,i-1 and ai,i?1 are sensitivity coefficient describing the

intensity of interaction, gi is the rate of generation or loss at

entrance or exit ramps (equal to zero for all internal lanes)

and ki0 is the equilibrium density of the ith lane. For the

first (last) lane, i - 1 (i ? 1) should be set as i. Here, the

exchange of vehicles between neighboring lanes is pro-

portional to the difference of deviation of their densities

from equilibrium values. In Munjal and Pipe’s [5] model,

lane changing flows are prioritized over through flows on

target lanes. Laval [9] overcame this limitation proposing a

multilane cell transmission rule.

A second approach was to divide the vehicle population

into different classes having different driving characteris-

tics. The interaction between these classes will give a better

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 159

123

Author's personal copy

description of heterogeneous traffic flow. Zhang and Jin

[10], Logghe and Immers [11] and Chanut and Buisson

[12] used this approach with Eulerian coordinate system.

These models used flow conservation equation separately

for each vehicle class and the class wise flows are formu-

lated as a fraction of the total flow from fundamental dia-

gram. In general, these models can be represented as [13]:

oki

otþ oqi

ox¼ 0 i ¼ 1; 2; . . .;N;

qi ¼ aiQe

ki

ai

� �;

ð8Þ

where N is the number of vehicle classes, Qe (.) is the

common fundamental diagram assumed for all classes and

ai is a fraction assumed for vehicle class i and is formulated

differently in the above models. Logghe [11] used the same

concept, but divided the flow regime into free flow, semi

congested and congested regimes. Class interactions are

modelled according to user equilibrium which states that

no vehicle can increase its speed any further without

influencing the speed of slower vehicles. Wong and Wong

[14] showed that multiclass LWR model can reproduce the

traffic phenomena such as discontinuity in density,

hysteresis and platoon dispersion. The model reveals that

the discontinuity in the flow–density diagram might not

necessarily be caused by the two operational regimes

(congested and uncongested), but is a result of user’s

interactions. The model is as shown below.

oki

otþXN

i¼1

cij

oki

ox¼ 0 i ¼ 1; 2; . . .;N;

cij ¼ Vidij þ ki

oVi

okj

;

ð9Þ

where N is the number of vehicle lass, cij is the kinematic

wave speed of vehicle class i in response to the presence of

class j, Vi = Vi(k1, k2,…,kN) is the equilibrium speed of class

i expressed as a function of different class wise densities and

dij = 1 if i = j; and dij = 0 if i = j. Few other multiclass

extensions of LWR model can be seen in [15–19].

Because of the simple formulation of equations and the

analytical solution techniques, modification and extension

of the LWR model is still an active area of research. But,

for a network level model, that includes different road

types and specific traffic situations, accuracy of flow pre-

diction by modified LWR model should be tested. Litera-

ture suggests higher order models can capture traffic

realism better. Though higher order models have compar-

atively higher computational effort, they could provide

more accurate results when applied at network level.

2.1.1.2 Cell Transmission Model When simulating LWR

model, outflow is typically specified as a function of

occupancy of the section from which it is emitted and

not as a function of downstream occupancy [20]. Such

approach does not converge to a desired solution and

cannot produce reasonable results [21]. Ensuring cell

occupancies between zero and the maximum possible do

not guarantee convergence, for example, stopped traffic is

predicted not to flow into an empty freeway [20]. These

issues are resolved when Daganzo [20, 22] formulated a

Cell Transmission Model (CTM) as a discretization of first-

order LWR model. The model divides the freeway stretch

into cells (cellular automation model [23] also use the same

concept, but a micro–macro approach) and uses a piece-

wise linear relationship between traffic flow and traffic

density (triangular fundamental diagram). In the model

formulation, given a time step, the length of the cells is

chosen such that under free flow conditions, all vehicles in

a cell will flow into the immediate downstream cell. The

model formulation is as follows:

Let Ni(t) be the maximum number of vehicles that can

be present in cell i at time t, Qi(t) be the maximum number

of vehicles that can flow into cell i when the clock

advances from t to t ? 1, ni(t) be the number of vehicles in

cell i at time t and yi(t) be the inflow to cell i in the time

interval (t, t ? 1), w is the disturbance propagation speed

(backward wave speed), and v is the free flow speed, then,

the recursive relationship by the CTM is given by:

niðt þ 1Þ ¼ niðtÞ þ yiðtÞ � yiþ1ðtÞ; ð10ÞyiðtÞ ¼ min ni�1ðtÞ; QiðtÞ; d NiðtÞ � niðtÞ½ �f g &d ¼ w=v:

ð11Þ

Daganzo [24] introduced a new version of the model

called lagged CTM (L-CTM) which enable variable cell

lengths and adapts a non-concave flow–density. Szeto [25]

modified L-CTM into an enhanced L-CTM removing few

of its drawback (negative densities or densities greater than

jam density). Laval and Daganzo [26] introduced lane

changing algorithms that can be incorporated in CTM. This

model consists of a discrete time formulation of the

multilane KW module by Munjal and Pipes [5] and a

module for the lane changing particle. This model is able to

reproduce the condition of drop in the discharge rate of

freeway bottlenecks when congestion begins and the

relation between the speed of a moving bottleneck and its

capacity. Hu et al. [27] introduced a variable CTM that

includes two parameters, namely, cell length and cell

density and obeys the flow conservation law.

The use of discrete numbers and the reliance of the

model on simple rules reduce the processing power when

compared to continuum models. In CTM, the number of

vehicles that enter a cell depend on the number of vehicles

from the previous cells, maximum flow and the occupancy

of the cell itself. Therefore the unusual situation where

160 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

there may be still inflow to a fully occupied cell, the

common problem in LWR model that needs complicate

constraints to rectify that, is avoided. In the location spe-

cific CTM by Chen et al. [28], a modified CTM is used to

take into account the model variability influenced by sensor

locations, geometry features and many other factors. Using

various shapes of fundamental diagrams, the phenomena

like capacity drops, lane-by-lane variations, non-homoge-

neous wave propagation velocities and temporal lags can

be reproduced. Long et al. [29] used the concept of CTM

for urban traffic jam simulation. The proposed model

includes two components: the cell inflow model and the

flow conservation model and used two measures, namely,

traffic jam size and congestion delay to explain the effect

of congestion.

Though the use of discrete cell in the model formulation

makes the model simple and reduces the processing power in

computation, the need for practically sized cells and clock

intervals inevitably generate errors in actual applications.

The model assumes uniform velocity for all the vehicles, and

hence cannot accurately predict the platoon dispersion phe-

nomena at light traffic conditions and the effect of moving

bottleneck. Also, in the model vehicles are assumed to have

instantaneous acceleration and deceleration. Despite the

above limitations, CTM is widely applied and studied.

2.1.1.3 Link Transmission Model (LTM) The LTM by

Yperman [30] is a model for dynamic network loading

(DNL) and computes time-dependent link travel times and

route travel times using cumulative arrival–departure

curves. In this model, the flow conservation equation holds

for the whole link and sending and receiving flows are

determined using Newell’s [21] simplified theory of kine-

matic waves. Instead of the triangular fundamental diagram

by Newell, the model uses a piecewise linear fundamental

diagram. Since the entire link is analyzed as a whole, the

computational effort for this model is less when compared

to CTM, but requires more memory for storing cumulative

arrival and departure volumes for each time step.

2.1.1.4 A porous flow approach for heterogeneous traf-

fic Most of the traffic flow model formulations are intended

to be used for homogeneous traffic with perfect lane disci-

pline. In this type of traffic, vehicles moves one after another

in each lane, and changes lanes or overtake vehicles only at

permitted zones. In the case of heterogeneous traffic lacking

in lane discipline, vehicles with widely varying sizes will

move to front according to the space availability. Even

though, the vehicle heterogeneity is considered in LWR

model (see in Sect. 2.1.1.1), the problem of lacking lane

discipline traffic is not addressed in the model. Inspired from

the work of Logghe [13], Nair et al. [31] used LWR frame-

work with a porous flow approach to model heterogeneous

traffic having no lane discipline. The vehicles on the network

define a network of spaces (pores) and each vehicle class will

have different network of pores since the smaller vehicle can

use some pores that larger vehicles cannot. The traffic state

variables for each type of vehicle stream are defined sepa-

rately and equilibrium speed–density relationship is deter-

mined using the available empty spaces; not the density. The

model formulation is as follows: assume a traffic stream with

n vehicle classes indexed by m (m = 1, 2,…,n). In addition to

the traffic state variables flow (q(m, x, t)), density (k(m, x, t))

and speed (u(m, x, t)), the model defines a pore–space dis-

tribution with probability density function fp(rp, x, t) where rp

is the pore size with

fp rp; x; t� �

� 0 and

Z1

0

fp rp; x; t� �

drp ¼ 1: ð12Þ

Here fp(rp, x, t) denotes the fraction of pores within size

rp and rp ? drp. The fundamental diagram applies to each

vehicle class independently and the LWR model is extended

to several vehicle classes as a set of partial differential

equations:

okðm; x; tÞot

þ oðkðm; x; tÞfeðKðx; tÞÞÞox

¼ 0

8m ¼ 1; 2; . . .; n;ð13Þ

where K(x, t) = [k(1, x, t), k(2, x, t),…,k(n, x, t)] is the

vector of densities for all vehicle classes and fe(K(x, t)) is

the speed from the equilibrium speed–density relationship.

To account for the significant lateral movement, the traffic

stream for each vehicle class is considered to have two sub-

streams of ‘free’ and ‘restrained’ vehicles.

uðm; x; tÞ ¼ urðm; x; tÞZrðmÞ

0

fp rp; x; t� �

drp

þ uf ðm; x; tÞZ1

rðmÞ

fp rp; x; t� �

drp

8m ¼ 1; 2; . . .; n;

ð14Þ

where

urðm; x; tÞ ¼ uf ðmÞ 1�ZrðmÞ

0

fp rp; x; t� �

drp

2

64

3

75

ar

;

uf ðm; x; tÞ ¼ uf ðmÞ 1�ZrðmÞ

0

fp rp; x; t� �

drp

2

64

3

75

af

8m ¼ 1; 2; . . .; n:

ð15Þ

Here r and f stands for restricted and free vehicles,

ar C af, and uf(m) is the free flow speed of the vehicle class

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 161

123

Author's personal copy

m. To solve the above system of equations, the finite

difference scheme cannot be applied directly, since the

pore–space distribution is not known and also the set of

n nonlinear partial differential equations have to be solved

jointly because of the K(x, t) term. The model’s algorithm

simulates pore size distribution using updated class wise

densities by Godunov method [32]. To calculate the

boundary flows, supply–demand analysis for each vehicle

class is performed independently. One drawback of this

model is that the number of vehicle class is restricted to

two since supply–demand analysis is cumbersome for more

number of classes. The model has not been calibrated with

real data.

