Research Article Mathematical Approach to Security Risk … · 2019. 7. 31. · Research Article Mathematical Approach to Security Risk Assessment RobertVrabel,MarcelAbas,PavolTanuska,PavelVazan,MichalKebisek,

Research ArticleMathematical Approach to Security Risk Assessment

Robert Vrabel Marcel Abas Pavol Tanuska Pavel Vazan Michal KebisekMichal Elias Zuzana Sutova and Dusan Pavliak

Faculty of Materials Science and Technology Slovak University of Technology in Bratislava Hajdoczyho 1 917 01 Trnava Slovakia

Correspondence should be addressed to Robert Vrabel robertvrabelstubask

Received 3 December 2014 Accepted 27 January 2015

Academic Editor Evangelos J Sapountzakis

Copyright copy 2015 Robert Vrabel et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The goal of this paper is to provide a mathematical threat modeling methodology and a threat risk assessment tool that may assistsecurity consultants at assessing the security risks in their protected systemsplants nuclear power plants and stores of hazardoussubstances explosive atmospheres and flammable and combustible gases and liquids and so forth and at building an appropriaterisk mitigation policy The probability of a penetration into the protected objects is estimated by combining the probability of thepenetration by overcoming the security barriers with a vulnerability model On the basis of the topographical placement of theprotected objects their security features and the probability of the penetration we propose a model of risk mitigation and effectivedecision making

1 Introduction

The term physical protection of safety-critical objects rep-resents a set of technical regime actions or organizationalactions necessary to prevent the unauthorized actions per-formed with or in the objects (intrusion and sabotage)of critical infrastructure such as nuclear facilities powerplants transmission grids drinking water supplies storagesof chemicals oil pipelines and related facilities and roads

The infrastructure of developed countries is highlyvulnerable and also highly interconnected As the criticalinfrastructure is an international phenomenon an attackon any state may result in the infrastructure failure at theregional level as well as at a broader international geographiclevel Thus various countries seek to harmonize their legalprocedures in this paper for example HR3696 NationalCybersecurity and Critical Infrastructure Protection Act of2014 (USA) [1] Council Directive 2008114EC on the identi-fication and designation of European critical infrastructuresand the assessment of the need to improve their protection [2]and the associated legal acts of member states and so forth

Currently an increased attention is being paid to thesafety of important objects In the literature we can findmanydifferent approaches to analyze and to solve the problem ofassessing the threat for critical infrastructure

For example in the paper [3] the author presents newmethodology and develops the strategy and solutions forvulnerability assessment to identify and understand thethreats to and vulnerabilities of critical infrastructure

In the work of Hromada and Lukas [4] the conceptualapproach and the possible ways of how to develop relevantframework for critical infrastructure protection to increasethe resilience of its functional continuity are discussed

Oyeyinka et al [5] develop an analytical methodologyfor physical protection systems evaluation and their effective-ness

The paper ofWoo [6] serves as a dynamical quantificationof the detection and action against the incidents using theVensim simulation software

As the testing and validating in real conditions are feasibleonly to a limited extent the computer technique allowssimulating different types of attempts to violate the protectedarea and thus revealing the hidden security vulnerabilities Acarefully designed model of the real examined environmentfilled with the correct data is inevitable

The aim of the study is to propose algorithms enabling theusers to analyze the probability of an intruder penetration tothe protected object located in the area bounded bymultilevelbarriers with transition gates (Figure 1)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 417597 11 pageshttpdxdoiorg1011552015417597

2 Mathematical Problems in Engineering

Intruderrsquos path

Gate

Target object

Zone

Figure 1 Sketch of the protected area with security features showingone of the possible intruderrsquos paths to the target object

The probability that the physical protection system pre-vents a hostile attack to finish an unwanted event is in theliterature generally calculated as

PE = PI lowast PN (1)

where PE is probability of total system effectiveness PI isprobability of interruption the overall probability of theattack detection during its duration including the criticaldetection point (CDP) based on the principle of earlydetection and the concept of critical point detection andPN is probability of neutralization the probability thatthe corresponding force can prevent the completion of themalicious act such as the theft of nuclear material or nuclearfacility sabotage [7]

The term to neutralize means that the correspondingforce stops the invader occupies the object or eliminates thehostile attack in another way (by causing the escape of theinvader)

The principle of early detection is as follows to interruptthe enemy attack before the requirement for the sabotage ortheft is terminated From the time of detecting the event thereaction of the defense forcesmust be shorter in time than thetime remaining for the completion of the enemy attack

