Realtimeoptimized trafficmanagement

International Journal of Computer Science & Information Technology (IJCSIT) Vol 6, No 4, August 2014
REAL TIME OPTIMIZED TRAFFIC MANAGEMENT
ALGORITHM
Partha Sarathi Chakraborty, Prajeeth Nair, Pranshu Raj Sinha and Ishan Kumar
Behera
Department of Computer Science and Engineering, SRM University, India
ABSTRACT
In the present scenario, research conducted is mostly based on determining the duration of green light.
Moreover the research papers published on Adaptive Traffic Management did not focus much on the
concept of handling Emergency Vehicles. This major role of this project is as a continuation to the
existing research papers published on this topic. Here we not only handle traffic effectively but also
elaborate on effective management of highly prioritized vehicles through all possible phases. In this
particular research paper, Wireless Sensor Networks (WSN) is assumed to be the source of input.
KEYWORDS
Wireless Sensor Networks, WSN, Traffic Management.
1. INTRODUCTION
The number of vehicles seen has been increasing rapidly over the past few years. The reason for
this phenomenon could be attributed to growing population and everyone’s desire to possess their
own personal mode of transport. To bear this burden is where Intelligent Transportation System’s
comes into play. ITS or Intelligent Transportation System, as name suggests, provides an
effective and intelligent traffic management solutions with a goal to minimize delay in traffic,
improvement in traffic, and efficient flow of traffic. Such goal could be achieved through
surveillance units like sensors (wired or wireless), Bluetooth, infrared and so on.
The communication system manages the communication between surveillance system and traffic
light control system. The traffic light control system implements the dynamic traffic signal
control algorithm and displays the corresponding traffic lights (red/green) to various lanes.
1.1. Responsibility of WSN
In order to achieve the desired objective the source of input we use is the Wireless Sensor
Networks (WSN) which effectively monitors traffic.
1.2. Need for Dynamic Traffic Signal Control
There are two main methods in traffic signal control:
i. Periodical signal control.
ii. Dynamic signal control.
DOI:10.5121/ijcsit.2014.6408 119

The periodic signal control method uses a pre-defined sequence and duration of green light for
various lanes. As name suggests, the first method is ineffective for real time traffic scenarios and
therefore has limited efficiency. The dynamic traffic signal control method adapts the traffic
signal in accordance with the dynamic traffic flow and thus works effectively for less as well as
busy intersections also.
1.3. Steps of proposed Dynamic Traffic Signal Control Algorithm
The proposed algorithmcontains four steps:
i. Determining queue length (volume) of traffic.
ii. Determining the presence of Emergency Vehicles as well as the scenario (best to worst
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case)
iii. Assigning green light to most suitable phase.
iv. Calculating green light duration for each phase.
2. LITERATURE SURVEY
One of the very first phases of our project was to collect all possible research papers
written in any of the National or International Journals for the concerned topic.
These research papers were thoroughly studied and any scope for further improvements was
pondered upon. Our project uses these two papers as its base:-
2.1. Literature Survey [1]
Title– Dynamic Traffic Signal Control Algorithm in Intelligent Transportation System through
Wireless Sensor Networks.
Author– Monica Johri ,Anurag Goel, Ashutosh Kumar Tiwari.
Journal/Conference– International Journal of Engineering& Science Research
Description –
i. The flow of traffic through the intersections decides the signal phase selection. The
four directions are marked as North, South, East and West, each direction having three
lanes, which are turn-left (L), go-forward (F) and turn-right (R).
ii. To formulate the problem, the paper assumes that the left-turn (L) is permitted all
the time. So each passing vehicle can have a direction d of {E, S, N, W} and lane l
of {F, R}.For the emergency vehicles, the proposed algorithm needs some prior
information about them such as their position, speed, direction of travelling
etc. all of which it receives from the GPS.
All the possible twelve phases are introduced in their proposed algorithm.

