Research Article Strong -Barrier Coverage for One-Way Intruders...

Research ArticleStrong 𝑘-Barrier Coverage for One-Way IntrudersDetection in Wireless Sensor Networks

Junhai Luo and Shihua Zou

University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Correspondence should be addressed to Junhai Luo; junhai [email protected]

Received 18 January 2016; Revised 6 April 2016; Accepted 8 May 2016

Academic Editor: Isaac Agudo

Copyright © 2016 J. Luo and S. Zou. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Intruders detection is one of the very important applications inWireless Sensor Networks (WSNs). Sometimes detecting intrudersis not sufficient; distinguishingwhether an intruder is legal or illegal is necessary. Since 𝑘-barrier coverage is widely used in detectingintruders, a barrier construction algorithm is needed, which can not only detect an intruder but also judge an illegal intruder. Anintruder is defined as illegal if and only if it crosses straightly through the monitored region from the special side to another side.On the contrary, it is a legal intruder. To detect an intruder and distinguish whether the intruder is legal or illegal, a strong 𝑘-barriercoverage algorithm is proposed.The strong 𝑘-barrier coverage is a local barrier constructing algorithm and can detect any intrudercrossing the 𝑘-barrier with a full probability. The strong 𝑘-barrier coverage detects all intruders penetrating the annular region for𝑘 times. What is more, the proposed strong 𝑘-barrier algorithm can provide a reliable judgement on whether an intruder is legalor illegal, and the constructed 𝑘-barrier coverage is different from the traditional one-way barrier coverage using binary barriersthat are intersected but not overlapped. Some simulation tests show that the proposed algorithm can construct strong 𝑘-barriercoverage very well.

1. Introduction

Wireless Sensor Networks (WSNs) have been studied foryears, which have many important applications for theirparticular capabilities in monitoring environment, such asin a battlefield or in an international border; all thesecircumstances need a comprehensive monitoring. Barriers inWSNs for intruders detection have been studied a lot, whichcan assure that all intruders crossing a belt annular can bedetected. In some applications, only one direction crossingthe region is illegal. For example, people can leave a theaterwithout their movie tickets being detected, whereas they arenot allowed to see a movie without tickets. In the aboveapplications, it is not appropriate to use the traditional barriercoverage models since they cannot distinguish the legal andillegal intrusion behaviors.

In most previous works, researchers only consideredwhether or not WSNs can form barrier coverage or whetherWSNs provide 𝑘-barrier coverage. The theoretical basis ofdesigning barriers of WSNs is studied in [1]. In one barrier

coverage, intruders may cross the barrier from the gap with-out being detected. Since one barrier coverage cannot meetthe requirements of intrusion detection inmany applications,high degree of coverage must be taken into account, andthe reasons of requiring 𝑘-barrier rather than 1 barrier arediscussed in [1]. Aweekly 𝑘-barrier coverage has been studiedin many literatures, such as [2]. For some applications ofWSNs, weekly 𝑘-barrier coverage is not enough; thus thestrong 𝑘-barrier coverage is needed, for example, aWSNwitha really high detecting probability. Detecting intruders withhighly probability relies on a strong barrier coverage.

In the study of barrier coverage, construction algorithmsfor barrier have been researched much. Generally speaking,there are two kinds of research directions, local barrierconstruction and the global barrier construction algorithms.Though global barrier coverage, such as the one in [3],which requires much fewer sensors than full coverage, is abefittingmodel of coverage for intrusion detection, it also haslimitations.Then a local barrier coverage algorithm is studiedintensively in [4–7].

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2016, Article ID 3807824, 16 pageshttp://dx.doi.org/10.1155/2016/3807824

2 International Journal of Distributed Sensor Networks

Our contributions in this paper are as follows:

(1) We find that traditional one-barrier coverage modelis not suitable for some applications of WSNs inintruders detection, especially in one-way intrudersdetection.

(2) Comparing the previously binary barriers models, weconsider using a 𝑘- (more than 2) barrier model toachieve the one-way intruders detection, which doesnot need to satisfy the conditions in binary barriermodel.

(3) We propose a 𝑘-barrier coverage algorithm, whichcan easily construct 𝑘 barriers in monitored areas;then we use the formed 𝑘-barrier coverage modelto realize the goal of one-way intruders detectionwithout satisfying the limitations of binary barrierscoverage.

The remainder of the paper is organized as follows. Section 2introduces the related work.Then the systemmodel, the bar-rier construction algorithm, and the barrier mending modelare presented in Section 3. Section 4 shows an importantapplication of ourmodel.The simulation results are presentedin Section 5. Finally, conclusion of this paper is presented inSection 6.

2. Related Work

The authors in [1] research the mathematical conditions ofbuilding a weak barrier coverage in randomly deploymentWSNs and devise a centralized algorithm to determinewhether a region is 𝑘-barrier covered. In literatures [8–11],the authors study barrier coverage intensively. The authorsin the literature [12] suppose the situation of one barrier,and they study an energy-efficient border intrusion detectionalgorithm, with measuring and guaranteeing the coveragequality of WSNs.

In [2], the authors introduce a weak 𝑘-barrier coverageproblem by at least 𝑘 sensors in a belt region and proposea simple algorithm to determine whether a belt region isweakly 𝑘-barrier covered. The strong barrier coverage ofWSNs is introduced in [9], which is a model with a reallyhigh probability to detect intruders. The authors in [13]describe the worst- and best-case coverage calculation forhomogeneous isotropic of WSNs, in which they combine theVoronoi diagram and graph search algorithm. In literatures[14, 15], the authors present a fundamental geometric datastructure of the Voronoi diagram, which present us with apreliminary knowledge of theVoronoi diagram. Sometimes itis not sufficient for weak 𝑘-barrier coverage. In the literatures[16–18], the strong barrier coverage has been researcheddeeply. And the authors propose the minimum and maxi-mum exposure path algorithms, respectively. The minimalexposure path is thought to be the worst-case coverage ofWSNs, which is a weak barrier coverage. On the contrary, themaximum exposure is best-case coverage of WSNs.