CDP is the last chance to detect the enemy attack Thetime for the action is shorter than the time remaining forterminating the invader requirement [7]

In order to perform the intervention effectively the earlyattack detection must be achieved at all possible paths to thetarget object

On the basis of an extensive use of the principles ofprobability and the graph theory the paper deals withthe proposal of a mathematical model suitable for furthercomputer processing The mathematical model describes allthe aspects of a real situation and creates an abstract view onthe issue

Based on the customer requirements three scenarioshave been developed

(120572) how far does the intruder penetrate into the objectuntil the desired level of detection is reached

(120573) What is the distance between the intruder and thetarget when only the given time to the target is left

(120574) Does the intruder get into the target and out of it untilthe desired level of detection is reached

All three scenarios allow for detecting the supposed positionof the intruder if the input condition of the given scenario ismet The algorithms called Alpha Beta and Gamma weredeveloped for individual scenarios and are listed in otherparts of the paper in Section 6

An application with a graphical user interface was devel-oped for the purposes of verifying the mathematical modeland algorithms

At the end of the study one of the series of tests carriedout for randommodels with different parameters such as thenumber of barriers gates detection probability is chosen

According to our best knowledge no paper focused onsolving the specific tasks mentioned above the scenariosAlpha Beta and Gamma

2 Definitions of Security Features andNomenclature

In this section we introduce the definitions of principalconcepts used in this paper as follows

(1) target object TO (eg nuclear reactor)

(2) barriers the continuous obstacles to penetrate intothe protected object (eg a fence) Bar

119894 119894 =

0 1 119896 Bar119896= an outer barrier and Bar

0= TO

The number of barriers Bar119894= 119896 with TO we have

(119896 + 1) barriers

(3) Zone the area between two consecutive barriers Z119894

119894 = 1 2 119896 The total number of the zones Z119894= 119896

(4) Gates inputs on the barriers G119894119895 119894 = 1 2 119896

119895 = 1 2 119897119894G1198941G1198942 G

119894119897119894fences on the barrier

Bar119894 The total number of the gates on barrier Bar

119894is

G119894= 119897119894 and the total number of the gates G

119894119895=

sum

119896

119894=1119897119894

(5) Rays half-lines 120588119894119895connecting the target object with

the barriers more precisely 120588119894119895is the half-line TO minus

G119894119895 The total number of the rays is equal to the total

number of the gates that is 120588119894119895= sum

119896

119894=1119897119894

(6) R-gates physical places on the barriers lying on therays 120588

119894119895 R119894119895119899

119894 = 1 2 119896 119895 = 1 2 119897119894 and 119899 =

1 2 119896 which is a physical place on the half-line120588119894119895lying on the barrier Bar

119899Thus if 120588

119894119895= TOminusG

119894119895 it

is the connection of the target object TO and the gateG119894119895 and then the R-gate R

119894119895119899lies on the intersection

of 120588119894119895

and Bar119899 The total number of the R-gates is

R119894119895119899= 119896 sdot 120588

119894119895

Remark 1 The reason for introducing the concept of R-gateis the need of implementing the calculations in real time byreducing the number of less probable paths of the intruder

Remark 2 Obviously R119904119895119904= G119904119895 119904 = 1 119896 Therefore in

the process of implementing the algorithms (Section 6) wewill denote the gates and the R-gates on the same barrier (say119904) consecutively and with two indexes only

Mathematical Problems in Engineering 3

3 Required Data

31 Location of Objects Let 119860 = [119860119909 119860119910] denote the coor-dinates of the object 119860 Thus one can see the following

(1) the coordinates of the target object TO [TO119909TO119910]after translation [TO119909TO119910] = [0 0]

(2) the coordinates of G119894119895 [G119894119895119909G119894119895119910]

(3) the coordinates of R119894119895119899 [R119894119895119899119909R119894119895119899119910]

(4) the rays 120588119894119895 119883 = G

119894119895119909 sdot 119905 119884 = G

119894119895119910 sdot 119905 119905 gt 0 (TO =

[0 0])

32 Probabilities of Detection during Penetration Let 119875+119889119860

denote the probability of detection of the subjects penetratingthrough the object 119860 in the direction to the target object TOand 119875minus119889119860 in the direction from the target object TOThen the

probability of the penetration through the object 119860 will be119875

+

119901119860 = 1 minus 119875

+

119889119860 and 119875minus

119901119860 = 1 minus 119875

minus

119889119860

(1) 119875119901TO (119875+

119901TO 119875minus

119901TO)

(2) 119875119901Bar119894 (119875

+

119901Bar119894 119875

minus

119901Bar119894)

