CELLULAR AUTOMATA MODELS FOR
TRAFFIC FLOW IN URBAN NETWORKS
by
Abdelhakeem A. Hammad, B.Sc. M.Sc.
A thesis submitted in fulfilment of the
requirements for the degree of
Ph.D. m Computer Applications
School of Computer Applications
Dublin City University
Dublin 9, Ireland
Supervisors
Dr. Heather Ruskin and Dr.Micheál O’hEigeartaigh
CELLULAR AUTOMATA MODELS FOR TRAFFIC FLOW
IN URBAN NETWORKS
Abstract
R ecently, traffic problem s have attracted considerable attention Num erical
sim ulations, hydrodynam ics m odels, and queueing theory are a few o f the
basic theoretical tools used to describe car traffic on highw ays
Com putationally, Cellular Autom aton m odels are sim ple and flexible and are
increasingly used in sim ulations o f com plex system s, providing considerable
insight on traffic behaviour A particular strength o f these m odels is fast
’’m inim al” m icroscopic sim ulation, w hich nevertheless can reproduce
important m acroscopic features
This m ethodology requires streets to be divided into sites (cells), w ith sites
linked into road segm ents and forming networks, punctuated by junctions,
traffic signals and so on Car m ovem ents are represented by “jum ps”, where
each jump represents the current car speed
M uch previous work has concentrated on flow o f cars under highw ay
conditions, but less effort has been concerned with urban networks and the
constraints, w hich apply in this context The research reported here, uses
Cellular Autom aton m ethodology to exam ine traffic patterns in urban and
inter-urban areas A three state determ inistic Cellular Autom ata M odel is
defined for the dynam ic process and networks o f various sizes are
investigated, with all nodes controlled and diverse traffic conditions
considered at each intersection Both transient m ovem ents o f cars through the
network and temporary lost to flow , through off-street parking, are exam ined
for im pact on traffic parameters
A stochastic feeding m echanism , in w hich car arrivals follow a Poisson
process, has been im plem ented throughout the sim ulation ’f or different arrival
rates Lane-changing rules for sim ulation o f tw o-lane traffic are also discussed
and, finally, a Stochastic Cellular Autom ata M odel for inter-urban areas is
presented K ey features o f traffic behaviour under the various network
conditions are analysed and com parisons with highw ay flow
Suggestions and future im provem ents on m odel realism are also given
Acknowledgements
I take this opportunity to thank m y advisors, Dr Heather Ruskin and Dr
M icheál O ’hEigeartaigh, for their invaluable advice and guidance throughout
m y research study Their constant support, encouragem ent and friendship
m ade m y stay at D C U an enjoyable
I would like to thank Prof M ichal Ryan and the School o f Computer
A pplications in general for providing m e with the opportunity and the
necessary funding to undertake this research
M any thanks to Rory Boland, the director o f the traffic section in Dublin, and
to all people in the traffic control room for helping m e to conduct m y survey
I also thank Prof Duarte for helpful discussions at an early stage and for
pointing m e to som e important references
Thanks for M ostafa Abou_Shaban for helping m e with the cod e used to plot
the space-tim e diagrams
Finally, I w ish to express m y heartfelt thanks to m y w ife, for her
understanding and her constant support, and to m y parents for their affection
and encouragem ent
Declaration
I hereby certify that this material, w hich I now submit for assessm ent on the
programme o f study leading to the award o f Ph.D. is entirely m y ow n work
and has not been taken from the work o f others save and to the extent that such
work has been cited and acknow ledged within the text o f m y work.
Signed
A »
Date
Table of Contents
Part I “Traffic Simulation on a Single-lane’
Chapter 1 “Introduction”------------------------
-0
1 0 Introduction
-1
---------------------------------
1 1 Literature R eview ----------------------------------1 1 1 M acroscopic Traffic Flow M odels1111
Car Following Theory—j---------------- ----------
1112
P sychological-P hysiological Spacing M odels-
1 1 2 M acroscopic M od els........
112 1
Continuum M odels—
1 1 2 11
1 1 2 12
Sim ple Continuum M odelsM odels with M om ent um —
-2
-2
-5
9
-11
-12
-14
-20
112 2
T w o fluid Theory.......
1 1 3 Cellular Autom ata M odels—
-2 3
-25
O ne-dim ensional Cellular Autom ata M odels---------------------26
113 1
113 2
T w o-dim ensional Cellular Autom ata M odels------------------- 28
------------------31
1 1 4 Thesis O utline-—
Chapter 2 “ Simple Cellular Automata Model for Urban Traffic Flow
-33
2 0 Introduction----------------------------------------------------------------------------------------------------- 34
2 1 The Cellular Autom ata M odel------------------------------------------------------------------------ 35
2 11 M odel Parameters------------------------------------------------------------------------------36
2 12 M odel D ynam ic--------------------------------------------------------------------------------- 37
2 2 Sim ulation and R esults------------------------------------------------------------------------------------ 39
2 2 1 The M odel with Periodic Boundary C onditions-------------------------------- 39
2 2 2 The M odel with Open Boundary C onditions--------------------------- ---------- 43
2 3 Road Traffic Sim ulations--------------------------------------------------------------------------------46
2 3 1 Road D escription---------------------------------------------------------------------------------46
2 3 2 Sim ulation and R esults--------------------------------- —------------------------------------48
2 4 Sum m ary and Concluding C om m ents------------54
Part II “Network Traffic Flow”
Chapter 3 “Traffic System and Transient Movement Simulation”----------------- 55
3 0 Introduction--------------------------------------------------------------------
56
3 1 N etw ork description---------------------------------------------------------------------------------------- 56
56
3 1 1 N od es-----------------------------------------------------------------------------3 1 2 L in k s-.............................— -...........— -------59
3 1 3 Segm ents------------------------------------------------------ ---------------- --------------------- 59
3 1 4 N etw ork geom etry------------------------------------------------------------60
3 1 5 Traffic Control and Traffic conditions at Junctions-----------------------------60
3 1 6 Traffic Parameters— ----------62
3 2 D escription o f Sim ulation Experim ents----------------------------------------------------------- 63
3 3 Sim ulation and Experim ents---------------------------------------------------------------------------- 66
3 3 1 The E ffect o f Arrival Rate on N etw ork Perform ance------------------66
3 3 1 1 The existence o f the “Jamming threshold??------------------------------------- 66
3 3 1 2 Arrival Rate vs N etw ork Size
-----------
3 3 2 Varying the Traffic C onditions-------------------- --------
69
73
3 3 3 Tim e-m terval Based Sim ulation---------------------------------------------------------- 77
3 3 4 The Influence o f the transient Period on the N etw ork Perform ance—82
3 4 Sum m ary and Concluding C om m ents--------------------------------------------------------------85
Chapter 4 “Network Performance under Various Traffic Events”....................—87
4 0 Introduction--------------------------------------
—88
4 1 L oss to F low Sim ulations.....................
—88
4 1 1 Sim ulation w ith L oss to R o w at Fixed rate------------------------------------------88
4 1 2 Sim ulation w ith L oss to R o w at Stochastic Rate----------------------------------94
4 2 Traffic System under Short and Long term Events-------------------------------------99
4 2 1 M odelling Short-term E vents---------------
99
4 2 2 M odelling L ong-term Events-------------------------r.
43
Sum m ary and C oncluding C om m ents
---------
104
109
Part HI “Two-lane Traffic Simulation”
Chapter 5 “ Simple Lane-changing Rules for Urban Traffic using Cellular
Automata”
-----------
111
5 0 Introduction------------------------------------ ------- --------------------------------------------------— 112
5 1 M otivations for lane changing-----------------------------------------------------------------------113
5 2 The T w o-lane M odel--------------------------------------------------------------------------------------114
5 3 Sim ulation and R esults----------------------------------------------------------------------------------- 119
5 3 1 Lane-usage Behaviour-------------------------------------119
532
533
F low B ehaviour-------------------------------------------------------------------------------- 126
Lane-changing Behaviour----------------------------------------------------------------- 130
5 4 M odel Validation------------------------------------------------------------------------------------------- 134
Part IV “ Inter-urban Traffic Simulations”
Chapter 6 “Stochastic Cellular Automata Model for Inter-urban Traffic”— 137
6 0 Introduction-
-------------------
138
6 1 The M odel-----------------------139
-------------139
6 1 1 D efinition o f the M odel
6 1 2 Im plem entation o f the M odel------------------------------------ ---------— 141
6 2 Sim ulation and R esults----------------------- ------143
6 2 1 M odel D ynam ics--------------------144
6 2 2 Fundam ental Diagram s----------------------------------------------------------------------148
6 2 3 The effect o f varying the m odel Parameters-------------------------------------- 150
6 3 Open System s-------------------------------------------------------------------------------------------------155
6 3 1 B ottleneck Situations-----------------------------------------157
6 3 2 Self-organisation o f the M axim um Throughput
-----6 4 Sum m ary and Concluding com m ents
.........................................
157
-.158
Chapter 7 “ Summary and Conclusion ”— ------
160
7 0 Research Contribution---------------------------
161
7 1 Future W ork--------------------------------------------------------------------------------------------------166
7 2 Concluding Rem arks------------------------------------------------------------------------------------- 167
B ib lio grap h y-------------------------------------------------------------------
168
A p p en d ices--------------
179
A
Statistical A nalysis----------------------------------------------------------------------
179
A 1
The Relationship betw een the netw ork size and the input parameters— 180
A l l N etw ork size vs traffic conditions....................
180
A 1 2 N etw ork size vs length o f transient period-----------------------------------182
A 1 3 N etw ork size vs interval length for calculating the output
Parameters-----------------------------------------------------------------------------------182
A 14
N etw ork size vs
the number o f the N etw ork-blocked exits-183
A 1 5 N etw ork size vs arrival rate--------------------------------------A2
184
Statistical analysis using three-input factor----------------------------------------------- 185
A 2 1 experim ents using a 17 node netw ork-----------------------------A 2 2 experim ents using a 41 node netw ork------------------------------------
185
-.........186
B
Sum m ary o f the sim ulation experim ents
188
C
Feeding M echanism -------------- ----------—----------------
196
D
Com putational Perform ance
-------— ------------
197
E
D iskette---------------------------------
— ----------------
199
List of Figures
CHAPTER 1
FIG URE (1 1) Perceptual thresholds in car-follow ing behaviour
FIG URE (1 2 ) V ehicles passed by tw o m oving observers
10
14
FIG URE (1 3) Fundam ental diagram obtained for real data
18
FIG URE (1 4 ) The use o f the flow -density curve to predict the local
conditions near a shock w ave
19
FIG URE (1 5 ) Schem atic illustration o f the CA m odel o f traffic flow with
tw o-level crossings
31
CHAPTER 2
FIG URE (2 1) Space-tim e diagrams o f the CA m odel defined on 100 site
lattice w ith closed boundary conditions(c b c)
38
FIG URE (2 2 a) Flow -density relation for the Sim ple M od el with c b c
41
FIG URE (2 2 b) V elocity-density relation for the Sim ple M odel w ith c b c 41
FIG URE (2 3) Space-tim e diagram s, where in (a) p = 0 3, (b) p = 0 8
42
FIG URE (2 4) Fundam ental diagram s o f the Sim ple M odel, with open
boundary conditions, w ith system o f size 60 cells
FIGURE (2 5) Fundam ental diagram s o f the Sim ple M od el with open
44
boundary conditions, with system o f size 80 cells
FIGURE (2 6) D ay-one o f our sim ulation using road o f size 77 cells
FIG URE (2 7) D ay-tw o o f our sim ulation using road o f size 77 cells
45
47
49
FIG URE (2 8) Day-three o f our sim ulation using road o f size 144 cells
50
FIGURE (2 9) Space-tim e diagram s, where in (a) the road o f size 77 cells
and in (b) the road o f size 144 cells
52
CHAPTER 3
FIG URE (3 1) Illustration o f the node object
58
FIG URE (3 2) Illustration o f the intersection types
58
FIG URE (3 3) Representation o f a section o f the real netw ork in D ublin by
'5
the proposed network geom etry
61
FIG URE (3.4). Flow-chart o f the sim ulation cod e
65
FIG URE (3 5) Num ber o f cars vs arrival rate for 17 nodes network
67
FIG URE
FIG URE
FIG URE
FIG URE
(3
(3
(3
(3
6) Num ber o f waiting cars vs arrival rate for four networks
67
7) W aiting cars vs arrival rate for various traffic conditions 68
8) Transported cars vs arrival rate for different networks
68
9) F low -density relation for a 1 7 ,4 1 node networks using the
the turning percentages 25% , 50% , 25%
71
FIGURE (3 10) V elocity-density relation for a 17, 41 node networks using
the turning percentages 25% , 50% , 25% and p = 0 55
72
FIGURE (3 11) Flow -density relation using tw o different traffic conditions 76
FIGURE (3 12) Flow -density relation for a network o f size 25 nodes using
p = 0 3 and using tw o different interval lengths
80
FIG URE (3 13) F low -density relation for a 1 7 ,4 1 node networks using tw o
different transient periods, 200 and 500 tim e steps
84
CHAPTER 4
FIG URE (4 1) Num ber o f cars vs arrival rate for 17 and 41 node networks 89
FIG URE (4 2) Number o f w aiting cars v s arrival rate for tw o different
N etw orks
90
FIGURE (4 3) Fundam ental diagrams for non-transient m ovem ent sim ulation
for tw o different networks with com parison to transient
m ovem ent sim ulation
92
FIGURE (4 4) Number o f w aiting cars vs arrival rate for two different
networks using different values for p i
FIGURE (4 5) System density vs tim e steps for 17, 41 node networks,
p = 0 15 and p i = 0 1, 0 3
FIGURE (4 6) System density vs tim e steps for 17, 41 node networks,
p = 0 50, p i = 0 1, 0 3
FIG URE (4 7) The effect o f introducing short-term events on the system
94
density for tw o different networks, 17 and 41 nodes
FIG URE (4 8) The effect o f increasing short-term events on the system
101
density for a 17 nodes netw ork
96
98
102
FIG URE (4 9) Fundam ental diagrams for the extrem e cases obtained in
m odelling long-term events
..
105
FIG URE (4 10 a) Fundam ental diagram obtained for the case o f highest
value o f q
107
FIG URE (4 10 b) Fundam ental diagram for observed road traffic
107
CHAPTER 5
FIG URE (5 1) Illustration o f conditions required for lane-changing
115
FIGURE (5 2) Flow-chart o f the lane-changing m echanism
118
FIG URE (5 3) Lane-usage frequency vs traffic density for different values
o f P_L obs
120
FIG URE (5 4) Left lane usage vs traffic density for different values o f
P_L obs and P_chg = 0 4, P_obs = 0 3
123
FIG URE (5 5) Left lane usage vs traffic density for different values o f
P_L obs and P_chg = 0 4, P_obs = 0 4
123
FIG URE (5 6) F low vs density for both single and tw o-lane M odels
127
FIG URE (5 7) The relation betw een the left-lane flow and the nght-lane
flow for tw o different sim ulations
128
FIG URE (5 8) The effect o f varying P_obs on the flow on both lanes
129
FIGURE (5 9) The effect o f varying P_obs on the flow -density relation 129
FIGURE (5 10) lane-changing frequency vs traffic density for different
values o f P_L obs
4
FIGURE (5 11) Com parison betw een lane-changing frequency to the
left lane and to the right lane
131
132
FIG URE (5 12) the effect o f lane-changing probability on the frequency
o f lane changes
133
FIG URE (5 13) lane-usage vs traffic density for our em pirical data
135
FIGURE (5 14) Lane-usage frequency vs traffic density, where
P_obs =o 3, P_Lobs = 0 5, 0 5, and P_chg = 0 4
135
CHAPTER 6
FIG URE (6 1) The tim e-evolution o f the Autom aton using the SC A M
144
FIG URE (6 2) Space-tim e diagram s for SCA M , where L = 200, vmax =3 146
FIG URE (6.3). Typical start-stop w aves obtained at higher d en sity (0 .5)....14 7
FIGURE (6 4) Fundam ental diagrams o f the SC A M using short-term
and long-term averages
FIG URE (6 5) F low vs density relation o f the SC A M , w hich obtained
by varying the m odel m axim um velocity
149
150
FIGURE (6 6) V elocity vs density relation o f the SC A M , w hich obtained
by varying the m odel m axim um velocity
151
FIG URE (6 7) The effect o f varying P_ac on the flow -density relation
151
FIG URE (6 8) The effect o f varying P_rd on the flow -density relation
152
FIG URE (6 9) The effect o f varying P_inc on the flow -density relation 153
FIG URE (6 10) Fundam ental diagram o f the SC A M with open boundary
conditions using different arrival rates, vmax = 3
FIG URE (6 11) Space-tim e plot o f the outflow from a traffic jam
156
158
APPENDICES
FIG URE (A 1) The interaction betw een the netw ork size and the traffic
conditions at low and high arrival rates
180
FIG URE (A 2) The interaction betw een the netw ork size and the transient
period at low and high arrival rates
182
List of Tables
TA BL E (2 1) The jam m ed tim e range, in tim e steps, for different road size
and light cycles
51
TA B LE (3 1) The influence o f varying the arrival rate on the m axim um flow
and its density, cars transported, and the queue length
69
T A BLE (3 2) The influence o f varying traffic conditions on the m axim um
flow and its density, cars transported, and the queue length 75
TA BL E (3 3) Sam e as TA BL E (3 2), but the arrival rate has increased
from p = 0 3 to p = 0 55
75
TA BL E (3 4) The influence o f varying the interval length on the sim ulation
outputs Sam ple o f the sim ulation outputs for 600 tim e steps for
a netw ork o f size 25 nodes at low p = 0 1
78
TA B LE (3 5) Sam e as TA BLE (3 4), but for a larger netw ork (41 nodes) 78
TA B LE (3 6) The influence o f varying the interval length on the sim ulation
outputs Sam ple o f the sim ulation outputs for 600 tim e steps for
a netw ork o f size 25 nodes at p = 0 3
TA B L E (3 7) Sam e as T A B L E (3 6), but for a larger netw ork (41 nodes)
81
81
TA BL E (3 8) The influence o f changing the transient period on the m axim um
flow and its density using different values o f p for two different
networks, where the turning percentages are 25% , 50% , 25% 82
TA BL E (3 9) Sam e as TA BLE (3 8), but the turning percentages are 12%,
76% , 12%
83
TA BLE (4 1) Com parison betw een the highest values and the low est values
o f the m axim um flow obtained for tw o different networks,
where som e o f the netw ork exits w ere blocked
108
TA BL E (5 1) The influence o f changing the lane-changing parameters on the
lane-usage inversion points using P_chg = 0 4, and different
values o f the other tw o parameters
TA BL E (5 2). Sam e as TA BLE (5 1), but with P_chg = 0 5
121
125
TA B L E (6 1). The relation betw een the velocity and its safe distance
.138
TA BL E (6 2) The influence o f varying the m odel parameters on the
m axim um velocity and the critical density
154
TA BL E (A 1) Statistical analysis for the relationship betw een the network size
and traffic conditions, transient period, interval length,
and the number o f netw ork-blocked exits
181
TA BLE (A 2) Statistical analysis for the relationship betw een the network size
and arrival rate
184
•s
TA BLE (A 3) Statistical analysis for the relationship betw een traffic
conditions, transient period, and interval length for a 17 nodes
netw ork
185
T A BLE (A 4) Statistical analysis for the relationship betw een traffic
conditions, transient period, and interval length for a 41 nodes
netw ork
187
Chapter 1
“Introduction”
o
1.0 Introduction
The m odelling o f traffic flow has never been an easy task in view o f the high
com plexity o f social networks U nlike physical networks, there are no
underlying dynam ics in traffic but rather dynam ical consequences that appear
as the result o f the interaction o f various vehicle-driver units, with one
another, with the road and with control networks
External parameters such as weather conditions, different road conditions,
different m otivations for the drivers have a great impact on traffic
perform ance T hese external parameters are changeable from one location to
another, and even for an individual stretch o f the road these conditions vary
over tim e
The initial developm ent o f traffic m odels w as started in the 1950s in order to
study the theoretical description o f traffic flow These m odels are based on a
set o f m athem atical equations or on an analogy to other physical system s
Traffic flow m odels, w hich have been developed recently, may be classified
into tw o classes
•
M odels w hich attempt to explain traffic phenom ena on the basis o f the
behaviour o f the individual elem ents, (single vehicles), are called
M icroscopic M odels In m icroscopic m odels each veh icle in the road
network m ay be described by its position, its actual velocity, its desired
velocity, its origin-destination route, its tendency to overtake other
veh icles and other characteristics o f the driver's behaviour and the vehicle
• In contrast, where traffic flow phenom ena are described through
1
parameters, which characterise the aggregate traffic properties, the
resulting m odels are called M acroscopic m odels
1.1 Literature Review
1.1.1 Microscopic Traffic Flow Models
These m odels are based on a m echanism , w hich describes the process o f one
car follow in g the other M icroscopic m odels are also know n as “ H eadw ay
M odels” because they relate the headw ay betw een tw o cars to the speed
M odels o f this kind allow for the characteristics and behaviour o f individual
cars to be distinct They are w ell suited for sim ulation studies in w hich
stochastic behaviour can be represented by using probabilistic techniques
In the transportation field this approach to m odelling has been applied to
queueing and gap acceptance process (Berilon, 1988, 1991), car follow ing and
lane changing m odels (Gipps, 1981, 1986) and m odels o f travel, route and
departure tim e (M acket, 1990)
M icroscopic m odels consist basically o f tw o main com ponents
a) A n accurate description o f the road network geom etry, including
traffic facilities such as traffic lights, traffic detectors, variable
m essage sign panels, etc
b) A very detailed m odel o f traffic behaviour, w hich reproduces the
dynam ics o f each individual car, distinguishing betw een different
types o f cars, with the possibility o f taking into account
behavioural aspects o f veh icles drivers
2
M icroscopic m odels are close to reality in that they reproduce the traffic
system s w ell T hey open up a w ide range o f traffic scenarios in w hich precise
description o f traffic control and traffic m anagem ent schem es can be explicitly
included B efore w e discuss m icroscopic m odels in detail one m ight ask for
the reasons behind m icroscopic sim ulations9 To answer this question w e need
to know their advantages and the disadvantages
■)
Microscopic advantages
Firstly, as a great m any parameters can be used to describe the individual
features o f driver behaviour it is easy to investigate very subtle changes that
m ay be induced through, for exam ple, changes in the driver education
Secondly, the sim ulation outputs are capable o f describing the m otion o f an
individual car which, in turn, allow us to study and interpret certain aspects o f
the traffic dynam ics
Microscopic disadvantages
A gainst these advantages o f m icroscopic sim ulations one should consider the
effort in m odel specification, data requirements, statistical analysis and
com putational requirem ents (tim e and m em ory)
Brackstone and M cD onald(1995) have m entioned other important factors
w hich m ay lim it the use o f m icroscopic sim ulations in the inter-urban area In
this case, cars have plenty o f opportunity to interact with each other, which
gives the chance for a shock w ave to form and propagate if the flow is high
enough, w hich in turn leads to flow breakdown
3
The first factor is the
lack of appropriate data, which can be dem onstrated as
follow s, the validity o f any m icroscopic sim ulation m odel has to be
m aintained at tw o levels
1
A m acroscopic level,(validation), to ensure the general perform ance o f
the m odel gives a good approxim ations to the observed traffic
n
A m icroscopic level,(calibration), with respect to interaction betw een
vehicles
The m acroscopic data can be obtained by recording the traffic parameters,
flow and average velocity, at regular specific tim e intervals In contrast,
m icroscopic data is not easy to obtain and investigate as m ost o f the
parameters required relate to “ the leader” car and hence cannot be sampled
sufficiently at regular tim e intervals by a m ethod w hich uses only one set point
for observations
The
second factor is the lack of knowledge o f the sensitivity o f the m odels
Since m icroscopic m odels are evaluated according to their ability to reproduce
traffic jam s and the flow density relations, the follow ing question arises
Can w e find a “perfect set o f parameters” to obtain the “best fit” data using
different com binations o f changes in parameters9
The reasons behind our inability to find the “perfect set o f parameters” are
1 a lack o f appropriate data
2 inability to optim ise a system with large number o f degrees o f
freedom per car
For exam ple, it is not possible to define how changes in one
parameter w ill affect the system perform ance both m acroscopically
4
and m icroscopically, follow ing interactions betw een a large
number o f cars over a long period o f time
The stability of the traffic model is our third factor If w e assum e that the other
problem s are solved, there is still this further problem nam ely, the variability
o f data due to different traffic patterns For exam ple, if w e m easure the flow density relation for a certain number o f days, w e find different breakdown
points o f the flow , despite the fact that all the m easurem ents w ere done under
the sam e conditions This is due to different configurations o f the traffic
stream and also to different events occurring and these w ill have alm ost
unpredictable chain reactions, within the traffic stream, that w ill cause the
speed o f successive cars to vary in a differing order in both space and time
l.l.l.l
Car following theory
The process in w hich one car in a stream o f traffic reacts to the behaviour o f
the preceding car is called “car following
theory” and it is based on a cycle o f
stim ulus and response
“Car following models” are used to describe the behaviour o f the dnver-car
system in a stream o f interacting particles,(the cars), and to provide the basic
com ponents o f m icroscopic traffic sim ulation m odels C ar-follow ing m odels
(G azis, 1974, Gabord, 1991) consist o f a differential difference equation,
w hich is used to m odel continuous-tim e system s with inputs and outputs,
w hich produces the acceleration at tim e instant (t+T) from
♦ car speed
♦ The relative distance, speed o f the car ahead
5
At tim e instant t, the general form o f the car follow in g m odels can be
written as
response (t+T) = sensitivity * stimulus (t)
The nature o f the response is acceleration or deceleration o f the follow ing car,
and the stim ulus is the difference in velocity betw een the lead car and the
follow er
The m ost com m on m odel w as introduced by G azis
,
™
aA ,+T)
=“
et al (1961)
Av(0
(*)'(»)
(1 «
where
vn_x( t) - vn(t)
A v(i) =
Ax(t)
= xn_y(jt)~ xn(t)
H ere the reaction tim e
T tries to m odel the delay betw een stim ulus and
reaction The parameters a , m and 1 m ust be evaluated from observations and
Av(t), Ax(t) are the changes in both the speed and the space
Sim ple “ car-following models” resem bles a feedback control process in w hich
oscillations m ay occur This leads to various kinds o f instabilities in the traffic
flow , w hich in turn can lead to collisions
There are tw o types o f instability
1
L ocal instability, w hich can be observed in situations in which
disturbance, (e g a change in a distance-headw ay resulting from the
change in speed o f the leading car), does not die out but rather
increases with tim e
6
ii
A sym ptotic instability, defined as the situation in w hich a disturbance
grow s in m agnitude as it propagates from car to car
In contrast, stability o f the traffic flow m odel m eans that changes in the
velocity by the lead car o f a traffic stream w ill not be am plified by successive
cars in the stream (until a collision occurs) A lso there are tw o types o f
stability
1
L ocal stability, w hich considers the response o f a car to the
change in m otion o f the car im m ediately ahead
u
A sym ptotic stability, w hich deals with the propagation o f a
fluctuation through a platoon o f cars
R ecently (G ipps, 1981, 1986) proposed a new car-follow ing m odel, w hich w as
designed to possess the follow ing features
i
The parameters in the m odel, a , m, and 1 should correspond to
obvious characteristics o f drivers and cars ,
n
The m odel should be w ell behaved when the interval betw een
successive recalculations o f speed and position is the sam e as
the reaction tim e
This m odel is based on the assum ption that the driver o f the follow ing car
selects lim its to his desired braking and acceleration rates For acceleration
_1
v n<t+T)
Z v „ (0 + 2 5 a n T
1
-
_v„(0 * f 0025 + -v„(0^2
^
K
V
K i
}
( 12 )
where
vn(t + T )
is the m axim um speed to w hich car n can accelerate during the
tim e m terval(t, t+T)
Vn
is the desired speed for car
n and an is the m axim um
acceleration for car n
For braking
v,(r+T) <b„ r+(i>„;rJ-J,(2u„_1(i)-v, -*„«)] -v,«)r-vri!w /s))’
( 13 )
where
vn(t + T)
is the m axim um speed for a car n with respect to car n -1
bn
is the m ost severe braking that the driver o f car n w ishes to
undertake, (bn < 0)
sn
is the effective size o f car n,(i e , the physical length plus a
margin into w hich the follow ing car is not w illing to intrude),
even at rest
B
is the estim ate o f
bn_{ used by the driver o f car n
If it is assum ed that the driver travels as fast as safety and the lim itation o f the
car permit, the m ean speed is given by com bining these equations, as follow s
vn(t + T)= m in(acceleration, braking)
(14)
to
The m odel has been used to sim ulate vehicular traffic in m ulti-lane arterial
roads with special attention devoted to the structure o f the lane changing
decisions
M ore recently (Bando et al, 1994, 1995) the foliow m g equation has been used
for calculating the acceleration
a (t ) = y[v ( gap(t )) - v(i)]
(1 5)
W here
V is the desired velocity function, w hich has approxim ately a linear
relationship with gap and also depends on som e other variables such as road
conditions
1.1.1.2
Psychological-Physiological Spacing models
Car follow ing equations assum e that the driver o f the follow ing car reacts, to
arbitrary sm all changes in the relative speed, even at a very large differences
in distances to the front car Therefore, car follow ing equations assum e that
there is no response as soon as speed differences disappear, w hich is not very
realistic
A significant new approach w as developed by W iedm ann(1974) based on
know ledge about human perception and reaction behaviour and w hich used
different perceptual thresholds O nly w hen these thresholds are reached w ill
the driver o f the follow ing car be able to perceive the change in the apparent
size o f the leading car and, subsequently, be able to react to the changes o f
acceleration or deceleration
Such thresholds are presented as parabolas in the relative distance vs relative
speed relation in Fig (1 1) It can also be seen from this picture how car
follow ing proceeds A vehicle with speed v (n + l), which is larger than the
speed v(n) o f the preceding veh icle w ill catch up with constant relative speed
Av U pon reaching the threshold, the driver reacts by reducing his speed One
such exam ple o f relative m otion w ith constant deceleration appears as a
parabola The m inim um o f the parabola lies on the A x-axis The driver tries to
decelerate so as to reach a point at w hich Av = 0 H e is not able to do this
accurately because, firstly, he is not able to perceive sm all speed differences
and, secondly, he is not able to control his speed sufficiently w ell
The result is that the spacing w ill again increase W hen the driver initially
reaches the opposite threshold he accelerates and tries again to achieve the
desired spacing (indicating in Fig (1 1) by the upper part o f the loop)
If one assum es that the relationship o f the perceptual thresholds for spacing
are the sam e for both positive and negative changes in relative speed, then the
resulting spacing behaviour resem bles a sym m etrical pendulum about its
equilibrium point
Ax
1 perceptual threshold
Fig(l 1)
Source
Perceptual thresholds in car-following behaviour,
Wiedemann
(1974)
10
1.1.2
Macroscopic Models
M acroscopic traffic flow m odels (Lightill and W hitham , 1955, G azis, 1974)
treat the traffic system as a continuous fluid T hey are concerned only w ith the
behaviour o f groups o f cars, ignoring the behaviour o f the individual
transportation units
B efore w e classify m acroscopic m odels, w e m ention som e o f their advantages
and disadvantages
M acroscopic advantages
i
They are useful in studying traffic behaviour under heavy traffic
conditions
u
D ue to the use o f the aggregate traffic variables, flow , density, and
m ean speed, the com putational requirem ents are m uch less than with
m icroscopic variables, allow ing real tim e sim ulation o f traffic flow in
large networks
M acroscopic w eaknesses
i
D o not incorporate driver, car, and roadway parameters in an explicit
w ay
u
M acroscopic m odels are not able to provide inform ation about fuel
consum ption or route choice for individual cars and som etim es show
poor results in the event o f m icroscopic phenom ena occurring e g like
queueing at traffic light, on-ramps, and others
In the next section w e classify m acroscopic m odels, giving a brief description
o f each
li
1.1.2.1
Continuum Models
The first contribution to the continuum m odels are due to Lightill and
W hitham (1955), w ho proposed that certain traffic phenom ena o f dense
highw ay traffic can be described in terms o f continuum variables, traffic flow ,
density, and m ean speed
The assum ption o f the theory is that
At any point o f the road the flow is a function o f the concentration o f cars
This assum ption im plies that sim ple changes in the flow rate are propagated
backwards through the traffic stream along a kinem atic w ave w hose velocity
relative to the road is the slope o f the flow -density curve
The flow q and the concentration k have no significance except as m eans The
purpose o f the theory is to determ ine how these mean values vary in space and
tim e This is done by considering the speed w ith w hich changes in q and k are
propagated along the roadway
A fixed observer sees a flow
N
q = — = uk where N is the number o f vehicles
T
passing him in tim e T, and u is the m ean speed at w hich the N vehicles pass
him A ssum ing that the observer m oves upstream with uniform speed c, then
he w ill pass additional veh icles say, c k w hich w ill be added to q so that
q + ck = —NT
h
and, if he m oves down stream this becom s
12
N
q - ck = —T
(1 6 )
N is the difference betw een the number o f veh icles passing the observer and
those, which he passes If he m oves at the mean speed o f the stream, then N
0 and
=
c=u
Consider now tw o observers m oving at a uniform speed c, the second starting
at tim e T and rem aining behind the first Suppose that the flow and
concentration are changing with tim e, but that the observers jointly adjust their
speed c so that the number o f veh icles w hich pass them m inus the number
w hich they pass, is, on the average, the sam e for each observer during a time
interval T This is illustrated in Fig ( 1 2 ) B y (1 6) the result o f their
observation w ould be
N
q\ - ck\ = — = q2 - ck2
(1 7)
if they w ere m oving dow nstream in the positive direction o f flow
Solving for c, w e find that
q2 - ql
c = -------- —
k2 - kl
(1 8)
If the changes in flow and concentration are sm all, then
Aq
c
-
i
'
(19)
Thus, when the difference in flow and concentration are sm all, they propagate
Aq
at a speed given by the tangent —— to the flow -concentration diagram
Za/C
13
AT
F ig (l 2) V ehicles passed by tw o m oving observers,
Source D Gazis
1.1.2.1.1
(1974)
Simple continuum models
Continuum traffic flow m odels are based on a fluid flow analogy w hich
regards traffic flow as a particular fluid process, with states characterised by
aggregate variables such as density (in V eh/km ), flow (in V eh/h) and m ean
speed (in km /h)
In fluid flow analogy, the traffic stream is treated as a one-dim ensional
com pressible fluid This leads us to describe the dynam ic evolution o f
m acroscopic traffic parameters by m eans o f a
1
Conservation or continuity equation
n
O ne-to-one relationship betw een speed and density or betw een flow
and density, w hich is know n as the fundam ental diagram o f traffic
engineering
14
The sim ple continuum m odel consists o f the continuity equation and the
equation o f state:
flow = density * mean speed.