The authors in [4–7] consider that the sensors cannotlocally determine whether the sensor deployment providesglobal barrier coverage. So the authors present a localized

sleep-wake-up algorithm for maximizing the network life-time. Barriers in the above literatures are all presented inone-dimensional space. In the literatures [19, 20], a 𝑘-barrierscheduling for the intruder detection in 2-dimensional spaceis introduced. The authors propose a distributed schemewhose target is providing low detection delay and low energy.Considering that the detection delay and low energy at thesame time are a challenge, it is essential to consider them inmultidimensional space. Besides, a new concept of one-waybarrier coverage with wireless sensors has been introducedin [21–23], and the authors propose the concept of one-way barrier and establish a necessary condition for WSNsto provide a one-way barrier coverage, which requires thatWSNs report illegal intruders but ignore the legal ones.However, the two barriers, which are intersected but notoverlapped, are limited in binary barriers.

Motivated byworks of the above literatures, we propose touse 𝑘-barrier coverage ofWSNs to replace the previous binarybarriers to achieve the one-way detection while achievingthe same goal in the literatures [21–23]. A strong 𝑘-barriercoverage algorithm is proposed, which is a local barrieralgorithm and can detect any intruders crossing the 𝑘-barrierwith a full probability, assuring that all intruders penetratingthe annular region can be detected 𝑘 times. Furthermore, theproposed 𝑘-barrier model can make a reliable judgement onwhether an intruder is legal or illegal, which is different fromthe traditional one-way barrier coverage. When comparedwith the algorithm proposed in [24] and another sensordeployment, the number of sensors is much less.

3. System Model

3.1. Network Model. This paper assumes that sensors aredeployed in an annular region to detect intruders whichtry to cross straightly through the monitored region and toreport the illegal intruders. An illegal intruder is an intrudercrossing through the monitored area from the special side tothe other side. If not, it is a legal intruder on opposite movingdirection. In this section, we give some basic definitions forthe annular region. The annular region is a special case ofa belt region whose two left and right boundaries coincidewith each other while its two upper and lower boundaries areparallel, as shown in Figure 1. In addition, some definitionsabout 𝑘-barrier coverage are given. In Figure 1, we can seethat it is combined with two concentric rings, with the radiusof outer ring being 𝑅𝑂 and inner ring being 𝑅

𝐼, respectively.

The center of the area is marked as 𝑂, and its coordinate isknown as (𝑥𝑂, 𝑦𝑂).

The sensors are deployed following the Gaussian deploy-ment, and the total number of sensors is 𝑁. Each sensorhas an identical sensing radius 𝑟𝑠, and the coverage modelof a sensor is called a disk model. When an intruder is inthe sensor’s sensing range, the intruders can be detectedwith full probability. On the contrary, the intruders cannotbe detected. The disk model usually adopts the Voronoidiagram to evaluate the detection possibility. In the literatures[14, 15], the Voronoi diagram has been well studied. Besides,the number of sensors in each single ring is different, and

International Journal of Distributed Sensor Networks 3

O

RI

RO

Figure 1:The initial wireless sensor network; sensors are distributedas the Gaussian distribution.

the number of sensors deployed in each ring increases inproportion to the circumference of each ring, which is shownin (3). The locations of sensors or their coordinates can beobtained by GPS. Sensors can communicate with each otherwhile the distance between them is in the communicationradius 𝑅

𝑐. 𝑅𝑐≥ 2𝑟𝑠, where 𝑟

𝑠is the sensing radius of a sensor.

Definition 1 (annular belt region). A region bounded withtwo up and down parallel curves and two overlapping rightand left boundary curves is called an annular belt region.

Definition 2 (barrier). A barrier is a sensor group withsensing area overlap to form a belt coverage region startingfrom the left boundary to the right.Theremight bemore thanone barrier in belt region.

Definition 3 (𝑘-barrier coverage). A circular region isreferred to as 𝑘-barrier coverage if and only if any intruderscrossing through the region can be detected 𝑘 times by theWSNs.

Definition 4 (traditional 2-barrier coverage model). It is amodel with two barriers whose lower boundary of one barrierand upper boundary of the other barrier overlap.

Definition 5 (global and local barriers). Global barrier meansthat sensors can know a global knowledge of WSNs to builda barrier; on the contrary, local barrier is defined as a barrier;the constructing sensors of the barrier only know a localbackground of WSNs.

Definition 6 (strong barrier coverage and weak barrier cover-age). A weak barrier has at least one gas and a strong barrierhas no gap, so that no intruders can cross the monitoredarea without being detected no matter what crossing pathsintruders would choose, but there is at least one safety path

A

B C O

k

RI

RO

.

.

.

1

Figure 2: The loop dividing figure, where the blue lines are thedividing line, and the black lines are the two borders, and 𝑘 is presetin our opinion, and 𝑘 ≥ 2.

that intruders could cross the monitored area without beingdetected.

Definition 7 (alarm line). An alarm line is the bottom line(or curve) of the innermost barrier, which will report analarm when an intruder crosses the alarm line from the otherbarriers to the inner of region but does not report any alarmwhenever an intruder crosses the line from the inner regionto the outer region.

Definition 8 (one-way barrier coverage). A circular regionover which sensors are deployed is defined as one-way barriercovered if and only if whenever an intruder crosses themonitored region from the outer ring to the inner ring, andat least one sensor reports an alarm; otherwise, there is notany.

In this paper, we consider a belt region with its twoapproximate vertical boundaries being overlapped, that is,a circular region in Figure 1. The network model shown inFigure 2 describes the loop dividing, and the monitored area𝑆 is divided into 𝑘 equal concentric loops, which can beexpressed as

𝑑𝑤 =(𝑅𝑂− 𝑅𝐼)

𝑘, (1)

where 𝑘 is the required parameter predesigned. Assume thatall sensors know their local coordinates, and the coordinateof the center point𝑂 is known. Each sensor can communicatewith its neighbor sensors within its communication range.Considering a sensor 𝑁

𝑢, whose coordinate is (𝑥

𝑢, 𝑦𝑢), a set

of its neighbor sensors is marked as 𝑁(𝑢) = {𝑢1, 𝑢2, . . . , 𝑢

𝑡},

0 < 𝑡 < 𝑛. As shown in Figure 2, the blue rings representthe inner and outer rings, and the black rings represent thedivided virtual rings.

3.2. Strong 𝑘-Barrier Construction Algorithm. In this section,we propose an algorithm to achieve 𝑘-barrier coverage, whichneeds much less sensors in building barriers. As mentioned


the part of network model, the monitored area is a circulararea, which is divided into 𝑘 equal parts. In addition, theway of building every barrier is identical in each part. Theconstruction of one barrier is introduced as follows, such asthe innermost circular loop, and others do the same.