2.1.1.5 Second-order LWR model A drawback in LWR

model is its inability to capture hysteresis effect. One of the

main reasons for the hysteresis phenomena observed in real

traffic is the driver’s response to frontal stimuli and inertial

effect (in the equilibrium speed–density relationship, speed

changes instantaneously with the density). In order to

account for these, Lighthill and Witham suggested second

order terms in the simple first-order model as shown

below [4].

ok

otþ c

ok

oxþ T

o2k

ot2� D

o2k

ox2¼ 0; ð16Þ

where T is the inertial time constant for speed variation, c is

the speed of small disturbances (dq/dk), and D is a diffu-

sion coefficient representing how vehicles respond to non-

local changes in traffic conditions. However, the simple

extension of first-order LWR equation to second-order

equation did not resolve the limitations of LWR model.

Following this second-order model, Payne [33], Ross [34],

Kuhne [35, 36] and Michalopoulos et al. [37] also proposed

higher order models of which Payne model aroused con-

siderable interest.

2.1.2 Higher order models

As mentioned in the above section, higher order models are

formulated to incorporate the inertial effect and effect of

drivers’ anticipation on vehicle speeds. Thus, differed from

the LWR model, these models contain additional equa-

tion(s) on velocity dynamics. The following models: Payne-

type model, Aw–Rascle model, Zhang model, Helbing’s

model and speed gradient (SG) model are presented in the

following subsections.

2.1.2.1 Payne-type models Second-order models remained

under-explored for some time until Payne [33] extended

LWR model using a simple car-following rule. One of

the main critiques of the simple LWR model is that the

mean velocity adapts instantaneously to the traffic density

(steady-state speed–density relationship). Along with the

LWR Eqs. (1) and (3), he proposed a partial differential

equation describing the dynamics of the velocity u. The

simple car-following rule is shown below.

uðxðt þ TÞ; t þ TÞ ¼ ueðkðxþ DÞ; tÞ; ð17Þ

where x(t) is the location of driver at time t, u(x, t) is the

velocity at x and t, ue is the equilibrium velocity expressed

as a function of density k, T is the reaction time and D is the

gross distance headway with respect to the preceding

vehicle. Applying Taylor’s expansion to both sides of the

above equation and substituting k = 1/D, the equation

becomes [1]:

ou

otþ u

ou

ox¼ ueðkÞ � u

T� c2

0

k

ok

ox; ð18Þ

where the constant c0 [ 0 is defined as the traffic sound

speed. The second term in the left hand side denotes con-

vection which describes changes in the mean velocity due

to in- and out-flowing vehicles. The first term in the right

hand side denotes relaxation which describes the tendency

of traffic flow to relax to an equilibrium velocity. The last

term constitutes the anticipation term that describes dri-

ver’s anticipation on spatially changing traffic conditions

downstream.

Kotsialos and Papageorgiou [38] used discretized Payne

model in the network model METANET along with net-

work relevant extensions (links are modelled separately as

motorway, origin, destination and store-and-forward links

and it also includes a node model). Whitham [39] also

proposed a second-order Payne-type model, the so called

Payne–Whitham model and in general, the velocity

dynamics of Payne-type model is given as [1]:

ou

otþ u

ou

ox¼ ueðkÞ � u

T� 1

k

oP

oxþ g

k

o2u

ox2; ð19Þ

P is the traffic pressure equal to c20k and g is the kinematic

traffic viscosity. The last two terms in the right hand side

stand for anticipation where qP/qx describes the local

anticipation behavior of the driver and the diffusion term

o2u=ox2 describes the higher order tendencies (immediate

acceleration or deceleration, stop-and-go behavior, etc.) of

drivers. Considering non-viscous flow and ignoring the

relaxation term, Eqs. (1) and (18) together can be expressed

in a nonlinear hyperbolic conservative form (see Eq. (4)).

The conservative variables U and F are given by:

U ¼ k

ku

� FðUÞ ¼ ku

ku2 þ c20k

� :

In Payne family of models, the state of the system at (x, t) is

determined using two characteristics (curves called Mach

lines) both emanating from t = 0 [1]. The speed of the

characteristics is the Eigen values of the Jacobian of the flux

162 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

matrix F and is given in Table 1. In a certain density range,

Payne model is meta-stable. In this region, small variations in

the traffic density yields regions of increasing traffic densities

leading to the occurrence of start–stop waves or localized

traffic jams. This is one of the important properties of these

models [40, 41].

Daganzo [42] critically reviewed Payne model pointing out

that the kinematic theory of fluid should not be applied to

model vehicular flow. First, the fluid flow is isotropic, but

vehicular flow (driver reacts only to the downstream vehicles)

is anisotropic. Second, in fluid flow slow particles get affected

by faster particles, but in vehicular flow, slow vehicles remain

unaffected by fast vehicles. Finally, the driver’s personality is

also an important factor in the vehicular flow which is irrel-

evant in the fluid flow modelling. Other criticisms for Payne

model are: speed of one of the characteristic is higher than the

vehicular speed (information travels faster than the vehicles’

speed), and the model cause negative speeds at tails of the

congested region. Later, Liu et al. [43] pointed out that the

upstream moving curve is an indication of speed difference

between vehicles and not negative speed. Next section dis-

cusses the Aw–Rascle [44] model that overcomes this nega-

tive vehicle speed phenomenon.

2.1.2.2 Aw–Rascle model Aw and Rascle [44] formu-

lated a new anisotropic continuum flow model, addressing

the drawbacks noted by Daganzo [42] for the Payne model,

using a convective derivative of pressure instead of spatial

derivative. Thus the drivers’ anticipation to the traffic

ahead depends not on the spatial change in density, but on

the spatio-temporal change in density. The model includes

Eqs. (1), (3) and the velocity dynamics equation given in

Eq. (20).

oðuþ pðkÞÞot

þ uoðuþ pðkÞÞ

ox¼ 1

TueðkÞ � uð Þ; ð20Þ

where p(k) is the traffic pressure expressed as an increasing

function of density. They proved that with a suitable choice

of the function p(k), the model addresses two criticisms of

the Payne-type models (anisotropy and negative speed of

vehicles). Moreover, the model nicely predicts the

instabilities near vacuum i.e. for very light traffic. The

functional form of p(k) chosen in [44] is as follows:

pðkÞ ¼ C2kc; ð21Þ

where C is a constant equal to 1. Multiplying Eq. (1) by dp/dk

and adding to Eq. (20), the velocity dynamics equation can

be rewritten as:

ou

otþ ðu� kp0ðkÞÞ ou

ox¼ 1

TueðkÞ � uð Þ: ð22Þ

For the nonlinear hyperbolic conservative system, the

conservative variables are given by:

Table 1 Macroscopic models for traffic flow

Models Disturbance propagation speed Nature of

model

Literature on

Extension to multilane/multiclass traffic Large/network model

LWR model dq/dk Anisotropic See [5–8, 10–15, 17–19] Lebacque and

Koshyaran [90]

Payne model u ? c0, u - c0 Isotropic Nil Kotsialos and

Papageorgiou [38]

Aw–Rascle model u, u - cp(k) Anisotropic Bagnerini and Rascle [78], Colombo [79] Garavello and Piccoli

[91]b

Zhang model u, u ? ku0e(k) with u

0e(k) \ 0 Anisotropic Nil Nil

Helbing

modelaHelbing

[50]

u

u ? f1(k, u, H) - f2(k, u, H)

u ? f1(k, u, H) ? f2(k, u, H)

Isotropic Hoogendoorn and Bovy [60], Tampere et al.

[80], Ngoduy et al. [72]

Helbing et al. [58]

Helbing

[77]

u ? (H/u2) ? uf3(k, u, H)

u ? (H/u2) - uf3(k, u, H)

Speed gradient

model

u, u - c0 Anisotropic Jiang and Wu [65], Tang et al. [66, 68, 69] Nil

Porous flow

approach

dq/dk, depends on pore size

distribution function

Anisotropic Nair et al. [31] Nil

CTM w = a constant (backward wave

speed)

Anisotropic Laval and Daganzo [26], Tuerprasert and

Aswakul [103]

Lo and Szeto [104]

Non-continuum

model

– Anisotropic Nil Nil

a f1, f2 and f3 are nonlinear functions of k, u and Hb Separate modelling of mid-blocks and intersection, not the network as a whole

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 163

123

Author's personal copy

U ¼ k

m

� FðUÞ ¼ m� kpðkÞ

m2

k� mpðkÞ

� ;

where m = k(u ? p(k)). The characteristic speeds of the model

as shown in Table 1 satisfy the anisotropic behavior of vehicles.

The model uses a simple pressure function satisfying few

qualitatively important conditions [44]. However, since the

model’s predictive ability depends on the assumed pressure

function, alternate formulations should be explored. A different

formulation of pressure term in the hybrid (macroscopic ?

microscopic) model by Moutari and Rascle [45]:

pðqÞ ¼uref

cqqm

�cc[ 0

�uref ln qqm

�c ¼ 0

8<

:; ð23Þ

Here q denotes the fraction of space occupied by vehicles (a

dimensionless local density), uref is a given reference

velocity, and qm is the maximal density (as fraction) equals 1.

2.1.2.3 Non-equilibrium model of traffic flow Defining

the concept of non-equilibrium traffic flow and to overcome

the limitation of ‘negative speed’ in Payne’s model, Zhang

[46] proposed a different model with a velocity dynamics

equation as in Eq. (24) and the LWR Eqs. (1) and (3).

ou

otþ u

ou

ox¼ 1

TueðkÞ � uð Þ � k u

0

eðkÞ �2ok

ox: ð24Þ

The model formulates the disturbance propagation speed

of non-equilibrium traffic flow as a function of the

equilibrium flow dynamics. LWR model is a special case

of this model where drivers do not anticipate traffic

conditions ahead (u = ue(k)). He showed that the higher

order terms in the model can be neglected if the temporal–

spatial scales are properly treated. However, this model

could not fully remove the isotropic behavior of vehicles.