(3) 119875119901G119894119895 (119875

+

119901G119894119895 119875

minus

119901G119894119895)

(4) 119875119901R119894119895 (119875

+

119901R119894119895 119875

minus

119901R119894119895)

(5) 119875119889Z119894 (119875

+

119889Z119894 119875

minus

119889Z119894) the probability of detection per

second of the stay in the zone Z119894

33 The Assumed Times Needed to Overcome the Security Fea-tures Let 119879+119860 denote the assumed time of the penetrationthrough the object119860 (in the direction to the target object TO)and 119879minus119860 (in the direction from the target object TO) Thenwe denote the following

(1) 119879TO (119879+TO 119879minusTO)(2) 119879Bar

119894 (119879

+Bar119894 119879

minusBar119894)

(3) 119879G119894119895 (119879

+G119894119895 119879

minusG119894119895)

(4) 119879R119894119895 (119879

+R119894119895 119879

minusR119894119895)

4 Required Inputs and Outputs

41 Inputs into theMathematicalModel Thenecessary inputsinto mathematical model are the following

(1) scenario selection 120572 120573 or 120574(2) specifying V+

119894and Vminus119894 119894 = 1 2 119896 the speed of the

penetrating subject through the zoneZ119894towardsfrom

the target object TO respectively(3) the probabilities and times of the penetration through

the protection elements(4) the possibility to switch off the selected security

features

(a) if the barrier Bar119904is switched off when moving

inwards then 119875+119901Bar119904= 1 119875+

119901G119904119895= 1 119875+

119901R119894119895119904=

1 119875+119901Z119904= 1 119879+Bar

119904= 0 and 119879+G

119904119895= 0

119879

+R119894119895119904= 0 forall119894 119895 analogously when moving

outwards 119875minus119901Bar119904= 1 119875minus

119901G119904119895= 1 119875minus

119901R119894119895119904=

1119875

minus

119901Z119904= 1 119879minusBar

119904= 0 and 119879minusG

119904119895= 0

119879

minusR119894119895119904= 0 forall119894 119895

(b) if the gate G119904119903

is switched off when movinginwards then 119875+

119901G119904119903= 1 119875+

119901Z119904= 1 119879+G

119904119903= 0

analogously whenmoving outwards 119875minus119901G119904119903= 1

119875

minus

119901Z119904= 1 119879minusG

119904119903= 0

(c) if the zone Z119904is switched off when moving

inwards then 119875+119901Z119904= 1 analogously when

moving outwards 119875minus119901Z119904= 1

42 Outputs fromMathematicalModel The required outputsfrom mathematical model are the following

(120572) For the given probability of the detection 119875119889 deter-

mine the set of points (and paths belonging to them)in which the probability level of the detection 119875

119889is

reached exactly(120573) For the given time 119879 determine the set of points

(and paths belonging to them) by which the time forachieving the target object TO is equal to the time 119879

(120574) For the given probability of the detection 119875119889 find

the return paths (if any) with the probability of thedetection lower than required (the return path isdefined as the path starting at some point on the outerbarrier Bar

119896 passing through TO and ending on the

outer barrier Bar119896)

5 Preliminary Calculations

Using the data specified in Sections 31 32 and 33 weput together a mathematical model of the whole protectedobject Obviously these sensitive data require a high degreeof confidentiality In addition to these data it is necessary todetermine and calculate the following values

(1) The target object TO is being translated into the origin[0 0] of the coordinate system

(2) Location of an arbitrary object 119860 is [119860119909 119860119910] =[119860119909 119860119910] minus [TO119909TO119910]

(3) The real position [119860119909 119860119910] for each object 119860 is beingcalculated using the map scale

(4) Let 119860 be the object from the set G119894119895R119903119895119894 119894 =

1 119896 fixed and let 119861 be the object from the setG119894minus11198951015840 R11990310158401198951015840119894minus1 The distance 119889 calculated using the

classical Euclidean norm is calculated for every suchpair (119860 119861)

(5) Let119863 = 119889(119860 119861)Then the time that the subject passesfrom the object 119860 to the object 119861 through the zone Z

119894

at the rate V+ is equal to 119879+119863 = 119863V+119894 Similarly the

time that the subject passes from the object 119861 to theobject 119860 through the zone Z

119894at the rate Vminus is equal to

119879

minus119863 = 119863Vminus

119894


R[36]

G[31]

R[37]

G[21]

G[23] = G[33]

G[22]

R[35]R[34] G[32]

R[25]G[11]