121
Figure 2.1 The twelve phases
This algorithm considers a number of traffic factors such as Lane waiting queue, Queue
passing time, Phase waiting queue, waiting time etc.
2.2. Base Research Paper [2]
Title -Intelligent Traffic Light Flow Control SystemUsing Wireless Sensors Networks.
Author -Khalil M. Yousef, Jamal N. Al - Karaki and Ali M. Shatnawi.
Journal/Conference- Journal of Information Science and Engineering 26, 753-768 (2010).
i. A running vehicle can be given as {P, D} which results in twelve cases { NR, NF, NL,
SR, SF, SL, ER, EF, EL, WR, WF, WL} where P represents the path and D represents the
direction.
ii. In this paper, an intelligent traffic light control system is presented which consists of two
parts:
a. Wireless Sensor Network (WSN).
b. Traffic control box (TCB).
WSN is nothing but a network of small nodes known as Traffic Sensor Nodes which
gathers information regarding the traffic and stores it in the Base Station.
iii. In this paper, two algorithms are proposed –
i. Traffic system communication algorithm (TSCA).
ii. Traffic signal time manipulation algorithm (TSTMA).

122
Figure 2.2 Model of Intersection and TSN Architecture
The TSCA controls the communication routes between all the TSNs and BS as well as interfacing
with TCB in a simple and efficient manner.
The TSTMA generates a traffic phase plan which is a sequence of four phases and in each phase,
two lanes are selected to pass the intersection, on the basis of dynamic traffic volume.
3. PROBLEM DESCRIPTION
3.1. Existing Problem
Need for detailed approach towards handling prioritized vehicles (also known as emergency
vehicles) like Fire Truck, Ambulance and Police Vans by taking into consideration all
possible scenarios from best case to worst case.
Reference paper [1] proposes a very basic approach towards handling these vehicles by simply
assigning green light for the direction where the emergency vehicle is present. Reference
paper [2] however does not consider emergency vehicles at all.
3.2. Proposed Technique
i. In our proposed algorithm we offer a solution to tackle the worst case (deadlock)
condition for management of emergency vehicles.
This is done as a 2 stage process:
Stage 1 : The first stage categorizes Emergency Vehicles on the basis of Standard
policies which may vary from one country to another. This assigns a priority value for
each type of Emergency Vehicle.

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Stage 2 : Considering the distance of each of the Emergency vehicle from the
intersection point.
Figure 3.1 Process Flow Chart
ii. As an addition we have also included a totally abstract approach for more Optimized
Emergency Vehicle Management by collaborating Code Messaging Service (CMS) and a
GPS Enabled Mobile Device. Being an abstract thought, this model serves the purpose of
base for possible future research.
4. PROPOSED METHODOLOGY
4.1 Problem Formulation and Assumptions
4.1.1 Problem Formulation
i. The flow of traffic through the intersections decides the signal phase selection.
ii. To formulate a solution, we assume that the left-turn (L) is permitted all the time.

iii. All 8 pairs of (d, l) combine in various combinations which results in twelve possible
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cases, as shown in figure 4.1.
4.1.2 Assumptions
i. Constant speed of all vehicles of similar type.
ii. It is assumed that all the vehicles have a vehicle node installed in them.
iii. The data regarding traffic volume approaching towards the intersection point from
various lanes, which is required for the implementation of proposed dynamic traffic
signal control algorithm, is assumed to be getting from the Wireless Sensor Network.
The working of WSN is not a part of our paper.

125
Figure 4.1. All possible phases for a 3 Lane intersection
4.2 Different Problem Notations Used
The notations depicting directions, lanes and cases are given as:
D= {North, South, East, West}.
L= {Forward, Right}.
C= {1, 2, 3, 4… 12}.
Here, D, L and C denote the set of directions, lanes and cases respectively.
4.2.1 Expectant Phase
Expectant Phase, denoted as E, of a vehicle is defined as the phase in which a vehicle pass
the intersection.
Case 01: Ef (N,F)=Ef (N,R)= fa.
Case 02: Ef (S,F)=Ef (S,R)= fb.
Case 03: Ef (E,F)=Ef (E,R)= fc.
Case 04: Ef (W,F)=Ef (W,R)= fd.
Case 05: Ef (N,F)=Ef (S,F)= fe.
Case 06: Ef (N,R)=Ef (S,R)= ff.
Case 07: Ef (E,F)=Ef (W,F)= fg.
Case 08: Ef (E,R)=Ef (W,R)= fh.
Case 09: Ef (N,F)=Ef (W,R)= fi.
Case 10: Ef (S,F)=Ef (E,R)= fj.
Case 11: Ef (N,R)=Ef (E,F)= fk.
Case 12: Ef (S,R)=Ef (W,F)= fl.