Zhang [47] modified the velocity dynamics as follows:

ou

otþ uþ ku

0

eðkÞ � ou

ox¼ ueðkÞ � u

T: ð25Þ

The conservative variables for Zhang [47] model are

given by:

U ¼k

m

" #

FðUÞ ¼mþ kueðkÞ

m2

kþ mueðkÞ

2

4

3

5;

where m = k(u - ue(k)). Aw–Rascle [44] and Zhang [47]

models, together known as ARZ model is extended by

Lebacque et al. [48, 49] using a variable fundamental dia-

gram (fundamental diagram varying as a function of traffic

state) with inverse equilibrium speed–density relationship.

2.1.2.4 Model based on kinetic theory of gases In addi-

tion to the criticism by Daganzo [42] on Payne model,

Helbing [50] introduced three more conditions which are to

be fulfilled by a macroscopic traffic flow model. These are

finite space requirement of vehicles, velocity variance, and

finite reaction time and breaking time of driver–vehicle

units. For incorporating velocity variance, finite reaction

time and breaking time, a new model derived from gas

kinetic equations was proposed. The concept of gas kinetic

theory was first introduced by Newell [21], but, models

based on this received greater attention when Prigogine and

Andrews [51] used this theory with a Boltzmann-like

approach [52]. Gas kinetic theory has been widely used in

mesoscopic traffic flow modelling [51–55]. The macro-

model based on this theory is proposed by Helbing [50] and

consists of Eqs. (1), (3) and (19) along with the following

equation of velocity variance.

oHotþ u

oHox¼ �2

P

k

ou

oxþ 2

He �HT

� 1

k

oJ

ox; ð26Þ

where H and He are the velocity variance and equilibrium

velocity variance, respectively, P is the traffic pressure

equal to k(x, t) H and J is the flux of velocity variance

which is defined as the product of density k(x, t) and

skewness of the velocity distribution C(x, t) as shown

below.

Jðx; tÞ ¼ kðx; tÞCðx; tÞ: ð27Þ

Also,

ueðk; u; HÞ ¼ u0 � Tð1� pÞP;Heðk; u; HÞ ¼ Z � Tð1� pÞJ;

ð28Þ

where u0 and ue are the expected desired velocity and

equilibrium velocity, respectively, T is the reaction time, Z

is the covariance between the velocity and the desired

velocity and p is the immediate overtaking probability. The

propagation of disturbance is analyzed by the model using

three characteristic curves (one straight line and two Mach

lines). By this model, the small disturbances are transported

along with the mean traffic flow as well as in the up- and

downstream directions with respect to this mean flow.

Conserved variables in the model when expressed as a

nonlinear hyperbolic system are given by:

U ¼ U1; U2; U3½ �T

FðUÞ ¼ U2;U2

2

U1þ U1U3�U2

2

U1ð1�U1�s0U2Þ ;U2U3

U1þ 2ðU1U2U3�U3

2Þ

U21ð1�U1�s0U2Þ

h iT

;

where

U1 ¼ k; U2 ¼ ku; U3 ¼ ku2 þ r2kH; r ¼ffiffiffiffiffiffiH0

p

uf

;

s0 ¼ kjuf T :

Here kj, uf and H0 are respectively the jam density, free

flow speed, and maximum velocity variance of traffic [56].

164 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

Helbing [57] modified the model for finite space requirement

of vehicles by replacing the P and J by:

P0 ¼ P

1� ksðuÞ J0 ¼ J

1� ksðuÞ ; ð29Þ

where s(u) = l ? uT, l is the average vehicle length, and

uT is the safe distance each driver should keep from the

next vehicle ahead. Helbing et al. [58] also proposed a gas

kinetic, non-local traffic model, MASTER that avoids

diffusion or viscosity terms and allows only forwardly

directed interactions. This model, similar to the other

higher order models (and differed from Helbing’s [50, 57]

original model), has two equations—flow conservation and

velocity dynamics equations—and hence two characteristic

curves. The main difference of this model from other

higher order models is in the velocity dynamics equation,

where, the equilibrium velocity also depends on the density

and average velocity at an interaction point that is

advanced by about the safe distance. This model allows

simulating synchronized congested traffic [59] that mainly

occurs close to on-ramps. The model is also extended to

multilane, heterogeneous traffic and successful calibration

with real traffic is reported [60]. Compared to the other

higher order models listed in previous sections, number of

parameters to be calibrated in this model is more, requiring

more computational effort at a network level.

2.1.2.5 SG model The SG model proposed by Jiang et al.

[61] is based on an improved car-following model. Helbing

and Trieber [62] pointed out that there exists in the real

world a common driver behavior that none of the earlier

car-following models can explain; when the distance

between the leader and follower become shorter than the

safe distance, the follower may not decelerate if the pre-

ceding vehicle drives faster than the follower. By com-

bining the classical car-following model [63] and the

optimal velocity model [64], the formulated improved car-

following model incorporating the effect of both the dis-

tance and the relative speed of two successive vehicles is as

follows [65]:

dunþ1ðtÞdt

¼ j UðDxÞ � unþ1ðtÞ½ � þ kDu; ð30Þ

where j is the reaction coefficient, U(Dx) represents the

legal velocity of the follower, un?1(t) is the speed of the

follower, k is the sensitivity coefficient and Du is the

relative speed of the vehicles. But this improved equation

represents the traffic flow condition in a microscopic point

of view. Transforming the discrete variables of individual

vehicles into continuous variables, the macro-approach for

the above formulation becomes [61]:

ou

otþ u

ou

ox¼ ue � u

Tþ c0

ou

ox; ð31Þ

where c0 is the propagation speed of disturbance. Thus, the

new model consists of Eq. (30) along with Eqs. (1) and (3).

The variables for the system of nonlinear hyperbolic

equations in the SG model are given by:

U ¼ k

u

� FðUÞ ¼ ku

u2

2� c0u

� :

Compared to other higher order models, the SG replaces

the density gradient in the anticipation term, and guarantees

the property that the characteristic speed is always less than

the macroscopic flow speed. Using hypothetical data, the

model proved to obtain shockwaves, rarefaction waves, stop-

and-go waves, and local cluster effects and is consistent with

the diverse nonlinear dynamical phenomena observed in the

freeway traffic.

Jiang et al. [65] extended the SG model for mixed traffic

consisting of fast cars and slow vehicles. The model for-

mulation is as follows:

oki

otþ oðkiuiÞ

ox¼ 0

oui

otþ ui

oui

ox¼ ueiðk1; k2Þ � ui

Ti

þ c0i

oui

ox

8>><

>>:

9>>=

>>;8i; ð32Þ

where i = 1, 2 represents fast cars and slow vehicle,

respectively, c0i is the kinematic wave speed of vehicle i,

uei is the equilibrium speed of the vehicle type i and Ti is

the reaction time for the vehicle type i. In the above

equation, effect of slow vehicles acting on fast cars is

neglected. To account for this, and also by considering the

difference in length of the two types of vehicles, Tang et al.

[66] modified the above expression as follows:

oki

otþ oðkiuiÞ

ox¼ 0

oui

otþ ui

oui

ox¼ ueiðk1; k2Þ � ui

Ti

þ c0i

oui

ox

� 1� k

kj

� �u1 � u2

si

km6¼i

kj

� �2

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

8i; m ¼ 1; 2;

ð33Þ

where k = k1 ? k2 and kj is the jam density for the mixed

traffic. The right most term indicates that the frictional

effects between fast cars and slow vehicles are proportional

to the speed difference of these vehicles with si as the

proportional coefficient. The double exponential form of

the speed–density relationship by Del Castillo et al. [67] is

adopted for ue and extended for uei. The jam density kj for

the mixed traffic is calculated as follows:

kj ¼kj1

a� ða� 1ÞR ; ð34Þ

where kj1 is the jam density under the homogeneous traffic

where there are only fast cars, a is the length ratio of the

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 165

123

Author's personal copy

slow vehicle to fast cars and R is the ratio of fast car

density to the total density.

Another extension to SG model is by Tang et al. [68]

where it is used to describe traffic flow on two lane free-

ways. Here, terms related to lane change are added into the

continuity equation and the velocity dynamic equation.

Tang et al. [69] extended the SG model for heterogeneous

traffic consisting of n different types of vehicle classes. The

new model, developed from an improved car-following

model, says that the acceleration of a vehicle depends on

the speed, spacing and the reactive coefficients of different

vehicle classes. The developed model is shown below.

oui

otþ ui

oui

ox¼ uieðkÞ � ui

Ti

þXN

j¼1

cijpj

ouj

oxþ pj

sij

uj � ui

� �� �;

ð35Þ

where N is the number of vehicle classes, uie is the equi-

librium speed for the vehicle class i, Ti and sij are the

reactive coefficients for vehicle class i with respect to a

leading vehicle of type j, pj is the proportion of leading

vehicle of type j, and cij is the disturbance propagation

speed of vehicle class j as result of vehicle class i. The

value of cij depends on the speed of the vehicle type j and

also on the reaction coefficient sij. SG model has not been

sufficiently validated with real data.

2.2 Non-continuum model

Darbha and Rajagopal [70] proposed a non-continuum

approach for macroscopic traffic flow modelling. They

suggested that the usage of the macroscopic variable

‘density’ has no intuitive meaning. In mechanics a section

of flow consists of infinite number of particles, so for

relating fluid flow to vehicular flow one must ensure there

are ‘sufficiently large’ number of vehicles. To have a

comparable number of vehicles in a section as particles (or

molecules) in a representative volume, the section lengths

to be considered must be at least millions of miles long

which is not practical.

In the non-continuum model, traffic is treated as a col-

lection of dynamical systems, with each vehicle in the

traffic treated as a dynamical system. A ‘representative

vehicle’ can be thought of as a limit of a collection of

dynamical systems of finite state space dimension. The

vehicle-following behavior of the representative vehicles

reflects the aggregate vehicle-following behavior of traffic.