R[24]

R[15]

R[16]

R[17]

R[14]R[13]

R[26]R[27]

G[12]

TO

Figure 2 Topographical placement of the target object and its security features

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO

Figure 3 Schematic placement of the target object and its security features

(6) The probability of the detection of the subject movingthrough the zone Z towards TO is 119875+

119889119863 = 1 minus (1 minus

119875

+

119889Z)119879+119863 and that in the direction away from TO is

119875

minus

119889119863 = 1 minus (1 minus 119875

minus

119889Z)119879minus119863

6 Algorithms

Based on the three algorithms there are three cases of theintrusion by intruding into the protected object proposedand analyzed in this section The Alpha analysis representsthe evaluation of the possibility of the intruder penetrationalgorithm based on a set of detection probability level TheBeta analysis evaluates the distance from the penetrationspot to the target with respect to time The Gamma analysisexamines the possibilities of the intruder penetration into thetarget and out of the protected object based on the desireddetection level

61 Recursive Procedure Path Alpha (Figure 10) This subsec-tion introduces the flowchart [8] for the Alpha analysis usedfor implementing the mathematical model into the softwareenvironment

Path characterization is as follows How far does theintruder penetrate into the object until the desired level ofdetection is reached

62 Recursive Procedure Path Beta (Figure 11) In this subsec-tion we propose the flowchart implementing the mathemat-ical model to the software environment with the purpose ofexamining the Beta path

Path characterization is as follows What is the distancebetween the intruder and the target when only the given timeto the target is left

63 Recursive Procedures Path Gamma (Figures 12 and 13)The flowchart presented in this subsection was designed forthe Gamma path and is supposed to examine the probabilityof the intruder penetration into and out of the objectsuccessfully

Path characterization is as follows Does the intruder getinto the target and out of it until the desired level of detection isreached

7 Application of Mathematical Model

In this section we apply the proposed methodology to thefictive model of the protected area

Figures 2 and 3 show the topographical and schematicplacement of the target object and its security featuresrespectively The symbols used in Figures 2ndash9 are explainedin Table 1


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO

Figure 4 Alpha analysis Path 1 (an example)

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

R[24]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO

Figure 6 Beta analysis Path 1 (an example)

Table 1 Explanatory notes to the schemes

TO Target object

G[11] Gate

R[11] R-gate

FencePath in the direction to target objectPath in the direction from target object

Tables 2 and 3 refer to the parameters of gates and zonesrespectively

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


71 Analysis Alpha The Selected Paths

711 Path 1 One can see the followingdesired probability of detection 119875

119889= 097015

path R[36] rarr R[26] (segment length 2433m) rarrG[12] (segment length 806m) rarr TO (segmentlength 3406m) (Figure 4)total time of penetration = 123 s


119889= 096000

path G[32] rarr R[24] (segment length 4518m) rarrR[17] (segment length 3740m of 4329m) (Figure 5)total time of penetration = 73 s


Table 2 The gate parameters (an example)

Zone number Gate number Gate typeProbability ofdetectioninwards

Probability ofdetectionoutwards

Inwardpenetrationtime [s]

Outwardpenetrationtime [s]

119883 [m] 119884 [m]

TO 0410 0400 26 25 207 114

1 1 G 0350 0340 29 27 213 71

1 2 G 0350 0340 29 27 209 148

1 3 R 0280 0300 33 31 185 101

1 4 R 0360 0340 31 31 194 100

1 5 R 0390 0370 41 43 186 148

1 6 R 0410 0450 35 38 180 135

1 7 R 0360 0340 36 38 181 114

2 1 G 0340 0360 35 32 145 155

2 2 G 0350 0360 31 34 161 72

2 3 G 0340 0360 35 32 159 188

2 4 R 0340 0360 29 27 186 71

2 5 R 0420 0390 31 32 220 71

2 6 R 0380 0340 28 24 213 155

2 7 R 0390 0340 40 38 179 155

3 1 G 0410 0380 35 34 32 115

3 2 G 0380 0350 29 28 165 31

3 3 G 0410 0380 35 34 159 188

3 4 R 0400 0390 34 36 123 32

3 5 R 0370 0360 28 26 233 36

3 6 R 0390 0340 28 23 217 179

3 7 R 0280 0290 26 28 101 182

Table 3 Zone parameters (an example)

Zone number Probability of detectioninwards

Probability of detectionoutwards

Inward penetrationspeed [ms]

Outward penetrationspeed [ms]

1 0100 0080 5600 55002 0130 0110 5400 55003 0150 0120 5400 5500

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO

Figure 8 Gamma analysis Path 1 (an example)