There are two steps in the barrier construction algorithm.The first step is to assign some basic information to sensorsand let them know the information of their neighbors’locations.Then the other step is to find the remaining sensorsused in building a barrier.

3.2.1. Information Assignment. In this step, we divide the 𝑆into 𝑘 equal parts, and, in every part, a barrier is going tobe designed, as shown in Figure 2. Since the total numberof sensors is 𝑁, we suppose that the number of sensors inthe innermost ring is 𝑛, and the number of other circularrings increases according to the circumference of each ring.Suppose that circumference of the innermost ring is 𝑐

𝑖, and

the circumference of adjacent outer ring is 𝑐𝑖+1

, and the radiiof sensors are 𝑟

𝑖and 𝑟𝑖+1

, respectively. Then the relationshipof the radii is given by

𝑟𝑖+1

= 𝑟𝑖+ 𝑑𝑤, (2)

where 𝑑𝑤 is described in (1).These variables get the followingformulation:

𝑛𝑖

𝑛𝑖+1

=𝑐𝑖

𝑐𝑖+1

=2𝜋 (𝑟𝑖)

2𝜋 (𝑟𝑖+1)=

𝑟𝑖

𝑟𝑖+1

, (3)

where 𝑛𝑖and 𝑛𝑖+1

are the numbers of sensors deployed in twoadjacent circular rings deployed as the Gaussian distribution.Moreover there is

𝑛1+ 𝑛2+ ⋅ ⋅ ⋅ + 𝑛

𝑘= 𝑁. (4)

Previous sections havementioned that the coordinates of sen-sors are known, which are marked as ((𝑥

1, 𝑦1), . . . , (𝑥

𝑛, 𝑦𝑛)).

One of the limitations is the Euclidean distance between twoadjacent critical sensors; that is,

𝑑 (𝑥1, 𝑥2) =√(𝑥1 − 𝑥2)

2+ (𝑦1 − 𝑦2)

2. (5)

And it must satisfy the following condition:

𝑑 (𝑥1, 𝑥2) ≤ 2𝑟

𝑠, (6)

where 2𝑟𝑠 is the critical condition for building an effectivebarrier; the ultimate goal of our paper is to detect foreigninvaders. If 𝑑(𝑥1, 𝑥2) > 2𝑟𝑠, there will be a gap in the barrier;then it has to miss some invaders. 2𝑟𝑠 ≤ 𝑅𝑐 is to keep theWSNs connecting and to ensure that adjacent sensors cancommunicate with each other.

3.2.2. Barrier Construction Algorithm. The last step is specificto constructing the barrier. Firstly, we randomly find astarting sensor and search the next satisfying sensor in turn.In order to simplify the operation, we do the search alongthe clockwise. Equation (6) describes the distance limitation

A1

rs

rs

dw

2rs

Nm+1

Nm

Ri Δ𝜃 O

Nz

Figure 3: Graph of the next sensor in the maximum polar angle.𝑟𝑠is the sensing radius of sensors, 𝑅

𝑖is the innermost radius of the

circular ring, and Δ𝜃 is the same in each single circular rings.

of two adjoining sensors, but it cannot help us find theideal candidate sensor. Then at least one more condition isreally needed, which is the angular restriction. Noting that,we consider the polar coordinates of sensors. The specificexpression of a sensor𝑁

𝑢is as follows:

𝑥𝑢= 𝜌𝑢cos 𝜃𝑢

𝑦𝑢 = 𝜌𝑢 sin 𝜃𝑢,(7)

where 𝜌𝑢is the polar radius and 𝜃

𝑢is the polar angle. Since

the coordinate (𝑥𝑢, 𝑦𝑢) of a sensor is known, the center point

𝑂 is supposed as the pole, whose rectangular coordinate isdescribed as (𝑥

𝑂, 𝑦𝑂), and we can transform the rectangular

coordinate to the pole coordinate.When finding the next sensor, it is not necessary to search

it in global scope because we have to construct 𝑘 strongbarriers, which has no gap between two adjoining sensors.To satisfy the distance limitation, that is, (6), it must be in thepolar angle Δ𝜃 interval, which is a constant value. As shownin Figure 3, the maximum polar angle to find the candidatesensor is described in detail.There are two tangent disks alongclockwise, whose centers are 𝑁

𝑚and 𝑁

𝑚+1, respectively.

Since radius of 𝑆 is big enough for the radii of sensors, thelength of arc between𝑁

𝑚and𝑁

𝑚+1is as approximate as the

Euclidean distance 2𝑟𝑠.

In the triangle Δ(𝑁𝑚𝑂𝑁𝑧), 𝑁𝑧is the intersection of

the center line 𝑁𝑚𝑁𝑚+1

and circle 𝑁𝑚+1

, and rectangularcoordinate is known as (𝑥

𝑧, 𝑦𝑧). Δ𝜃 = 4𝜋/⌈𝜋𝑟(𝑚)/𝑟

𝑠⌉ is the

approximate maximum polar angle interval, which can beobtained by using the Cosine theorem; that is,

cosΔ𝜃 =𝑑2

𝑁𝑚𝑂+ 𝑑2

𝑁𝑧𝑂− 𝑑2

𝑁𝑚𝑁𝑧

2𝑑𝑁𝑚𝑂𝑑𝑁𝑧𝑂

. (8)


Then we can get Δ𝜃 as follows:

max(Δ𝜃 = arccos𝑑2

𝑁𝑚𝑂+ 𝑑2

𝑁𝑧𝑂− 𝑑2

𝑁𝑚𝑁𝑧

2𝑑𝑁𝑚𝑂𝑑𝑁𝑧𝑂

) , (9)

where 𝑑𝑁𝑚𝑁𝑧

= 3𝑟𝑠and 𝑑

𝑁𝑚

, 𝑑𝑁2𝑂can be calculated by

using (5), and in every circular ring Δ𝜃 is a constant value,which is just different from the polar radius. Equation (9)is another condition to find the next candidate. The angleinterval diagram is shown in Figure 3. Then the remainingsensors can be found to construct barriers in the same wayas before. At the same time, we will implement this methodin other rings. But while constructing the barrier, there maybe a situation that such a satisfying candidate sensor cannotbe found regardless of any efforts, so we have to dissolve thisbarrier.