The variables used to describe the flow of traffic are: the

number of vehicles in the section at any given time (N), the

aggregate following distance (D*), the aggregated speed of

traffic (v*), the number of vehicles entering the section

from upstream (Nen), the number of vehicles exiting from

the section to downstream (Nex), net inflow to the section

from ramps (dn*/dt), the length of the section (L), and the

speed correction factor (b) for travelling to the downstream

section. For the ith section of a freeway, the model of

traffic flow by the non-continuum approach is given by the

following system of equations:

dNi

dt¼ dNen

i

dt� dNex

i

dtþ dn�i

dt; ð36Þ

dD�idt¼ �ðL

�car þ D�i Þ

2

Ls;i

dNi

dtþ bi;i�1 þ v�i�1 � v�i ; i [ 1;

ð37Þ

dD�1dt¼ �ðL

�car þ D�1Þ

2

Ls;1

dN1

dt; ð38Þ

dv�idt¼ f v�i ; D�i ;

dD�idt

� �; ð39Þ

dNexi

dt¼ v�i Ni

Ls;i; ð40Þ

dNeni

dt¼ dNex

iþ1

dt: ð41Þ

Equation (36) is for vehicle balancing, Eqs. (37) and

(38) is for evolution of aggregate following distance,

Eq. (39) is an approximation to vehicle speed dynamics

and Eqs. (40) and (41) are to ensure compatibility. One of

the advantages of this approach is that the aggregate

vehicle following behavior is integrated in the macroscopic

flow model that replicates the effects of microscopic level

control on the macroscopic dynamics. However, this model

has not been studied in detail probably because of the

number of equations involved and the strong biased

towards continuum theory for traffic flow modeling.

2.3 Solution to macroscopic link models

This section discusses mainly the solution procedures of

continuum models. Macroscopic continuum traffic flow

models can be solved either analytically or by using

numerical simulation. Analytical method for solving LWR

model uses a set of characteristics that are straight lines

while higher order models contains more than one set of

characteristics which are curves. Hence, except for LWR

model, the analytical approaches for other models are

complex and numerical schemes are the widely used

solution procedure. Three important questions for picking a

numerical simulation technique are: the type of the scheme

to be used, time–space discretization, and the initial and

boundary conditions of the simulating domain.

166 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

Numerical schemes can be classified as explicit or

implicit. Explicit methods calculate the state of a system at

a later time from the state of a system at the current time,

while implicit methods find a solution by solving an

equation involving both the current and later states of the

system. In realistic traffic simulations, data is continuously

fed into simulations for varying boundary conditions and

hence explicit methods are more useful. These methods are

more flexible for the simulation of on- and off-ramp or

entire road networks [62]. There also exist finite difference

methods (FDMs) and finite volume methods (FVMs) in

numerical schemes. Macroscopic continuum traffic flow

models are expressed as system of nonlinear hyperbolic

partial differential equations. In practice, solution to these

equations will not be smooth, but contain discontinuities

such as shockwaves. Classical FDM, in which derivatives

are approximated by finite differences (point wise

approximation at grid points), can be expected to break

down near discontinuities in the solution where the dif-

ferential equation does not hold. Thus, to solve continuum

traffic flow models, FVMs are more preferred. In FVM, the

domain or space is divided into grid cells and approximate

the cell average of the state of the system (integral of state

of the system divided by the cell volume). These values are

modified in each time step by the boundary fluxes of the

grid cells. The information that can be used to compute the

boundary fluxes can be obtained by solving the ‘Riemann

problem’. Riemann problem is simply a hyperbolic equa-

tion with piecewise constant initial data having a single

discontinuity at some point. An example is the hyperbolic

system (4), along with the following initial data:

Uðx; 0Þ ¼ UL; x\0

UR; x [ 0

; ð42Þ

where x = 0 is the point of discontinuity that separate two

different traffic states UL and UR (respectively on the left

and right of x = 0). The solution to the Riemann problem

consists of a finite set of waves that propagate away from

the point of discontinuity with constant wave speeds. A

detailed explanation of the available FVM methods can be

seen in [32].

Two simple schemes commonly used are upwind

scheme and Godunov scheme [32]. Zhang et al. [71] used

weighted essentially non-oscillatory numerical scheme in

multiclass LWR model and simulated results for a signal

control problem that are in good agreement with the ana-

lytical counterparts. Higher order schemes (if Dx and Dt

are simultaneously decreased by a factor of e, upper bound

of the local error will be proportional to e2), such as

Lax–Wendroff and flux splitting scheme [32] are not

necessarily more accurate than simple schemes and

sometimes even cause numerical instabilities [62]. Thus, it

is always recommended to implement different numerical

methods and compare their simulation results. Ngoduy

et al. [72] compared three different numerical schemes for

Payne model, namely, Steger–Warming flux splitting

scheme, Mac Cormack scheme, and Harten–van Leer–

Lax–Einfeldt (HLLE) scheme and obtained minimum total

relative mean square error in flow and mean speed by

HLLE scheme. There also exist high resolution schemes

where first-order scheme (upper bound of the error is

proportional to e) can be used for simulating congested

regime and higher order scheme can be used for uncon-

gested regime. Selection of the numerical scheme also

depends on the type of the problem to be solved and the

traffic flow model used. In the SG model, the system of

hyperbolic equations are not expressed in conservative

form (differed from other higher order models, when

expressed as a system of hyperbolic equations, the velocity

dynamics equation has not rewritten in terms of conserved

variables). Thus the numerical schemes for conservation

laws should not be applied directly for this model.

Time–space discretization for any numerical schemes

should be chosen to ensure two different stabilities,

namely, convective stability and relaxational stability.

Convective stability is ensured by the Courant–Friedrichs–

Levy (CFL) condition that the discretized time interval

Dt should be less than or equal to the minimum time for the

fastest vehicle to cross the discretized space interval

Dx. Relaxational instability can occur if Dt is greater than

the relaxation time of the vehicles. Traffic flow models

involving second-order viscosity term (for example, in

Payne model), may cause also a diffusion instability which

can be removed by a diffusional CFL condition [62].

Cremer and Papageorgiou [73] verified through simulation

that lowering discretization intervals does not create

amplified accuracy in macroscopic traffic flow models, and

in fact, increasing the discretization interval leads to much

lower computational effort. The choice of time–space

discretization also depends on the type of the numerical

scheme to be used. Mohan and Ramadurai [74] showed that

as grid size becomes smaller, LWR, Payne and Aw–Rascle

models produce more accurate results. However, as stated

by Papageorgiou [75], there exist optimum discretization

levels beyond which further improvements in accuracy

cannot be achieved.

Initial conditions for simulation is specified as the state

of the system in each grid cell at time t = 0. For boundary

conditions, different existing options are: Dirichlet condi-

tions, homogeneous von Neumann conditions, free and

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 167

123

Author's personal copy

periodic boundary conditions [32]. Dirichlet boundary

conditions are given by the empirically measured values at

the boundaries and are best for realistic traffic simulations

with measured values of speed and flow at the boundary.

Homogeneous von Neumann and free boundary conditions

assume that the state of the system remains unchanged at

the boundaries and can be used if traffic situation outside

the boundaries is not of interest. Periodic boundary con-

ditions assume that the density and flow at entry and exit of

the considered segment are equal for considered time

period. Helbing and Trieber [62] pointed out few disad-

vantages of using Dirichlet boundary conditions. They

showed that if the imposed upstream boundary flow is

higher than the equilibrium flow or imposed downstream

flow is higher than the flow arriving from the simulated

section, continuity equation yields a lower equilibrium flow

or negative densities, respectively. To solve the above

problem they suggested hybrid boundary conditions, where

at boundaries Dirichlet condition is used if the direction of

information wave propagation pointed towards the simu-

lation stretch, and homogeneous von Neumann condition is

used otherwise.

Macroscopic traffic flow models other than continuum

models use first-order FDM in flow conservation or a

system of simultaneous equations for problem solution.

The CTM by Daganzo [20, 22] uses first-order Godunov

scheme and determine appropriate traffic flow between

two cell boundaries using supply–demand restriction. The

porous flow approach by Nair et al. [31] uses a hybrid

solution procedure where the densities are simulated using

finite difference (first order) scheme on conservation

equation and the supply side of the conservation equation

is determined by the incremental transfer principle [76].

The simulated densities are used to determine the pore

size distribution which in turn yields equilibrium speeds

of each vehicle class. The flow variables are then updated

using the fundamental equation of traffic flow. The non-

continuum model by Darbha and Rajagopal [70] is solved

as a simultaneous system of equations (given in Sect.

2.4).

2.4 Summary of link modelling

Link modelling has a strong and vast base in continuum

theory. The simple LWR model initiated traffic flow

modelling using the kinematic wave theory with the

explicit representation of shockwave formulation under

equilibrium state. Major limitations of the LWR model

(instantaneous speed–density relationship, prediction of

instability, stop–start waves, platoon dispersion, etc.) have

led to development of higher order models. These higher

order models are derived from simple follow-the-leader

rule and a review can be seen in [77]. These models seem

to have complex system of equations because of the type

and number of variables involved. However, the intuitive

conceptual basis, good analytical formulation, and ease of

implementation lead researchers to continue with the

continuum theory and to develop (or improve) new

(existing) models. Also, to capture the complex wave

interactions observed in real traffic, higher order models

are required. The CTM which is the discretized version of

the original LWR model was one of the milestones in

traffic flow modelling giving the concept of division of

road into spaces or cells and finding out the traffic state

using simple equations. Even though it holds the same

limitations as the LWR model, the advantage of less

computational effort when using this model led to wide

spread applications even up to network level. These models

use a direct mapping of micro-variables to macro-level

which is only an approximation even for ‘ideal’ equilib-

rium traffic. Two differed attempts in link modelling are

the non-continuum model and the porous flow approach.

Among the higher order models, two well known models

used at the network level are METANET in which the

underlying model is Payne model and MASTER in which

the underlying model is the Helbing’s [57] model. Exten-

sion of higher order models to multiclass traffic is limited

[60, 78–80], and is not explored to lacking lane disciplined

traffic. The discussed models are summarized in Table 1.

Numerical simulation is the widely used solution pro-

cedure in continuum macroscopic models where explicit

FVMs with hybrid boundary conditions are well proved.

Though the porous flow approach gave a good platform

for heterogeneous traffic flow modelling, the difficulty to

identify an appropriate pore distribution function and the

supply function analysis that restricts the number of

vehicle classes are limitations of this method. The non-

continuum model questioned the basic premise of stating

vehicular density analogous to that of fluid density and

assumes vehicles as a collection of dynamical systems.

Apart from the number of parameters involved, another

limitation of this model is the involvement of aggregate

following distance that restricts its extension to multive-

hicle traffic.

3 Node modelling

A node is a junction or intersection of two or more links in

the transport network, or can also be a point of abrupt

change in road characteristics (for example, a change in

number of lanes). Flow modelling for a node is difficult

168 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

mainly because of the stochastic directional distribution of

traffic and also due to the existence of the various levels of

control for the node. Probably because of this complexity,

existing literature for node modelling in the macroscopic

level, is less when compared to the link modelling. Liter-

ature on DNL models discuss about two types of modelling

approach for intersection—point-like modelling and spatial

modelling. Many of the node models are point-like, i.e.

without physical dimensions, combing all the external and

internal constraints into a strongly coupled set of equations.