If these equations (1.1 0.a and b) are solved together, then w e can obtain speed,
flow , and density at any tim e and point o f the roadway:
q = uk
dq
*
(l.lO .a )
dk
+ ¥
=
0
(U O b)
B y know ing these traffic variables, w e know the state o f the traffic stream and
can derive m easures o f effectiveness, such as delay stops, total travel tim e and
others that help engineers to evaluate the perform ance o f the traffic system s.
Equation of continuity
Equation (l.lO .b ) expresses the law o f conservation o f a traffic stream (cars)
and is known as the conservation or continuity equation. It has the sam e form
as in fluid flow .
If entries/exits exist within the stretch o f the roadway, then the equation takes
the form
dq
dk
§ + ¥ - *(*,)
W here
(l id
k(x ,t) and q(.x, t) are the traffic density and flow resp ectively, at the
space-tim e point (x,t). The generation term g(x,t) represents the number o f cars
entering or leaving the traffic flow in a roadway section with entries/exits. The
equation o f continuity relates two fundam ental variables, density and flow rate
with tw o independent ones (space and tim e).
15
The solution o f equation (1 10 b) is im possible without an additional equation
The first possibility considers the m om entum equation described later The
second option uses the fundam ental diagram
Fundamental diagram
The m acroscopic parameters o f the traffic flow are related by equation (1 10 a)
where the equilibrium speed
u(x,t) = u(k) must be provided by a theoretical or
em pirical equation o f state, that can take the form
ue = uf
1-
(1 12)
Vk jam
W here
)
Ue is the equilibrium speed, Uf is free flow speed
Equation (1 10 b) sim ply states that flow ,
q, is a function o f density, k, l e q -
f (k) U sing this relation one can also obtain the relation, w hich relates the
m ean speed and the density i e
u = f (k) This, how ever, is only valid at
equilibrium Equilibrium can hardly be observed in practice, so that obtaining
a satisfactory speed-density relationship is a task that is hard to achieve and is
always assum ed theoretically
The flow -density relation presented in Fig ( 1 3 ) reveals tw o extrem e points
K=0
K = ^ max
^
U= 0
=>
q =0
=>
q= 0
The cloud o f points betw een these tw o extrem es is based on measurem ents
perform ed over specific tim e intervals This cloud o f points represents an area
o f m axim um flow , w hich is considered to be an important feature o f the road
section under concern
16
The fundam ental diagram can be divided into tw o regions
I
The free flow area, in w hich all cars are m oving with their desired
speed
II
T he dense traffic area, in w hich all cars are alm ost stopped
A ccording to H all (1993) the fundam ental diagram can be characterised by the
follow ing
a) Free flow traffic area, w hich can be represented by sections o f linear
approxim ations
b) the dense traffic regim e
c) M easurem ent points found outside these tw o areas o f traffic conditions
represent transition situations, such as traffic leaving the head o f a queue
This traffic flow cannot be larger than the capacity o f the congested area,
but it can be faster than the congested flow
The solution o f the sim ple continuum m odel leads to the form ation o f shock
w aves as illustrated m Fig (1 4 ) The shock w ave is show n as a heavy line on
the space-tim e diagram, ahead o f it the flow is denser and the w aves are drawn
parallel to the tangent to the flow -density curve at A, which represents a
situation where traffic flow s at near capacity im plying that speed is w ell below
the free-flow speed
Behind it the concentration is less and the w aves travel faster, they are drawn
parallel to the tangent to the curve at B , w hich represents an uncongested
condition where traffic flow s at a higher speed because o f the low er density
Real Traffic
oc cu pa nc y [%]
Fig. (1.3): Fundamental diagram obtained for real data;
Source: Hall, L. et al
(1986)
Lighthill and W hitham (1955) have used the flow -density curve to predict
conditions near a shock w ave.
Since the sim ple continuum m odels do not consider acceleration and inertia
effects, they do not faithfully describe the non-equilibrium traffic flow
dynam ics. These are taken into account in the higher-order continuum m odels.
T hese m odels add a m om entum equation that accounts for the acceleration and
inertia characteristics o f the traffic mass. In this manner, shock w aves are
sm oothed out and the equilibrium assum ption is rem oved.
18
X
V
ik
Fig
the
(1 4)
local
The use of the
conditions
flow-density
near
a
shock
curve to predict
wave,
source.
G a z i s (1974)
Disadvantages of simple continuum models
1- K inem atic m odels contain stationary speed-density relations (l e the mean
speed should adjust instantaneously to traffic density) M ore realistically,
it is adapted after a certain tim e delay and to reflect traffic conditions
downstream .
2- K inem atic w ave theory show s shock w ave form ation by steeping speed
jum ps ( îe increases linearly) to infinite sharp jum ps A m acroscopic
theory is based on values, w hich are average values from an aggregate o f
veh icles A verages are taken either over tem poral or spatial extended
areas Infinite jum ps, therefore, are in contradiction to the basics o f
m acroscopic description
3- Unstable traffic flow is characterised, under appropriate conditions, by
regular stop-start w aves with am plitude-dependent oscillation tim e
19
4- The dynam ics o f traffic flow result in the hysteresis phenom ena This
consists o f generally retarded behaviour o f veh icle platoons after em erging
from a disturbance com pared to the behaviour o f the sam e vehicles
approaching the disturbance Sim ple continuum m odels cannot describe
such phenom ena
5- B esides hysteresis, the crucial instability effect is bifurcation behaviour
(1 e
, traffic flow becom es unstable beyond a certain critical traffic
density) A bove the critical density, the traffic flow becom es rapidly more
congested without any obvious reason
i.1.2.1.2
Models with Momentum (inertia)
The extension o f the sim ple continuum m odels, in order to explain the
dynam ic effects in the preceding section, w as first pointed out by W hitham
(1974) and Payne (1979)
The actual speed
u(x, t) o f a sm all ensem ble o f veh icles is obtained from the
equilibrium speed-density relation, using a delay tim e x, and from an
anticipated location
x + Ax
u(x,t + z) = Ue(k(x + Ax,t))
M uller and Eerden (1987) have studied this recursive equation in detail
20
(1 13)
Expanding, using Taylor series with respect to x and
Ax, assum ing that both
quantities can be kept sm all, yields the substantial acceleration o f a platoon o f
vehicles
(1 14 a)
In a fixed co-ordinate system this transforms into
du
dt
du(x(t),t))
-------dt
dx
= ur —dt + u, - uru + u.
(1 14 b)
And from equations (1 14 a) and (1 14 b) w e get
du
du
1 r. ,
-|
, 1
dk
(1 15)
where
x is the relaxation tim e, the tim e in w hich a platoon o f cars reacts to the speed
alternations
Ueis the fundam ental diagram
ca2 is constant, independent o f density k, and is called the anticipation term
The L H S in equation (1 15) is decom posed into a
convention term, the
second term, indicating the acceleration due to spatial alternation o f the
stream lines and a local acceleration w hich is tim e dependent
The first term on the R H S is the equilibrium term, that is the effect o f the
drivers adjusting their speed to the fundam ental diagram and the second term
represents the anticipation on the dow nstream density (1 e the effect o f drivers
reacting to the dow nstream traffic)
21
dq dk
ox + at— = 0
T he continuity equation
—
from ( 1 1 0 b) and
the
m om entum equation
du
du
1 r
"I
, 1
T
dk
U
^
(115)
form a set o f first order, partial differential equations w hich describes the
dynam ic effects associated with the traffic flow , such as stop-start w ave
form ation, bifurcation into unstable flow and transience, and behaviour at
bottlenecks
To investigate these equations, the road section is discretised in tim e and
space N um erical m ethods used in com putational fluid dynam ics can be
applied to solve these equations
22
1.1.2.2 Two fluid theory
The tw o fluid theory o f tow n traffic w as proposed by Herrman and Prigogine
(1979), Herrman and Arkekani (1984) Cars m a traffic stream can be view ed
as two fluids, the first consisting o f m oving cars and the second o f cars stopped
due to congestion, traffic lights, stop signs, and obstructions resulting from
constructions, accident and reasons other than parked cars, w hich are ignored
since they are not com ponents o f the traffic
The tw o fluid theory provides a m acroscopic m easure o f quality o f traffic
service in a street netw ork which does not depend on the density
The tw o fluid m odel is based on the assum ptions
i
The average m oving speed in a street network is proportional to the
fraction o f m oving cars
n
The fraction o f stopped tim e o f a test car, circulating in a network, is
equal to the average fraction o f the cars stopped during the sam e period
The first assum ption represents the relationship betw een the average speed o f
the m oving cars and the fraction o f m oving cars w hich can be form ulated as
follow s
V = ym(X~fs)nH
(116)
where
Vm is the average m axim um m oving speed
n
is an indication o f the quality o f traffic service in the
network
23
V
is the average speed, w hich can be written as
fs
is the fraction o f stopped cars
V =Vrf r
The second assum ption m eans that the netw ork conditions can be represented
by a single car appropriately sam pling the network
Equation (1 1 6 ) can be written as
=
(117)
f, = J
(1 18)
T
Com bining the Equations (1 17) and (1 18) w e get
T
=
T
-
T
n +i
T n+1
(1 19)
w hich represents the tw o fluid form ulation
Equation can be written as
Ts = T - Tr
(1 20)
where
T r
=
T m TTT
T 0 +1
( 121 )
w hich represents the relation betw een the trip tim e per unit distance, T, and the
running tim e per unit distance, T r
Taking the natural logarithm o f both sides in Equation (1 21) yields
In T r =
n + 1 In T m
— Tr
+
—
n 7+ —1 In T (122)
w hich provides a linear expression for the use o f least squares analysis
24
V.
1.1.3 Cellular Automata Models
Cellular autom ata (CA) are discrete dynam ical system s, w hich are defined on a
one-dim ensional lattice (or m ulti-dim ension grid) o f k identical cells
The global behaviour o f the system is determined by the evolution o f the states
o f all cells as a result o f m ultiple interactions
CA, w hich w ere developed to m odel sim ple m athem atical system s, are
increasingly used in the sim ulation o f com plex system s In this approach, the
traffic system is regarded as an interacting particle system , w hich show s a
transition betw een tw o phases
L ow -density phase m w hich all cars m ove sm oothly w ith m axim um speed, and
H igh-density phase in w hich cars are alm ost stopped
The C A evolves in discrete tim e steps The state o f each site at the next tim e
step is determ ined from the state o f the site itself and its nearest sites at the
current tim e step
The CA m odels sim ulate the m ovem ent o f each individual car according to a
number o f sim ple rules, essentially m oving each forward by an integer number
o f increm ents at an integer speed
25
1.1.3.1 One-dimensional Cellular Automata Models
V ery recently, N agel and Shreckenberg (1992) have introduced a stochastic
Cellular Autom ata m odel to sim ulate freew ay traffic T hey found that there are
tw o regions in the traffic flow
1
Free-phase, w hich is dom inant at low density
n
Congested traffic in w hich traffic jam s appear at a high density
The m odel is defined over a lattice o f k-identical sites, each o f length 7 5 m
representing the length o f the car plus distance betw een cars in jam, and each
site can be either em pty or occupied by a single particle
Each particle can have an integer velocity betw een 0 and vmax, where v max
^
in general G iven the configuration o f the particles at tim e step t the
configuration at tim e step t+1 is com puted by applying the follow ing rules ,
w hich are done in parallel for all particles
♦ Calculate the headw ay distance(= gap)
♦ D eceleration
If v> gap (the particle is running too fast), then slow dow n to
v = gap
(rule 1)
♦ A cceleration
If v < gap and v< vmax then accelerate by one
v = v+ 1
(rule 2)
♦ Random isation
If after these steps, the velocity v is greater than zero, then w ith probability
p reduce v by one v = v-1
(rule 3)
26
♦ Car advance
Each particle is advanced v sites ahead
(rule 4)
N agel and Schreckenberg (1992), show ed that the start-stop w aves, (traffic
jam s), appear in the congested traffic region A lso, they obtained good
agreem ent w ith realistic fundam ental diagrams
B asically, they tried to m odel tw o properties o f the road traffic
•
Cars travel at som e desired speed, unless they are forced to slow dow n in
order to avoid collision s with other vehicles
•
Interactions are short ranged and can be approxim ated by being restricted to
nearest neighbours
Im perfections in the w ay drivers react is m odelled as noise
The continuous lim it o f the CA m odel has been investigated by Krauß
(1996), this is obtained by letting vmax —>
oo a n d
Pmax
et al
0
The generalised version o f the C A m odel allow s for continuous values o f the
velocities and spatial co-ordinates In the N -S m odel noise, is introduced by
random ly decelerating car velocity by one with probability
pbreak This,
how ever, is generalized to an equipartition betw een zero and the m axim um
acceleration in the generalized version
The C A rules for the interm ediate m odel are defined as follow s
27
V*.
= n u n (v(i)
v(f + 1 ) = m ax(0,
+ a(t), vmax, gap(t)),
vdes - o randQ),
(1 23)
x(t + 1) = x(t) + v (i + 1)
W here gap(t) is the free headw ay distance, a max is the m axim um acceleration ,
rand() is a random number in the interval (0, 1) and a is the m axim um
deceleration due to noise
N um erical sim ulations show ed that the transition, leading from the free flow
regim e to the congested flow regim e, bears strong sim ilarities to a first order
phase transition in equilibrium therm odynam ics An additional advantage o f the
continuous m odel is that it is m uch easier to calibrate with real data, despite the
slight decrease in the num erical efficien cy
1.1.3.2 Two-dimension Cellular Automata Models
Tw o-dim ensional problem s, (city traffic), are m ore com plicated com pared to
one-dim ensional ones and m turn they are less realistic
Biham and M iddleton (1992), have introduced a sim ple determ inistic tw odim ensional m odel Three variants o f the m odel w ere investigated, the first tw o
variants o f the m odel use three-state Cellular Autom ata defined on a square
lattice Each site contains either an arrow pointing upwards, an arrow pointing
to the right, or is em pty
In the first variant (M odel I) the traffic dynam ic is controlled by traffic lights,
such that the right-arrows m ove only in even tim e steps and the up arrows
m ove in odd tim e steps
28
On even tim e steps, each arrow m oves one step to the right unless the site on its
right-hand side is occupied by another arrow If it is blocked by another arrow
it does not m ove, even if during the sam e tim e step the blocking arrow m oves
out o f that site Sim ilar rules apply to the up arrows, w hich m ove upwards
The m odel is defined on a square lattice o f N x N sites with periodic boundary
conditions
In m odel II, the traffic light is rem oved and all arrows m ove in all tim e steps
(unless they are stopped) If both an up and a right arrow try to m ove to the
sam e site, one o f them w ill be chosen randomly, w ith equal probabilities
M odel II is considered to be the non-determ inistic variant o f the m odel
E xtensive num erical sim ulation show s a sharp jam m ing transition w hich
separates the low -density dynam ical phase in w hich all cars m ove at m axim al
speed and the high-density jam m ed phase in w hich they are all stopped
The third variant o f the m odel is a four-state C A defined on a square grid Each
site contains either an arrow pointing upwards, an arrow pointing to the right,
an arrow pointing left, or is em pty In this four-state m odel all arrows try to
m ove at every tim e step If both an up arrow and a right arrow try to m ove to an
em pty site at the sam e tim e step they both m ove in and overlap On the other
hand no arrow can m ove into a site w hich is already occupied
The sim ulations o f B iham and M iddleton(1992) show that the m odel exhibits a
continuous transition, w hich is qualitatively sim ilar to the one-dim ensional
case
Nagatam (1993)
has extended the C A m odel proposed by B iham and
M iddleton (1992) (the BM L m odel), to investigate the effect o f the tw o-level
29
crossing on the traffic jam in the original m odel The m odel, as in Fig (1 5), is
defined on a disordered square lattice w ith tw o com ponents
•
The first com ponent is the site o f three states representing the onelev el crossing
•
The second com ponent is the site o f four states representing the tw olev el crossing
Nagatani (1993) show ed that the dynam ical jam m ing transition does not occur
w hen the fraction c o f the tw o-level crossings becom es larger than the
percolation threshold, w hich gives rise to the critical behaviour H ow ever, the
dynam ical jam m ing transition occurs at higher density o f cars with increasing
fraction c o f the tw o-level crossing below the percolation threshold
Torok and K ertesz (1996) have studied the sequential update version o f the
B M L m odel called the G reen W ave M odel (GW M )
The mam difference betw een the tw o m odels is that in the B M L m odel single
cars m ove w hile in the Green W ave M odel, w hole con voys (line o f sam e type
o f cars with no em pty space betw een them) travel together
In the BM L m odel, tw o cars cannot m ove together because if they becom e
neighbours the second car is not able to m ove until the first is m oved aw ay In
the G W M the tw o neighbouring cars alw ays stay together if the first m oves the
second w ill m ove in the sam e tim e step and there is no effect that could
separate them The G W M show s tw o types o f transition the free flow -jam
transition and a structural transition in the jam m ed region
30
î
î
t
| Two-level
crossing
î
Car moving up
-h >
*
—>
î
Fig
(1 5)
Car moving
to the right
>
Schematic illustration of the cellular automata
model of traffic flow with two-level crossings,
source
T
N a g a t a n i (1993)
1.1.4
Thesis Outline
i
The rem aining chapters o f this thesis are organized as follow s In Chapter 2 w e
present a three state determ inistic cellular automata m odel for urban traffic,
then the m odel is adapted to sim ulate road traffic In Chapter 3 w e m ove
towards netw ork traffic flow and w e investigate using the sim ple m odel,
developed in Sec (2 1), the netw ork perform ance under different parameters
that govern the traffic flow , whereas in Chapter 4 w e investigate the traffic
behaviour along the network with loss to flow and under short and long-term
traffic events
31
In Chapter 5 a set o f lane-changing rules for cellular automata is presented and
a stochastic cellular autom ata m odel for traffic flow in inter-urban areas is
presented in Chapter 6 Finally, Chapter 7 concludes this thesis by highlighting
the main contributions and discussing directions for future research
32
Chapter 2
“Simple Cellular Automata Model for Urban Traffic Flow”
33
2.0 Introduction
Num erical sim ulations, hydrodynam ic m odels, and queuing theory are a few
o f the basic theoretical tools, used to describe traffic flow on freew ay networks
(Leutzbach, 1988, Kuhne, 1984) Traffic control has been intensively studied
also from the point o f view o f Operations Research (Improta, 1987) A s one
m ight expect, the theory o f traffic flow is related to telecom m unication and
com puter netw ork theory Equally, m any basic concepts in traffic flow theory
have their origins in physics, (Lighthill and W hitham , 1955, Herrmann
et al,
1959)
Cellular automata, w hich w ere developed to m odel sim ple m athem atical
system s, are increasingly used in the sim ulation o f com plex physical system s
(W olfram , 1986), and have helped dem onstrate phenom ena w hich are o f
practical interest
C ongestion is a sim ple phenom enon cars can not m ove w ithout sufficient
space betw een them (Nagatam , 1993a,b, 1994a,b , N agel and Herrmann,
1993) In order to sim ulate freew ay traffic flow , N agel and Schreckenberg
(1992) extended the ID asym m etric sim ple exclusion m odel by taking into
account car velo city ,(S ec(l 1 3 1)) They show ed that a transition from
laminar traffic flow to start-stop w aves occurs with increasing car density, as
observed in real freew ay traffic
Control o f traffic flow in cities is a m ore com plex endeavour as it involves
m any degrees o f freedom such as local densities and speeds Cremer and
Ludw ig (1986) have developed a fast sim ulation m odel for m odelling traffic
34
r
flow through urban netw orks T hey sim ulate the progression o f cars along a
street using bit m anipulation com puter programs W hen compared with
standard m icroscopic m odels, the com putational tim e needed is less by a ratio
o f 1 / 150 The m odels sim ulate, accurately, m acroscopic phenom ena o f traffic
flow , w hile at the sam e tim e reproducing the main m echanism s o f m icroscopic
m odels
In this chapter w e present a Cellular A utom aton (CA) to sim ulate traffic flow
in urban networks, w hich extends this approach
2.1 The Cellular Automata Model
Our m odel is a 3-state cellular autom aton defined on a one-dim ensional lattice
L , o f k identical sites with at m ost one particle per site, where each site can
take one o f the states 0, 1 and 2 State 0 is the em pty site, state 1 represents a
ik
s.