In our 𝑘-barrier coverage algorithm, we can calculatethe minimal amount of nodes that can be used to createa 𝑘-barrier deployment. It is a variable changing with the

perimeter of ideal barriers and the sensing radius of sensornodes. Let us take 𝑛min as the minimal amount of nodes, and𝑟(𝑖) is the radius of the ideal barriers, 𝑖 = 1, 2, . . . , 𝑘; then theformula is as follows:

𝑛min = ⌈𝜋 ⋅ 𝑟 (𝑖)

𝑟𝑠

⌉ . (10)

And the degree of redundancy of deployed nodes in eachcircular is at least six times of the minimal amount ofnodes. When the degree is larger than six, our algorithmcan construct 𝑘-barrier coverage more easily. The number ofsensors nodes deployed in each single circular rings ismarkedas 𝑛𝑖, 𝑛𝑖 = 6 ⋅ 𝑛min, 𝑖 = 1, 2, . . . , 𝑘. The six times is not literallydesigned; it is achieved by many times of simulation.

The computational complexity of the proposed algorithmis marked as 𝑂(𝑓(𝑛)), which is relative with 𝑘, 𝑍 and 𝑋𝑍. 𝑍is the maximal times of each searching in theory; 𝑋𝑍 is theactual searching number:

𝑂 (𝑓 (𝑛)) =

𝑘

∑

𝑚=1

{𝑐1+ 5𝑐2+ 3𝑐3+

𝑍

∑

𝑝=1

[𝑐6+

𝑋

∑

𝑞=𝑚+1

(4𝑐1+ 𝐴1𝑐2+ 9𝑐3+ 𝐴2𝑐6+ 𝐴3⋅ 𝑐7+ 2𝑐8+ 𝑐9)]} . (11)

𝑍1, 𝑍2, 𝑍3are variables satisfying some conditions. 𝑐

1repre-

sents an identity matrix. 𝑐2represents a variable assignment.

𝑐3represents a constant assignment. 𝑐

4represents a compari-

son. 𝑐5 represents a multiplication. 𝑐6 represents a addition. 𝑐7represents a square root. 𝑐8 means taking the absolute value.The details of our 𝑘-barrier are shown in Algorithm 1.

3.3. Barrier Mending Model

3.3.1. Energy Model. After running for a certain time, nodesfailure may be an unavoidable problem by running out ofenergy or by some physical damage. In previous works, mostof them consider the remaining energy of an individualsensor, which can be a good way to decide whether a barrierneeds to be mended. As one sensor fails, the barrier willbe destroyed. In this paper, we consider a strong 𝑘-barriercoverage, which should detect all intruders permeating themonitored area almost being detected 𝑘 times. Once a holeappears in a barrier, intruders may pass through the barrierwithout being detected. Gap is the crack without any barrierdetecting an intruder’s incursion.

To maintain the monitoring quality, a barrier mend-ing model is required. We can get the remaining energy(described as 𝐸

𝑟) of sensors as follows:

𝐸𝑟=

𝑛𝑖

∑

𝑖=1,𝑗=1,𝑖 =𝑗

𝐸𝑖− [(𝑒𝑅

𝑖+ 𝑒𝑇

𝑖) ⋅ 𝑑𝑅

𝑖𝑗− 𝑒𝑆] ⋅ (𝑡𝑖) , (12)

where 𝑒𝑅𝑖and 𝑒𝑇𝑖are energy for receiving and transmitting one

unit data, respectively; they all decline with the distance 𝑑𝑖𝑗between two adjacent sensors 𝑖 and 𝑗, 𝑖, 𝑗 = 1, 2, . . . , 𝑛. And𝛼 is the parameters of signal decline. 𝑒𝑆 is energy required

for sensing one unit data. 𝐸𝑖is the initial energy reserve of

sensor𝑖. 𝑡𝑖is the network lifetime of barrier

𝑖; when (12) is

equal to zero, the network lifetime of barrier is achieving theupper bound. We can get the distance of two neighboringsensors by the GPS. We quote (12) from the literature [24].We can use (12) to calculate the lifetime of a single barrier.And the lifetime of a barrier is related to both the number ofoverlapping sensors (𝑃𝑂𝑖) in a barrier and the coverage ratio(𝑃𝐶𝑖) of a barrier:

work time of barrier𝑖= 𝑃𝑂𝑖⋅ 𝑃𝐶𝑖.

𝑃𝑂𝑖 =the number of overlap sensor

𝑖

the total number in barrier𝑖

.

𝑃𝐶𝑖=a certain barrier coverage area

maximum coverage area.

(13)

Then we can get relationship between network lifetimeand coverage ratio. Assuming that 𝑇 is the work time ofnetwork, we can get the relationship as follows:

𝑡𝑖=

𝑃𝑂𝑖⋅ 𝑃𝐶𝑖

∑𝑘

𝑖=1𝑃𝑂𝑖 ⋅ 𝑃𝐶𝑖

⋅ 𝑇. (14)

Once the remaining energy of a sensor is equal to zero,it means that the sensor is dead. Two neighboring sensorsaround the sensor in the same barrier can detect its failure.The two neighboring sensors can transmit information toeach other. The previous sensor before the dead one in thebarrier has to find another sensor to replace the dead one.At the same time, the replacement sensor connecting withthe latter sensor of the dead sensor in the barrier mending


Require: 𝑟𝑠= the sensing radius of a sensor; 𝑅

𝑐= the communication radius between two adjacent nodes;𝑚

𝑖= the number

of sensors deployed following the Gaussian deployment in the 𝑖th circular ring; 𝑅𝑂 = the outer radius; 𝑅𝐼= the inner

radius; 𝑑(𝑛𝑖, 𝑛𝑗) = the Euclidean distance between sensor 𝑛

𝑖and 𝑛

𝑗; Δ𝑥 = the searching radius, which is a positive

discrete number; 𝑘 = the barrier number.Ensure: 𝑚

1is the number of sensors deployed in first circular ring according to the Gaussian distribution, set as

𝐴 = (𝑥1, 𝑥2, . . . , 𝑥

𝑚1

)

while 𝑧 = 1 do𝑛1is the starting sensor, its polar coordinates is

(𝑥1, 𝑦1), 𝑥1= 𝑟(𝑖) cos 𝜃

𝑚, 𝑦1= 𝑟(𝑖) sin 𝜃

𝑚. 𝑖 = 1, 2, . . . , 𝑘, 𝑟(𝑖) ∼ 𝑁(𝑟(𝑚) + 𝑤/2, delta), 𝜃

𝑚=

𝜃𝑖∼ 𝑈(0, 2𝜋), 𝑟(𝑖) = 𝑟𝑖 + (𝑖 − 1) ∗ 𝑤, 𝑤 = (𝑅

𝑂− 𝑅𝐼)/𝑘.