In spatial modelling, intersections and conflict zones are

represented through dummy links and nodes. This paper,

however, uses a different classification for node models,

i.e. first generation models that include simple merge or

diverge models and models based on constraint flow

maximization.

3.1 First generation models

First generation of the node models started with the merge

and diverge models [22, 81, 82]. The merge model of

Daganzo [22] consist of two incoming links (flows: q1 and

q2) and one outgoing link (total flow q = q1 ? q2) with an

objective function to maximize q subject to the following

demand and supply constraints:

qi� Si;X

i

qi�R 8i ¼ 1; 2; ð43Þ

where Si (demand) denotes the maximum flow that the

incoming link i could transfer if the node and the outgoing

link impose no constraints on the outflow of link i and R

(supply) is the maximum inflow that the outgoing link can

receive if the node and incoming link impose no

constraints. The model also uses ‘distribution factors (di)’

for the flow assignment to each link. Jin and Zhang [81]

modified the distribution factors in the model as:

di ¼SiPi Si

: ð44Þ

Later [83], they introduced a ‘level of reduction (a)’ to

represent the flows as a function of upstream variables.

a ¼ RP

i Si

; qi ¼ minð1; aÞSi: ð45Þ

Ni and Leonard [84] introduced the merge model in the

same way as that by Daganzo [22], but represented the

distribution factors in terms of capacities (Ci) of incoming

links, i.e.

di ¼CiPi Ci

: ð46Þ

The simple diverge model of Daganzo [22] contains one

incoming link (total flow: q) and two outgoing links (flows:

q1 and q2). The objective function is to maximize the total

flow, q subjected to the constraints as follows:

qi ¼ fiq�Ri; q� S; ð47Þ

where R and S are as defined for the merging model and fiis the turning fraction. The level of reduction, as in merge

model is given as:

ai ¼Ri

fiS; qi ¼ min 1; a1; a2ð ÞfiS: ð48Þ

Lebacque [85] proposed an exchange zone for

intersection flow modelling, where this zone is like a cell

with several entry and exit points. The traffic inside the cell

is disaggregated according to the entry and exit points. Let

N be the total number of vehicles, NIi be the number of

vehicles through the entry point i, NOj be the number of

vehicles through exit point j, Nij be the number of vehicles

entering through i and exiting through j, then the following

conservation holds:

NIi ¼X

j

Nij; NOj ¼X

i

Nij; N ¼X

i

NIi

¼X

j

NOj ¼X

ij

Nij:ð49Þ

A global fundamental diagram Qe(N) is defined for each

zone yielding total demand and total supply at the zone as

SN and RN, respectively. Let bi be the fraction of lanes at

entry i, then the split of SN and RN among upstream and

downstream links are:

Si ¼ biSN ; Rj ¼NOj

NRN : ð50Þ

If the downstream link has a supply of Rdj, the zone

outflow through exit point j is given by:

Qj ¼ min Sj; Rdj

� �ð51Þ

Similarly, if the demand of the upstream link i is Sui, the

zone inflow through entry point i is given by:

Qi ¼ min Sui; Rið Þ: ð52Þ

3.2 Models based on constraint flow maximization

Holden and Riserbo [86] were the first who developed a

mathematical model of traffic flow on a network with

unidirectional flow and solved a Riemann problem of a

highway intersection with m upstream links and n

downstream links. All links in the network have the same

fundamental diagram and traffic dynamics are described

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 169

123

Author's personal copy

by the LWR model. The Riemann problem is solved by

introducing an entropy condition that maximizes a con-

cave function of node inflow (Qi; i = 1, 2,…,m) or out-

flow (Qj; j = 1, 2,…,n) subject to the following

constraints:

0�Qi� di; 8i; 0�Qj� rj; 8j;X

i

Qi ¼X

j

Qj;

ð53Þ

where di is the demand of the upstream link i, and rj is the

supply of the downstream link j. Few drawbacks of this

model are the lack of physical interpretation to the entropy

condition, oversimplification of the model by the assump-

tion that the drivers will choose their destination link based

only on the principle of least resistance, and the uniform

speed–density assumption for all the links. Herty and Klar

[87] followed the same assumptions and formulated a

multilane model. Coclite et al. [88] solved the Riemann

problem to maximize the total flux with an additional

constraint on turning proportions. They too assume a uni-

form fundamental diagram for all the links. Further, the

Riemann problem cannot be uniquely solved for a junction

with fewer downstream links, hence unable to apply to a

merging junction [85].

Durlin and Henn [89] proposed a delayed flow inter-

section model that captures three important effects of sig-

nal alteration at intersections. The effects, namely, cyclic

delay, bottleneck effect, and the shockwave propagation

are illustrated using a wave tracking resolution scheme.

The calculation of wave propagation (the waves separate

zones of different densities) is made possible by the

approximation of the fundamental relationship by a

piecewise linear functions. The flow through the intersec-

tion can be replaced by the use of a fictive arc (an artificial

arc between upstream and downstream arc), which has its

own fundamental diagram and the length of which can be

varied with respect to the inflow rate to the intersection.

The cyclic delay at intersection (s) can be computed using

the equation for uniform delay given by the Highway

Capacity Manual. The two bottleneck restrictions that are

used at the entry and exit of the fictive arc are as follows

[89]:

Qin ¼ min S; k � R; k � Rfictive

� �and

Qout ¼ min Sfictive; k � R� �

;ð54Þ

where Qin and Qout are respectively the inflow and outflow

for the fictive arc, k is the green ratio at the intersection, S

is the demand at the entry of the intersection, R is the

supply of the downstream arc, and Sfictive and Rfictive are the

demand and supply at the exit and entry of the fictive arc,

respectively. The above formulation of Qout holds only for

the constant inflow to the intersection where the length of

the fictive arc is constant. If inflow varies, length of the

fictive arc also changes; the speed with which the

downstream frontier of the fictive arc moves (Vfrontier)

and corresponding Qout are given by:

Vfrontier ¼ Vf 1� Qout

Qf

� �and Qout ¼ Qf 1� Vfrontier

Vf

� �;

ð55Þ

where Qf is the flow leaving the fictive arc and Vf is the

speed of vehicles on the fictive arc.

Lebacque and Koshyaran [90] introduced the ‘Invari-

ance Principle’ that any dynamic network model should

satisfy. It states that: ‘‘If the flow on any link i is supply

constrained, then link i enters a congested regime. After an

infinitesimal increment in time, because of the traffic flow

dynamics, Si (demand in link i) increases to the capacity of

the link, Ci. Then the flow in link i (qi) should not be

increased because of change in Si to Ci. Similarly, if qj is

demand constraint, then the solution should be invariant to

an increase in Rj (supply of link j) to Cj’’. Instead of linking

demand and supply directly, the new model incorporates

node demand and node supply constraints derived from a

global zone fundamental diagram for each node in the

network.

Among macroscopic continuum models, LWR model

(discretized or continuous) is the widely used one for

intersection [82, 85, 90]. However, there exists literature

[91–93] on intersection modelling using Aw–Rascle model.

They [91] used Aw–Rascle model without relaxation term

and solved the hyperbolic nonlinear equation system for

each node n as follows:

oUi

otþ oFðUiÞ

oUi

oUi

ox¼ 0; ð56Þ

where

U ¼ki

mi

" #

FðUÞ ¼mi � kipi kið Þ

m2i

ki

� mipi kið Þ

2

64

3

75

mi ¼ ki ui þ pi kið Þð Þ:

Here i represents the incoming and outgoing links of

node n.

Gentile et al. [94] introduced a node model capable of

the explicit representation of vehicle column formation

and dispersion, but sets no limits to the length of the

queue, therefore may well exceed the length of the arc.

Gentile et al. [95] extended this model to a network per-

formance model for modelling the spill back congestion.

170 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

The new extended model eliminates the drawback in [94]

by translating the interaction among flows on adjacent

arcs by time varying arc entry and exit capacities. Spill-

back phenomenon is reproduced here as a hyper-critical

flow state either propagating backwards from the final

section of the arc or originating from the initial section of

the arc that reduces the capacities of the arc behind the

current arc.

Bliemer [96] formulated a node model that maximizes

the total flow subject to the constraints of turning fraction

[81] and flow conservation [87]. The model incorporates a

level of reduction aj equal to the ratio of Rj and the total

demandP

Sij towards j. The flow qij (flow from upstream

link i to downstream link j) is given by:

qij ¼ min Sij; a�j ðiÞSij

�a�j ðiÞ ¼ min

8j;Sij [ 0aj

� �; 8i; ð57Þ

when several links j impose a reduction on i.

Several alternatives have also been used to model flow

at intersections. Yuan et al. [97] developed a Hybrid Petri

Net model for unsignalized T-intersections, in which one

part represents the levels of traffic priority (Discrete Petri

Net) and the other part for the computation of the corre-

sponding time varying quantities (Timed Petri Net). The

model is capable of representing the relationship between

traffic flows, passing capacity and queuing vehicles for

dynamic traffic flows. Petri Net modelling of traffic flow

is well explained in Tolba et al. [98]. Liu and Dai [99]

came up with an interrupted traffic flow model which

consists of two parts: intersection buffer having inter-

rupted traffic flow and links having continuous traffic

flow. The link flow dynamics can be calculated using

LWR model and for the interrupted flow at junctions, two

additional equations are used. One is conservation of

vehicle at the intersections and the other is an entropy

condition that minimizes the uniform delay at junctions.

For the delay calculation, accumulation diagram with the

effect of physical queue is used. Jin [100] developed a

systematic approach for the kinematic wave solutions to

the Riemann problem of merging traffic flow in supply

demand space.