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



LengthFenceZoneSpeedIn

= ProbabPenetNewPathProbabPenet

Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect

Path = null= 0PathsCount

Sections = SectionsBeginGate(Gate)

gt 0PathsCount

geGateProbabDetect

PathsAlpha(Gate2 TargetProbabDetect)

= LengthTargetLength

=NewPathProbabPenet GateProbabPenetIn

NewPathDataCopy(Path)

Section = Sections[i]

gt PathCount)(PathsCountgt 0) and(PathCount

= TargetLengthNewPathTargetLength

= GateTimeInNewPathPenetTime

(LengthFenceZoneSpeedIn)


Length = SectionLengthFence = SectionFence

Gate2 = SectionGate2

NewPathAddSection(Section)TargetProbabDetect

TargetProbabDetect

TargetProbabDetect

TargetLengthFenceZoneSpeedIn

= PathsCountPathsCount

le1 minus ProbabPenetPathPathsDeletePath( Count minus 1)

1 minus ProbabPenet lt

Path = Paths(PathsCount minus 1)

NewPath = newPathsAddPath(NewPath)

i = 0 SectionsCount minus 1

PathProbabPenet GateProbabPenetInlowast=

NewPathPenetTime = NewPathPenetTime +

TargetLength = Targetlength minus 005

ProbabPenet = NewPathProbabPenet lowastFenceProbabPenetZoneIn and (TargetLength minus FenceZoneSpeedIn)


ProbabPenet = NewPathProbabPenet lowast FenceProbabPenetZoneIn and

PathPenetTime GateTimeIn+=

Figure 10 The flowchart of recursive procedure path Alpha


TargetTime

GateTimeIn

Time le TargetTime

= TimeNewPathPenetTime


=PathPenetTime TargetTime


PathsBeta(Gate1 TargetTime)

PathTimeEndGate = TargetTime minus PathPenetTime

Time = NewPathPenetTime +lePathPenetTime GateTimeIn+=

PathPenetTime +=

GateTimeIn gt TargetTime

Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime


Sections = SectionsEndGate(Gate)

gt 0PathsCount


= PathsCountPathsCountPath = Paths(PathsCount minus 1)

PathPathsDeletePath( Count minus 1)



NewPathAddSection(Section)

=NewPathProbabPenet GateProbabPenetIn= GateTimeInNewPathPenetTime



+Time NewPathPenetTime= TargetLengthFenceZoneSpeedIn

(TargetLengthFenceZoneSpeedIn)









TargetLength = Length minus (Time minus TargetTime ) lowast FenceZoneSpeedIn

Figure 11 The flowchart of recursive procedure path Beta


Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount













GateTargetProbabDetect

TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type

PathsGamaIn(Gate2 TargetProbabDetect) PathsGamaOut(Gate2 TargetProbabDetect)





le1 minus ProbabPenet



TO

(TargetLength minus FenceZoneSpeedIn)




Figure 12 The flowchart of recursive procedure path Gamma (inwards)


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount

geGateProbabDetectTargetProbabDetect

(LengthFenceZoneSpeedOut)

=NewPathProbabPenet GateProbabPenetOut= GateTimeOutNewPathPenetTime


TargetProbabDetect




TargetProbabDetect



TargetLengthFenceZoneSpeedOut

PathsGamaOut(Gate1 TargetProbabDetect)


lt1 minus ProbabPenet



LengthFenceZoneSpeedOut


NewPath = NewPathsAddPath(NewPath)


PathProbabPenet GateProbabPenetOutlowast=




ProbabPenet = NewPathProbabPenet lowast Fence andProbabPenetZoneOutPathPenetTime GateTimeOut+=

ProbabPenet = NewPathProbabPenet lowast Fence andProbabPenetZoneOut(TargetLength minus FenceZoneSpeedOut)

Figure 13 The flowchart of recursive procedure path Gamma (outwards)


72 Analysis Beta The Selected Paths721 Path 1 One can see the following

time needed to reach TO 119879 = 120 s-CDPpath G[31] rarr R[26] (segment length 11517m of18537m) rarr R[14] (segment length 5819m) rarr TO(segment length 1910m) (Figure 6)probability of detection = 099886

722 Path 2 One can see the following

time needed to reach TO 119879 = 120 s-CDPpath G[32] rarr R[26] (segment length 13297m) rarrG[12] (segment length 806m) rarr TO (segmentlength 3406m) (Figure 7)probability of detection = 099885