end whilerepeat

search the next satisfying sensor, 𝑛𝑧∈ 𝐴

until 𝑧 ≤ 𝑚1, max 𝑑(𝑛

𝑧−1𝑛𝑧) ≤ 2𝑟

𝑠, max Δ𝜃,

Δ𝜃 =4𝜋

⌈𝜋𝑟(𝑖)/𝑟𝑠⌉

if 𝑑(𝑛𝑧𝑛𝑧+1

) ≥ 2𝑟𝑠, or max Δ𝜃 ≥ 4𝜋/⌈𝜋𝑟(𝑖)/𝑟

𝑠⌉ then

break, fail to find sensor 𝑛𝑧

end ifif max Δ𝜃 ≤ 4𝜋/⌈𝜋𝑟(𝑖)/𝑟

𝑠⌉, 𝑧 ≤ 𝑚

1then

turn to next stepif 𝑑𝑛2𝑛1≤ 2𝑟𝑠then

successfully find the next sensorend if

end ifgo back to repeat to find the next valid sensor;while the number of sensors 𝑛

𝑠in one barrier is equal to

the minimal number 𝑛min = ⌈2𝜋𝑟(𝑖)/2𝑟(𝑠)⌉ doif 𝑑𝑛min𝑛1

≤ 2𝑟𝑠, then

succeed in building a barrier, break;else

𝑑𝑛min𝑛1

≥ 2𝑟𝑠, continue to find the next valid senor, then

go back to step before two steps;end if

end while

Algorithm 1: A strong k-barrier construction algorithm.

model must satisfy some requirements. In other words, thenew barrier canmonitor one line of the field and no intruderscan pass through the line without being detected. In otherwords, the new barrier can monitor one line of the fieldand no intruder can pass through the line without beingdetected. There are some limitations needing to be satisfied.The limitations are the Euclidean distance between the deadsensor and the replacement sensor and the distance betweenthe replacement sensor and the latter sensor. These distancesall must be shorter than two times the sensing range, whereasthe Euclidean distance between the dead sensor and the lattersensor should be shorter than the communication range 𝑅

𝑐.

The mathematical expressions are 𝑑N2N3 ≤ (2𝑟𝑠), 𝑑N3N5 ≤

(2𝑟𝑠), 𝑑N2N5 ≤ 𝑅

𝑐,𝑅𝑐= 2𝑟𝑠. As shown in Figure 4, sensors N0,

N1, N2, N3, N5, and N9 belong to the same barrier, and thesesensors monitor one of the circular rings of the interestedarea. Suppose that sensorN3 is dead, and both sensorsN2 andN5 can detect its failure. Since N2 is the previous sensor, it isresponsible for finding the replacement sensor N4 accordingto the following rules. Then N2 sends N4 a flag message, tellsN4 the remaining energy of N3, and judges whether it should

A B ON0

N1

N2

N3

N4

N5 N9

N11

N12k1

Figure 4: An example of barrier reconstruction.

wake up immediately or later. If yes, the sensor N4 sets theflag = 1 and sends it to senor N2. If no, the sensor N4 setsflag = 0, and it keeps sleeping. After N2 and N5 receive themessage flag = 1, they will know that the replacement sensorwill join the barrier to keep it working effectively. The newbarrier becomes N0, N1, N2, N4, N5, and N9.


Since the remaining energy of each sensor can be cal-culated according to (10), there is 𝐸

𝑟= 𝜂𝐸

𝑖, where 𝜂 is

a small mathematical variable less than 1, and it can becalculated at every working time. 𝜂 change with the timebeing spent on communication with the neighbor sensors isworth discussion. As our barrier constructing algorithm isa real-time communicating algorithm, the communicationtime can really be ignored; then 𝜂 can be set as zero or nearingzero.

The procedure of finding a replacement sensor shouldsatisfy the following rules.

Rule 1. Theprevious sensor should select the replacement oneamong the nearest neighborhoods in its ring, while they arefree to use.

Rule 2. If such a suitable sensor is not found, we can borrowa suitable replacement one to form the other rings.

Rule 3. If no sensor can be found based on the above tworules, the current barrier cannot be mended; then we releasethe remaining sensors, which become redundant sensors andare free to help other barriers in case some sensors fail.

3.3.2. A Barrier Health System. The remaining energy of thedying sensor𝑖 is 𝐸𝑟

𝑖

; when it satisfies the condition that 𝐸𝑟𝑖

=

𝐴 ⋅ 𝐸𝑖, its neighbor nodes need to search an alternative nodefor it. 𝐴 is a least coefficient of the initial energy reserveof sensor𝑖, which is relative with the distance (𝑑𝑖𝑗) betweentwo neighbor nodes, and the speed of data transmission. TheEuclidean distance between two neighbors can be calculatedaccording to their coordinates, and the coordinates can beobtained by GPRS.

Suppose that the speed of data transmission is equal,marked as V. Then the time of data transmission between thedying node and its neighbor nodes is 𝑡

𝑖𝑗. 𝑡𝑖𝑗= (𝑑𝑖𝑗/V)𝑡𝑗𝑘

=

𝑑𝑗𝑘/V; they describe the time of date transmitting from two

neighboring nodes 𝑖, 𝑗, and 𝑗, 𝑘. And sensor 𝑖 represents thedying sensor node:

𝐴min =[(𝑒𝑅

𝑖+ 𝑒𝑇

𝑖) ⋅ 𝑑𝑖𝑗− 𝑒𝑆] ⋅ (𝑡𝑖𝑗+ 𝑡𝑗𝑘)

𝐸𝑖

. (15)

The remaining energy of the dying node is at least 𝐴mintimes the initial energy of sensor

𝑖; its neighbor sensor nodes

need to find its alternative node. Only, in this way, we mayensure that the neighbors can find an alternative node beforethe dying node dies.

And an actual protocol that can be used to send thealerts to the base station is in reference [6]. In [6], nodes ina cooperative node group will be considered as opponentsto each other; therefore, each node will maintain a 𝑄-valuewhich reflects the payoff that would have been received if thatnode selects one action and the other nodes jointly selectedthe other action. After that, the node with the highest totalpayoff will be elected to forward the data packet to the nextcooperative node group towards the sink node.