Tampere et al. [83] illustrated the violation of invariance

principle in the Bliemer [96] model and pointed out a set of

requirements from literatures, that need to be satisfied by

any first-order macroscopic node model. They are listed as

follows: (a) general applicability irrespective of the number

of incoming and outgoing links, (b) flow maximization,

(c) non-negative flow, (d) vehicle flow conservation,

(e) satisfying demand and supply constraints, (f) conser-

vation of turning fractions, and (g) satisfaction of invari-

ance principle. Along with this they also proposed some

node supply constraints and a supply constraint interaction

rule (SCIR—mandatory only for specific node model

instance) separately for unsignalized and signalized inter-

sections. The formulation of the model is as given below:

MaxX

i

X

j

qij

s:t: qij� 0 8i; j;

qi ¼X

j

qij� Si 8i

qj ¼X

i

qij�Rj 8j;

fij ¼Sij

Si

¼ qij

qi

8i; j;

if 9ijqi\Si; qi is invariant to Si ! Ci;

if 9jjqj\Rj; qj is invariant to Rj ! Cj;

SCIR constraints

optional ðnode supply constraintsÞ;

ð58Þ

where fij is the turning fraction. Node supply constraints Np

(defined through an internal node supply formulation) set

an upper bound to a selection Ap of all flows that make use

of the internal infrastructure p (e.g. green phase, conflict

point, arc on a roundabout).X

Up

f qij

� ��Np: ð59Þ

The SCIR is for answering the following important

questions: (a) each flow qij is limited by which of the

constraints (Si, Rj, Np), (b) for each supply constraint (Rj,

Np) how is the supply distributed to each of the competing

flows towards j or through p, and (c) if exactly one

constraint can be identified as the most restrictive one for

each link, define the composition of the sets Ui, Uj and Up,

each of which denotes the set of links that are demand

constrained, supply constrained and internal node supply

constrained, respectively.

Flotterod and Rohde [101] modified the generic node

model (GNM) of Tampere et al. [83] as an incremental

node model (INM) using Daganzo et al.’s [76] incremental

transfer (IT) principle. In this model flow transmissions are

computed in a multistage process with stage index

k = 0,…,K. The system of equations in the INM is given

by:

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 171

123

Author's personal copy

Here ‘ represent one particular node, D(k) is the vector

of upstream and downstream nodes at stage k, u(k) is the

constant flow transfer rate at stage k, h(k) is the length of the

stage k, and bij is the non-negative turning fractions from

i to j.

The main limitation of Daganzo’s IT principle [76] and

the INM [101] is that it fails to explain the situation where

the increase in one flow leads to the decrease of another

flow. This situation occurs mainly when one traffic stream

yields to another and an increase in the high priority stream

decreases the throughput of the low priority stream. To

overcome this limitation, Flotterod and Rohde [101] further

extended the INM into the INM with node supply con-

straints. Corthout et al. [102] proposed additional supply

constraints in node models due to conflicts within the

intersection and analyzed the solution non-uniqueness for

point-like and spatial modelling of nodes.

3.3 Solution to node models

First generation models, as explained in the previous sec-

tion, are solved using simple supply–demand boundary

conditions (demand at upstream and supply at downstream

of the node) but can be applied only for merge or diverge

junctions. Node models formulated as directional flow

maximization problem is increasingly capturing attention

for further research, implementing additional constraints or

modifying the existing ones. Aw–Rascle model for inter-

section is solved in literature analytically as Riemann

problem with piecewise constant initial data at upstream

and downstream of the node. The model also incorporates

turning fractions which are obtained through constraint

downstream flow maximization problem [83]. Numerical

schemes as for link models can be used to solve spatial

node models where the concepts of grid points or cells

exist. However, representation of intersection as cells to

account for turning maneuvers is a difficult task following

restrictive assumptions, and thus analytical methods are

commonly used as a solution procedure.

3.4 Summary of node modelling

First generation node models (merge and diverge models)

provide a good base for node modelling and relies on

simple constraints of demand and supply. However, a

major limitation is that it cannot be applied for a general

case where there exist m upstream and n downstream links.

The problem remained unresolved until Holden and Ris-

erbo [86] formulated a constraint flow maximization

problem for intersection modelling. Following this model,

several other mathematical models [87, 88] are also

introduced giving better physical interpretation of the

entropy condition to the Riemann problem. The general

requirements for a node model are first introduced by

Lebacque and Koshyaran [90] and introduction of the

‘invariance principle’ became an inevitable condition for

the convergence of the solution by numerical schemes.

Incorporating all the reviewed requirements of a node

model, Tampere et al. [83] created a good algorithmic

platform for a GNM along with the supply constraint

interaction rule for modelling specific node instances.

Flotterod and Rohde [101] used GNM with the IT principle

and formulated an INM with node supply constraints.

Analytical methods are commonly used in literature for

qðkÞ ¼ qinðkÞ

qoutðkÞ

!

and qð0Þ ¼ 0;

qðkþ1Þ ¼ qðkÞ þ hðkÞu qðkÞ �

;

For any stage;

DðkÞ ¼ ‘

1� ‘� I ðupstream nodesÞ : qðinÞ‘ \D‘ and 8j; b‘j [ 0; q

ðoutÞ‘ \

X

j

I\‘� I þ J ðdownstream nodesÞ : qðoutÞ‘ \

X

‘

and 9i 2 DðkÞ; bi‘ [ 0

���������

8>>><

>>>:

9>>>=

>>>;

hðkÞ ¼ min‘2DðkÞ

D‘ � qinðkÞ‘

�=uin

‘ qðkÞ �

for ‘ upstream

X

‘

�qoutðkÞ‘

!

=uout‘ qðkÞ �

for ‘ downstream

8>>><

>>>:

9>>>=

>>>;

ð60Þ

172 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

solving node models rather than solving numerically

(constrained optimization problem or a Riemann problem

with constrained optimization).

4 Synthesis

The models discussed above are synthesized in this section

based on four aspects as follows.

(1) Ability to map the real world traffic,

(2) Applicability to different flow regimes,

(3) Solution techniques and computational effort, and

(4) Scale of application.

Among link models, LWR model as well as its dis-

cretized model (CTM) limitedly captures traffic realism

when compared to higher order models. Anticipation and

inertial effect by drivers are two basic real traffic phe-

nomena which are lacking in the LWR model. When using

higher order models, formulation of velocity dynamics

leads results closer to real traffic. When applied to a

hypothetical mid-block section, LWR and higher order

models produce similar results, but for specific traffic sit-

uations such as lane reduction, higher order models dom-

inates to give better results [74]. In higher order models, as

the number of equations increase, the accuracy also

increases. Most of the continuum models discussed are

address vehicle heterogeneity and multilane effect in their

extensions, however, the lack of lane discipline behavior,

which is common in developing countries, should also be

studied to ensure transferability of these models. Porous

flow approach considered vehicle heterogeneity but

restricted the number of vehicle classes for tractability.

LWR model cannot model congestion dissipation well

and is more suited only for uncongested regime. Except

Payne model, all higher order models model congested as

well as uncongested regime equally well. In fact, Payne

model when applied to congested regime redraws the

cluster formation accurately; only at the tails of the con-

gestion, the model causes negative speed for vehicles.

Porous flow approach is also suited to model both the two

regimes, but not proven with actual data.

Easier analytical solution and less computational effort

are the two major highlights of LWR model. Though

analytical solution to higher order models is tiresome, the

computational effort when using simulation is comparable

with that of LWR model. Because of varying traffic con-

ditions on links and at boundaries, a hybrid approach of the

numerical schemes as well as of the boundary condition

will be the best option for accurate simulation results. The

high computational effort needed for porous flow approach

(because of the number of supply–demand conditions) and

for the non-continuum model (because of the number and

type of the variables involved) is a restriction for their

application in heterogeneous traffic flow.

Because of its simplicity, LWR model can be used for

large scale networks, but the errors may be large in con-

gested flow conditions when compared to other macro-

scopic models. For dynamic traffic assignment, when

applied to a single origin destination pair, higher order

models showed better convergence when compared to

LWR model [74]. There is a reasonable chance this effect

may hold at network level too. Not all higher order models

are used at a network level and is still an open research

area. Accuracy versus computational effort when applied to

a large scale network should be compared among higher

order models and also with LWR model.

Among node models, first generation models can only

be used for cases without any specific traffic instances or

constraints. The computational effort for these models are

less when compared to constraint flow maximization

approach, but the latter method will be the best choice for

modelling real flow behavior at intersections to avoid sig-

nificant compromise in the level of accuracy that can be

achieved. The GNM and its extensions should be explored

to large scale network and results should be compared with

that by existing network level node models both qualita-

tively and quantitatively. This promises to be an important

area of further research.

5 Conclusion

Links and nodes are the two major elements in a transport

network. Hence modelling of the network traffic flow

comprises these two elements either through their explicit

modelling and combination, or through any specific

approach that can simultaneously redraw the flow charac-

teristics on these elements. An efficient network model

should capture bottleneck effect and wave propagation

correctly and should also model the intersection flow

behavior well. Among the two major levels of models’

aggregation (micro- and macro-levels), macroscopic mod-

els are best suited for network wide applications where the

overall stream features (congestion, delay, etc.) are of

prime interest with comparatively less computational cost.

A clear understanding of the existing macro-level traffic

flow models will help in modelling of varying traffic sce-

narios with accurate qualitative and quantitative results.

The paper gives a broad over view of important, mac-

roscopic link and node models till date, and are described

and summarized separately in two main parts. Macroscopic

link models have a strong and vast base in continuum

theory. Earlier macroscopic link models were originally

formulated for homogeneous traffic conditions with perfect

lane discipline and hence the flow dynamics at stream level

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 173

123

Author's personal copy

captures real conditions accurately. Research in the area of

modelling heterogeneous traffic lacking lane discipline is

insufficient. Non-continuum and porous flow approaches

for link modelling are complex because of the number and

type of variables involved. Literature in node modelling is

less when compared to link models because of the com-

plexity in modelling turning behavior. Compared to the

first generation node models, the constrained optimization

problem formulation for node modelling is gaining

increasing attention by researchers in this area. Numerical

schemes are the most used solution techniques in link

models where as analytical procedures are commonly used

for node models.

There exists numerous models of traffic flow in the

macroscopic level; yet not all the models are shown to be

used (or can be used) for network wide applications. Fol-

lowing an extensive review of the macroscopic traffic flow

models, the paper also presents a synthesis on the advan-

tages and disadvantages of the discussed models. The paper

provides the reader an understanding of the current status

of the macro-level modelling of traffic flow, highlighting

the positive features and hopefully will encourage efficient

research in this area to overcome drawbacks and witness

more applications particularly at network level.

Acknowledgements The authors thank the Ministry of Urban

Development, Government of India, for sponsoring the Center of

Excellence in Urban Transport at Indian Institute of Technology (IIT),

Madras that enabled this research work. The second author also

thanks the New Faculty Grant provided by IIT Madras that partially

funded this research work. All findings and opinions in the paper are

the authors and do not necessarily reflect the views of the funding

agencies.