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4.2.2 Lane Waiting Queue
Lane Waiting Queue, denoted as Q (d, l), is defined as the number of vehicle waiting on
the path P= {d, l}, d Î D and l Î L.
4.2.3 Phase waiting Queue
Phase Waiting Queue, denoted as Qx, is the length of queue for phase x
Qx = MAX (TQ (d, l)), (1)
where (d, l) Î {Ef (d, l) = fx};
d Î D ;
l Î L ;
x Î {a, b, c,…………,l}.
4.2.4 Queue passing Time
Queue passing time, represented as TQ (d, l), is the time taken by all waiting vehicles to
pass the intersection.
TQ (d, l) = T1 + (τ * (Q (d, l) -1)) (2)
T1 is the time taken by first vehicle to cross the intersection.
τ is the time taken by a vehicle to move to the place of front vehicle.
Phase Queue passing time, denoted as TQx, is calculated as -
TQx = MAX (TQ (d, l)) (3)
where (d, l) Î {Ef (d, l) = fx};
d Î D ;
l Î L ;
x Î {a, b, c,…………, l}.
4.2.5 Emergency condition
Emergency condition, EC (d, l), determines the presence of any kind of Emergency
vehicle.
4.2.6 Waiting Time
Waiting Time, WT (d, l),gives the waiting time for the vehicle in front.
4.2.7 Threshold Waiting Time
Threshold waiting time, Tthreshold, sets the maximum waiting time.
4.2.8 Phase Time
Phase time, TPx, defines duration of green light for phase x
TPx = MIN (TQx , TPmax) (4)
4.2.9 Maximum Phase Time
Maximum phase time, denoted as TPmax, gives the maximum time for which green light
is allotted.
4.3 The main steps in Dynamic Traffic Signal Control Algorithm
The proposed dynamic traffic signal control algorithm contains mainly four steps:
i. Determining the queue length (volume) of traffic.
ii. Check for the presence of Emergency Vehicles as well as deciding the current
scenario (best or worst).

127
iii. Assigning green light to the most suitable phase.
iv. Calculating how much time must be allotted to the phase decided in (iii).
4.3.1 Traffic Volume Detection
In the first step of our algorithm the queue length (Qx) for each phase is computed. This
gives us a fair idea regarding the number of vehicles present in each phase. The source of input
for this stage of our algorithm is WSN.
4.3.2 Check for Emergency Vehicle Presence
In the second step of the algorithm we make a routine check in order to ensure whether an
Emergency vehicle is present at the intersection or not.
An Emergency vehicle is a vehicle which has to be assigned a greater priority than other
motorized vehicles.
For this purpose we first need to assign a priority for different types of emergency vehicles. For
example Fire Trucks can be assigned priority 0(highest) followed by ambulance and so on.
Assigning priority for the Emergency Vehicles is totally dependent on the standard policies,
taking into consideration multiple parameters, and may vary from country to country.
Considering all probable scenarios, we classify the management of Emergency Vehicles on the
basis of 4 cases:
CASE 1 : Only One Emergency Vehicle is Present
In this case we simply assign green light to the lane where the
Emergency Vehicle is present.
CASE 2 : Two Emergency Vehicles each having different Priority
In this case we simply allow the vehicle with the higher priority to pass
through first followed by the vehicle with lower priority.
CASE 3 : Two Emergency Vehicles each having same Priority
This is a typical scenario since both the Emergency Vehicles have the
same priority. In order to tackle such a situation we need to be fair to
both these vehicles. Therefore we assign green light priority first to the
Emergency Vehicle present at the shortest distance from the
intersection.
CASE 4 : Four Emergency Vehicles (all different priorities) placed at each of
the four lanes [DEADLOCK]
Such a situation can be rightly tackled by assigning Green light to the
Emergency Vehicle on the basis of Highest Priority First (HFS)
Scheduling.
4.3.3 Assigning Green Light to most suitable phase
In this step, the queue length (Qx) is calculated as per the input given by the Wireless Sensor
Network.
4.3.4 Determination of Green Light Duration
The third step is to determine as to how much time green light should be assigned to a phase.
The maximum phase time should not be more than 120 seconds.