Clockwise

A k1k2kk O

s3

s1 s2

E

Figure 5: Application of one-way barrier in detecting differentintruders. s1 and s2 are the trajectories of crossing straight throughthemonitored area and s3 is the trajectories of incompletely crossingline.

4. Practical Application

This part presents the application of our WSNs in one-way intruders detection. By using the barrier constructionalgorithm proposed in this paper, a 𝑘-barrier coverage can befinally achieved as an ideal situation, as shown in Figure 5.Since the sensors in our network are distributed followingthe Gaussian distribution when 𝜇 = 0, 𝜎 = 1, we canconstruct the preset 𝑘-barrier coverage very well by usingour algorithm. In the former section, it shows that thebarriers are almost constructed in the center of every circularring. After 𝑘-barrier bridges being constructed perfectly, theywill be ultimately used in a practical application which iscalled one-way barrier coverage. That is to say every singleintruder crossing straightly through the monitored area canbe detected to be legal or illegal.

Previous works using barriers to provide one-way barriercoverage have been studied a lot. A new coverage modelcalled one-way barrier was firstly proposed in [6]. It proposedthe concept of neighboring barriers and designed differentprotocols to provide one-way barrier coverage for differentsensor models based on neighboring barriers. In addition, aone-way barrier coverage has been proposed in [19], and thesingle intruder has been considered in [19]. But when used todetect multiple intruders, it is not so effective.

In this section, the model is used to detect the illegaland legal intrusion. We also need to set an alarm line likein [6, 7, 19]. But there is little different between the alarmline in this paper and in literatures [6, 7, 19] because thebarriers in [6, 7, 19] are two neighboring barriers whichhave a continuous overlapping region. There are two specialbarriers. However, in ourWSNs, the above limitations are notnecessary conditions. This is an improvement we have made.Our detection system is shown in Figure 5. In the monitored


Intruder a

Intruder b

k1 k2 kk OA

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(a) Invasion way 1 of multiple intruders

Intruder a

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Clockwise...

(b) Invasion way 2 of multiple intruders

Intruder a

Intruder bk1 k2 kk OA

Clockwise

.

.

.

(c) Invasion way 3 of multiple intruders

Figure 6: Three kinds of invasion ways.

area, it has formed 𝑘-barrier coverage being described by theblack rings with arrows. This situation is the ideal situationand the direction of arrows is the direction of buildingbarriers. In fact, it does not matter how 𝑘-barrier coverageis constructed by using our algorithm; 𝑘 has to be morethan two. When 𝑘 is equal to two, our algorithm is just theneighboring barriers proposed in [6, 7, 19]. That is to say, it isa special situation of our model.

4.1. One Intruder. Obviously, WSNs with only one barriercannot furnish one-way barrier coverage. As mentionedbefore, two barriers can provide one-direction barrier cov-erage, but it must follows some special assumptions, such asthe limitations mentioned in literatures [21–23]. This paper

concentrates on using 𝑘-barrier coverage to achieve the samegoal. Here we suppose that there is only one intruder crossingthe annual region. As shown in Figure 5, there are threepossible paths that one intruder may follow. For example, thethree red dotted lines with arrows are some possible invasionpaths of the intruders in Figure 5, described as s1, s2, and s3,respectively.

We assume that intruders penetrating straightly throughthe circular monitored area from outside to inner regionare illegal intruders, which need to be detected. In practicalsituation the circular monitoring area is like an isolationbelt, and only the intruders thoroughly penetrating the areafrom outside to inside need to be broadcast, such as s1.On the contrary, intruders not thoroughly penetrating themonitoring area or penetrating the area from the inside to


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(b) 𝑘-barrier coverage constructed by our algorithm with sensorsfollowing Gaussian deployment, 𝑘 = 4

Figure 7: Extended application in rectangular area.

outside are thought to be legal, like the paths s2 and s3 inFigure 5. In order to distinguish the illegal intruders, we quotethe notion of alarm line in literature [6]. And the alarm linein [6] is a little different from to the one in this paper. Analarm line here is a line or curve along the belt with thefollowing property: the network reports an alarm wheneveran intruder crosses the alarm line from outside to inside, butit does not report any alarm when an intruder crosses theline from inside to outside. The bottom line of the innermostbarrier of WSNs is chosen as the alarm line.

Here we take 𝑘-barrier as an example. When an intruderthoroughly penetrates the monitoring area and enters intothe protected region, the intruder will be detected 𝑘 timesand will trigger the alarm line. It is illegal that the intruder isdetected 𝑘 times at first and then triggers the alarm line. Onthe contrary, it is legal. For example, the path s1 in Figure 5is illegal, and the path s3 is legal, which will not report anyalarm. When the alarm is reported, we can easily distinguishthe illegal one. Since some intruders are not going to enter theinner region of ourmonitored area, such as the path s3 shownin Figure 5, an intruder enters into themonitoring area and isdetected several times being less than the maximum numberof barriers constructed in our model, but does not trigger thealarm line; then the intruder is also thought to be legal. It isobvious to see that our model can effectively detect the illegalintruders, and it does not require two neighboring barriers,and it can be used more widely. Above all, there are somecommon situations most likely happened in our monitoredarea.

4.2. Multiple Intruders. When considering more than oneintruder at the same time, our 𝑘-barrier coverage can provideone-way barrier coverage if the intruders cross straightthrough the region satisfying some constrains. AS shownin Figures 6(a), 6(b), and 6(c), there are three possiblepenetrating paths of intruders.

When intruders cross the circular area along the samedirection, such as in Figures 6(a) and 6(b), intruders 𝑎 and 𝑏either penetrate the circular rings from the outer boundary tothe inner boundary or from the inner boundary to the outerboundary. In this case, intruders are detected 𝑘−1 times first;

then they trigger the alarm line or trigger the alarm line first;then they will be detected 𝑘 − 1 times. Obviously, when thereare multiple intruders, we can detect the illegal intrudersdepending on the order of triggering the alarm line first orbeing detected 𝑘 − 1 times first. There is no effect on eachintruderwhen intruders cross the circular ringswith differentdirections, as shown in Figure 6(c).There exits a situation thatintruder 𝑏 triggers the alarm line while intruder 𝑏 penetratesthe circular rings and triggers the alarm line at the same; thenthe detecting systemmay ignore the behavior of intruder 𝑎. Soour 𝑘-barrier coverage cannot detect multiple intruders wellunder this circumstance.