References

1. Hoogendoorn, S.P., Bovy, P.H.L.: State-of-the-art of vehicular

traffic flow modelling. In: Proc. Inst. Mech. Eng. I. 283–301

(2001)

2. Lighthill, M.J., Whitham, G.B.: On kinematic waves. II, A

theory of traffic flow on long crowed roads. Proc. R. Soc. A 229,

281–345 (1955)

3. Richards, P.I.: Shockwaves on the highway. Oper. Res. 4(1),

42–51 (1956)

4. Transportation Research Board: Revised Monograph on Traffic

Flow Theory: A State-of-the-Art- Report. Transportation

Research Board, National Research Council, Washington, DC.

http://www.tfhrc.gov/its/tft/tft.htm (1997). Accessed 14 March

2011

5. Munjal, P.K., Pipes, L.A.: Propagation of on-ramp density

perturbations on unidirectional two and three lane freeways.

Transp. Res. 5(4), 241–255 (1971)

6. Munjal, P.K., Hsu, Y.S., Lawrence, R.L.: Analysis and valida-

tion of lane drop effects on multilane freeways. Transp. Res.

5(4), 257–266 (1971)

7. Holland, E.N., Woods, A.W.: A continuum model for the dis-

persion of traffic on two-lane roads. Transp. Res. B 31(6),

473–485 (1997)

8. Greenberg, J.M., Klar, A., Rascle, M.: Congestion on multilane

highways. SIAM J. Appl. Math. 63(3), 813–818 (2003)

9. Laval, J.A.: Some properties of a multi-lane extension of the

kinematic wave model. Working paper, Intelligent Transporta-

tion Systems. University of California, Berkeley (2003)

10. Zhang, H.M., Jin, W.L.: A kinematic wave traffic flow model for

mixed traffic. Transp. Res. Rec. 1802, 197–204 (2002)

11. Logghe, S., Immers, L.H.: Heterogeneous traffic flow modelling

with the LWR model using passenger car equivalents. In: Pro-

ceedings of the 10th World Congress on ITS, Madrid.

http://www.kuleuven.be/traffic/dwn/P2003E.pdf (2003). Acces-

sed 13 March 2012

12. Chanut, S., Buisson, C.: A macroscopic model and its numerical

solution for a two flow mixed traffic with different speeds and

lengths. Transp. Res. Rec. 1852, 209–219 (2003)

13. Logghe, S.: Dynamic modelling of heterogeneous vehicular

traffic. PhD Thesis, KU Leuven, Belgium (2003)

14. Wong, G.C.K., Wong, S.C.: A multi-class traffic flow model—

an extension of LWR model with heterogeneous drivers. Transp.

Res. A 36(9), 763–848 (2002)

15. Gavage, S.B., Colombo, R.M.: An n-populations model for

traffic flow. Eur. J. Appl. Math. 14(5), 587–612 (2003)

16. Lebacque, J.P.: A two-phase bounded acceleration traffic flow

model. Analytical solutions and applications. Transp. Res. Rec.

1852, 220–230 (2003)

17. Leclercq, L., Laval, J.A.: A multiclass car following rule based

on the LWR model. In: Rolland C.A., Chevoir F., Gondrte P.,

Lassarre S., Lebaque J.P., Schreckenberg M. (eds.) Traffic and

Granular Flow ’07, pp. 151–160. Springer, Heidelberg (2009)

18. Li, X., Lu, H.: An extension of LWR model considering slope

and heterogeneous drivers. In: International Conference on

Computer and Information Science, pp. 1236–1240. http://ieee

xplore.ieee.org/xpls/abs_all.jsp?arnumber=5709505 (2010).

Accessed 12 Nov 2011

19. Zhu, Z., Zhang, G., Wu, T.: Numerical analysis of freeway

traffic flow dynamics under multi-class drivers. Transp. Res.

Rec. 1852, 201–208 (2003)

20. Daganzo, C.F.: The cell transmission model: a dynamic repre-

sentation of highway traffic consistent with the hydrodynamic

theory. Transp. Res. B 28(4), 269–287 (1994)

21. Newell, G.F.: Comments on traffic dynamics. Transp. Res. B

23(5), 386–389 (1988)

22. Daganzo, C.F.: The cell transmission model, Part II: network

traffic. Transp. Res. B 29(2), 79–93 (1995)

23. Nagel, K., Schreckenberg, M.: A cellular automation model for

freeway traffic. J. Phys. I 2, 2221–2229 (1992)

24. Daganzo, C.F.: The lagged cell transmission model. In: Ceder A.

(ed.) Transportation and Traffic Theory, Proceedings of the 14th

International Symposium on Transportation and Traffic Theory,

pp. 81–104. Pergamon, Amsterdam (1999)

25. Szeto, W.Y.: Enhanced lagged cell transmission model for

dynamic traffic assignment. Transp. Res. Rec. 2085, 76–85

(2009)

26. Laval, J.A., Daganzo, C.: Working paper, Intelligent Transpor-

tation Systems. University of California, Berkeley (2004)

27. Hu, X., Wang, W., Sheng, H.: Urban Traffic flow prediction

with variable cell transmission model. J. Transp. Sys. Eng. Inf.

Technol. 10(4), 73–78 (2010)

28. Chen, X., Shi, Q., Li, L.: Location specific cell transmission

model for freeway traffic. Tsinghua Sci. Technol. 15(4),

475–480 (2010)

29. Long, J., Gao, Z., Zhao, X., Lian, A., Orenstein, P.: Urban traffic

jam simulation based on the cell transmission model. Netw.

Spat. Econ. 11, 43–64 (2011)

30. Yperman, I.: The link transmission model for dynamic network

loading. PhD Thesis, KU Leuven, Belgium (2007)

174 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy

31. Nair, R., Mahmassani, H.S., Hooks, E.M.: A porous flow

approach to modelling heterogeneous traffic in disordered sys-

tem. Transp. Res. B (2011). doi:10.1016/j.trb.2011.05.009

32. Leveque, R.J.: Finite volume methods. In: Leveque, R.J. (ed.)

Finite Volume Methods for Hyperbolic Problems, pp. 64–348.

Cambridge University Press, Cambridge (2002)

33. Payne, H.J.: Models for freeway traffic and control. In: Bekey,

G.A. (ed.) Mathematical Models of Public Systems, pp. 51–61.

Simulation Council, La Jolla (1971)

34. Ross, P.: Traffic dynamics. Transp. Res. B 22(6), 421–435 (1988)

35. Kuhne, R.D.: Macroscopic freeway model for dense traffic—

stop-start waves and incident detection. In: Volmuller J., Ha-

merslag R., Technische Universiteit Delft (eds.) Proceedings of

the 9th International Symposium on Transportation and Traffic

Theory, pp. 20–42. VNU Science Press, Utrecht (1984)

36. Kuhne, R.D.: Freeway control and incident detection using a

stochastic continuum theory of traffic flow. In: Proceedings of

the 1st International Conference on Applications of Advanced

Technologies in Transportation Engineering, San Diego, CA,

pp. 287–292 (1989)

37. Michalopoulos, P.G., Yi, P., Lyrintzis, A.S.: Continuum mod-

elling of traffic dynamics on congested freeways. Transp. Res. B

27(4), 315–332 (1993)

38. Kotsialos, A., Papageorgiou, M.: Traffic flow modelling of

large-scale motorway networks using the macroscopic model-

ling tool METANET. IEEE Trans. Intell. Transp. Syst. 3(4),

282–292 (2002)

39. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York

(1974)

40. Jin, W.L., Zhang, H.M.: The formation and structure of vehicle

clusters in the Payne–Whitham traffic flow model. Transp. Res.

B 37, 207–223 (2003)

41. Li, T.: Nonlinear dynamics of traffic jams. Physica D 207, 41–51

(2009)

42. Daganzo, C.F.: Requiem of second-order fluid approximations

of traffic flow. Transp. Res. B 2(4), 277–286 (1995)

43. Liu, G., Lyrintzis, A.S., Michalopoulos, P.G.: Improved high-

order model for freeway traffic flow. Transp. Res. Rec. 1644,

37–46 (1998)

44. Aw, A., Rascle, M.: Resurrection of ‘‘Second Order’’ models of

traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000)

45. Moutari, S., Rascle, M.: A hybrid Lagrangian model based on

the Aw–Rascle traffic flow model. SIAM J. Appl. Math. 68(2),

413–436 (2007)

46. Zhang, H.M.: A theory of non equilibrium traffic flow. Transp.

Res. B 32(7), 485–498 (1998)

47. Zhang, H.M.: A non-equilibrium traffic model devoid of gas-

like behavior. Transp. Res. B 36(3), 275–290 (2002)

48. Lebacque, J.P., Haj-Salem, H., Mammar, S.: Second order

traffic flow modelling: supply–demand analysis of the inhomo-

geneous Riemann problem and of boundary conditions. In:

Proceedings of the 16th Mini-EURO Conference and 10th

Meet, pp. 108–115. EWGT. http://citeseerx.ist.psu.edu/viewdoc/

summary?doi=10.1.1.65.682 (2005). Accessed 22 Feb 2012

49. Lebacque, J.P., Mammar, S., Haj-Salem, H.: The Aw–Rascle

and Zhang’s model: vacuum problems, existence and regulari-

ties of the solutions of the Riemann problem. Transp. Res. B 41,

710–721 (2007)

50. Helbing, D.: Gas-kinetic derivation of Navier–Stokes-like traffic

equations. Phys. Rev. E 53(3), 2266–2381 (1996)

51. Prigogine, I., Andrews, F.C.: A Boltzmann-like approach of

traffic flow. Oper. Res. 8(6), 789–797 (1960)

52. Paveri-Fontana, S.L.: On Boltzmann-like treatment of traffic

flow: a critical review of the basic model and an alternative

proposal for dilute traffic analysis. Transp. Res. 9, 225–235

(1973)

53. Hoogendoorn, S.P.: Multiclass continuum modelling of multi-

class traffic flow. PhD Thesis, T99/5, TRAIL Thesis series, Delft

University Press (1999)

54. Hoogendoorn, S.P., Bovy, P.H.L.: Modelling multiple user-class

traffic flow. Transp. Res. B 34(2), 123–146 (2000)

55. Prigogine, I., Herman, R.: Kinetic Theory of Vehicular Traffic.

American Elsevier, New York (1971)

56. Li, S.F., Zhang, P., Wong, S.C.: Conservation form of Helbing’s

fluid dynamic traffic flow model. Appl. Math. Mech. (Engl. Ed.)