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4.4 The Proposed Dynamic Traffic Signal Control Algorithm
Dynamic Traffic Signal Control Algorithm
//Initialization – Here parameters like Max Time and threshold time is initialized
1. Initialize Tmax = 90, Tthreshold=120.
Set Q(d,l) " (d,l), where d Î D, l Î L.
Q(d,l) contains the number of vehicles on the path P={d,l}
//Traffic Volume Detection – Computing the queue length for each lanes
2. Compute Qx = MAX(Q(d,l)) " x, where x Î {a,b,…,l}, d Î D, l Î L.
Qx contains the maximum queue length of each of the 12 phases
/*Emergency Check and Traffic Phase Selection – Here check for the presence of emergency
vehicles is made so as to assign green light to the most suitable phase */
3. Check emergency condition " d, d Î D.
4. If direction d¢has greater priority then goto 5 else goto 7
5. If(Emergency vehicle only in one direction d¢)
Assign green light to phase fx next such that {Ef(d¢,l1)= Ef(d¢,l2)}, l1,l2 Î L.
d¢= direction of emergency vehicle
l1, l2 = all the 2 possible lanes
6. Else If (Emergency vehicle in more than one direction but of different priorities)
a) Assign count= number of emergency vehicles in different direction
b) For k= 1 to count
Loop
Int priority[k] = Priority of emergency vehicle for each direction
Char dir[k] = direction of presence of Emergency Vehicle
End Loop
c) Sort the array priority[] in ascending order of priority and correspondingly
change the value of dir[]
d) For I = 1 to count
Loop
Assign green light to Ef(dir[i],l1)= Ef(dir[i],l2)=fx
End Loop
7. Else
a) Assign count= number of emergency vehicles
b) For I=1 to count
Loop
Int arr[i] = distance from intersection of emergency vehicle for each direction
Char dir[i] = direction of presence of Emergency Vehicle
End Loop
c) Sort the array arr[] in ascending order of distance and correspondingly change the value
of dir[]
d) For I = 1 to count
Loop
Assign green light to Ef(dir[i],l1)= Ef(dir[i],l2)=fx
End Loop
8. Compute WT(d,l) " (d,l), where d Î D, l Î L.
9. If there exists a path (d¢,l¢) with WT(d¢,l¢) ³ Tthreshold then
10.Compute Qx = MAX(Q(d′,l′),Q(d,l)), such that {Ef(d¢,l¢)= Ef(d,l)=fx},
for every possible x, xÎ{a, b,c,…,l}, d Î D ; l Î L.

129
11. Assign green light to phase fx having maximum value of Qx computed
above, next.
12. Else
Assign green light to phase fx having maximum value of Qx computed
in step 2, next.
13. End if.
14. End if.
//Determination of Green Light Duration – Calculating the duration of green light
15. Compute TQx= T1 + (τ * (Qx -1)).
16. Compute TPx = MIN(TQx, TPmax).
17. Set the green light time for phase fx for time TPx.
Note : Qx as well as TQx are changed dynamically according to the traffic flow.
4. RESULT
The efficiency of our solution to the existing shortcoming is expressed on the basis of
theoretical and physical factors.
On theoretical basis, our algorithm is a success because we have tried to provide solution to a
very new problem of Emergency Deadlock condition, which opens a new dimension for further
research and exploration of more complex traffic situations and development of their solutions.
On the basis of physical factors like intersection throughput factor etc. our algorithm is nearly
similar to the one presented in our base research paper[1]. But due to the introduction of new
concept in the algorithm, there has been some negligible effects on the time complexity.
In computer science, the Time Complexity of an algorithm quantifies the amount of time
taken by an algorithm to run as a function of the length of the string representing the input.
For the above proposed algorithm, the entire time complexity relies on the sorting
technique used in order to sort the distance of emergency vehicles from the
intersection as well as to sort the pre-defined priority. In order to maximize the efficiency, the
sorting technique considered is Merge Sort [11] with best case, worst case and average case as O
(nlogn).
Therefore the overall time complexity for the proposed algorithmis O (nlogn).
5.1 Comparing Efficiency with Base Algorithm
Keeping the time complexity as the medium of consideration we compared the time taken by both
these algorithms for different values of ‘n’.
Table 5.1 Time Complexity for different values