73 Analysis Gamma The Selected Paths


desired probability of detection 119875119889= 099938

path R[36] rarr R[26] (segment length 2433m) rarrR[15] (segment length 2789m) rarr TO (segmentlength 3996m) rarr R[16] (segment length 3421m)rarr R[25] (segment length 7547m) (Figure 8)total time of penetration = 229 sprobability of detection = 099938



path G[32] rarr R[24] (segment length 4518m) rarrR[13] (segment length 3002m) rarr TO (segmentlength 2555m) rarr R[15] (segment length 3996m)rarr G[23] (segment length 4826m) rarr G[33] (seg-ment length 000m G[23] = G[33]) (Figure 9)total time of penetration = 260 sprobability of detection = 099937

8 Conclusions

The submitted study analyzes the alternatives of the intruderpenetration into the protected area by processing the datadescribing the detection capabilities in overcoming the tran-sition gates and barriers or moving through the area Thesolution relevance is closely related to the accuracy of theinput data

A mathematical view of the studied issue created anabstraction serving as a basis for the model and algorithmproposal The computer technology must be involved due tothe number of combinations arising in the model transitionTherefore the user interface suggesting the design of applica-tion assisting in the processing of the issue was proposedThesubsequent implementation was necessary in order to verify

the correctness of the mathematical model the functionalityof the proposed algorithms and the applicability and intu-itiveness of the designed user interface

Emerging from the performed tests it can be concludedthat the proposed algorithms are functional and are ableto achieve the desired results The tests also highlight theproblem of an exponential increase of road alternativesafter increasing the number of barriers and gates It will benecessary to establish criteria filtering out the uninterestingintrusive ways A significant reduction in the total paths isrequired for the postprocessing of results by man

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] HR3696mdashNational Cybersecurity and Critical Infrastruc-ture Protection Act of 2014 httpwwwcongressgovbill113th-congresshouse-bill3696text

[2] CouncilDirective 2008114ECon the identificationand designationof Europeancritical infrastructures and the assessment of the needto improve their protection httpeur-lexeuropaeuLexUriServLexUriServdouri=OJL200834500750082ENPDF

[3] V Kostadinov ldquoDeveloping new methodology for nuclear powerplants vulnerability assessmentrdquoNuclear Engineering andDesignvol 241 no 3 pp 950ndash956 2011

[4] M Hromada and L Lukas ldquoCritical infrastructure protectionand the evaluation processrdquo International Journal of DisasterRecovery and Business Continuity vol 3 pp 37ndash46 2012

[5] O D Oyeyinka L A Dim M C Echeta and A O KuyeldquoDetermination of system effectiveness for physical protectionsystems of a nuclear energy centrerdquo Science and Technology vol4 no 2 pp 9ndash16 2014

[6] T H Woo ldquoSystems thinking safety analysis nuclear securityassessment of physical protection system in nuclear powerplantsrdquo Science andTechnology ofNuclear Installations vol 2013Article ID 473687 5 pages 2013

[7] ITC Physical Protection of Nuclear Facilities andMaterials ITCAlbuquerque NM USA 2010

[8] D Madison Process Mapping Process Improvement and ProcessManagement Paton Press 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Intruderrsquos path

Gate

Target object

Zone

Figure 1 Sketch of the protected area with security features showingone of the possible intruderrsquos paths to the target object

The probability that the physical protection system pre-vents a hostile attack to finish an unwanted event is in theliterature generally calculated as

PE = PI lowast PN (1)

where PE is probability of total system effectiveness PI isprobability of interruption the overall probability of theattack detection during its duration including the criticaldetection point (CDP) based on the principle of earlydetection and the concept of critical point detection andPN is probability of neutralization the probability thatthe corresponding force can prevent the completion of themalicious act such as the theft of nuclear material or nuclearfacility sabotage [7]

The term to neutralize means that the correspondingforce stops the invader occupies the object or eliminates thehostile attack in another way (by causing the escape of theinvader)

The principle of early detection is as follows to interruptthe enemy attack before the requirement for the sabotage ortheft is terminated From the time of detecting the event thereaction of the defense forcesmust be shorter in time than thetime remaining for the completion of the enemy attack

CDP is the last chance to detect the enemy attack Thetime for the action is shorter than the time remaining forterminating the invader requirement [7]

In order to perform the intervention effectively the earlyattack detection must be achieved at all possible paths to thetarget object

On the basis of an extensive use of the principles ofprobability and the graph theory the paper deals withthe proposal of a mathematical model suitable for furthercomputer processing The mathematical model describes allthe aspects of a real situation and creates an abstract view onthe issue