32(9), 1109–1118 (2011)

57. Helbing, D.: Derivation and empirical validation of a refined

traffic flow model. Physica A 233(1–2), 253–282 (1996)

58. Helbing, D., Hennecke, A., Shvetsov, V., Trieber, M.: MAS-

TER: macroscopic traffic simulation based on a gas-kinetic,

non-local traffic model. Transp. Res. B 35(2), 183–211 (2001)

59. Kerner, B.S., Rehborn, H.: Experimental properties of phase

transitions in traffic flow. Phys. Rev. Lett. 49, 4030–4033 (1997)

60. Hoogendoorn, S.P., Bovy, P.H.L.: Multiclass macroscopic traffic

flow modelling: a multi-lane generalization using gas-kinetic

theory. In: Ceder A. (ed.) Transportation and Traffic Theory,

Proceedings of the 14th International Symposium on Transporta-

tion and Traffic Theory, pp. 27–50. Pergamon, Amsterdam (1999)

61. Jiang, R., Wu, Q.S., Zhu, Z.J.: A new continuum model for

traffic flow and numerical tests. Transp. Res. B 36, 405–419

(2002)

62. Helbing, D., Trieber, M.: Numerical simulation of macroscopic

traffic equations. Comput. Sci. Eng. 1(5), 89–98 (1999)

63. Gazis, D.C., Herman, R., Rothery, R.W.: Nonlinear follow-the-

leader models of traffic flow. Oper. Res. 9, 545–567 (1961)

64. Bando, M., Hasabe, K., Nakayama, A., Shibata, A., Sugiyama,

Y.: Dynamical model of traffic congestion and numerical sim-

ulation. Phys. Rev. E 51, 1035–1042 (1995)

65. Jiang, R., Wu, Q.S.: Extended speed gradient model for mixed

traffic. Transp. Res. Rec. 1883, 78–84 (2004)

66. Tang, C.F., Jiang, R., Wu, Q.S., Wiwathanapataphee, B., Hu,

Y.H.: Mixed traffic flow in anisotropic continuum model.

Transp. Res. Rec. 1999, 13–22 (2007)

67. Del Castillo, J.M., Pintado, P., Benitez, F.G.: A formulation for

the reaction time of traffic flow models. In: Daganzo C. (ed.)

Transportation and Traffic Theory, Proceedings of the 12th

International Symposium on Transportation and Traffic Theory,

pp. 387–405. Elsevier, Amsterdam (1993)

68. Tang, C.F., Jiang, R., Wu, Q.S.: Extended speed gradient model

for traffic flow on two-lane freeways. Chin. Phys. 16(6),

1570–1575 (2007)

69. Tang, T.Q., Huang, H.J., Zhao, S.G., Shang, H.Y.: A new

dynamic model for heterogeneous traffic flow. Phys. Lett. A

373, 2461–2466 (2009)

70. Darbha, S., Rajagopal, K.R.: Aggregation of a class of nonlinear

interconnected dynamical systems. Math. Probl. Eng. Theory

Methods Appl. 7, 379–392 (2001)

71. Zhang, P., Wong, S.C., Shu, C.W.: A weighted essentially non-

oscillatory numerical scheme for a multi-class traffic flow model

on an inhomogeneous highway. J. Comput. Phys. 212(2),

739–756 (2006)

72. Ngoduy, D., Hoogendoorn, S.P., Van Zuylen, H.J.: Cross-com-

parison of numerical schemes for macroscopic traffic flow

models. Transp. Res. Rec. 1876, 52–61 (2004)

73. Cremer, M., Papageorgiou, M.: Parameter identification for a

traffic flow model. Automatica 17(6), 837–843 (1981)

74. Mohan, R., Ramadurai, G.: A Case for advanced traffic flow

models in DTA. In: Presented at the 4th International Sympo-

sium on Dynamic Traffic Assignment. Martha Vineyard, Vine-

yard Haven (2012)

75. Papageorgiou, M.: Some remarks on macroscopic traffic flow

modelling. Transp. Res. A 32(5), 323–329 (1997)

Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 175

123

Author's personal copy

76. Daganzo, C.F., Lin, W.H., Del Castillo, J.M.: A simple physical

principle for the simulation of freeways with special lanes and

priority vehicles. Transp. Res. B 31(2), 103–125 (1997)

77. Helbing, D.: Derivation of non local macroscopic traffic equa-

tions and consistent traffic pressures from microscopic car-fol-

lowing models. Eur. Phys. J. B 69, 539–548 (2009)

78. Bagnerini, P., Rascle, M.: A multi-class homogenized hyper-

bolic model of traffic flow. SIAM J. Math. Anal. 35(4), 949–973

(2003)

79. Colombo, R.M.: A 2 9 2 hyperbolic traffic flow model. Math.

Comput. Model. 35, 683–688 (2002)

80. Tampere, C.M.J., van Arem, B., Hoogendoorn, S.P.: Gas Kinetic

traffic flow modelling including continuous driver behavior

models. Transp. Res. Rec. 1852, 231–238 (2003)

81. Jin, W.L., Zhang, H.M.: On the distribution schemes for deter-

mining flows through a merge. Transp. Res. B 37(6), 521–540

(2003)

82. Lebacque, J.P.: The Godunov scheme and what it means for first

order traffic flow models. In: Lesort J.B. (ed.) Transportation

and Traffic Theory, Proceedings of the 13th International

Symposium on Transportation and Traffic Theory, pp. 647–677.

Pergamon, Tarrytown, New York (1996)

83. Tampere, C.M.J., Corthout, R., Cattrysse, D., Immerse, L.H.: A

generic class of first order node models for dynamic macro-

scopic simulation of traffic flows. Transp. Res. B 45, 289–309

(2011)

84. Ni, D., Leonard, J.D.: A simplified kinematic wave model at a

merge bottleneck. Appl. Math. Model. 29(11), 1054–1072

(2005)

85. Lebacque, J.P.: Intersection modelling, application to macro-

scopic network traffic flow models and traffic management. In:

Hoogerdorn SP., Luding S., Bovy P.H.L., Schreckenberg M.,

Wolf D.E. (eds.) Traffic and Granular Flow ’03, pp. 261–278.

Springer, Heidelberg (2005)

86. Holden, H., Riserbo, N.H.: A mathematical model of traffic flow

on a network of unidirectional roads. SIAM J. Math. Anal.

26(4), 999–1017 (1995)

87. Herty, M., Klar, A.: Modelling, simulation and optimization of

traffic flow networks. SIAM J. Sci. Comput 25(3), 1066–1087

(2003)

88. Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on a road

network. SIAM J. Math. Anal. 36(6), 1862–1886 (2005)

89. Durlin, T., Henn, V.: A delayed flow intersection model for

dynamic traffic assignment. In: Proceedings of the 16th Mini-

EURO Conference and 10th Meet, pp. 78–81. EWGT. http://

citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.77.2228 (2005).

Accessed 14 March 2011

90. Lebacque, J.P., Koshyaran, M.M.: First order macroscopic

traffic flow models: intersection modelling, network modelling.

In: Mahmassani H.S. (ed.) Transportation and Traffic Theory,

Proceedings of the 16th International Symposium on Transpor-

tation and Traffic Theory, pp. 365–386. Elsevier, Amsterdam

(2005)

91. Garavello, M., Piccoli, B.: Traffic flow on a road network using

the Aw–Rascle model. Commun. Partial Differ. Equ. 31(2),

243–275 (2006)

92. Herty, M., Rascle, M.: Coupling conditions for a class of

‘‘second-order’’ models for traffic flow. SIAM J. Math. Anal.

38(2), 595–616 (2006)

93. Herty, M., Moutari, S., Rascle, M.: Intersection modelling with a

class of ‘‘second-order’’ models for vehicular traffic flow. In:

Gavage, S.B., Serre, D. (eds.) Hyperbolic Problems: Theory,

Numerics, Applications, pp. 755–763. Springer, Heidelberg

(2008)

94. Gentile, G., Meschini, L., Papola, N.: Fast heuristics for con-

tinuous dynamic shortest paths and all-or-nothing assignment.

In: Proceedings of the 35th Annual Conference of the Italian

Operation Research Society. http://citeseerx.ist.psu.edu/viewdoc/

summary?doi=10.1.1.157.2983 (2004). Accessed 5 May 2011

95. Gentile, G., Meschini, L., Papola, N.: Spillback congestion in

dynamic traffic assignment: a macroscopic flow model with time

varying bottlenecks. Transp. Res. B 41, 1114–1138 (2007)

96. Bliemer, M.J.: Dynamic queuing and spillback in an analytical

multiclass network loading model. Transp. Res. Rec. 2029,

14–21 (2007)

97. Yuan, S.X., An, Y.S., Zhao, X., Fan, H.W.: Modelling and

simulation of dynamic traffic flows at non-signalized T-inter-

sections. In: Proceedings of the 2nd International Conference on

Power Electronics and Intelligent Transportation Systems.

http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5406

951 (2009). Accessed 16 May 2011

98. Tolba, C., Lefebvre, D., Thomas, P., El Moudni, A.: Continuous

and timed Petri nets for the macroscopic and microscopic traffic

flow modelling. Simul. Model. Pract. Theory 13, 407–436

(2005)

99. Liu, X., Dai, S.: Macroscopic models for interrupted traffic flow

of signal controlled intersections. In: ISECS International Col-

loquium on Computing, Communication, Control and Manage-

ment, pp. 78–81. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=

&arnumber=5268043 (2009). Accessed 15 May 2011

100. Jin, W.L.: Continuous kinematic wave models of merging traffic

flow. Transp. Res. B 44, 1084–1103 (2010)

101. Flotterod, G., Rohde, J.: Operational macroscopic modelling of

complex urban road intersections. Transp. Res. B 45, 903–922

(2011)

102. Corthout, R., Flotterod, G., Viti, F., Tampere, C.M.J.: Non-

unique flows in macroscopic first order intersection models.

Transp. Res. B 46(3), 343–359 (2012)

103. Tuerprasert, K., Aswakul, C.: Multiclass cell transmission model

for heterogeneous mobility in general topology of road network.

J. Intell. Transp. Syst. Technol. Plan. Oper. 14(2), 68–82 (2010)

104. Lo, H.K., Szeto, W.Y.: A cell-based dynamic traffic assignment

model: formulation and properties. Math. Comput. Model.

35(7–8), 849–865 (2002)

176 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176

123

Author's personal copy