site occupied by a stopped car and state 2 corresponds to a site with a m oving
car The CA evolves in discrete tim e steps The state o f each site at the next
tim e step is determined from the state o f the site itself and those o f its two
nearest neighbour sites, at the current tim e A pplying a parallel update rule for
all sites in the m odel, the transition rules m ay be described as follow s
If a site is
1- O ccupied by a m ovm g car and the neighbouring site in the direction of
m ovem ent is em pty, then the car is advanced one site
(Rule 1)
2- O ccupied by a m ovm g car and the next site m front is occupied by a
stopped or m ovm g car, the car can not advance
35
(R ule 2)
3- O ccupied by a stopped car and the nearest site in front is em pty , then it
w ill advance by one site
(Rule 3)
4- Em pty, and a car w ishes to enter it from an adjacent cell, then the current
site is occupied by a m oving car if the next site ahead is either em pty or
occupied by a m oving car, and by a stopped car otherw ise
(Rule 4)
2.1.1 Model parameters
The fundam ental diagram s in traffic flow m odels are (1) flow v s density and
(n) average velocity vs density The follow ing parameters play a central role in
this analysis
The mean velocity o f cars in a unit tim e interval x is defined to be the number
o f m oving cars divided by the total numbers o f cars
Vx =
(2 1 )
n
where
n_^ = number o f m oving cars in x
and
n = total number o f cars
The average m ean velocity over an interval o f tim e T is
(2 2)
where
vt = system velocity at t tim e step
T = Length o f the tim e interval
36
The density of cars in the system, p, and the flow, q, are defined by
p = n /k
(2 3)
q = <v> p
(2 4)
where
n = number o f cars
and
k = number o f sites in the m odel
2.1.2 Model Dynamics
The tim e-evolution o f the autom aton follow s the sim ple rules given earlier for
applying periodic boundary conditions, (as illustrated on the space-tim e
diagram, Fig (2 1))
W hen a periodic
51
boundary condition is im posed on the m odel, cars that exit
the system (on the right hand side) are fed back into the system on the lefthand side, as our traffic m oves from left to right
Each site on the lattice can take one o f the states 0 (em pty), 1 (stopped car) or
2 (m oving car) If one follow s the m ovem ents o f individual cars in the spacetim e diagram F igs (2 1 a and b), cars m oving freely are characterised by
diagonal lines with the sym bol 2, w hile stationary cars are characterised by
vertical lines with the sym bol 1
Lines are configurations at consecutive tim e steps A t low density, these lines
show laminar traffic, with the system m oving w ith m axim um velocity subject
to one gap ahead indicating
“free phase” traffic
In contrast, at high density, w e find congestion clusters, each congestion
represent a traffic jam , indicating a
“jammed phase” traffic
37
Space
000000000 1020000020000020002020202 102202020010202
2 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 0 0 0 0 0 2 0 00 2 0 2 0 2 1 0 2 2 0 2 0 2 0 2 0 0 2 0 20
0200000000020200000200000200020210220202020200202
2 0 2 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 2 1 0 2 2 0 2 0 2 0 2 0 2 0 2 0 0 20
020200000000020200000200000200 1022020202020202002
2 0 2 0 2 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 0 0 0 0 0 20 0 2 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0
0202020000000002020000020000020202020202020202020
0020202000000000202000002000002020202020202020202
Time
2002020200000000020200000200000202020202020202020
0 2 0 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 20 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2
2020020202000000000202000002000002020202020202020
0202002020200000000020200000200000202020202020202
2 0 2 0 2 0 0 2 0 2 02 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 2 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 20
0202020020202000000000202000002000002020202020202
2020202002020200000000020200000200000202020202020
(a)
Space
►
111102010111102111020211111111102111110101111111
111020101111021110202111111111021111101011111111
110201011110211102021111111110211111010111111110
Tim e
▼
102010111102111020211111111102111110101111111102
020101111021110202111111111021111101011111111020
201011110211102021111111110211111010111111110201
010111102111020211111111102111110101111111102011
101111021110202111111111021111101011111111020111
011110211102021111111110211111010111111110201111
111102111020211111111102111110101111111102011110
111021110202111111111021111101011111111020111102
110211102021111111110211111010111111110201111021
102111020211111111102111110101111111102011110211
021110202111111111021111101011111111020111102101
211102021111111110211111010111111110201111021010
111020211111111102111110101111111102011110210102
11020211111111102111110101111111 1020111102101020
102021111111110211111010111111110201111021010201
020211111111102111110101111111102011110210102010
202111111111021111101011111111020111102101020102
021111111110211111010111111110201111021010201020
211111111102111110101111111102011110210102010201
(b)
F i g . (2.1):
defined
on
Space-time
a
100
diagrams
site
of
Lattice
the
Simple
with
closed
CA
boundary
conditions, where in (a) p = 0.3 and in (b) p = 0.8
38
Model
N ote that when traffic is stopped, the traffic jam w ave m oves backwards
relative to the traffic flow
2.2 Simulation and Results.
2.2.1 The model with periodic boundary conditions
W e have im plem ented com puter sim ulations in Borland C ++ for a
determ inistic C A m odel, starting with random initial configuration o f cars with
density p and velocity = 0 D ifferent lattice sizes (k = 60, 80, 100, 120, 140)
,k
w ere used In each case, the C A m odel ran for 1000 tim e steps and this process
w as repeated 300 tim es to generate the average case statistics
M ost road-traffic observers have concentrated on m easuring flow ,
q, and
average velocity, v, as being the quantities o f greatest practical interest The
density p can be obtained from these m easurem ents
The relation betw een the three m acroscopic variables, flow , average velocity,
and density, is w id ely know n as
the, fundamental diagram In Sec ( 1 1 2 1 1),
w e have studied in detail the flow -density relation, w hich represents the first
com ponent o f the fundam ental diagram Another important relation is the
velocity-density relation This is regarded as the second com ponent o f the
fundam ental diagram, w hich show s how the system density affects mean
speed at different levels o f concentrations The fundam ental diagram s are also
know n as “ Empirical relations’’
U sing the fundam ental diagram s, our sim ulation results reveal that the system
reached its critical state when the density p c = 0 5
39
■>
The m axim um flow recorded w as
q = 0 5, after a transient period t0 = f ,
w hich does not depend on the system size.
Fig (2 2 a) show s the fundam ental diagram (flow vs density) w hich is
sym m etrical about the critical density
pc = 05, w hich marks the boundary
betw een free flow ing and congested traffic
In Fig (2 2 b) w e plot the average velocity against density, w hich indicates that
the system m oves w ith a constant average velocity <v> = 1 until p = 0 5 and
then decreases as w e increase the density
The space-tim e diagram s in F igs (2 3 a) and (2 3 b) are very useful in
visualizing traffic and traffic jam s In these figures,(2 3 a, b), each black p ixel
represents a car Space direction is horizontal, tim e is pointing
downwards,
i
cars are m oving from left to right and from top to bottom .
In free flow ing traffic, Fig (2 3 a), the system reaches a steady state
characterised by each car advancm g one site to
the
contrast, in congested traffic, Fig (2 3 b), the traffic
left at each tim e step
40
right at each tim e step In
jam m oves one site to the
Fundamental Diagram
05 0 45 04 0 35 _ 0 33
o 0 25 “■
02 0 15 010 05 ------------- 1
------------- f
------------- 1
----------0 ----------------- F
C
02
04
06
08
1
Densi ty
Fig
80
(2 2 a) flow-density relation for a system of size
Data are averaged over long time periods
time steps) using closed boundary conditions
(1000
It is
easy to see that the phase transition between the two
regimes occurs at the critical density 0 5
densi t y
Fig
(2 2 b) velocity-density relation,
boundary conditions,
using closed
again the System moves with constant
velocity until the density of 0 5, then the transition
between the two phases take place
41
42
2.2.2 The Model with Open Boundary Conditions
The rules o f the Sim ple M od el w ith open boundary conditions are identical to
the periodic case except that w e work with average densities and need to
consider the input to the system as w ell as the duration o f each sim ulation run
Different rates o f injecting cars into the system w ere investigated These
correspond to peak-tim e rates, such as those that occur at, say, 8 00 hrs or
17 00 hrs and off-peak tim e, corresponding to 11 00 hrs or 20 OOhrs
In our m odel, exit from the road (system ) is controlled by a traffic light,
operating under a green and red light regim e Cars can leave the system only
during the green hght phase, by deleting the last site o f the road, and form a
queue during a red hght phase In this m odel the user fixes the duration o f the
green and red phases
D iffering lattice sizes (60, 80) w ere selected, data \yas collected after
transience k / 2, and averaged over 200 sim ulation runs Each sim ulation run
consisted o f 600 tim e steps using various regim es for the green and red phases
The fundam ental diagram s for the tw o different array sizes, k =60, 80, are
presented in F igs (2 4) and (2 5) Comparing these diagram s with those
obtained for closed system (closed boundary conditions), w e make the
follow ing observations
i
Im posing open boundary conditions has led to low er values for the
m axim um flow and its density, <?max has decreased from 0 5 (at pc =
0 5) to 0 3525 (at
pc = 0 424)
43
ii.
The sim ulation reveals that the m axim um flow and its density depend
on the system size. U sing a system o f size 60 sites has led to a
m axim um flow o f <7 max ~ 0.36 at density
low er value for
pqmn ~ 0.48, whereas a
qmiX = 0.3525 w as obtained at low er density o f 0.42
in the case o f larger size netw ork(k=80).
0.345
0.46
0.48
0.5
0.52
0.54
0.56
0.58
density
(a)
(b)
F i g .(2.4):Fundamental diagrams for the Simple Model with open
boundary conditions;
relation,
flow-density relation, velocity-density
for array size 60 cells. Data were averaged over 200
runs, where each run is 600 time steps
44
in
A pplying open boundary conditions w as also able to give, roughly, the
characteristic shapes o f the fundam ental diagrams, flow -density
relation and velocity density-relation
04
0 45
05
0 55
06
0 65
06
0 65
density
(a)
09 H
08 >. 0 7 o
o 06 -
4>
05 04 03 04
0 45
05
0 55
density
(b)
Fig
(2 5)
Fundamental diagrams for the Simple. Model with open
boundary conditions,
relation,
flow-density relation, velocity-density
for array size 80 cells
Data were averaged over 200
runs, where each run is 600 time steps
45
2.3 Road Traffic Simulation
W e now extend our sim ulation to include road traffic flow , where each “road”
is formed by linking m ore than one segm ent, where a segm ent is a one­
dim ensional array o f k sites.
In this section w e introduce a new parameter
jammed, time t-, which
represents the waiting tim e period cars require to leave the road. In real traffic,
this parameter depends on the number o f traffic lights, w hich are installed
betw een the road segm ents, and the duration o f both light cycles, red and
green, as w ell as the traffic density. The aim o f this section is to study and
analyse the effect o f the number o f road segm ents and the light cycle duration
on the jam m ed tim e parameter.
2.3.1 Road Description
The m odel looks at traffic flow for a seven-day period on tw o designs: a road
consisting o f 77 cells and three segm ents and another consisting o f 144 cells
and five segm ents. The choice is made in order to avoid com plex situations
with large number o f intersections.
The sim ulation runs allow ed for:
1- Road input and output
Cars were generated at random and input into the system according to two
different injection rates: peak-tim e and off-peak time rates. Cars were allow ed
to leave the road only during the green cycle and the first site outside the road
w as assum ed to be free throughout the sim ulation run.
46
2- Specified duration o f the light cycle
\,
The sim ulation w as perform ed using different periods for both the red and
green cycles
Fig
(2 6)
cells,
Day-one
of
with green cycle
steps and Where
our
simulation
= 20
(a) , (b) , and
using
time steps,
(c)
road
of
red cycle
represents
size
the relations
the time-steps against jammed time, density, and(velocity
47
77
= 15 time
of
2.3.2 Simulation and results
Each sim ulation run is calculated using up to 10,000 tim e steps, where the
peak-tim e period is considered to be inside the interval 20 00 -8 000 tim e steps
Fig (2 6) describes the traffic behaviour throughout day-one traffic sim ulation,
where the roads size = 77 cells and the green tim e cycle is greater than the red
tim e cycle by a factor o f 4/3 The jam m ed tim e vs tim e steps relation,
presented in Fig (2 6 a), show s the sharp transition in the jam m ed tim e
parameter
t] at tw o points, (2000 & 8000 tim e steps respectively), w hich
separates the tw o traffic regim es, l e
dem onstrates how
free and dense traffic It also
rapidly increases over the dense traffic period and the
w ay in w hich it oscillates during the tw o regim es These' oscillations, w hich
m ay occur due to the density fluctuations and the duration o f the traffic light
cycle, have considerable im pact on the traffic velocity This is dem onstrated in
Fig (2 6 c) To observe the effect o f the light cycle duration on the
parameter, w e have increased the red cycle period to be the sam e as the green
cycle, see Fig (2 7) This change in turn decreased the outflow B y com paring
F igs (2 6 a and 2 7 a), w e find that increasing the red cycle period has result in
no significant changes in the jam m ed-tim e parameter over the dense traffic
period, w hich m y be noticed by very sm all oscillations in the
t} parameter as
it can be seen from Fig (2 7 a) This, also, w as the case w hen the green cycle
w as decreased to be the sam e as the red cycle period
N o w w e investigate the effect o f the number o f traffic lights on the jam m edtim e parameter i 7by increasing the road segm ents to five segm ents
48
Fig (2 7)
Day-two of our simulation using road of size 77
cells, with the green cycle and the red cycle the same and
Where (a), (b), and (c) represents the relations of fundamental
measures vs time steps
Fig (2 8) illustrates the traffic flow behaviour throughout a one-day traffic
sim ulation, w hich w as perform ed to investigate jam m ed-tim e behaviour after
the number o f road segm ents w as increased to five and with green tim e cycle
= 20 tim e steps and red tim e cycle = 1 5 tim e steps
49
tim e steps
(a )
tim * s t o p s
(b)
tim s <
(c)
Fig(2.8):
Day-three
of
our
simulation using
cells, and the green cycle = 20 time steps,
steps and Where
(a) , (b) , and
(c)
road
of
size
144
red cycle = 15 time
represents
the relations
of
the time-steps against jammed time, density, and velocity.
Com paring the results obtained in Figs (2.6 and 2.8), w e m ay observe the
follow ing:
Increasing the number o f road segm ents has generated the sam e traffic features
as for the three segm ent road, and has also increased the jam m ed tim e as
expected, see Table (2.1).
50
Green Cycle =20 &
Red cycle increased
Green Cycle reduced
Red Cycle =15
to Green cycle
to red Cycle
77 cells
5-80
15-120
40-100
144 cells
35-135
50-140
55-140
Road Size
Table
(2 1) The jammed-time range,
m
time steps,
for
different road size and light cycles
Table (2 1) show s that the jam m ed-tim e range, for both roads, is affected by
the changes in the light cycle duration for both red and green tim es For both
roads, the i parameter attains its m inim um value w hen the green cycle is
greater than the red cycle, w hereas the m axim um value o f
t] is obtained at
higher value for the red cycle
This can be seen also from the space-tim e diagrams in Fig (9 (a) and (b))
From these, it is easy to visualise the traffic behaviour for a certain number o f
tim e steps
In Fig (2 9), where Borland C++ is used to plot our space-tim e diagrams, cars
are m oving to the right from top left to bottom right Straight lines indicate
that cars are m oving freely, w hile vertical lines m eans that cars are blocked
(stopped)
Traffic jam s at the traffic lights are represented by the vertical trajectories,
w hich are m oving backwards against the traffic.
51
Fxg.(2.9.a):Space-time diagram at off-peak time, where the road
size is 77 cells
the road size is 144 cells
In the Figures, these jam s are m ore noticeable, and also last for longer periods
as w e increase the number o f the traffic lights from 3 to 5, hence the observed
increase sim ilarly in jam m ed tim e
Sim ilar phenom ena w ere apparent w hen different light cycle periods were
apphed, as in the three-segm ent road case From the above results, w e
conclude that the jam m ed-tim e parameter for the sim ulation depends on the
number o f traffic lights and the duration o f each light cycle, red and green, and
as w ell as the traffic density, as found for real traffic
The choice o f the different cases for the light cy cle’s duration w as destined to
create three different traffic patterns as follow s
•
B y increasing the green light cycle over the red cycle, w e consider our
sim ulated traffic to be m ore important than the traffic w hich passes the
sam e traffic lights on the other roads,
•
This situation is no longer dom inant when the traffic on the other roads
becom es heavier in this case w e increase the red cycle for our sim ulated
traffic, 1 e m ore green light for the other roads
•
£
In the third choice all roads are o f the sam e im portance, but the traffic
ahead is m ore congested In this case decreasing the green cycle w ill take
som e pressure at the front junctions
53
2.4 Summary and Concluding Comments
i
A three state determ inistic Cellular Autom ata Sim ulation M odel for the
dynam ic process o f traffic flow in urban N etw orks is presented in this
chapter, with space and tim e discrete
The tim e-evolution o f the automaton follow s sim ple rules, the state o f
each site at the next tim e step is determ ined from the state o f site itself
and those o f the nearest neighbour sites,
11
W e have perform ed our sim ulation, applying a parallel update strategy,
on sm all size lattices w ith open and closed boundary conditions
in
Sim ulations have been extended to include roads, where each road is
form ed by linking a finite number o f segm ents separated by traffic
¥
lights
iv
Results for the jam m ed tim e parameter
tJ show ed that this parameter
depends on the number o f traffic lights and the duration o f each light
cycle
v
Since the update rules treat segm ents o f the road networks, rather than
individual vehicles, com putational tim e does not depend on the number
o f the veh icles within the segm ents, but only on the segm ent size
54
Chapter 3
“Traffic System and Transient Movement Simulation”
55
3.0 Introduction
T he developm ent o f the netw ork level traffic approach w as based on the tw ofluid theory o f tow n traffic (Herman and Prigogine, 1979, Herman and
Ardekani, 1984), w hich relates the average speed o f m oving cars to the
fraction o f running cars in a street netw ork Further extensive studies have
been carried out on the basis o f this theory (M aham assani
W illiam s
et al, 1990,
et al, 1987), w hich has m any advantages but also involves som e
effort in m odel specification and high com putational requirem ents H ow ever,
the behaviour o f the various netw ork variables at high concentration levels o f
cars rem ains to be understood Other relevant w ork in the sam e area has
involved developm ent o f a fast sim ulation m odel for progression o f cars along
a street through bit m anipulation programs (Cremer and Ludw ig, 1986) In this
chapter w e m ove from a single-lane traffic m odelling to m odelling networks
using the Sim ple Cellular Autom ata M odel described in Sec (2 1)
3.1 Network description
Road networks are represented by nodes, segm ents (lanes), and links This
structure allow s the traffic sim ulation in integrated networks o f urban tw o-lane
carriageways The netw ork objects m ay be described by com ponents as
follow s
3.1.1 Nodes
Each node can be view ed as an intersection or a T-junction. N od es are points
in the netw ork where traffic enters or leaves the sim ulated network, or traffic
56
from one link is distributed am ong other links in the netw ork The entry links
o f the netw ork are located at all the peripheral nodes w ith the exception o f the
T junction nodes Each node object, Fig (3 1), has the follow ing data item s
1-Node number
Each node has a unique identification number
2-Node offset
For each node the offset is defined as the number o f tim e steps at which the
node w ill change its default start, (red tim e traffic), to green tim e traffic for the
duration o f the green cycle
The offset refers to the tim e relationship betw een the adjacent signals at
netw ork nodes The pattern o f offsets in a series o f signal aim s to m inim ize the
stops and delay associated with travel through netw ork o f signals
The node offset is calculated as follow s
i
the offset is set to zero for nodes, w hich are considered as the
offset-start nodes
n
for all other nodes, the offset is considered to be the shortest path
from the nearest entry and exit node up to the given node
3 -Node type
The node type specify whether the node is an intersection, a T junction, or
entry and exit node
4-Node green cycle
The green cycle is the number o f tim e steps, w hich allow s the traffic to flow
from a specific direction at a specified node
57
Node Object
Node number:
15
Node offset
20
Node type
I (intersection)
Node red cycle
15 (time steps)
Node green cycle
20 (time steps)
Node signal
0 (red)
Node entries
12, 34, 45, 20, 44, 21
F i g (3 1)
I
link 2 link 3
link 1
L
i
i
l
^
link 5
t
i
1
i
t
1
i
i
l
Intersection
F i g (3 2)
58
T junction
link 4
5-Node red cycle
Is the duration o f the opposite traffic flow at a specified node
6-Node signal
T w o signals only are introduced in our netw ork traffic sim ulation, red (0) and
green (1), assum ing that the default setting for each node signal is 0 (red)
1-Node entries
These are the segm ents, which link the give node to other network nodes
3.1.2
Links
A link is a set o f tw o adjacent lanes, each o f w hich has a different direction,
w hich m ay connect any tw o nodes, see Fig (3 2)
N etw ork roads are connected using links, all links in the sim ulated network
consist o f 2-lanes, with all feasible m ovem ents allow ed at all intersections
3.1.3 Segments
Segm ents are road sections, w hich connect nodes Each segm ent object,
dem onstrated m Fig (3 2), has the follow ing characteristics
1-
length
This is the number o f cells in each segm ent, where each cell is 7 5 m and
represents the car length plus distance betw een cars in dense traffic
2-
configuration
This indicates the initial state for the segm ent, l e the w ay m w hich the
n
veh icles are initially placed at random
3-
transient period
This specifies the number o f tim e steps, w hich w e discard before starting the
59
collection o f our data and is based on the segm ent size It represents the tim e
needed for relaxation to equilibrium
4- starting and ending nodes
The
starting and ending nodes o f any segm ent tell us the position o f the
segm ent inside the sim ulated network
3.1.4 Network geometry
The
geometry o f the netw ork w as chosen to be a circular grid with radius r,
w hich indicates the number o f nodes from the centre node o f the simulated
netw ork to the surface node A pplying this geom etry, each segm ent object o f
the netw ork can be either a straight line or a true arc, w hich is characterised by
the co-ordinates o f the start and end points (nodes) Increasing the radius o f
the grid w ould also increase the number o f segm ents by a fixed number, w hich
allow s different size netw orks to be sim ulated Fig (3 3) show s how the
proposed geom etry is used to represent real networks
3.1.5 Traffic control and traffic conditions at junctions
A ll nodes w ere controlled, (signalised), in our simulated netw ork Our signal
operating condition w as assum ed to be a pre-tim ed signal, (fixed tim e control)
B y fixed tim e control, both the green tim e cycle and the red tim e cycle are
fixed w ithout any consideration to the veh icle arrivals and departures at
junctions
60
'V
Fig (3 3)
representation of a section of a real road
network m
Dublin by the proposed network geometry
61
Three determ inistic traffic conditions w ere em ployed in our sim ulation o f the
number o f cars passing an intersection or junction, (1) 25% turn left, 50% go
straight, 25% turn right and (n) 20% turn left, 60% go straight, 20% turn right,
and (m )12% turn left, 76% go straight, 12% turn right T hese conditions apply
at all intersections where all three m ovem ents are allow ed, and reduce to 50% 50% at junctions with tw o m ovem ents only
3.1.6 Traffic parameters
In this chapter w e also extend the definition for the traffic parameters, density,
flow , and velocity, to the netw ork level
Network velocity is obtam ed at every tim e step as the ratio o f the total number
o f m oving vehicles, inside the network, to the total number o f veh icles
Network density is also calculated at every tim e step as the ratio o f the total
number o f vehicles, inside the network, to the total number o f sites
The corresponding
relation
network flow, at every tim e step, is calculated using the
Network flow = Network velocity * network density
Since the density varies dram atically w ith tim e in dynam ic traffic networks,
the sim ulation tim e is split into intervals, w hich corresponds to observation
periods The user can set the interval length and different values have been
applied in our sim ulations This is because the interval length, as w e w ill see
later, has a significant effect on the traffic parameters The choice o f the
interval length turned out to be influenced by the dom inant traffic regim e, free
or dense The tim e dependent density, velocity and flow are exam ined by
taking averages every 30-150 tim e step throughout the sim ulation run
62
3.2 Description of Simulation Experiments
It is the aim o f this chapter to investigate and study the behaviour o f urban
traffic networks and to characterise then- perform ance using the proposed
sim ple Cellular A utom aton m odel
In this section, our purpose is to study the physical and operational features o f
the sim ulated network through the analysis o f the fundam ental diagrams, for
relations such as density vs flow and density vs velocity
In general the follow ing factors m ay influence the networkiperform ance
i
car arrivals to the network ( at fixed or stochastic rate)
u
the number of the entry! exit nodes
ill
traffic conditions at intersections
iv
network geometry
v
external parameters such as weather conditions, different road
conditions, different motivationsfor the drivers
V ehicles w ere random ly distributed along the network, then thy generated at
random and input into the netw ork through all nodes o f entry In our sim ulated
traffic, veh icles did not have " K nowledge" about their com plete path along
>j'
the network, but only about their next tim e step m ovem ent N etw ork nodes
can be classified as entry/exit nodes, T-junction nodes and intersection nodes
In the follow in g sim ulations four different netw orks w ere considered, w hose
radu varied from r=3,(8 K m length for a network o f 17 nodes and 28 links), to
r=6,(24 K m length for a netw ork o f 41 nodes and up to 76hnks) Each link is a
tw o-lane carriageway
63
A stochastic feeding m echanism , see A ppendix (C), in-¡which car arrivals
tH
follow a P oisson Process, w as im plem ented, (using different rates o f arrivals),
throughout the sim ulation experim ents It is also assum ed that the network
neighbourhood can take any number o f veh icles that might leave the network
Traffic conditions m ay also affect the network perform ance For exam ple if
w e increase the probabilities for turning left or right, then netw ork density w ill
increase, since the number o f vehicles circulating inside the netw ork w ill
increases relative to those leaving it
In this chapter w e do not consider varied netw ork geom etry, but assum e a
sim ple geom etry for our sim ulated netw ork throughout, keeping the number o f
the entry/exits are constant
'
64
65
3.3 Simulation and Results
U sing a stochastic feedm g m echanism , the mam factors m the follow ing
experim ents are
;
i
arrival rate
u
traffic conditions
in
interval length
iv
transient period
In this Chapter, w e consider only the transient m ovem ent o f the cars along the
network, whereas, in Chapter 4 w e allow cars to park inside the network, so
that they are tem porarily lost to the flow
In the follow ing sim ulation each run consists o f 5000 iterations Network
parameters are calculated every 30- 150 tim e steps after a discard period o f
200-500 tim e steps with traffic conditions applied as m section (3 1 5) and
usm g different arrival rates A sum m ary o f the sim ulation outputs can be
found m Appendix B , Tables (B 1-B 4) W e start by lookm g at the effect o f
the arrival rate, p, on the traffic behaviour along the network
3.3.1 The effect of the arrival rate on the network
parameters
3.3.1.1 The existence of jamming threshold
Fig (3 5) show s the number o f cars passing through the netw ork for a 17-node
network, together with the number o f cars waitm g outside Arrival rate (p)
varies betw een 0 1 and 0 6 It is easy to observe the “jamming threshold?’ for
¡i greater than the critical value }i = 025 the network is notable to cope with
the traffic, which has results in long queues outside
66
The sim ulation also reveals that the critical arrival rate p is independent o f the
netw ork size (Fig (3 6)), where the netw ork size has extended from 17 nodes
up to 25 nodes, 33 nodes and 41 nodes, and also the traffic conditions, Fig
(3 7)
Fig
(3 5)
Number of cars vs arrival rate relation for a 17
node network and turning percentages 20% left,
60% through, and
20% right
8000
<2 7 0 0 0
O)" 6 0 0 0
B 5000
— # - - 1 7 nodes
M w ^ im v
2 5 nodes
- 3 3 nodes
— X - - 4 1 nodes
02
04
0 6
0 8
a rriv a l ra te (|j)
Fig
(3 6)
Waiting cars vs arrival rate for four different
networks, and turning percentages 20% left,
20% right
67
60% through, and
8000
(0o
o>
c
5n
a>
JQ
E3
C
♦
70 0 0 -
2 5 % le ft, 5 0 %
t h r o u g h , 2 5 % righ t
■»” i§$”“ - 2 0 % le ft, 6 0 %
t h r o u g h , 2 0 % righ t
6000 5000 -
th ro u g h , 1 2%
4000
righ t
3000
2000
10 0 0
0
-fSMfi*'?;
0.2
0.6
0.4
0.8
arrival rate (|j)
F i g . (3.7): Waiting cars vs arrival rate for various traffic
conditions
F i g . (3.8): Transported cars vs arrival rate for different size
networks and turning percentages are:25% left,
50% through, and
25% right
It is how ever, worth noting that the number o f cars transported through the
network is affected by the network size, especially for p greater than the
critical value p = 0.25, Fig (3.8), this is obtained for different network
sizes; 17, 25, 33, and 41 nodes.