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Figure 5.1: Algorithm Efficiency Graph
It is evident from above that with increasing number of emergency vehicles (n) the overall time
consumption by our algorithm would increase quite a lot in comparison to the existing
(base) algorithm. The results are not surprising since we have been giving a lot of emphasis on
priority vehicles, so a considerable deviation of time complexity is expected.
However, throwing some positive light, this deviation may not be taken as concern since even
in worst case scenario the number of emergency vehicles present at the intersection would be
limited. So there is no need to consider higher values of ‘n’.
Remaining graphs generated are in close reference to Base Paper [1] since our objective was to
provide a meaningful improvement or an “add-on” to the existing base algorithm. We have
successfully accomplished our desired objective, the result of which has been verified after
showcasing the technical implementation.
5. CONCLUSION AND FUTURE WORK
6.1 CONCLUSION
Wireless sensor networks offer a promising platform for traffic monitoring that can compete
with current technology in terms of accuracy and lifetime. In this paper, we have given an
extension to the existing dynamic traffic signal control algorithm keeping in mind the concrete
objective of minimizing the average waiting time. Here we also include a scenario involving
the worst case (deadlock) condition for management of emergency vehicles. This algorithm
offers a solution to the above mentioned problem which has not yet been approached in any of the
published research papers. This project concludes with a concept solution for internal
prioritization of ambulances which offers a wide scope for future development.
6.2 FUTURE WORK
In our future research, the proposed dynamic traffic signal control algorithm can be extended by
eliminating the above made assumptions such the vehicles approaching towards the intersection
from all the directions are ideal sized vehicles are assumed to be ideal sized. This assumption

can be removed by adding the detection of heavy vehicles. Moreover a better and much more
feasible solution to the problem of Internal Priority for Ambulances could be proposed in
future.
APPENDIX
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A. Why WSN?
i. Here we observe what all kinds of properties different surveillance units possess:-
Table A.1 Characteristics of different Surveillance Systems [3]
Y-Available N- Not Available
ii. Inductive loop detector is often considered to be one of the most accurate
surveillance system.
Table A.2 Error Rates for Different Surveillance Systems [3]

132
iii. Environmental factors that affect these traffic surveillance technologies
Table A.3 Environmental Factors affecting various Surveillance Technologies [3]
iv. Calculating the Life Cycle Cost
Formula Used to calculate Lifecycle Cost:
LC = ((device cost*quantity) +installation cost + maintenance cost)*
(5)
I=4.1 (considered to be inflation rate)
n=Life Time in (given as number of years)
Table A.4 Life Cycle Cost Calculation [3]
The values calculated as the life cycle cost does not include the traffic delay cost caused by
disrupting the traffic during future maintenance. Overall the Wireless Sensor Network has proved
to be better than compared to other traffic surveillance technologies.

133
B. Network Architecture
The proposed sensor layout is given as:-
Figure B1: Sensor Layout
There are 3 Incoming Sensor Nodes (ISN) placed on each of the incoming lanes
ISN-0 are the sensors placed closest to intersection
ISN-1 are the sensors placed at a particular distance from the intersection
Overall there are 24 sensors placed along the intersection.
Three Layered Network Architecture
To implement the designed algorithm we assume a Three Layer WSN Architecture (W3
Architecture).

Here we introduce a sensor like node to be installed at each and every vehicle. The
primary objective of introducing W3 Architecture is to solve the problem of vehicle
identification that can be implemented in a variety of applications like vehicle speed
detection, tracking in case of thefts etc.
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Figure B2 : 3 Layer Architecture
The three layers are given as:
a) LAYER 1 : ISN0  Access Point
b) LAYER 2 : ISN1  ISN0
c) LAYER 3 : Vehicle Node  ISN1
a) Just like a handshaking signal in network communication, ISN1 broadcasts REQ
(Request) packet. Upon receiving the signal the vehicle node which is installed inside
the vehicle grants its identification (V_ID) packet which contains information regarding
its Id number, speed, type etc.
b) The sensor node ISN1 then relays the information to its successor, ISN0.
c) ISN0 forwards the information to the Access Point where it is stored in the database
using the vehicle id as the primary key in order to avoid repetition of vehicles. The
Access point is the Base Controller where our algorithm will run, by using information
stored in the database.
A vehicle node is usually in a sleeping state. When a vehicle comes into the road where
sensor nodes are installed, vehicle node switches to the active state [6].
C: INTERNAL PRIORITY TO EMERGENCY VEHICLES
Measuring the Level of Emergency (For Ambulances)
There is a need to assign a first degree priority to emergency vehicles, most notably; an
ambulance on the basis how critical the patient is. This is needed since a person suffering from
a critical illness may need to reach the hospital more quickly than a person whose condition is
less severe.