5. Extended Application

In previous sections, we studied 𝑘-barrier coverage of WSNsin a circular area; here we are talking about its extendedapplication used inmore generalized environment, for exam-ple, with rectangular area. As seen in Figures 7(a) and 7(b),the length is 100m, and width is 80m. In Figure 7(a), therectangular area has been divided into four width partsequally, and sensors are deployed in the middle line of eachpart following Gaussian deployment (Δ = 1, 𝜇 = 𝑦 + Δ𝑦).

When sensors are deployed randomly in monitored area,there are risks of sensors being deployed in some place withhigh density; on the country, some places are with very fewsensors. Since the strong 𝑘-barriers mean that no intruderscan penetrate the monitored area without being detected,no gaps are allowed in barriers. To make sure of buildingstrong 𝑘-barriers in WSNs with sensors randomly deployed,we should deploy more than enough sensors in monitoredarea.

The minimal amount of nodes in each barrier calculatedaccording to formula (10) is 5, 5, 5, 5. And we can see inFigure 7(b), which costs 6, 6, 6, 6 sensors in practical, our 𝑘-barrier coverage can be used well in rectangular area, sincethe error can be ignored in terms of the whole WSNs. Aboveall, the proposed 𝑘-barrier algorithm can not only be use ina circular area, but also can be used in a more generalizedenvironment where the upper and lower boundaries areparallel.


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Figure 8: Initial Gaussian deployment of sensors with 𝜇 = 0, 𝜎 = 1, 𝑟𝑠= 10m, and 𝑅

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6. Performance Evaluation

6.1. Simulation Methodology. The simulation results areobtained in MATLAB. The objectives of the evaluation arethreefold: (1) testing the feasibility and effectiveness of ouralgorithm in constructing 𝑘-barrier coverage; (2) testing oursensor network in detecting whether an intruder is legal orillegal; and (3) studying the performance of our algorithmunder different system parameters, such as different sensingradii of sensors, different barrier constructing models, anddifferent sensor distributions.

In the simulations, deploy 640 nodes in an annularregion. The distribution of sensors follows the Gaussiandistribution and radius of the annual region ranges from20m to 100m. The sensing radii of all sensors are identical.Intruders try to cross the region from one ring to the otherring. Here the annular region is divided into four partswith equal width. Without loss of generality, we assume thatthe intruders’ always move from any of the two edges ofthe annular region. We study different influence of sensorsensing radius in constructing our barriers.Thenwe study theimpact of system parameters and intruders characteristics.The sensing range is set as 8m and the communication rangeis set as 16m.The number of 𝑘 is assumed to be 4 in advance.Since the sensors are deployed as the Gaussian deployment,the real value of 𝑘maynot exactly be 4.Thenumber of sensorsin each part of the four annular regions is 64, 128, 192, 256,respectively, which are calculated according to the perimeterof each ring.

6.2. Construction of the WSNs. To test the feasibility andeffectiveness of our algorithm, Figure 8 shows the initialGaussian deployment diagram, and we take 𝑘 = 4. Theinitial number of sensors is set as 56, 104, 152, 208 from theinnermost circular ring to the outermost, respectively. Thesensors are almost deployed around the middle virtual line.Here we take 𝜇 = 0, 𝜎 = 1.

Figure 9 shows the simulation results of our algorithm.We take 𝑟s = 8m, 𝑟

𝑐= 2𝑟𝑠= 16m, and 𝜇 = 0, and

𝜎 = 1. Figure 9(a) shows the barriers formed in the modelin Figure 8. We can see that there are four barriers in ourannular region. We do 20 times in MATLAB, and we get fourbarriers more than 14 times. But there is also the situationlike in Figure 9(b) and there is only three barriers in ourmonitored region, and the number of 𝑘 is smaller than thepreset value. But it just appears about 4 times in the 20times simulations. In Figures 9(a) and 9(b), the blue smoothcircular rings are our detected annular region and the reddotted line is our generated barrier. Moreover, the points aremarked as sensors. In Figure 9, it is just shown the sensorsconstructing barriers in each annular ring.

In the innermost circular ring of Figure 9(a), there is anincomplete dotted line, which does not construct a completebarrier, and the incomplete barrier is enlarged in Figure 9(b).It shows that the barrier cannot find a next suitable sensorunder our constraints. Such a barrier is not needed, then wewill dissolve it. And the sensors in the incomplete barrier cango to sleep until they are waken up.

6.3. Impact of Sensing Radius. To evaluate the impact of sens-ing radius on the performance of our barrier constructingalgorithm,we set the initial numbers of sensors as 56, 104, 152,208 from the innermost annular ring to the outermost one,and sensors are deployed following the Gaussian deploymentin every annular ring. In addition, if we can build one barrier,the number of sensors is enough.The sensing radii of sensorsare set to be 8m and 10m for each evaluation, respectively.Different sensing radii are shown in Figure 10. In Figure 10,we can see that the larger the sensing radius is, the less thesensors are needed to build a barrier in the same situation. Forexample, when 𝑟𝑠 = 8m, it needs 13, 25, 34, 45 sensors in everybarrier from the innermost to the outermost like Figure 9(c),but it only needs 13, 16, 28, 38 sensors, when 𝑟

𝑠= 10m. If we


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Figure 9: 𝑘-barrier constructing model with 𝜇 = 0, 𝜎 = 1, 𝑟𝑠= 8m, and 𝑅

𝑐= 16m.

want to build a barrier to cover the same region, we wouldbetter choose the sensors with larger sensing radius.

6.4. Comparison with Another Model. As the purpose of our𝑘-barrier coverage inWSNs is to detect intruder invasion, wecan compare it with [21], which uses a two-neighbor barriercoverage to detect one-way intruders. We construct such atwo-neighbor barrier coverage using our 𝑘-barrier coveragealgorithm. As shown in Figure 11(a), in Figure 11(b), it isa 2-barrier coverage constructed by our 𝑘-barrier coveragealgorithm, where 𝑘 = 2. They are different in nature. Thetwo neighboring barriers in Figure 11(a) are continuouslyintersected but not anastomosed. But the 2-barrier coveragein Figure 11(b) is separate. And the circular area is decidedinto twowidth parts.The environment of these two situationsis the same. Sensor nodes are deployed following Gaussian

deployment in the middle line of each circular ring. Whenthe total number of sensor nodes deployed in the monitoredarea is identical, we can see from Figure 11(c) that the numberof sensor nodes used in building two barriers between Figures11(a) and 11(b) is basically the same because the error isnegligible in terms of the whole WSNs. The comparison onthe number of sensor nodes used in building two barriersbetween Figures 11(a) and 11(b) shows that our 𝑘-barrieralgorithm can achieve a good performancewhen realizing thesame purpose.