68
This graph also suggests that the number o f cars transported via sm all size
networks is greater than those transported via larger size netw orks This is due
to the increm ent o f the traffic lights number, 1 e m ore delay at intersections,
and also car path through the netw ork becom es longer
3.3.1.2 Arrival rate vs Network size
In the next sim ulation runs w e start by changing the arrival rate and keeping
other factors the sam e, where different netw ork sizes w ere used The output
parameters, flow , density, and velocity, are averaged every 30-tim e steps after
a discard period o f 200-tim e steps to let transience die out
Performance Measure
Max Flow
Arrival rate
Density of Max Flow
Transported cars
Queue Length
Table (3 1)
41 nodes
&14S92
0
\83timr
-■- - - "
>1 = 01
17 nodes
mm
' i'
.
'
............. F i S ® 1........................<" C S M " " '..................... .
1*
t
\
0 180459
0143406
p=03
0 482386
0 430268
p = 0 55
0 414131
0 494114
p=0 1
2748
2430
»«03
m i
fi = 0 55
6215
5810
H = 01
2
3
H= 0 3
m
552
H = 055
mm
5769
The influence of varying the parameter u on the
maximum flow and its density, cars transported via the network
and the queue length outside it for different s\ze networks
69
At all intersections the traffic conditions, as in Sec (3 1 5) case (n), were
applied The fundam ental diagram s, presented in F igs (3 9) describe the flow
behaviour for two different netw ork sizes, 17 nodes and 41 nodes
The effect o f changing the arrival rate (p) can be sum m arised as follow s
i
A higher flow w as obtained in the case o f sm aller size netw ork and
this w as obtained for different values o f p , see Table (3 1) The
table also show s that increasing the arrival rate from p = 0 1 to p =
0 3 has significantly increased the m axim um flow (an increm ent o f
66% w as observed for the smaller netw ork and 91% for the larger
network) H ow ever, when p w as greater than the
“jamming
threshold” the flow behaviour w as very similar, irrespective o f the
netw ork size Statistical analysis, presented in A ppendix A (Table
(A 2)), reveals that the arrival rate strongly influences the flow , with
a significant interaction with the netw ork size at a = 0 01 level o f
significance
u
A lso Table(3 1) show s that the m axim um flow occurred at much
higher density w hen a high rate o f arrival w as used to inject the
netw ork with cars
m
Cars transported through the sim ulated netw ork are influenced by
both netw ork size and arrival rate The number o f the transported
cars increases significantly when p increases from 0 1 to 0 3 and
this increm ent becom es less significant at high rate o f arrivals (p =
0 55)
70
flow-density relation
5o
0
0
0
0
0
0
0
0
0
0
0
0
27
25
23
21
19
17
15
13
11
09
07
05
♦ M= 0 1
5)J = 0 3
M = 0 55
/
02
04
0 6
density
(a)
Flow-density relation for a 17 nodes network using various
arrival rates and the turning percentages 25% left, 50%
through, and 25% right
(b) Flow-density relation for a 41-node network using various
arrivals rates with turning percentages are 25% left,
through, and 25% right
F i g (3 9 )
71
50%
It can be seen from Table (3 1) that cars transported via the netw ork w as
m uch higher in the case o f sm aller netw orks irrespective o f the values o f p
as seen in Fig (3 8)
iv
A s the value o f p increases the “jamming threshold”, a w ell know n
M acroscopic phenom ena “Q ueues outside the netw orks” w ill occur
Our sim ulation reveals that the queue length increased significantly as
the arrival rate increased from p = 0 3 t o p = 0 55 with the queue
length longer in the case o f sm aller networks
The effect o f varying the netw ork size can also be seen from Fig (3 10), w hich
show s the velocity-density relation for tw o different netw orks using a
stochastic feeding m echanism , (in w hich car arrivals follow a P oisson process
with p = 0 55) A lso, Fig (3 10) show s that the m axim um velocity obtained in
the case o f larger size netw ork is greater than the m axim um velocity obtained
for the sm aller size netw ork
Fig
(3 10) :
Velocity-density
relation
for
two
different
networks using arrival rate of 0 55 and same traffic conditions
m
F i g (3 9)
72
3.3.2 Varying of Traffic Conditions
Tw o different sets o f traffic conditions w ere applied, ( 1) 25% turn left, 50% go
straight, 25% turn right and (n) 12% turn left, 76% go straight, 12% turn right
The influence o f changing the traffic conditions on som e o f the network
perform ance m easures are presented in Tables (3 2) and (3 3) A lso, in Fig
(3 11), w e present the fundam ental diagrams for four different networks,
applying tw o different traffic conditions using a m oderate arrival rate p = 0 3
B y com paring Tables (3 2) and (3 3) and studying the flow -density relations in
fig (3 11), w e m ake the follow ing observations
l
The netw ork m axim um flow w as increased by increasing the headway
traffic percentages from 0 50% to 0 76% and this increm ent was
obtained for arrival rates (p = 0 3, 0 55) A lso, Fig (3.11) show s that
the effect o f increasing the headw ay traffic percentages w as more
noticeable in the case o f sm aller size netw orks Statistical analysis,
presented in A ppendix A (Table (A 1)), show s a significant effect o f
the traffic conditions on the flow at low and high arrival rates at a =
0 01 level o f significance A lso, the analysis reveals a significant
interaction betw een the netw ork size and traffic conditions, using low
p, at the sam e level o f significance
n
The change in the traffic conditions also affected the number o f cars
transported through the networks Tables (3 2) and (3.3) show s that for
higher arrival rates, p = 0 3 ,0 55.
Increasing the headw ay traffic percentages has decreased the cars
transported through the network and this could be seen clearly m the
case o f sm aller sized networks This is because the probability for a car
to use the main roads rather than the sub-roads increases by increasing
the headway traffic, w hich, in turn, affect the number o f cars entering
the network
111
The sim ulations also reveal an important role for traffic conditions on
the queue length outside the sim ulated networks For smaller size
netw orks, increasing the headw ay traffic percentages increased the
queue length and this increm ent becam e m ore noticeable at higher
arrival rates This is because the system density changes rapidly,
increasing with every increm ent o f p
A s the network becom es more congested, the turning percentages play
an important role at the entry and exit nodes So, by increasing the
headw ay traffic, cars tended to queue longer rather than go straight
through In contrast, by increasing the netw ork size, the network was
able to cope with the incom ing traffic until m uch latter in the
sim ulation H ence the turning percentages w ere m ore effective for a
longer period and the queue length decreased as w e increased the
headw ay traffic percentages
74
Traffic
17 Nodes
33N «tes
Measare
25%, 50%, 25%
0 247978
0.2334
0.236816
Max Plow
lZ%t 76%t U%
0,260762
3,241085
0,245792
Dis&sitJ i)F
25%, 50%, 25%
0.482386
0.419332
0.430268
Mssaafiow
12%, 76%, 12%
0.455271
0.439291
0.459454
mt
574«
5351
12%, 76%, 12%
5781
5467
5314
2$%> so%,
MS
739
mi
12%, 76%, 12%
847
672
400
j tVaasported i
Cars
1 Qaewe length ;
Table
(3 2)
The Table shows the network parameters, maximum
flow and its density,
cars transported via the network and the
queue length outside it, obtained using different traffic
conditions and p=0 3
:
FwfeFtttance
:
Measure
Traffic
17 Nodes
33N<Kles
itliSixtes
Conditions
25%, 50%, 25%
0.250189
0.236959
0.236763
MaxFiew
12% , 76% , 12%
&26t315
0J242875
C250S23
D&itsitj of
25%, 50%, 25%
0.414131
0.433432
0.494114
12%, 76%, 12%
0.453458
0.458928
0.426943
6355
5989
mo
12%, 76%, 12%
6215
5796
5740
i Qaeue Length j 25%, 50%, 25%
6063
5854
5769
12%, 76% , 12%
6292
5744
5706
i MaximIPtow
:
Transpwted i z$ % ,sm a s%
Cats
Table
(3 3)
The table contents are the same as m
but using higher arrival rate
(y=0 55)
75
Table
(2),
flow-density relation
0 2 7 -|
0 25 0 23 5
o 021
M—
* 17 nodes
-
* 25 nodes
«
&
A
0 190 17-
* 33 nodes
' 41 nodes
*
0 15-
i
1
01
0 2
0 3
0 4
0 5
06
0 7
density
(a)
flow-density relation
0 27 0 25 0 23 1
• 17 nodes
?
0 21 -
» 2 5 node s
« <
a 3 3 nodes
0 19-
' 41 node s
0 170 15 -
A
.
01
,. ,
—*-----------02
03
04
05
06
0 7
density
(b)
Fig
(3.11)
. Flow-density relation using two different traffic
conditions
(a)25% left,
50% through, and 25% right and
(b)12% left, 76% through, and 12% right
76
3.3.3 Time-interval Based Network Simulation
In order to observe the changes in the traffic behaviour throughout the
sim ulation run, our sim ulation results w ere averaged over tw o different tim em tervals
To quantify the effect o f the tim e-interval, (interval length),, on these averages,
sim ulations w ere carried out using tw o lengths, 30 and 150 tim e steps (short­
term and long-term averages respectively)
The output values o f the traffic parameters for duration period o f 600 tim e
steps, after a discard period o f 200 tim e steps, applying a low arrival rate for
tw o different networks are presented in Tables (3 4) and (3 5)
Tables (3 6) and (3 7), on the other hand, present similar outputs, but with a
higher arrival rate (ji=0 3) From studying these output averaged values, w e
m ake the follow ing com m ents
i
Tables (3 4) and (3 5), w hich com pare traffic parameters that gathered
using both short-term and long-term averages at low
\i, demonstrate
the follow ing
A s long as the free traffic w as dom inant there w ere no significant
changes in the network density and consequently, in the network flow
This can be seen from Table (3 4), where the changes in the
network
density did not exceed 0 0148 during the short-term averages and 0 013
over the long-term averages
A s a consequence o f this traffic pattern, increasing the number o f data
points or averaged values w ill not give a better idea o f the fundam ental
diagram s,( flow -density relation here).
77
'i
Long-term averages
Short-term averages
velocity
0 863204
0 855269
0 879504
0 868557
0 869728
0 870116
0 86928
0 861815
0 853949
0 867702
0 862757
0 872121
0 862006
0 855352
0 870685
0 876735
0 850307
0 868917
0 863169
0 871927
density
0148
0 * li(9
0.1465
0,1405
0.1457
0.1464
0"l476
0.1424
0.1302
0*t378
0*1427
Qll'471
0144
0*1463
01459
0.1423
0.1303
0.1401
0.1309
0.1332
flow
velocity
01273
0 1214:
01288
01220
0,126? 0 888803
01274
0128&'
01227
G1188
0<1195 0 88667
01231
01283'
01241
01252
0,127 0 883948
01245
0118$
01217
01207
0,1162 0 894586
density
flow
0*1306 0.116
01383 0,1228
0*1436 0,1269
01383 0,1238
Table (3 4): Sample of the simulation outputs for 600 time steps
for a network of size 25 nodes at low arrival rate ]i = 0 1
Short-term averages
velocity
0 862791
0 870016
0 867203
0 879202
0 873163
0 875453
0 866876
0 860863
0 863462
0 866894
0 857361
0 877931
0 878355
0 866478
0 872744
0 880963
0 876164
0 871459
0 871158
0 8788
Table
(3 5)
density
0*1390
0*1384
04303
0,134
0.1364
0.1356
0*1354
0*1356
0*1345
0.1315
0*1334
01315
01318
0.1316
0.1301
0*1272
0*1266
0*1272
0*1293
Long-term averages
Flow
01199
01205
0,1208
0117?
0,1191
01165
01175
0,1166
0,1171
$.1166
01128
01172
0,1155
0,1142
01147
01146
01115
0,1103
0,1108
01136
velocity density
flow
0 893192 0*1415 01264
0 893607 0.1393 01245
0 888003 0*1384 01229
0 902559 01391 01256
Same as Table (3 4), but for a larger network (41 nodes)
78
This w as m ore noticeable in the case o f large sized (Table (3.5))
networks, with sm aller changes in the
network density, (0.0127,
0.0035), during short-term averages and long-term
respectively. This result in very sm all changes in the
averages
network flow,
(0.0105, 0.0035), over short-term and long-term averages respectively
The statistical analysis in A ppendix A (Table (A .l)) supports our
findings and reveals that the interval length does not have a significant
effect, at a = 0.01 level o f significance, on the network flow at low
arrival rate. H ow ever, a significant interaction betw een the network
size and the interval length at the sam e level o f significance was
obtained.
ii.
W hen a higher arrival rate was used to feed the sim ulated networks, the
network density increased rapidly especially in the case o f smaller size
networks. Everyday observations confirm that as long as the dense
traffic is dom inant, only sm all changes (increm ent or decrem ent) in the
network density and in turn its flow can be obtained.
The sim ulation results presented in Tables (3.6) and (3.7) show s that
sm all changes in flow behaviour, (i.e. flow -density relation), can be
observed w hen short-term averages were used to describe the traffic
behaviour, because this w ill generate more data points throughout the
sim ulation run. In contrast using long-term averages has m inim ized the
number o f the data points, which describe the flow behaviour, and in
turn sm all changes in the flow behaviour can not be observed. This can
be seen also in Fig (3.12). The statistical analysis, presented in
Appendix A (Table (A .l)), also reveals a significant role for the
79
interval length at higher rate o f arrivals at a = 0 01 level o f
significance
The sim ulations suggest that, for congested traffic, the smaller the
interval length the better in order to observe sm all changes in the
traffic behaviour through the network
Fig
(3 12)
nodes using
Flow-density relation for a network of size 25
= 0 3 and the turning percentages <25%, 50%, 25%
80
Short-term averages
velocity
0.810784
0.797033
0.798958
0.799456
0.787616
0.789998
0.768539
0.760912
0.780104
0.771537
0.759717
0.753567
0.754227
0.740113
0.747635
0.726546
0.724729
0.729201
0.751688
0.732638
density
0.2358
0 2395
0 249
0 249
0.2506
0.2562
0 2607
0.2661
0.2646
0.2714
0.2732
0.2835
0.284
0.2896
0.2966
0.3005
0 3007
0.3059
0.3089
0 3072
Long-term averages
flow
0.1912
0.1909
0.1989
0.1991
0.1974
0.2024
0.2004
0.2025
0 2064
0.2094
0.2075
0.2137
0.2142
0.2144
0 2217
0.2183
0.2179
0.223
0.2322
0.2251
velocity
flow
density
0.81997 0.2579 0.2115
0.775212 0.2873 0 2227
0.742689 0.3047 0.2263
0.722203 0.3242 0 2341
Table (3.6): Sample of the simulation outputs for 600 time
steps for a network of size 25 nodes,O.y = 0.3
Short-term averages
velocity
0.847112
0.83655
0.833052
0.829258
0.832169
0.832335
0.826608
0.821905
0.8083
0.803471
0.805983
0.802133
0.799815
0.804715
0.788325
0.780426
0.787022
0.791823
0.794961
0.773795
Table
(3.7):
Same
Long-term averages
density
0.1984
0.2057
0.2086
0.2122
0.2173
0.2199
0.2232
0.2286
0.2354
0.2374
0 2422
0.2445
0.245
0.249
0.2549
0.2588
0.2582
0.2616
0.2652
0.2651
Flow
0.1681
0.1721
0.1738
0.176
0.1809
0.183
0.1845
0.1879
0.1903
0.1908
0 1952
0.1961
0 196
0.2004
0.201
0.202
0,2032
0,2072
0.2108
0.2052
as T a b l e
(3.6),
81
velocity
density
flow
0.852978 0.2178 0.1858
0.833971
0.241
0.809594 0 2606
0 201
0.211
0.794252 0.2768 0 2198
but
for
a network
of
size
41
nodes
3.3.4 The influence of Changing the Transient Period
on Network Performance
il
N ext w e look at the effect o f the transient period on the sim ulation output by
using tw o different transient periods, 200 and 500 tune steps In the follow ing
experim ents, the traffic parameters w ere averaged every 30-tim e steps by
applying the follow ing traffic conditions as in case (l), Sec (3 1 5) for a
duration o f 5000 tim e steps using tw o different arrival rates (p = 0 1 & p =
0 55 ) This w as carried out for tw o different netw ork sizes, with radu r =
3,(17 node) & r = 6,(41 node)
The results o f the above sim ulations w ere as follow s
A t first glance, Table (3 8) reveals that the effect o f increasing the transient
period from 200 to 500 tim e steps has no significant^ influence on the
m axim um flow obtained at both rates o f arrivals, no matter what network size
is used in the sim ulation This is because no significance changes in the
m axim um flow occurred, for either network, or for either p
Performance Network
Measure
Maximum
p =0 1
size
Trans Pd = 200
17 JiOdôS
&14892
p = 0 55
Trans Pd = 500
0,14046
Trans Pd = 200
Q>25&i8£
Trans Pd = 500
Flow
41 nodes
0 123815
011531
0 236763
0 238576
Density of
17m éss
0.180459
Q jm r n
0<4i4131
0 528179
Max Flow
41 nodes
0143406
0 131372
0 494114
0 490861
Table
(3 8) The
influence
of changing the transient
the maximum flow and its density,
and traffic conditions 25% left,
82
period
on
using different values for p
50% through,
25% right
On the other hand, statistical analysis presented in A ppendix A (Table(A 1))
show s the significant influence o f the transient period on the netw ork flow at
both rates o f arrivals
A closer look at Table (3 8) show s the influence o f transient period on the
m axim um flow as the statistical analysis confirm s
This can be dem onstrated as follow s
Since no significance changes w ere observed in the m axim um flow for either
length o f transient period, the increm ent o f the transient period has not
correctly reproduced the density o f m axim um flow This is m ore noticeable in
the case o f sm aller size netw ork at higher p
Fig (3 13) show s how the fundam ental diagram s are affected by the length o f
transient period, especially for smaller networks The effect o f changing the
transient period for different traffic conditions w as also noted (Table (3 9))
Measure
size
Maximum
p = 0 55
o
II
3-
Performance Network
l?BOde$i
Trans Pd = 200
0J32537
Trans Pd = 500
0,133296
Trans Pd = 200
0,261315
Trans Pd = 500
0,264442
Flow
41 nodes
0137141
0122812
0 250523
0 250281
Density of
17 nodes
GJ5784
0154583
0 453458
0 504902
Max Flow
41 nodes
0157631
0139498
0 426943
0 415698
Table
(3 9): The influence of changing the transient period on
the maximum flow and its density, using different values for u
and traffic conditions
12% left, 76% through,
83
12% right
The sim ulation reveals that using 200-tim e steps as a discard period is good
enough to let the transience die out, where the largest road segm ent in our
sim ulated netw orks did not exceed 60 sites, Fig (3 13)
0 27
0 25
0 23 H
• r = 3 , tr a n s pe rio d = 2 0 0
0 21
« r = 3, tr'ans perio d = 5 0 0
0 19
0 17 H
0 15
01
02
03
04
05
06
07
d e n s it y
(a)
0 26
-|
0 24
-
0 22
-
I
♦ r=6,transperiod=200
»r=6,transperiod=500
/
0 2 0 18
-
0 16
-
0 14
-
I
|
0
♦
♦
1
C
01
I
1
I
1
1
-V
u
r
1
0 2 0 3 0 4 0 5 0 6 0 7
density
(b)
Fig
of
(3 13)
25%
Flow-density relation using the traffic conditions
left,
50%
through,
and
0 55
(a)for a 17 node network
(b)for a 41 node network
25%
right
and
arrival
rate
of
3.4. Summary and Concluding Comments
In this chapter w e m oved from single lane traffic flow towards road networks
D ifferent network sizes have been tested, where all nodes w ere controlled by
traffic hghts and the signal operating conditions w ere assum ed to be pre-timed
signal Various traffic conditions w ere considered at the network junctions A
stochastic feeding m echanism , in w hich car arrivals follow a Poisson process
with parameter p , has been im plem ented throughout the sim ulations using
different values for the parameter p
.v>
The w ork presented in this chapter suggests that parameters governing
perform ance o f urban netw orks m ay be investigated in som e detail using the
sim ple cellular automata M odels
The results obtained in this chapter indicate the follow ing
• It appears that there is a critical arrival rate
”jamming threshold ’ above
w hich the transportation through the netw ork is not efficient any m ore and
this rate is independent o f the netw ork size
• In a free traffic regim e, the traffic conditions, defined in S ec(3 1 5), seem
to involve significant interaction with netw ork size,(oc = 0 01), but for
dense traffic this interaction w as not significant
• The arrival rate is the principal factor o f im portance for larger netw orks A
significant interaction,(a = 0 01), also exists betw een the arrival rate and
the network size
85
•
The sim ulation also reveals that the interval length used to gather the
traffic parameters, average velocity, density, and flow , has a significant
ro le,(a = 0 01), on the flow rate at higher arrival rate A significant
interaction betw een the netw ork size and the interval length w as obtained
at a = 0 01 lev el o f significance The sim ulation reveals that the smaller
the interval length the better, in order to calculate the traffic parameters
•
The length o f the transient period, warm up period, also has influenced the
traffic param eters at both, low and high, arrival rate, where using 200-tim e
steps as a transient period w as good enough to let the transience die out A
significant interaction ( a = 0 01) betw een the transient period and the
netw ork size obtained at low arrival rate
86
(
/
Chapter 4
“Network Performance under Various Traffic Events”
87
4.0 Introduction
In Chapter 3, w e w ere concerned “only” w ith the transient features o f
vehicle m ovem ents in sim ulating passage through the netw ork under
sustained flow
In this chapter, w e investigate the traffic behaviour along the network with
loss to flow Four underground parks, each o f capacity up to 300 cars, were
introduced in order to exam ine the effect on the transient behaviour o f the
system In the follow ing experim ents, each sim ulation run consists o f 5000
tim e steps After a discard period o f 200-tim e steps, short-term averages
w ere used to gather the output parameters, flow , density, velocity, input o f
cars to the network, output o f cars o f the network, and the queue length
outside each entry/exit node The turning percentages w ere 25% left, 50%
through, 25% right T hese sim ulations w ere perform ed for tw o networks o f
different sizes, 17 and 41 nodes Each sim ulation run w as divided into two
stages In the first stage (1000-3000 tim e steps) veh icles m ay enter the
underground parks at fixed or stochastic rate, w hile in the second stage
i
(3000-5000 tim e steps) they leave the underground parks,’ also by the sam e
manner, and hence pass through the network
4.1 Loss to Flow Simulations
4.1.1 Simulation with Loss to Flow at Fixed Rate
In the follow in g sim ulation a car m ay enter or leave the underground park
with fixed probability (p = 0 5)
The m ost remarkable aspect o f introducing these parks is a different
“jamming threshold!” for different network sizes, as can be seen from
Fig (4 1)
(a)
V)
CO
o
total
transported
““■sfe— waiting
a>
Q
3
E
01
0 2
0 3
M
04
0 5
(b)
F i g (4.1) Number of cars vs arrival rate for
(a)
17 node network
(b)
41 node network
89
0 6
B y increasing the netw ork size its ability to cope with the traffic im proves,
see (Fig (4 2)) This can also can be illustrated by studying the fundamental
diagrams (Fig (4 3)) o f one o f the sim ulation runs, where p = 0 3 and the
data w ere averaged every 30-tim e step after a discard period o f 200 tim e
steps
Fig
(4 2)
Number
of
waiting
cars
vs
arrival
rate
for
two
different size networks
The first part o f the flow -density relation, F ig(4 3 (a)), represents the free
phase traffic, w hich is characterised by high velocities, see F ig(4 3 (b)), and
low densities
B y contrast the second part represents the jam m ed phase traffic, w hich lasts
for a longer period in the case o f sm all size network
90
(a)
(b)
(c)
91
Fig
(4.3) : The fundamental diagrams for simulation described
a b o v e , where m
(a) flow-density relation,
(c) velocity-flow relation,
(b) velocity-density relation
(d)velocity-flow relation for the
transient movement simulation,
see Chapter 3
The above scenario can be explained as follow s
Initial conditions o f low density o f cars and the application o f arrival rate ( p
= 0 3) did not m uch influence the netw ork density, as long as veh icles may
leave the network through the different exits and m ay also disappear into
one o f the underground parks This situation w as dom inant until the second
stage o f traffic m ovem ent w hen veh icles started to leave the underground
parks, causing traffic jam s to spread everywhere along the netw ork due to
the high density and low flow , characterised by decrease in netw ork
velocity, see Fig (4 3 (b)) This w as m ore noticeable in the case o f the
sm aller network, where the netw ork density increases significantly from p ~
0 1, during the first stage, to p~ 0 81, within the second stage
92
In contrast the m axim um netw ork density obtained in the second stage for
the larger netw ork w as ~ 0 6
T hese F igs (4 3 (a)) and (4 3 (b)) indicate strong sim ilarities o f the
fundam ental diagram s to those obtained for individual road sections when
compared to those obtained in transient m ovem ent sim ulations (sec 2 3 1 1 )
The flow vs velocity relations presented in Figs (4 3 (c)) and (4 3 (d)) show
the variation o f velocity with flow for the tw o types o f sim ulation,
sim ulation with loss to flow and transient m ovem ent sim ulation These
figures m ay be com pared and explained as follow s
The upper parts o f the fundam ental diagrams represent the freely flow ing
traffic, where each car travels at the desired speed, w hich lasts for a longer
period in the case o f loss to flow system s
A s the traffic becom es heavier the decrease in the average velocity begins
slow ly Then the continuous increase in the system density w ill lead to a
continuous decrease in the average speed and, in turn, the average flow ,
w hich is m ore noticeable in the case o f sm aller netw ork A lso, this w as more
noticeable in the case o f loss to flow system s as the density increases
considerably due to the existence o f the underground parks
93
4.2.1 Simulation with Loss to Flow at Stochastic Rate
In the next experim ents, at every tim e step, within the specified periods for
entering and leaving the underground parks, w e assum e that a car m ay enter
or leave any o f these parks according to a Poisson distribution with
parameter p 1
Fig (4 4) show s that increasing the parameter p 1 from 0 1 to 0 3 has led to a
low er values for the jam m ing threshold, w hich depends on the network size
In the case o f the sm aller size network, the jam m ing threshold has decreased
f r o m 0 2 t o ~ 0 1 8 and also the number w aiting cars has increased
M1=0 1, 17 nodes
------|J1=0 3, 17 nodes
|j1=0 1, 41 nodes
—
M1=0 3, 41 nodes
F i g . (4.4) : Number of waiting cars vs arrival rate for two
different networks using different values for the parameter
Ul
In contrast, using a larger size netw ork has increased the jamming
up to 0 25 and also above the jamming
threshold the number o f w aiting cars
has decreased considerably com pared to the sm aller size network
94
threshold
In order to study the effect o f varying the parameter p i on the network
perform ance, in the follow ing sim ulation w e restrict ourselves to studying
the behaviour o f the system density over the sim ulation run A s usual, in the
next sim ulations each run w as conducted up to 5000 tim e steps, then data
w ere gathered every 30 tim e steps after a relaxation period o f 200 tim e steps
The turning percentages w ere 25% left, 50% go through, 25% right, this
was perform ed for tw o different networks, 17 and 41 nodes, and using tw o
arrival rates, p = 0 15 and p = 0 5
At low arrival rate (p = 0 15), Fig (4 5) dem onstrates the influence o f the
parameter p i on the system density, both in (a) the system o f size 17 nodes
and in (b) the system o f size 41 nodes
The sim ulation results can be summarised as follow s
• W ithin the parking perm itted period, ( i e in the first stage o f the
sim ulation), increasing the parameter p i from 0 1 to 0 3 has increased
the probability for a car to disappear into one o f the parks This, in turn,
has decreased the system density within this period and this was
observed for both networks
• In contrast, the influence o f increasing the parameter p 1 on the system
density w as m ore noticeable during the off-parking period ( le the
second stage traffic) This increm ent has increased the probability for a
car to leave the park, which, in turn, increased the outgoing traffic from
the parks within a sm all tim e-interval A s a consequence o f this
dynam ic, the system density responds rapidly, l e
is considerably
increased
4
95
p 1= 0 1
M1=0 3
time steps
(a)
p1 = 01
M1= 0 3
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
time steps
(b)
Fig
(4.5)
(■
The influence of introducing underground parks on
the system density over the simulation run at low arrival
rate(u = 0 15) using different values for the parameter ul,
where m
(a) The network of size 17 nodes
(b) The network of size 41 nodes
96
This w as m ore noticeable in the case o f the sm aller size network, where
the m axim um density obtained w as ~ 0 41 compared to ~0 36 obtained
in the case o f the larger network, see Fig (4 5)
This situation w as dom inant until all cars left the parks, w hich last for a
longer period when p 1 = 0 1
At high arrival rate (p= 0 5), the influence o f increasing the Poisson
parameter p i on the system density w as observable only w hen the
underground parks w ere able to cope with the incom ing traffic This is
because the probability for a car to disappear into one o f the parks has
m creased, which, in turn, w ill lead to congestion inside these parks at early
stage
H ow ever, this period did not last for long and the system density m creased
rapidly, irrespective o f the value o f p 1 until the off-parking period started
At this stage the parameter p i , again, retains its influence on the system
density and the outgoing traffic from the underground parks increases by the
increm ent o f p 1, see Fig (4 6)
97
(a)
(b)
Fig
(4
6)
The influence of introducing underground parks on
the system density over the simulation run at high arrival
rate(u = 0
5) using different values for the parameter ul,
where m
(a) The network of size 17 nodes
(b) The network of size 41 nodes
98
4.2 Traffic System under Short and Long-term
events
\
In Chapter 3, w e assum ed that the netw ork neighbourhood w as able to cope
with any number o f cars that m ight leave the sim ulated network This means
that the first site outside any o f the netw ork exits w as alw ays free
In real traffic, this is not always true It som etim es happens that one (or
m ore) o f the network exits is (are) closed for a short-term, due to a car
accident or traffic jam s, etc, or for a long-term due to road works,
etc
The short-term events occur at random and last for a certain period o f time,
whereas the long-term events m ay last for days or even w eeks
To approach reahty these events, stated above, w ere m odelled in our
sim ulations In the follow ing experim ents each sim ulation w as conducted for
6000 tune steps and data w ere collected after a relaxation period o f 200 tim e
steps applying traffic conditions 25% turn left, 50% go straight, 25% turn
right This w as perform ed for a netw ork o f size 17 nodes by applying two
arrival rates, p = 0 15 and 0 35
4.2.1 Modelling Short-term events
The short-term events in our sim ulation w ere m odelled by blocking som e o f
the network exits at random, (l e they are exponentially distributed with
parameter
X), and each event m ay last only for a number o f tim e steps,
w hich w as also chosen at random, (using the function
rand()). H ow ever, the
duration o f any short-term event did not exceed 50 tune steps
The number o f occurrences o f such short-term events w as fixed in our
sim ulation and tw o values w ere used, 15 and 25
To quantify the effect o f introducing such events, sim ulations w ere runs by
blocking one or tw o o f the netw ork exits at random tim e steps, w hich were
exponentially distributed w ith parameter X = 0 005
1
Fig (4 7) show s the influence o f introducing these events on the system
density during the sim ulation run for a netw ork o f size 17 nodes and using a
low arrival rate (p= 0 15) In (a) 15 short-term events w ere im posed
throughout the sim ulation and in (b) the number o f these events w as
increased to 25
In Fig (4 7) the changes in the system density can be interpreted as follow s
the w ay in w hich the system density responds to the short-term
events(obstructions) depends on the follow ing factors
i
the number o f occurrences o f such events, how frequently they w ill
occur9
11
and if they occur for how lon g9
m
the percentage o f the traffic leaving the netw ork at the tim e o f
im posing these events
iv
ht e number o f the netw ork exits w hich are influenced by
these events
In ( i) the tim e-interval, w hich separates betw een any tw o consecutive
obstruction occurrences, depends on the choice o f the parameter
higher the values o f
the value o f
X
X the m ore frequent the obstructions
X
The
In our sim ulation
w as chosen to accom m odate the number o f the obstruction
occurrences over the sim ulation period
100
Increasing the number o f obstruction occurrences, provided that a longer
period for the obstruction duration w as random ly chosen, w ill increase the
system density This can be noticed from F igs (4 7 (a)) and (4 7 (b)), where
the number o f obstruction occurrences has increased from 15 to 25 tim es
(a)
(b)
Fig
(4 7)
system
The effect of introducing short-term events on the
density,
for
a
network
of
17
nodes,
over
the
simulation run at low arrival rate(p = 0 15) where m
(a)
The number of occurrences
of these events was
(b) the number increased to 25
101
15 and m
To get a clear insight into the im portance o f ( 111), let us assum e that an
obstruction exists outside an entry and exit node and its duration period is 20
tim e steps N ow if 25% o f the traffic is leaving the network, then there is a
probability o f P = 0 25 that car w ill be blocked O therw ise it w ill turn left
(right) or go straight through
This m eans that the obstruction has influenced the traffic leaving the
netw ork only for 5 tim e steps and not 20 tim e steps’ This explains w hy a
higher densities w ere obtained in case o f blocking only one exit in
Fig (4 7 (b)) Further sim ulations, not show n here, reveal a considerable role
for the traffic conditions on the system density
The influence o f factor
(i v )
becam e m ore noticeable w hen the arrival rate
was increased to p = 0 35, see Fig (4 8)
Fig
(4 8)
system
The effect of introducing short-term events on the
density,
simulation
run
for
at
a
high
network
arrival
of
17
rate(y
=
number of occurrences of these events was 15
102
nodes,
over
the
0 3*5)
where
the
A s the traffic becom es heavier, blocking m ore o f the netw ork exits, even for
a short period o f tim e, has significantly influenced the system density as can
be seen from Fig (4 8)
103
4.2.2 Modelling Long-term Events
In this section, long-term events have replaced the short-term ones, ( le
blocking som e o f the netw ork exits over the duration o f the sim ulation run)
A lso, four underground parks, each o f capacity up to 300 cars, were
introduced in order to sim ulate loss to flow (at fixed rate with P = 0 5)
The sim ulations w ere perform ed for tw o different network sizes, 17 and 41
nodes, by using different arrival rates, p = 0 1, 0 3, and 0 55 Each
sim ulation run w as conducted for up to 5000 tim e steps and data were
gathered every 30-150 tim e steps after a relaxation period 200-500 time
steps The traffic conditions applied as in Chapter 3, Sec (3 1 5) cases (i) &
(m ) A sum m ary o f the sim ulation results can be found in Appendix B
(Tables (B 5) and (B 6))
In the case o f the sm aller size netw ork our sim ulation reveals that the
highest value o f the m axim um flow (0 2598) w as obtained, w hen only one
o f the netw ork exits w as blocked and data w ere averaged every 150-tim e
steps after a warm-up period o f 200 tim e steps The turning percentages
w ere 12% left, 76% through, 12% right and the arrival rate used w as p = 0 3
In contrast, the low est value o f the m axim um flow (0 2097) w as obtained
when tw o o f the netw ork exits w ere blocked and usm g the sam e sim ulation
parameters as above but at low er arrival rate (p = 0 1)
To get a clear insight into the effect o f introducing both the long-term events
and the underground parks on the netw ork perform ance, w e therefore study
the above tw o extrem e cases (i e.