Based on the customer requirements three scenarioshave been developed

(120572) how far does the intruder penetrate into the objectuntil the desired level of detection is reached

(120573) What is the distance between the intruder and thetarget when only the given time to the target is left

(120574) Does the intruder get into the target and out of it untilthe desired level of detection is reached

All three scenarios allow for detecting the supposed positionof the intruder if the input condition of the given scenario ismet The algorithms called Alpha Beta and Gamma weredeveloped for individual scenarios and are listed in otherparts of the paper in Section 6

An application with a graphical user interface was devel-oped for the purposes of verifying the mathematical modeland algorithms

At the end of the study one of the series of tests carriedout for randommodels with different parameters such as thenumber of barriers gates detection probability is chosen

According to our best knowledge no paper focused onsolving the specific tasks mentioned above the scenariosAlpha Beta and Gamma

2 Definitions of Security Features andNomenclature

In this section we introduce the definitions of principalconcepts used in this paper as follows

(1) target object TO (eg nuclear reactor)

(2) barriers the continuous obstacles to penetrate intothe protected object (eg a fence) Bar

119894 119894 =

0 1 119896 Bar119896= an outer barrier and Bar

0= TO

The number of barriers Bar119894= 119896 with TO we have

(119896 + 1) barriers

(3) Zone the area between two consecutive barriers Z119894

119894 = 1 2 119896 The total number of the zones Z119894= 119896

(4) Gates inputs on the barriers G119894119895 119894 = 1 2 119896

119895 = 1 2 119897119894G1198941G1198942 G

119894119897119894fences on the barrier

Bar119894 The total number of the gates on barrier Bar

119894is

G119894= 119897119894 and the total number of the gates G

119894119895=

sum

119896

119894=1119897119894

(5) Rays half-lines 120588119894119895connecting the target object with

the barriers more precisely 120588119894119895is the half-line TO minus

G119894119895 The total number of the rays is equal to the total

number of the gates that is 120588119894119895= sum

119896

119894=1119897119894

(6) R-gates physical places on the barriers lying on therays 120588

119894119895 R119894119895119899

119894 = 1 2 119896 119895 = 1 2 119897119894 and 119899 =

1 2 119896 which is a physical place on the half-line120588119894119895lying on the barrier Bar

119899Thus if 120588

119894119895= TOminusG

119894119895 it

is the connection of the target object TO and the gateG119894119895 and then the R-gate R

119894119895119899lies on the intersection

of 120588119894119895

and Bar119899 The total number of the R-gates is

R119894119895119899= 119896 sdot 120588

119894119895

Remark 1 The reason for introducing the concept of R-gateis the need of implementing the calculations in real time byreducing the number of less probable paths of the intruder

Remark 2 Obviously R119904119895119904= G119904119895 119904 = 1 119896 Therefore in

the process of implementing the algorithms (Section 6) wewill denote the gates and the R-gates on the same barrier (say119904) consecutively and with two indexes only


3 Required Data





(4) the rays 120588119894119895 119883 = G

119894119895119909 sdot 119905 119884 = G

119894119895119910 sdot 119905 119905 gt 0 (TO =

[0 0])




+

119901119860 = 1 minus 119875

+

119889119860 and 119875minus

119901119860 = 1 minus 119875

minus

119889119860

(1) 119875119901TO (119875+

119901TO 119875minus

119901TO)

(2) 119875119901Bar119894 (119875

+

119901Bar119894 119875

minus

119901Bar119894)

(3) 119875119901G119894119895 (119875

+

119901G119894119895 119875

minus

119901G119894119895)

(4) 119875119901R119894119895 (119875

+

119901R119894119895 119875

minus

119901R119894119895)

(5) 119875119889Z119894 (119875

+

119889Z119894 119875

minus




(1) 119879TO (119879+TO 119879minusTO)(2) 119879Bar

119894 (119879

+Bar119894 119879

minusBar119894)

(3) 119879G119894119895 (119879

+G119894119895 119879

minusG119894119895)

(4) 119879R119894119895 (119879

+R119894119895 119879

minusR119894119895)








features


inwards then 119875+119901Bar119904= 1 119875+

119901G119904119895= 1 119875+

119901R119894119895119904=

1 119875+119901Z119904= 1 119879+Bar

119904= 0 and 119879+G

119904119895= 0

119879



119901G119904119895= 1 119875minus

119901R119894119895119904=

1119875

minus

119901Z119904= 1 119879minusBar

119904= 0 and 119879minusG

119904119895= 0

119879

minusR119894119895119904= 0 forall119894 119895

(b) if the gate G119904119903


119901G119904119903= 1 119875+

119901Z119904= 1 119879+G

119904119903= 0


119875

minus

119901Z119904= 1 119879minusG

119904119903= 0







119889is
















119894




119879


119894


R[36]