6.5. Impact of Sensor Distribution. To evaluate the impactof sensor distribution on the performance of our algorithm,we adopt the Gaussian development model in Figure 8and calculate the number of sensors needed to build abarrier in Figure 9(a). Here we discuss another sensor


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rs = 8

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Figure 10: Number of sensors constructing 𝑘-barrier with 𝑟𝑠= 8m and 𝑟

𝑠= 10m, 𝑘 = 4, 𝜇 = 0, and 𝜎 = 1.

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(c) The comparison on the number of sensor nodes used in buildingtwo barriers between (a) and (b)

Figure 11: Relevant comparison with our 𝑘-barrier coverage.


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Gaussian, rs = 10, sum = 87

Random, rs = 10, sum = 96

(c) The number of sensors for building barriers; the blue dotted one israndom deployment; the total number of sensors is 96; the red dottedone is Gaussian deployment; the total number of sensors is 87; the radiusof each sensor is 𝑟𝑠 = 10m, no matter sensors are deployed by Gaussianor random

Figure 12: Comparing our sensor distribution with random deployment.

distribution, which is the Uniform distribution, and com-pare the sensor number used in building barriers with theGaussian deployment model. The reason why we choosethe Uniform distribution is that it is widely used in pastresearches. The result is shown in Figures 11(a), 11(b), and11(c). Figure 12(a) shows the Uniform distribution generatedin MATLAB, and we can see that sensors are deployedrandomly and with no rules in the monitored region. Tocompare the influence of sensor distribution with Gaussiandistribution, the number of sensors distributed in every singleannular region is the same as the number in Figure 8.

To build the barriers by using the algorithm proposedin our paper, it is obvious that more sensors are required.Figure 12(b) is a simulation result, and there are also fourbarriers in the annular region. The sensing radius, the rangeof our annular region, and the simulation times are thesame as the previous Figure 9. Comparing Figure 12(b) withFigure 9(b), it is easy to see that it takes more sensors tobuild barriers. In Figure 12(b), it shows 4 barriers constructedby using sensors with the Uniform distribution. Figure 12(c)shows the contrast figure of the Gaussian distribution andUniform distribution, which plots the number of sensors


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.318

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Figure 13: The relationship between the percentage of nodes utilization and nodes redundancy.

in every barrier. In Figure 13, we can see more about therelationship between randomly deployed sensor redundancyand the number of strong 𝑘-barrier coverage. Since sensorsare deployed following random deployment, it is more hardto construct barriers than Gaussian deployment. In our 𝑘-barrier coverage algorithm, we can calculate the minimalamount of nodes that can be used to build a 𝑘-barrierdeployment according to formula (10).

The degree of redundancy changes from six times to ninetimes the minimal amount of sensor nodes in every barrier.As can be seen in Figure 13, the higher the redundancy, thelower the utilization rate of the nodes. It means that thenumber of sensor nodes used in building barriers is inverseto the nodes redundancy. When the degree is lower than six,it is very difficult to complete 𝑘-barrier coverage.

6.6. Impact of Delta Value (i.e., 𝜎). In this part, we evaluatethe impact of parameter 𝜎 on the performance of ouralgorithm. As seen in simulations, 𝜎 varies from 1 to 8. Theresults are shown in Figures 14(a) and 14(b). After 20-timesimulations, we note that the number of sensors used inbuilding barriers is related to the delta value 𝜎. When 𝜎 isgetting bigger, the performance is getting worse. And when 𝜎is big enough, the Gaussian distribution tends to be Uniformdistribution. In Figure 14(a), it is shown that there are almostfour overlapped lines, and when 𝜎 is equal to 1, 2, 3, 4,respectively, the number of sensors used in building barriersis nearly equal. As shown in Figure 14(b), the total number ofthe sensors used to build the barriers is different. It is shownthat when 𝜎 = 1, the performance is the best because thenumber of sensors used in building 4-barrier coverage is theleast. In addition, when 𝜎 is smaller than 5, their performanceis similar to each other.Therefore the bigger the 𝜎 is, themorethe total number of sensors is needed. For example, when𝜎 is more than 9, it is hard to get 4-barrier coverage, sincesensors are always developed outside the annular region, as

seen in Figure 14(c). At this time, the distribution of sensorsapproximates the Uniform distribution adopted in Figure 12.

7. Conclusion

In this paper, we propose a barrier constructing algorithm inWSNs for intruder detection, which constructs 𝑘-barrier in acircular area, and apply our algorithm in one-way intrudersdetecting.When an intruder is detected 𝑘 times at first, and itis broadcasted at last, it is an illegal intruder. On the contrary,if it is broadcasted at first, then it will be detected at least onetime; it is a legal intruder. The goal of detecting whether anintruder is legal or illegal is achieved well. On one hand, thesimulation results prove that our protocol can build 𝑘-barriervery well. On the other hand, the simulation results provethat the total number of sensors used in building 𝑘-barrieris small. Therefore the Gaussian distribution with low 𝜎 is agood choice.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported in part by the Program for Scienceand Technology Innovative Research Team for Young Schol-ars in Sichuan Province, China (Grant no. 2014TD0006),National Natural Science Foundation of China (Grant no.61001086), the Fundamental Research Funds for the Cen-tral Universities (Grant no. ZYGX2011X004), the OverseasAcademic Training Funds, University of Electronic Sci-ence and Technology of China (OATF, UESTC) (Grant no.201506075013), and the Program for Science and TechnologySupport in Sichuan Province (Grant nos. 2014GZ0100 and2016GZ0088).


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100Gaussian

(c) The Gaussian deployment of sensors with high delta (i.e., 𝜇 = 0,delta = 8), and the number of sensors deployed in themonitored regionis the same as in Figure 9

Figure 14: Comparing with different delta (i.e., 𝜇 = 0, delta).

References

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