q„m = 0.2598 and qm.a =0.2097).
104
The traffic behaviour for these sim ulation runs com pared with the case o f
m axim um flow obtained in the transient m ovem ent sim ulation, Chapter 3,
m ay be described with the help o f the fundam ental diagram s presented in
I
Fig (4 9)
03 n
0 25
A£
e
02H
jp
O 015
01
^
a no
blocking
• 1 blocked
2 blocked
0 05
0
I
0
I
!
I
I
I
I
I
I
I
01 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9
1
density
(a)
11
>»
**
08-
A*-l
Ô 06o
d> 0 4 >
02 00
f-
• no blocking
* 1blocked
2 blocked
I
01
I
I
I--------------- 1--------------- 1---------------1--------------- 1--------------- 1--------------- 1
02 03 04 05
06 07 08 09
1
density
(b)
Fig
(4
9) Fundamental
diagrams
of
the
above
two
extreme cases, where m
(a)flow-density relations
105
(b)velocity-density relations
From Fig (4 9), w e can observe the effect o f blocking one or more o f the
netw ork exits as follow s
•
The first striking feature o f Fig (4 9), flow -density relation, is this gap,
w hich isolates the dense traffic regim e This occurs due to the increment
o f the netw ork density from 0 330629 to 0 5969, w hen tw o exits w ere
blocked and from 0 4963 to 0 7059, when only one exit w as blocked
This in turn results in dropping the netw ork velocity from 0 63 to 0 33
and from 0 5 to 0 28 for the above cases respectively, w hich can be seen
from Fig (4 9 (b))
•
This discontinuity in the fundam ental diagram has been obtained in road
traffic m easurem ents, (Edie, 1974, H all, 1986), w hich lead to the
assum ption, that there is a point o f discontinuity in the fundamental
diagram around the m axim um traffic flow , betw een the left part o f the
diagram, representing the free traffic, and the right part o f the diagram,
i
representing the dense traffic The fundam ental diagram obtained for the
case o f highest value o f gmax (Fig (4 10 a) gives a good approxim ation to
the experim ental data in F ig(4 10 b)
•
A lso, introducing long-term events such as underground parks and
blocking som e o f the netw ork exits has decreased the netw ork m axim um
flow from <7 max = 0 2702, w hich obtained in the case o f transient
m ovem ent sim ulation in Chapter 3, to
qmax = 0 2097, w hich obtained by
blocking tw o o f the netw ork exits at low arrival rate(]i = 0 1)
106
0 28
0 26 0 24 -
0 22
-
02
-
0 18 0 16
0 14 -
0 12
01
-
0 08 0 06 0 04 -
0 02
-
0
-
-l-------- 1-------- 1-------01
02
03
04
05
-i------------ r06
07
08
09
density
Fig (4 10 a) The fundamental diagram obtained for the case of
highest value of
= 0 2598
( cars
hour)
k(cat'sAnile)
Fig (4 10 b)
Fundamental
source' Edie
(197 4)
diagram
107
for
road
traffic
flow,
•
Fig (4 9) reveals that introducing long-term events has resulted in low er
densities for the m axim um flow In the case o f transient m ovem ent
sim ulation, g max w as observed at density
p q^ - 0 50, whereas
progressively blocking netw ork exits has decreased the density o f
p q^
from 0 41, which w as observed in case o f blocking one exit, to 0 33
w hen tw o o f the netw ork exits w ere blocked
•
Increasing the netw ork size to 41 nodes has results in low er densities for
the extrem e values o f
qmax The results are summarised in Table (4 1)
It can be seen from Table (4 1) that, in the case o f larger size network,
slightly low er values for the m axim um flow w ere obtained at low er
densities The statistical analysis presented in A ppendix A (Table (A 1))
reveals a significant interaction betw een the netw ork size and the
number o f the blocked exits at a = 0 5 level o f significance
Perform ance
41 nodes network
17 nodes network
M easure
H ighest
L ow est
H ighest
L ow est
M axim um flow
0 2598
0 2097
0 254654
0 2054
D ensity o f m ax flow
0 4439
0 3309
0 4199
0 3168
Table
lowest
(4 1)
values
Comparison
between
the maximum
flow
the
highest
obtained
values
for
two
and
different
networks, where some of the network exits were blocked
108
the
4.3 Summary and conclusion
-t
In first part o f this chapter w e have studied the non-transient m ovem ent o f
cars along the netw orks by introducing underground parks and allow cars to
park inside the network, so that they are temporarily lost to flow This w as
performed at fixed rate (1 e car m ay leave (enter) the park with probability
P) and also at stochastic rate (1 e car arrival or departure to any o f the parks
follow a P oisson distribution w ith parameter p i)
In the second part o f this chapter, w e investigated, in brief, the influence o f
im posing short and long-term events on the netw ork perform ance
Short-term events in our sim ulation w ere m odelled by blocking som e o f the
netw ork exits at random, they w ere exponentially distributed w ith parameter
X, and this blocking lasts only for a certain number o f tim e steps, w hich did
not exceed 50 tim e steps
In contrast, blocking som e o f the network exits during the sim ulation run
have sim ulated the long-term events Our findings can be summarised as
follow s
1
The m ost remarkable aspect from introducing underground parks
w as a different
"jamming threshold" for different netw ork sizes, and
sim ulation with lo ss to flow at stochastic r a te ' show s that the
"jamming threshold” for each netw ork depends on the parameter p 1
n
The sim ulation also reveals that sm all size netw orks respond rapidly
to external changes, especially for networks with loss to flow
109
The sim ulation reveals that the w ay in w hich the system responds to
the sim ulated short-term events depends on their number and how
frequently they w ill occur and the duration o f each occurrence and
also the arrival rate used to feed the system w ith cars The influence
o f these events w as m ore noticeable at higher rate o f arrivals M ore
w ork needs to be done to investigate the influence the traffic
conditions and the change in the parameter X, w hich determ ine how
frequently these events w ill occur, on the netw ork perform ance
B y m odelling long-term events such as blocking one or more o f the
netw ork exits, over the sim ulation run, a significant interaction
betw een the netw ork size and number o f blocked exits w as obtained,
( a = 0 01), at high lev el o f concentration o f cars
D espite the com plex interactions in urban networks, the
characteristic shape o f the fundam ental diagrams w as also obtained
at network traffic level
Chapter 5
“Simple Lane-Changing Rules for Urban Traffic Using
Cellular Automata”
111
5.0 Introduction
To approximate reality the sim ulation tool has to be efficient and represent
som e o f the basic traffic features, which can be observed or tested Using
single-lane netw orks, important features like lane changing and its effect on
the traffic stream behaviour cannot be observed and studied
The basic idea behind the lane changing m echanism is either to maintain a
regular, (desired), speed or to im prove the current speed, where in both cases
the driver is hindered from doing so in his current lane
This situation is not alw ays the case inside urban traffic areas, where drivers
m ay change their lane due to different m otivations, such as perform ing a right
turn at intersections, to avoid obstructions and so on
In the first regim e, (highw ay traffic), one set o f lane-changing rules was
show n to be able to reproduce som e o f the m acroscopic aspects o f traffic flow
on highw ays such as density inversion betw een the tw o lanes, w hich take
place long before the density o f the m axim um flow (Rickert
W agner
et al, 1996,
et al, 1997)
D ue to the com plexity o f the second regim e (urban traffic), w e m ay raise the
follow ing question
Can w e find a perfect set, “super set”, o f
lane-changing rules in order to
accom m odate all the different m otivations for lane changing inside the urban
areas9
>r
112
In this chapter, w e present three different sets o f rules for lane changing as
follow s urgent lane-changing conditions, minimal and maximal conditions for
lane-changing each o f w hich is used in order to accom m odate the driver need
to change lane (Gipps, 1986)
5.1 Motivations for Lane Changing
Lane-changing w ill occur in one o f the follow ing cases
I
T o gain som e speed advantage
If the driver on the left lane, (slow lane), evaluates the other lane traffic
to be better, (1 e the second lane has a low er flow and higher speed),
then with probabihty P_chg he w ill change lane and w ith 1-P_chg he
w ill not
II
To avoid obstruction
Obstructions, such as cars m aking delivery, B us stops, and cars
intending to perform a left, (or right), turn, have been im posed at
random w ith probability P _obs This obstruction is located on the left
lane w ith probability P_lobs and with probability 1- P_lobs on the right
lane Cars w ill m ove to overtake the obstruction as they enter the
obstruction zone
III
T o turn at the next intersection
If the driver on the left lane intends to perform a right turn or the driver
on the right lane intends to perform a left turn, then
lane-changing w ill
occur and this depends on how close the car is to the intersection zone
113
IV
T o avoid slow platoons on the fast lane
Since all cars evaluate the fast lane to be the better one, this w ill result
in clusters o f slow m oving cars At this stage cars can change lane in
order to pass those platoons provided that the slow lane allow this
change
5.2 The two-lane Model:
The tw o-lane m odel consists o f tw o parallel single-lane m odels and three sets
o f rules for
lane-changing defined below , (Equations (5 l)-(5 3)
Our lane changing rules are based on the follow ing steps
(a)- check for vehicles that are qualified for lane-changing according to
the specified lane-changing rules
(b)- m ove the chosen veh icles for lane-changing to the neighbouring
sites in the second lane
( c )- use the m odel rules to update each single lane independently
In step (a), norm ally, cars m ust overtake on the right lane but overtaking on
the left lane is permitted1 W hen the driver intends to turn at the next intersection
2 W here the traffic is m oving more slo w ly on the right lane than the
left lane
Step (b) is needed for the sim ulation purpose only, as in real traffic the car w ill
advance to the destination site directly
Let us assum e that, in reference to F ig.(5.1):
114
dx
is the headw ay distance to the leading car on the destination lane
dy
is the backwards distance to the follow ing car on the destination lane,
where both dx and dy are measured from the em pty neighbouring site
on the destination lane
v
is the speed o f the car w hich intends to change lane
vb
is the speed o f the follow ing car on the destination lane
dz
is the headw ay distance to the car ahead in the current lane
vb
dy
dx
Fig
(5 1)
Then, the necessary conditions for lanes-changing are
dx> v + ki
1
[0, vmaJ
and in order not to disturb the traffic on the second lane
d y > v b + k 2,
2
If w e set
k2 e [0, vmaJ
kl = k2= 0, then w e obtain the minimal conditions for lane changing
as follow s
dx> v,
dy > vb
(5 1 )
115
H ow ever, as the driver gets closer and closer to the given intersection or
obstruction, he is prepared to accept less headw ay distance on the other lane
(1 e dx
= v) This w ill result in the urgent conditions for lane-changing as
follow s
dx> v,
(5 2)
dy > vb
Setting kr
=
i'
k2= vmax yields the maximal conditions for lane-changing as
follow s
d x>v+ vn
max
dy>vb+ vmax
(5 3)
To apply Equation (5 3), w e m ust have dz < a, where a is a parameter w hich
defines the headw ay distance on the left lane that satisfies the driver in order
to stay in his current lane
<■
In order to sim ulate the stochasticity elem ent in the lane-changing m echanism ,
with probability P_chg the driver w ill accept the maximal or minimal
conditions to change his lane and with probability 1- P_chg he w ill not
In case (I), Sec (5 1), the intention o f the driver to im prove his speed, by
changing his lane, is dependent on whether the traffic in the present lane or the
target lane is m ore likely to affect his speed
116
If the driver is hindered in his current lane, (left lane), he requires the minimal
conditions to change lane O therwise, the driver w ill m ove to the second lane
if it is worthw hile to do so, which m eans that the driver is not satisfied with
his current headw ay distance ( a < dz) and evaluates the second lane as the
better lane In this case, the maximal conditions for lane-changing are
required Fig (5 2) presents the flow -chart for the lane-changing mechanism .
In case (II), Sec(5 2), the obstruction zone is defined as being 3 sites distance
from the obstruction A s the driver m overs nearer to the obstruction, he is
ready to accept the minimal or urgent condition to change lane, in order to
avoid being blocked by the obstruction
In case (III), Sec(5 1), as long as the driver is outside the turning zone, it has
no influence on the lane-changing decision and the driver is concerned only
about im proving his speed, case (I)
W hen the driver enters the turning zone, he ignores any attempt to change lane
in order to im prove his speed and looks for minimal conditions to change lane
A s the driver m oves closer to the junction and enters the urgent zone for lanechanging he looks for the urgent conditions for lane-changing in order to be in
the right lane to perform the required turn
In case (IV ), Sec (5 1) to avoid slow platoons on the fast lane the driver looks
for the m inim um requirem ents to change his lane A platoon is defined as a
block o f 5 or m ore occupied sites, for m ost o f w hich are stopped
117
no
yes
*
kw
Get positionrelativeto
intersection (d)
no
no
yes
>
w
Get positionto
obstruction (k)
no
F ig (5 2)
118
5.3 Simulation and results
In the follow in g sim ulation experim ents, each sim ulation is conducted for 50
runs, where each run is calculated up to 2000 tim e steps to generate the
average case statistics Im posing periodic boundary conditions, w e simulated
a system o f size 2x100 sites, tw o-lane, w hich is large enough for an urban
street The turning probabilities at the intersections are 0.4 left and 0 30 right
The intersection zone is considered to be 10 sites (or less) under minimal
conditions and 5 sites or less for urgent conditions at the sam e intersection.
H eadw ay parameter, a = 3 (i e headway distance = 22 5m ), w hich is
reasonable for urban traffic The basic lane-changing parameters in our
sim ulation are
i
lane-changing probability (P_chg)
n
obstruction probability (P_obs)
in
lane obstruction probability (P_Lobs)
5.3.1 Lane-usage behaviour
In order to get clear insight about the im pact o f the lane-changing parameters
on the lane-usage frequency, w e have perform ed tw o different sim ulations by
changing the value o f the above parameters as follow
Experiment I:
In the first sim ulation experim ent, w e have fixed the lane-changing
probability at P_chg = 0 4 and m odified the values o f the other tw o
parameters (P_obs = (0.3, 0.4) andP _L obs = (0 .7 ,0 .3 ), (0 .6 ,0 .4 ), (0.5, 0.5)).
119
(a)
(b)
Fig
(5 3)
different
P_obs = 0 3
Lane
usage
simulation
of
frequency
size
2xk,
vs
traffic
k=100,
density
where
P_chg
for
=
two
0 4,
and m
(a)
P_Lobs = 0
5, 0 5 for left and right lane respectively
(b)
P_Lobs = 0
6, 0 4 for left and right lane, respectively
120
Simulation P_chg
P_obs
P_Lobs
Low density
Left
Right Inversion point
high density
Inversion point
R unl
04
03
07
03
0 075
0 475
Run2
04
03
06
04
0 115
0 505
Run3
04
03
05
05
0 295
0 625
Run4
04
04
07
03
0 065
0 485
Run5
04
04
06
04
0 105
0 515
Run6
04
04
05
05
0 275
0 715
Table
(5 1)
Simulation Experiment I
each "run'" refers to an
average of 50 Experiments over 2000 time steps each to obtain
average case statistics
The relationship betw een the lane-usage frequency and the sim ulation density
using different values for P_obs and P_L obs is presented in Table (5 1) and
Fig (5 3) The observations can be sum m arized as follow s
i
It can be seen from Fig (5 3) that both lanes have the sam e lane usagefrequency at three different points The first tw o points are called
“ density inversion” points At these points, high traffic-density sw itches
from one lane to the other lane
n
The first “density inversion” point has occurred at density p = 0 295 in
case (a) and at p = 0 115 for case (b), w hile the second one has occurred
at m uch higher density p = 0 625, 0 505 for both cases (a) and (b)
respectively
in
The location o f these points depends on the lane-changing parameters,
w hich from Table (5.1) m ay be summarised as follow s
121
(a) The effect of lane obstruction probability(P_Lobs)
This parameter has considerable effect on lane-usage frequency For
exam ple, using PJL obs = 0 6 for the left lane and P_Lobs = 0 4 for the
right lane has influenced the lane-usage frequency for both lanes at an
early stage, (low density) The first “ density inversion” point has
occurred at
p{ = 0 115, where usage o f the right lane becom es higher
than usage o f the left lane, and again the tw o lanes sw itch traffic at p2
= 0 505, where the left lane becom es m ore crow ded than the right one
In contrast, using both lanes w ith the sam e P_L obs = 0 5 has increased
the above densities ( px,p 2) to a m uch higher values ( p t = 0 295 and
p2 = 0 625), see Fig (5 3)
This can be explained as follow s
Increasing P_L obs for the left lane w ill disturb its traffic, even at low
i,
density, due to the frequent, random ly occurring obstructions
This results in the right lane again becom ing more crow ded at p =
0 505 A t this stage, a car w ill again sw itch lanes in order to avoid
traffic jam s H ow ever, applying the sam e P_L obs to both lanes has
increased the low -density inversion point from 0 115 to 0 295 This
increase is to be expected as the m otivation for lane-changing reduces
Fig (5 4) show s the location o f the inversion points for different values
o f P_L obs
122
Fig
(5 4) left
values
of
lane
P_Lobs
usage
and
intersection between
vs
traffic
P_chg
the
=
0 4,
lane-usage
density-, for
and
curve
P_obs
and
=
the
different
03
Each
horizontal
line represents an inversion point
Fig
(5 5) left
values
of
lane
P_Lobs
usage
and
intersection between
vs
P_chg
the
traffic
=
0 4,
lane-usage
line represents an inversion point
123
density
and
curve
for
P_obs
and
=
the
different
04
Each
horizontal
(b) The effect of obstruction probability(P_obs)
The
obstruction probability also influences the lahe-usage frequency
for both lanes, especially w hen there is no considerable difference
betw een the values o f P_L obs or w hen the tw o lanes are used with
sam e P_Lobs, (see Run3 and Run6 in Table (5 1)) For these runs,
increasing P_obs has resulted in low er value for the low density
inversion point and higher value for the higher density inversion point,
see F igs (5 4) and (5 5)
This can be explained as follow s
Increasing the value o f P_obs w ill result in m ore obstruction
occurrences on both lanes, which, in turn, increases the usage o f the
right lane over the left lane at density (0 27) This is to be expected as
the chance for a car to overtake on the right lane is greater than its
chance for overtaking on the left lane This increase in the usage o f the
right lane, w ill continue, follow ed by decrease in the left lane usage,
until p ~ 0 5, where the usage o f the right lane attain its m axim um
value over the left lane At this stage traffic starts to sw itch gradually
from the right lane to the left lane and due to the increase in
obstruction probability (P_obs), the second inversion point occurs at
higher density (0 71)
124
Experiment II
To exam ine the effect o f the
lane-changing probability on the lane usage
behavior, in the second experim ent, presented in Table (5 2), w e have
increased the lane-changing probability to P_chg = 0 5 and also used different
values for the other tw o
lane-changing parameters In general the sam e
m acroscopic phenom enon, “density inversion” points, w as observed
Table (5 2) show s the effect on the “ density
j
inversion” points o f increasing
P_chg to 0 5 This m eans that the “density inversion’' is m odified by the
lane-
changing parameters and m ore sim ulations are needed in order to determine
if there is a lim it for such changes For exam ple, Table (5 2) dem onstrates
that increasing
lane-changing probability as w ell as obstruction probability
resulted in a smaller density for the first inversion point (from p = 0 295 to p =
0 275) and higher density for the second inversion point (from p = 0 635 to
p = 0 735)
Simulation
P_chg
P_obs
Run7
05
03
P_Lobs
Left Right
07
03
Run8
05
03
06
Run9
05
03
Run 10
05
Run 11
Run 12
Table
Low density
Inversion point
0 085
high density
Inversion point
04
0 135
0 505
05
05
0 295
0 635
04
07
03
0 065
0 475
05
04
06
04
0 115
0 519
05
04
05
05
0 275
0 735
(5 2)
Simulation Experiment II
0 465
each " run" refers to
an average of 50 Experiments over 2000 time steps each to
obtain average case statistics
125
5.3.2 Flow Behavior
i
D espite adding another set o f lane-changing rules, the m axim um flow
for the tw o-lane m odel is m uch higher than the m axim um flow for the
single-lane m odel This can be seen clearly from Fig (5 6) These results
in Fig (5 6) are based on sim ulating tw o system s, the first o f size k=100
representing
a single-lane road, the second o f size k = 2x100
representing a tw o-lane road In both cases data w ere averaged over 50
runs and each run is conducted up to 2000 time steps
n The tw o-lane flow reaches its m axim um , approxim ately, at p ~ 0 5,
w hich is the density o f m axim um flow for the single-lane m odel, (see
F ig(5 6))
in In the case o f the single-lane m odel, there is a sharp bend in the flow density curve, w hich is not found in the tw o-lane m odel This bend
m eans that the dynam ics below and above the density o f m axim um
flo w (p ?max) for the single-lane m odel are different from those o f the
tw o-lane m odel, i e flow breaks-down quickly in sm gle-lane m odel
iv
The change in the
lane obstruction probability (P_Lobs) seem s to have
a great im pact on the flow behavior on both lanes as follow s
a) A pplying P_L obs=0 7 for the left lane and P_L obs = 0 3 for the right
lane has decreased flow on the left lane com pared to the right lane
E ven at low er densities The flow is correspondingly increased on the
right lane, see Fig (5 7 (a))
126
b) W hen P_L obs =0 5 for both lanes, flow for both w as similar at low
densities but, at higher densities, (p > 0 5), w e observed a
considerable change in the traffic flow betw een the tw o lanes, see
Fig (5 7 (b)) The first intersection point o f the tw o curves obtained
at low density
"inversion point ", w hile the last one occurred at the
high density
"inversion point" The graph also show s a third
inversion point w hich does not appear in lane-usage v s traffic
density relation presented in F ig(5 3)
For both cases (a) and (b) w e obtained similar traffic patterns by using
different values for P _chg,(0 4 ,0 5, 0 6), but m axim um flow on both lanes
decreased w hen the value o f P_obs w as increased, as can be seen from
Fig (5 8)
v
Fig (5 9) show s that as w e increased the
obstruction probability, the
m axim um flow for the tw o-lane m odel decreased from q ~ 0 77 for
P_obs = 0 3 to q = 0 68 for P_obs = 0 5, (i e a decreases o f 11 6 %), no
matter whether the obstacle w as located in the left or right lane
—
tw o lane
————sing le lane
d en s ity
Fig
(5 6)
flow vs density relation for both single-lane
and two-lane Models
127
0.45
0.4
0.35
0.3
S 0.25
£ 0.2
flow, left lane
flow, right lane
n
-
0.15 0.1
-
0.05 -
0
0
density
(a)
0.45
0.4
0.35
flow, left lane
flow, right lane
0.3
5 0.25
è
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
density
(b)
F i g . (5.7):
flow-density relation for two different simulations
of size k = 2x100;
periods
two lane
.Data were averaged over long time
over 50 runs using P_chg=0.4,
P_Lobs = 0 .7, 0.3
and
in
(b)
P_Lobs = 0.5,
right lanes respectively.