135
We begin with certain assumptions as follows:-
i. First of all, we begin with the assumption regarding the passengers in the ambulance
vehicle.
a) Driver : The person who drives the ambulance
b) Patient : The patient who may/may not be critically ill.
c) Nurse : The nurse acts as knowledgeable person who can make the most
accurate judgment regarding the condition of the patient. He / She is needed since a
doctor cannot make appropriate decision over the phone.
d) Family Members (optional).
ii. The nurse inside the ambulance is equipped with a standardized mobile device. This
mobile device has functionalities of GPS (Global Positioning System) and Code
Messaging Service (CMS).
HOW IT WORKS
Figure C1: A Proposed Solution for Internal Priority to Emergency Vehicles
iii. Initially we need a set of condition code for all possible conditions the patient may suffer
from.
iv. These condition codes are nothing but a set of integers (starting from 0) which depicts the
severity of the patient.
v. Every ambulance must have a chart depicting the condition code along with its
description.

136
For example
POSSIBLE MEDICAL CHART
Condition Code Description
0 Heart Attack(critical)(Highest Priority)
1 Heart Attack(mild)
: :
:
:
:
N Fever (Mild) (Lowest Priority)
vi. 0 depicts the most critical condition followed by 1, 2, 3… N, where N depicts the least
emergency.
vii. After making the necessary check up the nurse enters the relevant condition code in the
GPS Enabled CMS Device.
viii. The Message body comprises of two sections : -
(i) The Location – This contains the GPS Coordinates and is automatically
entered by the device.
(ii) Condition Code – An integer from 0 to N which depicts the condition of the
patient.
ix. The Central Database collects all information regarding each and every Access Points.
The Central Database processes the GPS Coordinates and inserts the value of the
condition code for that particular vehicle across all the Access Points present en route to
the hospital.
This way we can have the Access Point grant the Green Light for the Emergency Vehicle with
highest priority (lowest condition code).
REFERENCES
[1] Monika Johri, Anurag Goel, Ashutish Kr. Tiwari “Dynamic Traffic Control Algorithm in Intelligent
Transport System through Wireless Sensor Networks”, IJESR 2012
[2] “Intelligent Traffic Light Flow Control System Using Wireless Sensors Networks.” Khalil M.Yousef,
Jamal N.Al - Karaki and Ali M.Shatnawi. JOURNAL OF INFORMATION SCIENCE AND
ENGINEERING 26, 753-768 (2010)
[3] “Traffic Surveillance by Wireless Sensor Networks – Final report” Sing Yui Chen, Pravin Varaiya –
California Path Research Report UCB-ITS-PRR-2007-4
[4] “A Wireless Sensor Network for Traffic Surveillance”, Sing Yiu Cheung, Sinem Coleri, Ram
Rajagopal,Pravin Varaiya
[5] “Wireless Sensor Networks - Wikipedia” http://en.wikipedia.org/wiki/Wireless_sensor_network
[6] “A Real Time Dynamic Traffic Control using Wireless Sensor Networks”
CHEN Wenjie, CHEN Lifeng, CHEN Zhanglong, TU Shiliang Department of Computer Science and
Engineering, Fudan University IEEE- 1530-2016/05 © 2005
[7] “The Research of WSN-based Vehicle Information Detection Technology” Shen Ming Yu , Wang
Wei. School of Computer and Information, Hefei University of Technology, Hefei 230009, China -
2012 International Conference on Electronics, Information and Communication Engineering
[8] “A Distributed Algorithm for Multiple Intersections Adaptive Traffic Lights Control using a Wireless
Sensor Networks” Sebastien Faye, Claude Chaudet, Isabelle Demuere - Institut Mines-Télécom
CNRS LTCI UMR 5141
[9] “Comparison of Simulators for Wireless Sensor Networks” Martin Sehlik Master Thesis – Spring
2011 Faculty of Informatics Masaryk University
[10] “Time Complexity” http://en.wikipedia.org/wiki/Time_complexity
[11] “Merge Sort” http://bigocheatsheet.com/

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