128
P_obs=0.4,
0.5
where
for both
in
(a)
left
and
left lane, P_obs = 0 3
right lane, P_obs = 03
left lane P_obs = 0 4
right lane P_obs = 0 4
density
Fig
(5 8)
size
k
where
=
flow-density
2x100,
P_chg
=
relation
two-lane,
04
and
using
P_Lobs
=
for
different
different
0 7,
0 3
simulation
values
for
for
both
of
P_obs,
left
and
right lanes respectively
Fig
(5 9)
comparison of
flows
for different values
where P_chg and P__Lobs are fixed as m
129
Fig
(5 8)
of P_obs,
5.3.3 Lane-changing behavior
T o understand lane-changing dynam ics, w e have to study the relations
betw een
lane-changing frequency and lane-changing parameters This study
can be sum m arized as follow s
1
The influence o f varying P_L obs on
lane-changing frequency is
presented in Fig (5 10) The m axim um value o f
lane-changing
frequency (0 001706) w as obtained at a traffic density o f 0 2 9 5,when
PJL obs = 0 5 for both lanes H ow ever, increasing PJL obs for the left
lane decreased the frequency o f lane changes to 0 001527 at the same
density This dynam ic is observed at different values o f P_obs = 0 3,
0 4, and 0 5
n
Fig (5 10), also show s that
lane-changing frequency increases linearly
at low densities, (p < 0 125 ), and this is observed for all values o f
P_L obs Further sim ulations, see below , also show that as w e exceed
the density p = 0 1 2 5 , different values o f the
lane-changing
parameters w ill influence the lane-changing frequency Fig (5 10)
show s tw o significant peaks when P_L obs =(0 5,0 5), the first at the
first inversion point w hile the second occurs before the second
inversion take place
m
The m axim um number o f lane changes occurred at density p ~ 0 3 ±
0 01, which is w ell below the density o f m axim um flow for the 2x100
site system (p = 0 5)
130
iv
The m axim um frequency o f lane-changes to the right lane (0 000889)
w as m uch higher than the m axim um frequency to the left lane
(0 000655) and w as obtained at density p = 0 295 This is higher than
that found for m axim um frequency o f changes to the left lane (p =
0 245) This can be seen from Fig (5 11), for different values o f
P_L obs and P_obs
S'
- P_Lobs =0 7 ,0 3
0 0018
- P_Lobs =06 0 4
0 0 0 16 -
-P Lobs =05 0 5
® 0 0014 -
o 0 0012 n 0001 § , 0 0008 ra 0 0 0 0 6 °
0 0004-
jj 0 00020
-
0
01
02
03
04
05
06
07
08
09
1
density
Fig (5 10) lane-changes
frequency vs
traffic
different values of P_Lobs while P_chg = 0 4
density
for
and P_obs =
o 4
v
Increasing the value o f P_obs from 0 3 to 0 5 has increased the
m axim um frequency o f lane changes to 0 0017 04 (at p = 0 275), from
0 001486 (at p = 0 285) This is to be expected, because drivers have
to change their lane m order to avoid being . halted behind an
obstruction and these lane changes increase, as the occurrence o f
obstructions becom es more frequent
131
vi
The effect o f
lane-changing probability (P_chg) seem s to be
influenced by the values o f the other tw o lane-changing parameters,
P_obs and P_Lobs, as follow s
A pplying P_chg = 0 4, P_obs = 0 4 and P_L obs = 0 5 , (for each lane),
the m axim um lane-changing frequency obtained w as 0 001706, see
lane-changing probability
Fig (5 12 (a)) H ow ever, w hen w e increased
to P_chg = 0 6 and reduced the value o f P_obs to 0 3, also increasing
P_L obs for left lane to 0 7, the m axim um
lane-changing frequency
dropped to 0 001447, see Fig (5 12 (b)) This exam ple dem onstrates
that increasing the value o f P_chg w ill not always yield a higher
frequency o f lane changes w ithout controlling values o f the other two
parameters H ow ever, further sim ulations are required in order to
quantify m ore the effect o f changing o f P_chg
0
02
04
06
08
1
density
Fig
lane
(5 11)
and
comparison between lane changing frequency to left
right
lane
When
p >
0 125
the
increase
changes over left changes becomes more noticeable.
132
m
right
0 00 18
P _ ch g = 0 4 P_obs = 0 3,
P_Lobs = 0 7, 0 3
0 00 16
P_chg=0 4 P_obs = 0 3
P_Lobs = 0 5,0 5
P_chg=0 4 P_obs = 0 4,
P_Lobs = 0 7 0 3
O,
0 001 H
c
5
0 0008 -
o
g
(0
0 00 06
c
j=
P_chg=0 4 P_obs = 0 4,
P_Lobs = 0 5 0 5
0 0004
(a)
“ P_chg=0 6, P_obs = 0 3,
P_Lobs = 0 7 0 3
- P_chg=0 6 P_obs = 0 3
P__Lobs s= 0 5,0 5
P_chg=0 6, P_obs = 0 4
P_Lobs = 0 7 0 3
- P_chg=0 6 P_obs = 0 4
P Lobs s 0 5 0 5
04
06
08
density
(b)
Fig
(5 12)
The
effect
frequency
of
effect
increasing
of
lane
of
changes
P_chg
lane-changing
It
on
is
the
easy
probability
to
frequency
observe
of
depends on the other parameters, P_Lobs and P_obs
133
lane
on
the
that
the
changes
5.4
Model validation
Unfortunately, there exists no source o f real data, w hich studies lane-usage
characteristics on tw o-lane roads in urban areas In this regard none o f the
available studies w ere devoted for urban traffic, but they w ere concerned only
with highw ay traffic So, in order to calibrate our
lane-changing rules, w e
have, perform ed a sim ple study on one o f D ublin’s tw o-lane streets This w as
attempted in M ay 1998, as follow s A video camera w as set up on a high
building, overlooking the street o f interest The street, w hich is 250m in
length, is controlled by traffic lights from both sides at tw o intersections and
has one pedestrian crossing W e w ere interested in calculating the lane usage
frequency for each lane D ata w ere obtained for 4-hours per day for four
different consecutive days One lim it o f this em pirical study w as the lack o f
required equipm ent for surveying a longer urban street with many
intersections Another, clearly, w as the lim ited period o f tim e for w hich the
survey could be considered and w e acknow ledge D ublin Corporation, Traffic
Section, for facilitating us on this Unfortunately, w e did not find any study o f
urban traffic in relation to
lane-changing to com pare our data with
On highw ay traffic, Sparmann (1978) has dem onstrated an important
m acroscopic phenom enon called “
lane-usage inversion” or “ density
inversion”, w hich occurs long before m axim um flow
The
lane-changing rules, presented in this chapter, appear to reproduce the
sam e phenom enon for urban traffic, within the lim itations o f the sim ulation
em ployed, c f F igs (5 13) and (5 14)
134
Fig
(5 13) lane-usage
vs
traffic
density
for
empirical
data
Data were averaged every 3 -minutes for a small Road segment m
Dublin City
Fig
(5 14)
lane-usage
frequency vs
traffic density
It can be
seen that the first "density inversion" has occurred also long
before maximum flow density
135
The first
“density inversion” in our sim ulation has occurred long before the
density o f m axim um flow This is in agreem ent with Sparmann (1978)
A s has been described in Sec (5 3), the “density inversion “ point is adjustable
by the
lane-changing parameters, P_chg, P_obs, and P_L obs The choice
P_chg = 0 4, P_obs = 0 3, P_L obs = 0 5 for both lanes left and right, gives a
reasonably good approxim ation to our observed data
The data, presented in Fig (5 13) indicate that the lane-usage frequency for
both lanes is not correctly m odelled at very large densities (p > 0 7) The left
lane flow is higher than the right lane flow This, for exam ple, m ay be due to
the fact that long veh icles w ere not m odelled in our sim ulations, w hich have a
great influence on the city traffic H ow ever, the lim itations o f this study in
regard o f street and tim e for data collection have been noted earlier clearly,
further experim ental or observational studies on larger roads are needed
136
Chapter 6
“Stochastic Cellular Automata Model for Inter-urban Traffic”
137
6.0 Introduction
In this chapter, w e m ove from one-speed determ inistic cellular automaton
\c
m odels, where stochasticity w as introduced through the random feeding
m echanism and probabilistic lane-changing rules, to a com pletely stochastic
cellular autom aton m odel, w ith m ore than one-speed
The need for such m odels becom es necessary as w e m ove from urban areas,
where links betw een junctions do not exceed 50-500 m, to inter-urban areas
w ith links, w hich m ay exceed 10-15 km
In the latter case, cars have a good chance to interact with each other,
generating a very com plex driving process or "traffic dynamics"
In this m odel w e try to sim plify the above, com plicated dynam ics by
concentrating on the m ost important elem ents o f driving, o f course without
losing the essentials o f the start-stop w aves dynamic
A ccording to the rules o f road traffic in Ireland, the follow in g Table (6 1)
dem onstrates the car velocity and the safe distance needed in order to brake
without colliding with the leading car (or car in front)
T ab le 6.1
V elocity
V= 1
v=2
v=3
Safe distance in sites
1 site
3 sites
5 sites
W here one velocity unit is equivalent to 13 5 mph
The follow ing relation can give the safe distance
velocity v relative to the front car
138
ds for a car m oving w ith
<fc = 2 ( v - l ) + l,
v = 1 ,2 ,3
(6 1)
Car acceleration or deceleration at the next tim e step is then dependent on the
above safe distance
6.1 The model
The m odel is defined on a one-dim ensional lattice, w hich is regarded as a road
segm ent, o f k identical sites (cells) The state o f every cell)> is either em pty or
occupied by a car, where each car has an integer velocity varied betw een 0 and
m axim um velocity vmax The number o f em pty sites ahead o f a given car is
called
gap
6.1.1 Definition of the Model
For any given car, the number o f em pty sites to the front car after securing the
safe distance for the current car speed is called
"free headway distance” and
denoted by “d;t”, w hich can be defined as follow s
dx = gap - ds
-
(6 2)
i
The m odel update rules depend on the relation betw een the car velocity and its
free headway distance This m eans that starting from an arbitrary
configuration o f cars, the follow ing rules are applied in order to update the cell
state at tim e instant
t+ 1, given its state at tim e t This is applied
sim ultaneously for all cars
139
Rule 1
"Irregularities in the driver behaviour”
Let v be the current car speed and its target speed at the next tim e step
v ia *
If ds (v ±a)
> gap(v ±a),
a = 0 ,1 ,2 ,3
Then, with probability P_inc, the driver w ill replace
gap (v ± a) + b by ds(v ± a),
Rule
b e Z, 1 < b < v ± a
2 "acceleration”
If v < vmax and ds (v + 1)
<gap (v + I)
Then, w ith probability P_ac, the car w ill accelerate to
S = v + 1,
Rule
3 ’’deceleration”
If v >
dbc(v) and ds(v-a) < gap(v-a),
Then the car w ill decelerate to
S = v-a,
Rule 4
a =1, 2, 3
"speed fluctuation”
If the car is m oving w ith constant velocity, then with probability P_rd
the car w ill reduce its speed to
S =v-r,
Rule 5
r e Z and r e [1, 3]
“car movement”
Every car advances d sites ahead where
d = m ax(v, S)
a = 1 in case o f acceleration
140
6.1.2 Implementation of the Model
In what follow s, w e chose vmax = 3 to confirm w ith traffic regulations in
Dublin, w hich allow a speed o f 40 M PH (~ 3 in the m odel)in inter-urban
areas
The use o f the above probabilities, P_inc, P_rd, and P_ac, is essential in order
to sim ulate realistic traffic flow in inter-urban areas T hey generate three
different traffic properties, w hich characterise the inter-urban traffic as
follow s
i. Irregularities in the drivers behaviour
In inter-urban areas its is obvious that not all the drivers w ill com ply with
the rules o f the road and keep the required safe distance to the car ahead
In Rule 1, our aim is to m odel this “ irregular behaviour” o f the driver
using the probability P_inc, w hich control the irregularities in the driver
behaviour T his can be dem onstrated as follow s
Example:
A ssum e that w e have the follow in g configuration o f cars, where each
number represents the current car velocity and "dots" are em pty sites
. . . . 2 ..........3. . . .
N o w the follow ing car has the velocity
v = 2, which m eans that the
required safe distance for this speed from Equation (6 1) is
ds(v=2) = 2 ( 2 - l) + l = 3
and the free
headway distance for this car from Equation (6 2) is
dx(v=2) = 6 - 3 = 3
141
As
dx(v) > v, the driver w ill try to accelerate to v + 1 and the requirement
for this acceleration, according to Rule 2, is that
ds(v + 1) <gap(v + I)
According to the above configuration
ds(v + I) = 5 and gap(v + 1) = 3
w hich im plies that
ds(v + I) > gap(v + I)
N o w Rule 1 applies, because
b = 2 < v+ 1 Therefore, with probability
P_inc the driver w ill replace
gap(v + I) + 2 by dsiv + 1) and with
probability 1- P_inc he w ill not
ii. D elay factor
Rule 2 reads as follow s
W ith a certain probability, P_ac, only, a car w ill accelerate to the new
speed v + 1 and with probability 1 - P_ac it w ill not, even if the
requirem ents for acceleration are com pletely fulfilled
The reason behind introducing the above probability is to m odel a very
important property o f realistic traffic flow in inter-urban areas, w hich is
the delay in acceleration For exam ple, this d ela y 1m ay occur in the
follow in g cases
i
The car is travelling with v < vmax and it has
enough free headway
distance to accelerate, but the car is com ing to a pedestrian or traffic
light
u
Other problem s not related to normal road features, e g bad weather,
Road works, and faulty car perform ance
142
iii. Speed fluctuations
Traffic through urban areas is controlled by m any factors such as; traffic
lights, pedestrians, zebra crossings
Therefore, the free
etc.
headway distance is not alw ays the main cause behind
the oscillation in the car m ovem ents betw een acceleration or deceleration.
The above factors lead to a stochastic sequence o f changes in car speed and
speed fluctuations, w hich vary, from one car to another. In our stochastic
m odel, a fluctuation in speed occurs, with probability P_rd, provided that a
dx(v) then,
car is m oving with constant speed as follow s: If v =
or v =
with probability P_rd, the velocity v is reduced to v -
r, where r is an
integer random ly chosen from the interval [1, vmax ].
U sing the above probabilities, P_inc, P_rd, and P_ac, w ill generate
stochastic sequences for the changes in the car speed as follow s:
0
—>
0
—>
1 1
—>
—>
2
—>
2
—>
2
—>
0 >1
—
.........
W here each integer number represents the current car velocity.
6.2 Simulation and Results
W e have perform ed sim ulations with the Stochastic Cellular Autom ata M odel
(SC A M ) with various initial conditions. The Lattice size = 200 cells and
closed boundary conditions have been applied. In our sim ulation, car velocity
ranges from v = 0 to vmax = 3 and different values for the m odel parameters,
P_inc, P_rd, and P_ac, were investigated, see Sec(6.2.3).
143
Each run w as calculated up to 3000 tim e steps, and the data averaged over 50
runs, after a discard period o f 500 tim e steps Each curve contains about 100
data points
6.2.1 Model Dynamics
M odel dynam ics are easier to follow by visualising the tim e-evolution o f the
Autom aton with the help o f the space-tim e diagrams e g (Fig (6 1))
Space
2
►
1
0 0 1 00 0 00
2
0 1
0 1 000 1 0 1
2
1 1
2
1 01 0 00 1 00 2
2 1 2
2 0 1 0 oo :1 0 1 2
1 1
1 0 00 00 00 1
3
1 2
1 1 00 00 0 1 1
1
3
1
2 1 000 0 1 0 0 2
2 1
1 0 000 0 1 1 1 2
1 2
1 0 000 1 1 0 1
3
2 1
2 1 0 00 1 1 0 0 1
1 1
3
2 00 00 1 0 0 1 1
1
1
1 00 00 1 0 0 1 1
2 1
000 0 1 00 0 1 2
2 1
3
000 0 1 oc1 1 1
2
1 1
2 000 0 1 0 1 1 1
3
0
0000 1 00 1 1
2
0
3
0000 1 0 1 1 1
3
3 0
000 1 0 0 0 0 1
0
0 1
1 000 1 0 0 0 1 2
1
0 1
1 000 1 0 1 0 1 2
1
2
000 1 0 1 0 1 1
0
2
2
Fig
a
2
1 0 1 00 0 00_______ 2_______ 1 1
2
(6 1)
system
P_rd=
system
space-time diagram at density
of
0 3,
size
P_ac
after
one
200
cells
and
Vmax
=0 5
Each
new
row
update
step
and
=
(p)
of 0 22 for
3,
P_mc=
shows
just
the
after
0 5,
traffic
the
car
motion
R ow s are configurations at consecutive tim e-steps D ots represent em pty sites,
and integer numbers indicate the car velocity located at the given site.
144
The m ost im pressive w ay to describe the traffic flow is by a diagram, which
show s the space-tim e curves o f a set o f cars over a road segm ent Such
diagrams have been produced by Treiterer (1975) in the United States by
aerial photography The typical patterns o f car configuration for the car
densities 0 16 and 0 22 are presented in Fig (6 2 a and b), where the first
density is slightly below the density o f m axim um flow , see below , and the
second is slightly above the density o f m axim um flow
In Fig (6 2) each black P ixel represents a car, with trajectory indicated by a
dotted curve Cars are m oving from left to right and from top to bottom , i e
from top left corner to bottom right corner Vertical trajectories m ean that cars
are com pletely blocked, whereas diagonal trajectories show car m ovem ent
Traffic jams or jam waves are identifiable as solid areas in the space-tim e
diagrams At low densities, slightly below the density o f m axim um flow , (as
in Fig (6 2 a)), w e find that laminar traffic is the dom inant pattern, despite the
appearance o f sm all jam s, w hich occurs randomly due to speed fluctuations
The appearance o f these sm all jam s becom es m ore noticeable as w e exceed
the density o f m axim um flow , see Fig (6 2 b), and their lifetim e becom es
relatively longer
On the other hand, the high-density regim e, (Fig (6 3)), is characterised by
random form ation o f congested clusters, due to the stochasticity elem ents in
our m odel The lifetim e o f these traffic jam s becom es continuous to increase
compared to those obtained at low densities (Fig (6 2 (a) and (b))
The Figure also show s the backward m ovem ent o f the traffic jam s, i e against
the traffic flow
145
► space (Road)
Fig
(6 2)
The typical patterns of car configurations up
to 1500 time steps, where L =200, Vmax = 3, P_inc= 0 5,
P_rd= 0 3, P_ac =0 5
(a) the pattern for density of 0 16 (b) the pattern for
density of 0 22
146
Fig (6 3) show s that these congested clusters represents a typical start-stop
w ave and, due to lack o f real data in our study, may be com pared to data
obtained for sim ulated highw ay traffic (N age and Schreckenberg(1992) On
this basis, the m odel appears to produce fairly realistic results
Road
►
Time
Y
Fig
(6 3)
0 5,
cars,
P_rd=
Space-time
0 3,
P_ac
diagram
=0 5,
and
consecutive horizontal
consecutive
time
steps
for
It
p
the
SCAM,
= 0 5
be
jams,(sold areas), are moving backwards
147
= 3,
Black dots
lines represent
can
V
seen
P_mc=
represent
configurations at
that
the
traffic
6.2.2 Fundamental diagrams
N ext w e study the fundam ental diagrams, flow -density relation
q-p and
velocity-density relation v-p, o f our SCA M
The definitions o f the above m acroscopic aggregate variables,
q, p, and v, for
which data are show n in the fundam ental diagram s, have been introduced in
Sec (2 1 1), using Equations (2 l)-(2 4)
The sim ulation reveals that the m axim um flow obtained w as
qmax ~ 0 24 ±
0 02 at the density p 9max = 0 18 ± 0 01 (Fig (6 4 a)) In contrast the m axim um
flow obtained by N a gel and Schreckenberge (1992), in highw ay traffic, was
0 318 ± 0 0005 and obtained at low er density (0 085 ± 0 004), where vmax = 5
Further sim ulations, (Fig (6 4 (b)) show s that the system size does not have a
significant effect on the above tw o extrem es,
q uax, p ?max, where data were
K
long-term averaged
The flow -density relation (Fig.(6 4)) can be characterised as follow s
l
L ow -density phase, where the drivers change their speed according to
rules o f the road and the traffic control system In this regim e, flow
increases linearly by increasing the traffic density
n
H igh-density phase, where the drivers change their speeds according to
the needs o f driving in a queue The high-density phase is characterised
by periodic start-stop w aves (Fig (6 3)) and the significant drop in the
system flow as in Fig (6 4)
In betw een the above tw o-regim es, the traffic dynam ic lsh a rd to explain In
Sec (1.1 2 1) w e outlined the explanation given by H all (1995)
148
Fig
(6 4 a )
over
short
size
=
flow-density
time periods
200
after
a
relation
(50 time
transient
where Vmax = 3, P _ m c =
Points
steps)
period
of
are
averaged
for
a
system
500
time
of
steps,
0 5, P_rd= 0 3, P_ac =0 5
k=200
k=400
density
Fig
(6 4 b)
over
long
size
=
flow-density
time
200,
periods
400
after
relation
(2000
a
time
Lines
steps)
transient
are
for
period
of
averaged
systems
500
steps, where Vmax = 3, P _ m c = 0 5, P_rd= 0 3, P_ac =0 5
149
of
time
6.2.3 The Effect of Varying the Model Parameters
Firstly, w e look at the influence o f changing the m odel m axim um integer
velocity on the tw o-extrem es, <?max, p9max
Our sim ulations, (Fig (6 5)), show that the position and the form o f the
m axim um flow depend strongly on the m axim um integer velocity Reducing
the value o f vmax in the m odel has shifted the m axim um flow g max to higher
values o f density p w hile decreasing its value This is m ore noticeable for
vmai= 1, where the value o f
qmm has decreased from 0 2209, in case o f
vmax =3, to 0 1286 obtained for vmax = l( i e a decrem ent o f 58% approximatly)
A lso, the fundam ental diagrams show the asym m etry seen for real data
obtained for highw ay traffic (H all and Gunter, 1986)
vmax=3
vmax=2
vmax=1
density
F i g (6 5)
Fundamental diagrams obtained by varying the model
maximum integer velocity, where P _ m c
= 05
150
= 0 5, P_rd= 0 3, P_ac
The system velocity as a function o f the traffic density for different values o f
vmax is presented in Fig (6 6)
•vm ax=3
~vm ax=2
-vmax=1
oo
4)
>
0 2
04
06
0 8
d ensity
Fig
(6 6)
model
Velocity-density
maximum
velocity,
relations
where
P_inc
obtained by varying
=
0 5,
P_rd=
0 3,
the
P_ac
=0 5
Fig (6 7) show s that the changes in the acceleration probability (P_ac) have
significantly influenced the flow behaviour and the m axim um flow obtained
P_ac=0 7
T — P_ac=0 5
P ac=0 3
d ensity
Fig
(6 7)
Different
fundamental
the value of P_ac, where VmM
diagrams
obtained by varying
= 3, P_inc = 0.5, and P_rd= 0.3
151
Increasing the value o f P_ac from 0 3 to 0 7 has increased the m axim um flow
from 0 1554 to 0 2557(i e an increm ent o f 65% approximately)
In contrast using higher values for P_rd has reduced the m axim um flow from
¿7niax = 0 2209 to £/max = 0 2008, w hich m eans a decrem ent o f 9%
approxim ately, see Fig (6 8)
Fig
(6 8)
fundamental diagrams obtained by varying the value
of P_rd, where Vmax = 3, P_inc = 0 5, and P_ac= 0 5
Fig (6 9) show s that varying the value o f P_inc has influenced both the
position and the values o f m axim um flow Increasing the value o f P_inc from
0 3 to 0 7 has shifted the m axim um flow to higher values o f p (0 22) and
slightly decreases its value from 0 2237 to 0 2181 A t this density, 0 22, the
tw o curves crossover and the flow becom es higher as P_inc increases from 0 3
to 0 7 This situation w as dom inant until p ~ 0 55, then the flow becom es
similar on both lanes A ll the above experim ents involve changing the m odel
parameters individually, l e varying one o f the m od el’s parameters w hile
keeping others at the sam e values
152
Fig
(6 9)
Flow-density relations obtained by varying the
value of P _ m c ,
where Vmax = 3, P_ac = 0 5, and P_rd= 0 3
This has resulted in different traffic flow patterns, w hich can be interpreted as
changes in the traffic behaviour, due to different circum stances, (e g speed
lim its, bad weather, road conditions, etc )
Secondly, by keeping vmax =3, w e look at different com binations o f the m odel
param eters(P_inc, P_rd, P_ac) The summary o f the sim ulation runs can be
found in Table (6 2), where three values for each parameters w ere used(0 3,
0 5, and 0 7)
Looking at the values o f <7 max, p ?max in Table (6 2), it is easy to see that the
m odel can be adapted to m odel a w id e range o f traffic circum stances, as above
The m inim um value o f g max is obtained at bad driving conditions (P_rd = 0 7,
P_ac = 0 3), with m axim um v elocity obtained 1 6447 (l e 22 2 mph)
153
P_inc
Table
Pqmax
P_ac
P_rd
^m ax
03
03
03
0 17
0 1565
03
03
05
0 18
0 2231
03
03
0 7
0 16
0 2521
03
05
03
0 17
0 1483
03
05
05
0 18
0 2093
03
05
0 7
0 16
0 2395
03
0 7
03
0 16
0 1427
03
0 7
05
0 17
0 2025
03
0 7
0 7
0 16
0 232
05
03
03
0 17
0 1557
05
03
05
0 18
0 2219
05
03
0 7
0 17
0 256
05
05
03
0 17
' 0i 1464
05
05
05
0 19
0 2083
05
05
0 7
0 18
0 2433
05
0 7
03
0 17
0 1404
05
07
05
0 19
0 2005
05
0 7
0 7
0 19
0 2352
07
03
03
0 18
0 1533
07
03
05
0 22
0 2181
0.7
&3
OJ
0.18
0.2600
07
05
03
0 18
0 1445
07
05
05
0 21
0 2054
07
05
0 7
0 20
0 247
0.7
0.7
0*3
0.18 |
0.1389
07
0 7
05
0 21
0 1981
0 7
0 7
0 7
0 20
0 2389
(6 2)
The influence of varying the model
parameters on the two extremes
154
In contrast, im proving the driving conditions by decreasing P_rd to a lower
value (P_rd = 0 3) and increasing the value o f P_ac to a higher value
(P ac=0 7) has lead to the m axim um value o f <?max (0 2609) and also increased
the system velocity to a higher value (2 2044)
Further, the tw o extrem e values o f q mmi,(m inim um and m axim um values),
were obtained, suprisingly, at the sam e traffic density ( p ?max = 0 18)
Table (6 2) show s that m ost values o f p ?max are obtained at the density o f 0 18
± 0 01, with som e cases where p ?max is obtained at a low er (higher) densities
than 0 18 ± 0 01 This is due to due to different com binations o f the extrem e
values o f the m odel parameters in order to obtain a realistic traffic patterns
6.3 Open Systems
In this section, w e apply different boundary conditions and apply the sam e
rules o f the stochastic cellular automata m odel The system inputs and outputs
are treated as follow s
•
At the left side o f the road, a stochastic feeding m echanism , in w hich car
arrivals follow a P oisson process, has been im plem ented throughout the
sim ulations for different arrival rates
•
On the other hand, cars m ay leave the system at the right side o f the road
This is achieved by ensuring that the last three sites o f the road are em pty
The above conditions for the system input and output w ill create open
boundary conditions
The fundam ental diagram for the stochastic m odel (Fig (6 10)) w as obtained,
starting with a random initial configuration o f density 0 2 and velocity v = 0,
by applying open boundary conditions and em ploying different rates o f
arrivals The sim ulations included a system o f size 300 sites for duration o f
10,000-tim e step After relaxation, 1000 tim e steps, the m axim um flow
obtained is
i
Where
*open=qaa,clo»d+C>
0
0
0 03
(6 3)
< C1i <0 04
(6
' 4)'
< C<
at density o f
A"?m a x
open
= Pq
, , “ Ci
where 0 0
“ m ax, d o se d
The above results mean that a higher flow is obtained at low er density,
com pared to the m odel w ith closed boundary conditions
Fig
(6 10)
Fundamental
open boundary
periods
(50
diagram of
conditions
time
steps)
the
stochastic model
Points
are averaged
for
system
a
of
over
size
=
using
short-time
300
after
discard period of 1000 time steps
Duration of system evolution
= 10,000
time
to different
where
=3, P _ m c = 0 5, P_rd=0 5, P_ac = 0 4
steps
Data
relate
156
arrival
rates,
6.3.1 Bottleneck Situations
One o f the important factors in traffic science is the existence o f bottlenecks,
1e
sections that have less capacity o f accom m odating traffic than do other
sections o f the road A very com m on exam ple o f a bottleneck situation is a
tw o-lane directional road m erging into one lane only
In order to obtain a bottleneck situation in the follow ing sim ulation, w e apply
a high rate o f arrivals (p = 0 6) starting from a high initial density Duration o f
the sim ulation is 10,000 tim e steps for a system o f size 300 cells
6.3.2 Self-organization of the Maximum Throughput
For the conditions specified in the previous section, our m odel show s that the
outflow from a jam appear to self-organize into m axim um throughput This
phenom enon o f self-organized criticality w as reported initially by B ak et al
(1987) using a one-dim ensional sand-pile cellular automata m odel as the
transport process, but our m odel indicates a state o f non-tnvial critical density
(le
p c * 0) in agreem ent with N agel (1994, 1995) This can be seen from
Fig (6 11), where the system size k=300 and the initial density is 0 56 with
P_inc = 0 5, P_rd = 0 5, and P_ac = 0 5 After a relaxation:o f 1000 iterations,
w e started to collect our averages every 50-tim e steps The m axim um flow
obtained is
qmiXopen = 0 2 1 8 5 , w hich is 0 0102 greater than the m axim um flow
obtained in closed system s, at low er density ( P ?max open = 0 15)
157
Fig (6 11) space-time plot of the outflow from a jam, where the
system size = 300,
starting from a dense traffic p
apply high arrival rate on the left boundary
Vmax =3, P _ m c = 0
= 0 56,and
= 0 6,
5, P_rd=0 5, P_ac = 0 5
6.4 Summary and Conclusion
A stochastic cellular automata m odel for the traffic in inter-urban areas is
presented in this chapter The m odel is defined on a one-dim ensional lattice o f
k-identical cells, where each cell is either em pty or occupied by a car, where
each car has an integer velocity varied betw een v = 0 and v = vmax
The m odel update rules depend on the relation betw een the car velocity and its
free headw ay distance and take into account som e o f the basic features, w hich
characterise the inter-urban traffic such as delay in acceleration, speed
fluctuations, and irregularities o f the driver behaviour only in term o f gap
acceptance
Our sim ulation results can be sum m arized as follow s
158
• The m axim um flow
qmm = 0 24 ± 0 02 w as obtained at density p max =
0 18 ± 0 01, further sim ulation reveals that the system size does not effect
these values
•
At low densities, w e found that laminar traffic w as the dom inant pattern
despite the appearance o f sm all jam s, whereas the high density regim e was
characterized by congested clusters, w hich represent typical start-stop
w aves
• The sim ulation conducted in Sec(6 3 3) show s that the m odel can be
adapted to m odel various traffic conditions such as bad weather, rush
hours, etc, by m odifing its parametrs
•
W hen open boundary conditions w ere applied, a higher values for the
m axim um flow w as obtained at low er densities
•
A pplying open boundary conditions with higher arrival rate, the m odel
indicates that the outflow from a traffic jam appears to self-orgam se to a
m axim um flow
159
Chapter 7
“Summary and Conclusions”
¡160
)
7.0 Research Contribution
i
The main contribution in this research lies in developing three cellular automata
models for traffic flow in urban anil inter-urban areas, which can be summarized
and discussed as follows
A simple cellular automata model for urban traffic is presented, the model is a
three state deterministic cellular automaton with both space and time, discrete
!
The state of each site at next time| step is determined from the state of the site
itself and those of the nearest neighbour sites, where vmax = 1 is the maximum
possible jump for each particle For an arbitrary configuration, one update of the
system consists of the transition rules described in Sec (2 2), which are performed
in parallel for all sites
It is hard to classify the cellular automata models in the frame of car following
I
theory, (described in Chapter 1, Sejc (1 1 1 1)) In our model the stimulus can be
regarded as the gap between adjacent sites in the automata
Car movements, (described by rulers, Rule 1, Rule 3, and Rule 4, defined in Sec
(2 1)), can be viewed as the response, in the case o f acceleration, whereas Rule 2
may be considered as the deceleration due to insufficient space ahead
Our simulation reveals that the system, (imposing periodic boundary conditions),
reaches its critical state at density lpc= 0 5 and the maximum flow obtained was
9max = 0 5 after a discard period
t0 = —, which does not depend on the system size
161
I
This was also obtained by Nagel and Schreckenberg (1992) in the deterministic
limit of their stochastic model using vmax= l, but after a much higher transient
j
period For densities less than the critical density (p <
p c), the kinematic waves
move forward with velocity 1 In contrast, for densities higher than
p c,
the
I
kinematic waves move backward with velocity -1 The open boundary conditions,
j
(described in Chapter 2, Sec (2 3 2)j), have led to lower values for maximum flow
and its density, compared to maximum flow obtained for closed systems, which
i
depends on the system size
{
|
The model, then, is adapted to model road traffic, where the road is formed by
I
linking a finite number of segments separated by traffic lights In urban traffic,
i(
our experience and every day observations says that cars have to wait for a certain
t
period of time in order to leave a specific road
\
This tune period depends on the number of the traffic lights installed between
I
road segments and the duration of both light cycles, red and green, as well as the
!
traffic density To examine the realism of our simulation model, a new parameter"
i
jammed time
t] " is introduced, whlich represents the number of time steps a car
has to wait in order to leave the road Our simulation results show that this
parameter, as in real traffic, depends on the number of traffic lights and the
duration of each light cycle
i
The real task for our "toy" model was its ability to simulate traffic flow at
|
network level Four different networks, varying from 17 nodes, (8 km length), to
41 nodes, (24 km length), have been used in the simulation, the limitations of this
;
162
range is due to the effort needed,to represent larger size networks However,
/
j
future work has to incorporate larger size networks
Each network node corresponds to a signalized intersection, timing of signals at
all nodes followed two-phase timing scheme At all nodes traffic movements
I
followed turning percentages as described in Chapter 3, Sec (3 1 5) Cars are
injected into the network according! to a Poisson process with parameter p, where
different values for u are used
I
I
i
Two types of simulation were ¡executed, the first considers the transient
1
i
movement of cars along the networks, whereas in the second cars were allowed to
1i
park inside the network, using underground parks, and the network behavior
under short and long-term events were also investigated
i
The simulation results strongly suggest that traffic flow in urban street networks
and the parameters governing its performance may be characterized using the
Simple Cellular Automaton Models
i
Our findings indicate the following \
•
i
^
The characteristic shape of the fundamental diagrams as observed for highway
traffic is, surprisingly, obtained at urban network level, despite the complex
I
interactions in urban traffic This is in agreement with Williams (1986), and
results obtained here appear to leflect realistic traffic behaviour
•
The simulation reveals that there is a critical arrival rate” jamming
threshold”
above which the network is not able to handle the incoming traffic any more,
which results in “
queues outside the network"
163
This
“jamming threshold?' is
independent of the network size in the case of transient movement simulation,
t
whereas different
“jamming thresholds”
for different network sizes were
t
obtained in the case of non-transient movement simulation
t
•
i
At low arrival rate and also| slightly above the “jamming threshold” a
!
i
significant interaction,(a = 0 01), between the traffic conditions, transient
period, and the interval lengtlj was obtained for smaller size network, 17
nodes This interaction became insignificant for larger size network, 41 nodes
•
Modelling long-term events, such as blocking some of the network exits, has
successfully reproduced the jamming effect typically seen in highly congested
networks at peak-times
Using single-lane networks, important features such as
lane-changing
and its
'«i
impact on the flow behavior in urban networks cannot be observed and studied
Compared to highway traffic,
lane-changing in urban areas is more complex
This
is because the decision to change ¡lane in urban areas depends on a number of
objectives and at times these may conflict
Despite this complexity, our aim is to find the minimal sets of lane-changing
rules, which are capable of reproducing important macroscopic features such as
lane-usage inversion, which is observed long before maximum flow in highway
traffic and also observed in urban traffic flow in our study, but at higher density,
1
see Sec (5 4)
!
164
In this regard a two-lane model is introduced, which consists of two parallel
single-lane model and three sets of rules for lane-changing defined as follows
urgent lane-changmg conditions, minimal and maximal lane-changing conditions
Our simulation reveals that the above lane-changmg rules were able to reproduce
the lane-usage inversion as obtained in both highway and urban traffic
The
“density inversion“
point is modified by the
lane-changing parameters,
P_chg, P_obs, and P_Lobs The choice P_chg = 0 4, P_obs = 0 3, P_Lobs = 0 5
for both lanes left and right, gives a good approximation to our observed data in a
study performed in Dublin city to calibrate our lane-changing rules
The
limitations of our study was mentioned in Sec (5 4)
However, the lane-usage frequency for both lanes is not correctly modelled at
very large densities, where left lane flow is higher than the right lane flow
In the last chapter we moved from deterministic cellular automata models to a
completely stochastic cellular automaton models in order to model traffic in mterurban areas with links that may exceed 5 km
The stochastic model, also, is defined on a one-dimensional array of k sites The
state of each site is either empty or occupied by a car, where each car has an
integer velocity between v = 0 and v = vmax The model update rules, described in
Sec (6 1 2) depend on the relation between the safe distance of the car velocity
i
and its free headway distance
The model correctly reproduces the start-stop waves and the flow-density relation
shows the asymmetry known from the real data obtained for highway traffic
165
The simulation reveals that the model can be adapted to model various road traffic
conditions, by modifying its parameters
In case of an open system the model gives a good approximation to the flowdensity relation obtained for highway traffic and indicates that the outflow from a
jam appears to self-orgamze to maximum flow, when a high arrival rate is
applied, starting from a dense traffic condition
Throughout this thesis Microsoft Excel was used to plot all our graphs, and a
C++ code, which uses bitmaps see Appendix E a:\ Space-time\ bc.c, was used to
plot all the space-time diagrams A bitmap is a powerful graphics object used to
create, manipulate and store images as files on a disk
7.1 Future Work
•
Despite the deterministic update rules of the Simple Cellular Automata
Model, it was able to reproduce important features of urban traffic
Adding the stochasticity element to the update rules might yield a more
realistic behavior o f the model
•
Buses and Trucks represents more than 50% of the urban traffic, so future
work has to be directed to incorporate long vehicles in the model (1 e nonhomogenous units)
•
Also, future simulation has to involve networks with complex intersections
(1 e intersections with complex geometry), and roundabouts
166
•
Extend the two-lane model to include a third lane, which is dictated as a buslane The bus-lane represents one of the important short-terms events in urban
traffic
•
Investigate the efficiency of the simulations in the case of non-homogenous
units by looking at the bit-wise coding and also the use of parallel computing
•
In case of the stochastic CA model, more investigations are required to study
and scale the formation of the spontaneous traffic jams, Also, the limits of the
model has to be investigated in some detail
•
Use the models developed in this research to develop a High Speed
Microscopic Simulation for Urban and Inter-urban Traffic (HSMSUIT)
simulation package
7.2 Concluding Remarks
In this research, we have seen the importance and the capabilities of the Cellular
Automata methodology in modelling traffic flow m urban and inter-urban areas
Despite the complexity of the traffic dynamics, the advances we have seen in the
last years are demystifying the idea that Cellular Automata are too simple to be
capable of simulating highly complex systems
167
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Appendices
Appendix A
Statistical Analysis
In this section we use factorial analysis to study how changing the input parameters
might effect the output parameters
The input parameters are length of transient period, interval length, traffic conditions,
the number of network-blocked exits, and the arrival rate
In contrast our output parameters or responses are traffic density, flow rate, average
velocity, number of cars input to the system, number of cars output from the system,
and the queue length at each of the network entries
f.
In the first section we study the significance for network performance of the
interaction between the network size and the input parameters We examine the flow
rate in each case for different arrival rates at the injection point
In section two, using different network sizes, we study the higher order interactions of
the three factors, transient period, interval length, and traffic conditions, and examine
their effect on the flow rate using two levels for each factor, again for various arrival
rates
179
A.I. The relation between the network size and the input
parameters
A.1.1 Network size vs traffic conditions
We have used factorial analysis to examine the realism of our simulation model, in
particular to investigate the interaction between network size and traffic conditions
In the first experiment, network size and traffic conditions at intersections have levels
B1 to B4 and A1 to A2 as follows
B Network size, B1 = 17 nodes, B2 = 25 nodes, B3 = 33 nodes, B4 = 41 nodes
A traffic conditions, A1 = 50% go straight through, 25% turn left, 25% turn right and
A2 = 76% go straight through, 12% turn left,; 12% turn right
Fig (A 1)
The experiment is performed using a stochastic feeding mechanism, in which car
arrivals at the injection points follow a Poisson process for different rates of arrivals
t
(P)
180
Fig (A l) shows that, at low arrival rate p = 0 1 for free traffic conditions, a
significant interaction between the traffic conditions and the network size was
obtained at the a = 0 01 level of significance Increasing the arrival rate to p = 0 55,
results in a weak interaction between the two parameters
Using the higher arrival rate, Table (A 1) demonstrates that both factors have a
significant effect on the flow rate
Factor
A
B
AB
Fcaic
at p = 0 1
Fcaic
10 32
14 18
(1, 312)*
(1, 312)
4 28
28 11
4 67
1 69
24 65
29 71 J-
50 85
5 32
(3, 312)
(3, 312)
(3, 232)
(3, 232)
(3, 312)
(3, 312)
(3, 232)
(3, 232)
30 13
1 02
(3, 312)
(3,312)
C
BC
17 16
36 07
(1, 312)
(1, 312)
6 60
2 32
(3, 312)
(3, 312)
D
DB
3 56
314 12
(1, 232)
(1, 232)
26 04
10 94
(3, 232)
(3, 232)
?
0 39
E
EB
Table
at p = 0 55
(A 1)
8 59
(1, 232)
(1, 232)
1 70
0 37
(3, 232)
(3, 232)
In this table the response variable represents the
flow rate for all the experiments, where A
B
network size,
C
transient period,
E
network-blocked exits, and
181
a =
0 01
D
traffic conditions,
interval
length,
and
A.1.2 Network size vs length of transient period
In experiment 2 we study the effect of the transient period on the network size factor.
The same levels for the network size parameter have been used as before and two
levels for the transient period; Cl = 200, C2 =500.
Using the low arrival rate, the interaction between the length of transient period and
network size seems to have a significant effect on the average flow rat at a = 0.01.
Again, this is not sustained when the high arrival rate is applied to feed the network
(Fig (A.2.b)). Both factors independently, however, have a significant effect on
network performance, (Table (A.l)).
(a)
(b)
F i g ( A . 2)
A.1.3 Network size vs interval length for calculating the output
parameters
Our simulation results reveals that, as long the free flow is dominant, there are no
significant changes in the traffic parameters, average flow, average density, and
182
average density, see Chapter 3, Sec (3 3 3) Hence, the interval length used to
calculate these averages does not have much influence on their values This is not the
case in the dense traffic regime period, where the interval length used to average these
parameters plays a significant role (
= 3 1 4 a ta = 001 level of significance)
This can be observed from the statistical analysis presented in Table (A 1) with
interval length having two levels, D l= 30 time step, D2 = 150 time steps
The analysis also reveals that there is a significant interaction, at a = 0 01, between
network size and interval length for low arrival rate, less so for high arrival rate For
high arrival rate, network size and interval length are also separately significant at the
same level of significance However, for low arrival rate, interval length does not
have a significant effect
t*
”t
A.1.4 Network size vs the number of the Network-blocked exits
When car arrivals to the network are relatively low and/or if underground parks exist
inside the network, then blocking one or two of the network exits will not have a great
impact on the network performance This situation is no longer true as the network
becomes congested and car arrival rate increases In this case, blocking any of the
network exits will strongly influence the network performance
To quantify the above performance, we have used two levels for the factor number of
network-blocked exits l e E l = 1 exit and E2 = 2 exits, and the same levels for the
network size parameter (Table (A 1)) It can be seen that when free flow traffic
£
operated through the simulated network, neither the two factors nor their interaction
have a significant effect on the flow rate
183
By contrast, under dense traffic conditions (îe a high arrival rate) both factors
(network size and number of blocked exits) have a significant role, (a = 0 01), in
terms of flow performance At this level of concentration, the flow rate is not affected
by interaction of the two factors
A.1.5 Network size vs arrival rate
One of the key factors in our simulation is the arrival rate To investigate how factors
such as the arrival rate, network size and their combination might affect the network
performance, factorial analysis was used to analysis the simulation output, where two
levels,’ HI = 0 1 and H2 = 0 55,’ have been used for the arrival ratéyand the same four
levels as previously for the network size
Statistical analysis in Table (A 2) shows that the arrival rate has a great influence on
flow performance and a significant interaction, at
a
= 0 01, also exists between the
two factors, H and B
Factor
Fcalc
H
738 58
( 1 ,2 3 2 )
49 04
(3 , 2 3 2 )
24 33
(3, 232 )
B
HB
Table
and
(A 2)
each
where H
The response variable represents the flow
order
pair
represents
arrival rate and B
184
the
Degrees
of
freedom,
network size and a = 0 01
A.2. Statistical analysis using three-input factors:
In this section we study the effect of the following factors, transient period, the
interval length, and the traffic conditions on the flow rate performance using different
networks size Each of these parameters has two levels as follows
A
Traffic conditions,
A1 = 25% turn left, 50% go straight,, 25% turn right and
A2 = 12% turn left, 76% go straight, 12% turn right
B
Transient Period, B1 = 200, B2 = 500
C
Interval Length, C l = 30, C2 = 150
A.2.1 Experiments using a 17 node network
Table (A 3) describes the statistical analysis performed using different arrival rates
using 17-node network
Table
A
Source
Fcalc at M=0 1
Fcaic at (J=0 3
Fcaic at M=0 55
A
34 62
87 22
143 62
B
12 07
22 78
11 46
C
1 18
291 20
191 70
AB
2 08
14 78
1 076
AC
188 30
0 013
7 07
BC
11 72
14 73
7 77
ABC
13 73
10 72
0 98
(A 3)
The table contents same as m
traffic conditions, B
table
transient period, and C
(A 2), but
interval length
This was obtained for network of size 17 nodes and results obtained
at a = 0 01 level of significance
185
From the above table it is easy to see that the interaction between the three factors
does have a significant effect, at a = 0.01, on the flow performance at low arrival rate.
However, this interaction was less significant as we increase the arrival rate and was
unimportant at high arrival rate of 0.55. In the case of the two-factor interactions we
find a significant interaction, (a = 0.01), between the length of transient period and
the interval length at various levels of arrival rates. In contrast there is no interaction
between the transient period and the traffic conditions except at a moderate arrival
rate, with Poisson parameter p = 0.3.
Also a significant interaction, (a = 0.01), between the traffic conditions and the
interval length was obtained at both low and high arrival rates, whereas for ¡u
moderate and equal to 0.3, interaction between the two factors was negligible. The
statistical analysis also shows a significant effect at different arrival rates for each
parameter with the exception of the interval length, where no significant effect was
observed at low arrival rate as expected.
A.2.2 Experiments using 41 node network
To study further the effect of network size on traffic parameters, the simulated
network was extended to include 41 nodes.
Table (A.4) shows that there is a significant interaction, a = 0.01, between the traffic
conditions and the transient period only when low arrival rate is applied whereas
negligible interaction between the traffic conditions and the interval length was
observed at all of arrival rates.
The individual parameters in this case have a significant effect, at the same a, on the
flow performance compared to previous networks, except in the case of the interval
length parameter at low arrival rate.
186
Table
A
Factor
Fcalc
A
151 89
8 55
14 11
B
116 74
25 00
19 36
C
4 88
321 26
179 06
AB
41 15
0 01
0 003
AC
0 43
0 89
1 16
BC
4 86
10 66
6 54
ABC
5 72
0 35
0 0001
(A 4)
at M=0 1
P eak
at M=0 3
Fcaic
at M=0 55
the response variable represents the flow, where
traffic conditions, B
transient period, and C
interval length
This was obtained for network of size 41 nodes and results obtained
at a = 0 01 level of significance
Using Table (A 4) it is easy to see that the interval length factor and its interactions
terms do not have a significant effect the flow at low arrival rate no matter what the
network size However, under dense traffic conditions, the interval length as well as
the other two factors are significant, (a = 0 01), in terms o f influencing flow, although
most o f the higher order interactions become less significant as the network becomes
more congested This appears to be true, irrespective o f the network size (Chapter 3,
Sec(3 3))
187
Appendix B:
Experiment
Summary of the simulation experiments
Run Duration
Arrival Rate
initial Oensity
M ax Flow
V
Oensity of
Velocity
Max Flow
at max Fiow
Queue outside
the network
Total Input
Total Output
Cars
Cars
N17D200S30H50%L25%R25%
5000 time step
0.1
0.159274
0.14892
0.180459
0.825226
3
2744
2748
N17D200S30H50%L25%R25%
5000 time step
0.3
0.147177
0.247978
0.482386
0.514065
808
6270
5917
N17D200S30H50%L25%R25%
5000 time step
0.55
0.148185
0.250189
0.414131
0.604131
6063
6616
6215
N17D200S30H76% L12% R12%
5000 time step
0.1
0.155242
0.132537
0.15784
0.839694
6
2540
2539
N17D200S30H76% L12% R12%
5000 time step
0.3
0.145161
0.260762
0.455271
0.572761
847
6144
5781
N17D200S30H76% L12% R12%
5000 time step
0.55
0.140121
0.261315
0.453458
0.576273
6292
6810
6355
N1 7D200S150H50%L25%R25%
5000 time step
0.1
0.144153
0.119422
0.137413
0.863076
III!!
2323
2327
N17D200S150H50%L25%R25%
5000 time step
0.3
0.132056
0.249845
0.490848
0.509007
918
6217
5865
N17D200S150H50%L25%R25%
5000 time step
0.55
0.15625
0.254043
0.539544
0.470846
6270
6736
6329
N17D200S150H76%L12%R12%
5000 time step
0.1
0.138105
0.133206
0.154579
0.861731
0
2461
2452
N17D200S150H76%L12%R12%
5000 time step
0.3
0.145161
0.263661
0.509714
0.517273
997
6210
5842
N17D200S150H7€%L 12%R 12%
5000 time step
0.55
0.144153
N17D500S30H76% L12% R12%
2
N17D500S30H76%L12%R12%
5000 time step
0.3
0.15625
N17D500S30H76%L12%R12%
5000 time step
0.55
0.157258
N17D500S30H76%L12%R12%
5000 time step
0.1
0.143145
N17D500S30H76%L12%R12%
5000 time step
0.3
0.137097
0.533292
0.839994
0.553374
0.468864
0.862293
0.53774
0.52375
0.874755
0.531867
6325
0.141129
0.50673
0.168032
0.451186
0.528179
0.154583
0.4774
0.504902
0.142374
0.468973
0.586972
0.179288
0.463274
0.54265
6772
0.1
0.270235
0.141146
0.249675
0.247644
0.133296
0.256717
0.264442
0.124543
0.249432
6193
5000 time step
2683
6073
6834
2478
6299
6721
2266
6064
2692
5756
6410
2497
5932
6283
2272
5761
6431
2714
5784
6255
N17D500S30H76%L12%R12%
5000 time step
0.55
0.142137
N17D500S150H50%L25%R25%
5000 time step
0.1
0.143145
N17D500S150H50%L25%R25%
5000 time step
0.3
0.167339
N17D500S150H50%L25%R25%
5000 time step
0.55
0.147177
N17D500S150H76%L12%R12%
5000 time step
0.1
0.153226
N17D500S150H76%L12%R12%
5000 time step
0.3
0.147177
N17D500S150H76% L12% R12%
5000 time step
0.55
0.149194
0.249719
0.151318
0.265604
0.26992
0.425437
0.843993
0.57332
0.49741
918
6243
3
782
5970
3
869
6370
7
914
5946
6880
2734
6123
6654
Table (B.l): Summary of results for a transient moment simulations for a 17 nodes network, where :D: transient period,
S: interval length used to calculate averages, H: headway percentages, L: turn left percentages, R: turn right percentages
188
J&in Duration
Arrival Rate initial Density
W
N25D200S30H50%L25%R25%
5000 time step
0.1
0.157419
N25D200S30H50%L25%R25%
5000 time step
0.3
0.141935
N25D200S30H50%L25%R25%
5000 time step
0.55
0.152903
N25D2O0S3OH 76%L1 2%R12%
5000 time step
0.1
0.144516
N25D200S30H76%L12%R12%
5000 time step
0.3
0.138065
Max Flow
N25D200S30H76%L12%R12%
5000 time step
0.55
0.149032
N25D200S150H50%L25%R25%
5000 time step
0.1
0.148387
0.133535
0.250938
0.257036
0.122532
0.251473
0.255588
0.130982
N25D200S150H50%L25%R25%
5000 time step
0.3
0.155484
0.261953
N25D200S150H50%L25%R25%
5000 time step
5 0 0 0 1me step
0.55
0.1
0.147742
N25D200S150H76%L12%R12%
N25D200S150H76%L12%R12%
5 0 0 0 1me step
0.3
0.14
N25D200S150H76%L12%R12%
5 0 0 0 1me step
0.55
0.150323
N25 D500S30H 76%L12%R 12%
5 0 0 0 1me step
0.1
0.150323
N25D500S30H76%L12%R12%
5000 t me step
0.3
0.145806
N25D500S30H76%L12%R12%
5000 t me step
0.55
0.145161
N25D500S30H76%L12%R12%
5000 t me step
0.1
0.14
N25D500S30H76%L12%R12%
5000 t me step
0.3
0.14129
0.268279
0.129295
0.25995
0.265441
0.125374
0.255886
0.261093
0.127269
0.254167
0.255759
0.125037
0.263499
0.262152
0.132389
0.257569
0.261592
0.139355
N25D500S30H76%L12%R12%
5000 t me step
0.55
0.154839
N25D500S150H50%L25%R25%
5000 t me step
0.1
0.147097
N25D500S150H50%L25%R25%
5000 t me step
0.3
0.141935
N25D500S150H50%L25%R25%
5000 t me step
0.55
0.14129
N25D500S150H76%L12%R12%
5000 t me step
0.1
0.151613
N25D500S150H76%L12%R12%
5000 t me step
0.3
0.140645
N25D500S150H76%L12%R12%
5000 time step
0.55
0.149032
T a b l e ( B .2): S u m m a r y o f
S: i n t e r v a l l e n g t h u s e d
Density of
Velocity
max Flow
at max Flow
0.154924
0.42474
0.50891
0.141974
0.460259
0.477692
0.145693
0.455198
0.516321
0.147715
0.46235
0.501034
0.14348
0.431914
0.441902
0.146585
0.422594
0.42253
0.140622
0.442829
0.86194
0.590803
0.505071
0.857421
0.546468
0.535047
0.89903
0.575471
0.519598
0.8753
0.562236
0.529786
0.873803
0.592447
0.590839
0.868228
0.601446
0.605305
0.88925
0.595035
0.514322
0.873958
0.56077
0.59697
0.509705
0.151483
0.459314
0.438199
r e s u l t s f o r a t r a n s i e n t m o m e n t s i m u l a t i o n s f o r a 25 n o d e s n e t w o r k , w h e r e
to c a l c u l a t e a v e r a g e s , H: h e a d w a y p e r c e n t a g e s , L: t u r n l e f t p e r c e n t a g e s ,
189
Queue outside
the network
2
995
6533
6
937
6230
4
912
6336
6
838
6275
3
959
6192
3
1028
6189
2
816
6502
3
952
6130
:D: t r a n s i e n t
R: t u r n r i g h t
Tata! input
Oars
Iota! Output
2405
6073
6458
2376
2467
5993
6392
2371
6030
¡« 1 1 1 1 !
2282
6033
6455
2338
6074
6501
2328
6060
6479
2297
5973
6413
2305
6014
6526
period,
percentages
Cars
5583
5876
2399
5467
5794
2406
5532
5879
2324
5523
5897
2372
5588
5859
2364
5566
5906
2340
5476
5856
2364
5498
5845
Run Duration
"ITxpefimW
Arrival Rate initial Density
M
0.145714
0.1
N33D200S30H50%L25%R25%
5000 time step
N33D200S30H50%L25%R25%
5000 time step
0.3
N33D200S30H50%L25%R25%
5000 time step
0.55
0.16
N33D200S30H76% L12% R12%
5000 time step
0.1
0.15381
N33D200S30H76% L12% R12%
5000 time step
0.3
0.143333
N33D200S30H76% L12% R12%
5000 time step
0.55
0.146667
N33D200S150H50%L25%R25%
5000 time step
0.1
0.148095
N33D200S150H50%L25%R25%
5000 time step
0.3
0.140952
N33D200S150H50%L25%R25%
5000 time step
0.55
0.141429
N33D200S150H76% L12% R12%
5000 time step
0.1
0.152381
N33D200S150H76% L12% R12%
5000 time step
0.3
0.147143
N33D200S150H76%L12%R 12%
5000 time step
0.55
0.152857
N33D500S30H76% L12% R12%
5000 time step
0.1
0.153333
N33D500S30H76% L12% R12%
5000 time step
0.3
0.143333
N33D500S30H76% L12% R12%
5000 time step
0.55
0.138095
N33D500S30H76%L12%R12%
5000 time step
0.1
0.150952
N33D500S30H76% L12% R12%
5000 time step
0.3
0.144286
N33D500S30H76% L12% R12%
5000 time step
0.55
0.140476
N33D500S150H50%L25%R25%
5000 time step
0.1
0.144286
N33D500S150H50%L25%R25%
5000 time step
0.3
0.14381
N33D500S150H50%L25%R25%
5000 time step
0.55
0.14619
0.152381
0.123954
0.2334
0.236959
0.138539
0.241085
0.242875
0.143204
0.242285
0.242092
0.132734
0.25041
0.26S046
0.119552
0.237264
0.23482
0.124745
0.242352
0.240965
0.123623
0.241518
0.242907
0.118459
0.249781
0.254398
Density at
Max Flow
0.144823
0.419332
0.433432
0.163858
0.439291
0.458928
0.164671
0.43769
0.435438
0.153399
0.481382
Queue Outside
Velocity
at Max Row the Network
0.855896
0.556812
0.433432
0.845484
0.548804
0.511926
0.869639
0.553554
4
739
5854
Total Input
cars
Totaf Output
2321
6491
2369
5740
5989
2528
5467
5796
2425
5588
5983
2552
5480
5826
2437
5512
6030
2371
6005
6815
2507
6279
6751
2361
6263
6836
2494
6273
6727
2377
6245
6888
2312
6284
6790
2380
6230
6821
2282
6257
6710
r e s u l t s f o r a t r a n s i e n t m o m e n t s i m u l a t i o n s f o r a 33 n o d e s n e t w o r k , w h e r e : D: t r a n s i e n t
to c a l c u l a t e a v e r a g e s , H: h e a d w a y p e r c e n t a g e s , L: t u r n l e f t p e r c e n t a g e s , R: t u r n r i g h t
period,
percentages
N33D500S150H76%L12%R 12%
5000 time Step
¡1 1 1 1
0.144762
N33D500S150H76%L12%R12%
5000 time step
0.3
0.137619
N33D500S150H76%L12%R12%
5000 time step
0.55
0.145238
T a b l e (B.3): S u m m a r y o f
S: i n t e r v a l l e n g t h u s e d
Max flow
190
0.468599
0.136254
0.43985
0.518609
0.148753
0.486614
0.508666
0.142087
0.405884
0.470925
0.134982
0.400047
0.462507
0.555973
0.874409
0.52019
0.544273
0.877418
0.539421
0.452788
0.838607
0.498037
0.473719
0.870053
0.595043
0.515808
0.877589
0.62438
0.550043
4
672
5741
5
807
5776
0
791
8163
3
688
5937
8
612
6068
1
787
5882
I I I
615
Cars
5523
5884
2398
5546
5978
2309
5502
5802
initial Density
MaxFiow
Density of
Max Flow
Velocity
at Max:Flow
Queue Outside
the Network
0.148791
0.123815
0.236816
0.236763
0.137141
0.245792
0.250523
0.126424
0.244922
0.143406
0.430268
0.494114
0.157631
0.459454
0.426943
0.141542
0.365846
0.445275
0.159056
0 451121
0.433729
0.863386
0.550393
0.441462
0.86986
0.534966
0.586784
0.893192
0.669466
0.562015
0.886
0.57403
0.595883
3
0.131372
0.430988
0.490861
0.139498
0.455698
0.415698
0.148948
0.421425
0.402293
0.145445
0.447167
0.441876
0,877736
0.547282
0.486036
0.88039
0.542872
0.602073
0.891513
0.58369
¡ ¡ liii
0.614285
0.891836
0.565193
0.584505
5652
Experiment
Run Duration
Arrival Rate
N41 D200S30H50%L25%R25%
5000 t me step
0.1
N41 D200S30H50%L25%R25%
5000 t me step
0.3
0.150961
N41 D200S30H50%L25%R25%
5 0 0 0 1 me step
0.55
0.140732
N41 D200S30H76%L12%R12%
5000 t me step
0.1
0.154991
N41 D200S30H76%L12%R12%
5000 t me step
0.3
0.150961
N41 D200S30H76%L12%R12%
5000 t me step
0.55
0.147861
1
N41D200S150H50%L25%R25%
5000 t me step
0.1
0.150031
N41D200S150H50%L25%R25%
5000 t me step
0.3
0.155921
N41D200S150H50%L25%R25%
5000 t me step
0.55
0.151891
N41D200S150H76%L12%R12%
5000 t me step
0.1
0.148411
N41D200S150H7fi%t 12%R 12%
5 0 0 0 1me step
0.3
0.147861
N41D200S150H76%L12%R12%
5000 t me step
0.55
0.152201
N41 D500S30H76%L12%R 12%
5 0 0 0 1me step
5000 t me step
l i i i
0.3
0.145691
N41 D500S30H76%L12%R12%
N41 D500S30H76%L12%R12%
5000 t me step
0.55
0.150961
N41 D500S30H76%L12%R12%
5 0 0 0 1 me step
0.1
0.144761
N41 D500S30H76%L12%R12%
5000 t me step
0.3
0.152821
N41 D500S30H76%L12%R12%
5000 t me step
0.55
0.156541
N41D500S150H50%L25%R25%
5000 t me step
0.1
0.153131
N41D500S150H50%L25%R25%
5000 t me step
0.3
0.148791
N41D500S150H50%L25%R25%
5000 t me step
0.55
0.155921
N41D500S150H76%L12%R12%
5000 t me step
0.1
0.154991
N41D500S150H76%L12%R12%
N41D500S150H76%L12%R12%
5000 t me step
0.3
0.146621
5000 t me step
0.55
0.154061
0.145071
0.250251
0.140923
0.258957
0.258452
0 11531
0.235872
0.238576
0.122812
0.247386
0.250281
0.132789
0.245982
0.247123
0.129713
0.252735
0.258279
552
5769
3
400
5700
1
629
5591
5
494
5697
521
5476
3
492
5529
1
516
4
445
5668
Total Input Total Output
: Cars ;;
Cars
2335
6388
7385
2484
6514
7357
2337
6331
7821
3739
6488
7295
2262
6353
7242
2342
6525
7293
2327
6358
7152
2315
6485
7266
T a b l e (B.4): Summary of r e sults for a t r a n s i e n t m o m e n t s i m ulations for a 41 nodes network, where: D: t r a n s i e n t
period, S: interval length u s e d to c a l c u l a t e averages, H: h e a d w a y percentages, L: turn left percentages, R: tu r n
right perce n t a g e s
191
2425
5351
5810
2522
5314
5740
2412
5341
6246
3710
5250
5713
i l l llll li i
5341
5894
2414
5312
5699
2423
5351
5765
2419
5279
5681
Experiment
Run Duration
Arrival Bate
initial Density M a x F lo w
V
5000 time step
0.1
0.147177
N17D200S30B2H50% L25% R25%
5000 time step
0.1
0.133
N17D200S30B1 H50%L25%R25%
5000 time step
0.3
0.145161
N17D200S30B2H50% L25% R25%
5000 time step
0.3
0.135081
N17D200S30B1 H50%L25%R25%
N17D200S30B1 H50%L25%R25%
5000 time step
0.55
0.143145
N17D200S30B2H50% L25% R25%
5000 time step
0.55
0.152218
N17D200S30B1 H76%L12%R12%
5000 time step
0.1
0.16129
N17D200S30B2H76% L12% R12%
5000 time step
0.1
0.131048
N17D200S30B1 H76%L12%R12%
5000 time step
0.3
0.137097
N17D200S30B2H76% L12% R12%
5000 time step
0.3
0.145161
N17D200S30B1 H76%L12%R12%
5000 time step
0.55
0.153226
N17D200S30B2H76% L12% R12%
5000 time step
0.55
0.140121
N17D200S150B1 H50%L25%R25%
5000 time step
0.1
0.148185
N17D200S150B2H50% L25% R25%
5000 time step
0.1
0.142137
N17D200S150B1 H50%L25%R25%
5000 time step
0.3
0.150202
N17D200S150B2H50%L25%R25%
5000 time step
0.3
0.140121
N17D200S150B1 H50%L25%R25%
5000 time step
0.55
0.131048
N17D200S150B2H50%L25%R25%
5000 time step
0.55
0.144153
N17D200S150B1 H76%L12%R12%
5000 time step
0.1
0.157258
N17D200S150B2H76%L12%R12%
5000 time step
0,1
0.143145
N17D200S150S1 H76%Lt 2%R12%
5000 time step
¡lili
0.147177
N17D200S150B2H76% L12% R12%
5000 time step
0.3
0.139113
N17D200S150B1 H76%L12%R12%
5000 time step
0.55
0.153226
N17D200S150B2H76% L12% R12%
5000 time step
0.55
0.15625
N17D500S30B1 H50%L25%R25%
5000 time step
0.1
0.154234
5000 time step
0.1
0.143145
N17D500S30B2 H50%L25%R25%
192
Density of
Velocity
max Flow
A;-
0.224163
0.2120
0.236931
0.230511
0.238341
0.226313
0.217166
0.21553
0.244921
0.230073
0.251618
0.232449
0.236138
0.210436
0.23791
0.233809
0.247727
0.492272
0.4719
0.50286
0.467281
0.463133
0.508597
0.454462
0.335317
0.236659
0.216008
0.389588
0.44391
0.209734
0.2598
0.242613
0.258875
0.23615
0.227848
0.209774
0.421416
0.444752
0.54804
0.476871
0.523876
0.553357
0.501119
0.443625
0.438452
0.330629
0.443933
0.468975
0.493992
0.502183
0.582783
0.409535
Queue outside
the n e tw o rk
Total input
Cars
Total Output
Cars
3
2278
2202
102
2868
2099
2280
5144
4551
0.493303
0.514627
0.444976
0.477852
3046
3208
4584
3309
2297
0.642765
0.581185
0.517306
0.459122
2
4143
5193
4279
2486
2291
5040
4274
5464
0.455363
0.4493
0.471167
7450
8790
2
1790
2812
7427
0.487445
0.450753
0.38029
8642
0.474758
0.527042
2063
0.565005
0.60746
0.486602
7464
0.634349
wmmmm
0.517326
0.526072
0.470247
0.390966
0.512225
2
2
2827
8772
2
19
1898
2832
7584
8473
2
4
1796
4404
3199
4774
3236
2310
4268
2407
2437
4862
4119
5244
4191
2485
1891
4294
3135
4622
3261
2316
2401
1868
5130
4298
5437
iiiSii
4252
2698
2314
3161
4726
3266
2497
1832
N17D500S30B1 H50%L25%R25%
5000 t me step
0.3
0.149194
N17D500S30B2 H50%L25%R25%
5000 t me step
0.3
0.140121
N17D500S30B1 H50%L25%R25%
5000 t me step
0.55
0.140121
N17D500S30B2 H50%L25%R25%
5000 t me step
0.55
0.149194
N17D500S30B1 H76%L12%R12%
5000 t me step
0.1
0.150202
N17D500S30B2H76% L12%R12%
5000 t me step
0.1
0.147177
N17D500S30B1H76%L12%R12%
5000 t me step
0.3
0.149194
N17D500S30B2H76%L12%R12%
5 0 0 0 1me step
0.3
0.139113
N17D500S30B1 H76%L12%R12%
5 0 0 0 1 me step
0.55
0.144153
N17D500S30B2H76% L12%R12%
5000 t me step
0.55
0.141129
N17D500S150B1 H50%L25%R25%
5000 t me step
0.1
0.148185
N17D500S150B2H50%L25%R25%
5 0 0 0 1me step
0.1
0.149194
N17D500S150B1 H50%L25%R25%
5 0 0 0 1 me step
0.3
0.140121
N17D500S150B2H50%L25%R25%
5000 t me step
0.3
0.149194
N17D500S150B1 H50%L25%R25%
5000 t me step
0.55
0.146169
N17D500S150B2H50%L25%R25%
5 0 0 0 1me step
0.55
0.154234
N17D500S150B1 H76%L12%R12%
5000 t me step
0.1
0.140121
N17D500S150B2H76%L12%R12%
5000 t me step
0.1
0.15625
N17D500S150B1 H76%L12%R12%
5 0 0 0 1me step
0.3
0.136089
N17D500S150B2H76%L12%R12%
5000 t me step
0.3
0.145161
0.139113
N17D500S150B1H76%L12%R 12%
5000 t me step
0.55
N17D500S150B2H76%L12%R12%
5 0 0 0 1me step
0.55
0.141129
0.241441
0.231121
0.233647
0.225178
0.502963
0.406189
0.510158
0.564867
0.2208
0.228135
0.251889
0.228963
0.252824
0.22991
0.232802
0.40343
0.393391
0.496906
0.451763
0.475676
0.592603
0.486852
0.405369
0.51085
0.484377
0.480037
2236
0.568998
0.457989
0.39864
2953
2308
5031
4128
4466
3184
4702
3289
2115
1772
4388
3161
7568
5370
4309
2378
2788
5035
4083
5216
4359
2499
2474
5008
4074
5424
4723
3330
2287
2098
4416
3180
4601
3266
2289
1936
4381
3133
4749
8628
4204
3198.
7447
8569
0.547308
0.579921
0.506915
0.506821
3
16
1964
0.531505
0.387966
0.478179
7431
8394
69
0.224858
0.24286
0.229888
0.246749
0.239664
0.219682
0.471898
0.491177
0.429647
0.215391
0.252152
0.233687
0.255734
0.367112
0.476765
0.536835
0.52874
0.554698
0.475403
0.474605
0.522887
0.487938
0.511308
0.586718
0.52888
0.435306
0.483666
0.239967
0.51328
0.467516
2862
1
2388
3214
7537
8475
1
71
2030
2997
5055
4116
5370
4219
2271
Table (B.5): Summary of results for a non-transient moment simulations for a 17 nodes network, where: D: transient
period, S: interval length used to calculate averages, H: headway percentages, L: turn left percentages, R: turn
right percentages
193
Experiment
Run Duratìon
Arrivai Rate
Inìtlai Density
Max Flow
Density of
max Fk>w
V
Vetocity
: ai max Flow
Total input
To'.ai Output
Cars
Cars
2
2667
2312
Queue iength
0.32573
0.668821
0.219951
0.363491
0.605108
5
2228
1581
0.316262
0.702356
1018
6044
4085
N41D200S30B1 H50%L25%R25%
5000 time step
0.1
0.148171
0.217855
N41 D200S30B2H50%L25%R25%
5000 time step
0.1
0.146311
5000 time step
0.3
0.147861
0.222129
N41 D200S30B2H50%L25%R25%
5000 time step
0.3
0.152821
0.223269
0.360969
0.618526
1836
5314
2989
N41D200S30B1 H50%L25%R25%
5000 time step
0.55
0.156851
0.233571
0.359306
0.650061
6400
6433
4395
N41 D200S30B1 H50%L25%R25%
N41 D200S30B2H50%L25%R25%
5000 time step
0.55
0.147861
0.22648
0.479256
0.472567
7490
5396
3065
N41D200S30B1 H76%L12%R12%
5000 time step
0.1
0.144761
0.217483
0.334089
0.650973
5
2481
2030
N41 D200S30B2H76% L12% R12%
5000 time step
0.1
0.151271
0.227134
0.422467
0.537638
10
2362
1490
N41D200S30B1 H76%L12%R12%
5000 time step
0.3
0.150651
0.237137
0.350863
0.6758
648
6368
4831
N41 D200S30B2H76% L12% R12%
5000 time step
0.3
0.153131
0.24677
0.416134
0.593004
1946
5135
2887
N41D200S30B1 H76%L12%R12%
5000 time step
0.55
0.146931
0.24966
0.431118
0.5791
6860
6950
4817
0.153131
0.246243
0.444347
0.554168
7560
5535
3119
N41 D200S30B2H76%L12%R12%
5000 time step
0.55
N41D200S150B1 H50%L25%R25%
5000 time step
0.1
0.147551
0.214345
0.338739
0.632774
6
2375
2085
N41D200S150B2H50%L25%R25%
5000 time step
0.1
0.152201
0.230547
0.387583
0.594834
12
2300
1611
N41D200S150B1 H50%L25%R25%
5000 time step
0.3
0.149101
0.231983
0.449962
0.515561
997
5897
3981
N41 D200S150B2H50%L25%R25%
5000 time step
0.3
0.140422
0.236279
0.393681
0.60018
1710
5221
2894
N41 D200S150B1 H50%L25%R25%
5000 time step
0.55
0.141971
0.242762
0.449484
0.540091
6573
6382
4321
N41D200S150B2H50%L25%R25%
5000 time step
0.55
0.154991
0.242762
0.449484
0.540091
7199
5438
3151
N41D200S150B1 H76%L12%R12%
5000 time step
0.1
0.144761
0.224637
0.345602
0.649987
1
2335
1986
N41D200S150B2H76% L12% R12%
5000 time step
0.1
0.15902
0.231568
0.41157
0.562645
5
2274
1509
0.3
0.147241
0.247035
0.400498
0.616819
997
5925
3943
0.25273
0.436416
0.579105
1784
5209
2887
N41D200S150B1 H76%L12%R12%
5000 time step
N41 D200S150B2H76% L12% R12%
5000 time step
0.3
0.145691
N41D200S15081 H76%L12%R 12%
5000 time step
0.55
0,151891
0.254654
0.416999
0.610683
lijÉÉi
N41D200S150B2H76% L12% R12%
5000 time step
0.55
0.143521
0.252916
0.421461
0.600093
7207
5551
3112
N41D500S30B1 H50%L25%R25%
5000 time step
0.1
0.153441
4
2192
1999
0.21754
194
0.403541
0.648396
flllilll!
4224
N41D500S30B2 H50%L25%R2S%
SQOO time step
01
0140112
0 205436
Q 316837
12
2404
1547
0 628598
0,539077
1018
6193
4206
N41D500S30B1 H50%L25%R25%
5000 time step
03
0146001
0 224838
0 357681
N41D500S30B2 H50%L25%R25%
5000 time step
03
0150651
0 222188
0 437351
0 50803
2045
5257
2963
N41D500S30B1 H50%L25%R25%
5000 time step
0 55
0148171
0 228898
0 41972
0 545359
6360
6387
4363
N41D500S30B2 H50%L25%R25%
5000 time step
0 55
0152511
0 232706
0 393686
0 591094
7403
5508
3174
N41D500S30B1 H76%L12%R12%
5000 time step
0 1
0148791
0 221385
0 341653
0 64798
4
2405
2022
N41 D500S30B2H76%L12%R12%
5000 time step
0 1
0153441
0 224695
0 454287
0 49461
11
2608
1602
N41D500S30B1 H76%L12%R12%
5000 time step
03
0143831
0 2378
0 335209
0 709408
1041
5980
3978
N41 D500S30B2H76%L12%R12%
5000 time step
03
0146311
0 240485
0 407518
0 590121
1822
5229
2918
N41D500S30B1 H76%L12%R12%
5000 time step
0 55
015096
0 247642
0 397962
0 622276
6511
6350
4343
N41 D500S30B2H76%L12%R12%
5000 time step
0 55
0153131
0 243118
0 424339
0 572933
7704
5343
3042
0 746076
2128
5000 time step
0 1
0152821
0 218188
0 292447
4
2330
N41D500S150B2H50%L25%R25%
5000 time step
0 1
0146621
0 221446
0 432387
0 512148
10
2305
1448
N41D500S150B1 H50%L25%R25%
5000 time step
03
0150031
0 235491
0 340413
0 691779
955
6016
4099
N41D500S150B2H50%L25%R25%
5000 time step
03
0142901
0 233617
0 387521
0 602849
1701
5280
2989
N41D500S150B1 H50%L25%R25%
5000 time step
0 55
0154371
0 24067
0 377373
0 63775
13170
6449
4450
N41D500S150B2H50%L25%R25%
5000 time step
0 55
0145071
0 241727
0 410919
0 588259
7009
5520
3110
N41D500S150B1 H76%L12%R12%
5000 time step
0 1
0155921
0 226911
0 353205
0 642433
5
2314
1961
N41 D500S150B2H76%L12%R12%
5000 time step
0 1
0143831
0 232345
0 437421
0 53117
12
2391
1469
N41D500S150B1 H76%L12%R12%
5000 time step
03
0143521
0 248775
0 416261
0 597642
1064
5859
3876
N41D500S150B2H76% L12% R12%
5000 time step
03
015902
0 250608
0 426361
-0.587783
1815
5171
2892
N41D500S150B1 H76%L12%R12%
5000 time step
0 55
0153131
0 25311
0 435938
0 580609
6351
6338
4340
N41D500S150B2H76% L12% R12%
5000 time step
0 55
0152511
0 25334
0 418776
0 604953
7273
5440
3059
N41D500S150B1 H50%L25%R25%
Ta b l e (6)
Summary of results for a n o n - t r a n s x e n t m o m e n t s i m u l a t i o n s for a 41 nodes network, w h e r e
D
t r a n sient
period, S
interval length u s e d to c a l c u l a t e averages, H
h e a d w a y percentages, L
turn left percentages, R
turn
right p e rcentages
Appendix C
Feeding Mechanism
For Poisson arrivals, inter-arrival time between two vehicles is randomly drawn
from the negative exponential distribution, 1 e
t n+ l = t n
In (1 ‘ V ) ,
---------------------
p > 0r,, 0n < r <^ l!
f*
where
tn+l
= inter-arrival time for next vehicle
tn = inter-arrival time for previous vehicle
u
= arrival rate
r
= random number uniformly distributed between (0, 1]
As the random number generator is fundamental in stochastic simulation, we have
used the linear congruential random number generator, because of its cycle
length
We compute the zth integer
Xt =
where
X i in the pseudorandom from
+c)
by the recursion
modm
a = 16807, c = 0, and m = 2147483647, which is widely known as the
“multiplicative congruential” generator.
196
Appendix
D
Computational Performance
Table (D 1) shows the running times for simulating two different networks at two
arrival rates for two different simulation runs, 5000 and 7000 time steps
17 nodes network
Run
Duration
p=0 1
41 nodes network
p=0 5
fi=0 1
H=0 5
CPU
Simulated
CPU
Simulated
CPU
Simulated
CPU
Simulated
tune
cars
time
cars
time
cars
tune
cars
5000
39
2,635
41
12,696
49
2,309
50
11,350
7000
54
3,785
58
17,045
77
3,292
80
16,509
Table
(D 1)
the relation between the run duration and
the computational time,
m
Seconds,
for two different
networks at two arrival rates
It can be seen from Table (D 1), that the computational time mainly depends on
the network size not the number of the simulated cars Also it can be seen from
the table that the complexity of the code is 0 (n 2), being the increasing factors the
network size and the run duration Also there is a slight increase in the
computational time as we increase the arrival rate, Table (D 2) this increase is due
to the time required to update the queues outside the simulated network
197
The limitation in the run duration above is due to the formation of long queues
outside the network especially when a high arrival rate is used to feed the network
with cars
Arrival Rate
Network size
CPU, 5000 time steps
CPU, 7000 time steps
Increasing Factor
17
39
54
1 38
41
49
77
157
17
41
58
141
41
50
80
1 60
p = 01
p=05
Table
(D 2)
The
table
computational time,
shows
that
in Seconds,
the
depends
increasing
factor
m
the
on both the network size
and the arrival rate
All computational performance measures described above is obtained on DEL PC
with 400 MHZ of speed and 64 MB of RAM
198
Appendix E : Diskette
The list of the diskette folders and files are presented in Table ( E l)
folder
files
Brief description
Short evt net
model 1 h
n odelh
segment h
model 1 cpp
Header file, define model 1 base class
Header file, define nodel class
Header file, define segmenl derived class
Contains the methods of the base class model 1, which create
roads of the network
Contains the methods for the class nodel, which initialize
the network nodes and also update the network light cycles
Contains the methods for the derived class segmenl, which
link the roads within the network and also initialize and
update the road segments
The main file, see flow-chart m sec(3 2)
Contains nodes data for 17 nodes network
Contains nodes data for 25 nodes network
Contains nodes data for 33 nodes network
Contains nodes data for 41 nodes network
Contains segments data for 17 nodes network
Contains segments data for 25 nodes network
Contains segments data for 33 nodes network
Contains segments data for 41 nodes network
Same as the above folder , but deals with long term events
Simulate traffic flow using two lanes
Header file contains the global variables
Simulate traffic flow for inter-urban areas using closed
boundary conditions
Simulate traffic flow for inter-urban areas using open
boundary conditions
Used to produce the space-time diagrams throughout the
thesis
nodel cpp
segmenl cpp
newnet cpp
nodl7 dat
nod25 dat
nod33 dat
nod41 dat
seg56 dat
Seg88 dat
Segl20 dat
Segl52 dat
Long evt net
Two lan
Stochm odel
2 lane model
tlgvh
Stoch_closed
Stoch_open
Spacetim e
be c
Table ( E l)
199