G[31]

R[37]

G[21]

G[23] = G[33]

G[22]

R[35]R[34] G[32]

R[25]G[11]

R[24]

R[15]

R[16]

R[17]

R[14]R[13]

R[26]R[27]

G[12]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



119889119863 = 1 minus (1 minus

119875

+


119875

minus

119889119863 = 1 minus (1 minus 119875

minus

119889Z)119879minus119863

6 Algorithms












G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

R[24]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



TO Target object

G[11] Gate

R[11] R-gate



G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO




119889= 097015



119889= 096000








119883 [m] 119884 [m]

TO 0410 0400 26 25 207 114

1 1 G 0350 0340 29 27 213 71

1 2 G 0350 0340 29 27 209 148

1 3 R 0280 0300 33 31 185 101

1 4 R 0360 0340 31 31 194 100

1 5 R 0390 0370 41 43 186 148

1 6 R 0410 0450 35 38 180 135

1 7 R 0360 0340 36 38 181 114

2 1 G 0340 0360 35 32 145 155

2 2 G 0350 0360 31 34 161 72

2 3 G 0340 0360 35 32 159 188

2 4 R 0340 0360 29 27 186 71

2 5 R 0420 0390 31 32 220 71

2 6 R 0380 0340 28 24 213 155

2 7 R 0390 0340 40 38 179 155

3 1 G 0410 0380 35 34 32 115

3 2 G 0380 0350 29 28 165 31

3 3 G 0410 0380 35 34 159 188

3 4 R 0400 0390 34 36 123 32

3 5 R 0370 0360 28 26 233 36

3 6 R 0390 0340 28 23 217 179

3 7 R 0280 0290 26 28 101 182






1 0100 0080 5600 55002 0130 0110 5400 55003 0150 0120 5400 5500

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO





Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect



gt 0PathsCount

geGateProbabDetect














TargetProbabDetect

TargetProbabDetect

















TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 Required Data





(4) the rays 120588119894119895 119883 = G

119894119895119909 sdot 119905 119884 = G

119894119895119910 sdot 119905 119905 gt 0 (TO =

[0 0])




+

119901119860 = 1 minus 119875

+

119889119860 and 119875minus

119901119860 = 1 minus 119875

minus

119889119860

(1) 119875119901TO (119875+

119901TO 119875minus

119901TO)

(2) 119875119901Bar119894 (119875

+

119901Bar119894 119875

minus

119901Bar119894)

(3) 119875119901G119894119895 (119875

+

119901G119894119895 119875

minus

119901G119894119895)

(4) 119875119901R119894119895 (119875

+

119901R119894119895 119875

minus

119901R119894119895)

(5) 119875119889Z119894 (119875

+

119889Z119894 119875

minus




(1) 119879TO (119879+TO 119879minusTO)(2) 119879Bar

119894 (119879

+Bar119894 119879

minusBar119894)

(3) 119879G119894119895 (119879

+G119894119895 119879

minusG119894119895)

(4) 119879R119894119895 (119879

+R119894119895 119879

minusR119894119895)








features


inwards then 119875+119901Bar119904= 1 119875+

119901G119904119895= 1 119875+

119901R119894119895119904=

1 119875+119901Z119904= 1 119879+Bar

119904= 0 and 119879+G

119904119895= 0

119879



119901G119904119895= 1 119875minus

119901R119894119895119904=

1119875

minus

119901Z119904= 1 119879minusBar

119904= 0 and 119879minusG

119904119895= 0

119879

minusR119894119895119904= 0 forall119894 119895

(b) if the gate G119904119903


119901G119904119903= 1 119875+

119901Z119904= 1 119879+G

119904119903= 0


119875

minus

119901Z119904= 1 119879minusG

119904119903= 0







119889is
















119894




119879


119894


R[36]

G[31]

R[37]

G[21]

G[23] = G[33]

G[22]

R[35]R[34] G[32]

R[25]G[11]

R[24]

R[15]

R[16]

R[17]

R[14]R[13]

R[26]R[27]

G[12]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



119889119863 = 1 minus (1 minus

119875

+


119875

minus

119889119863 = 1 minus (1 minus 119875

minus

119889Z)119879minus119863

6 Algorithms












G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

R[24]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



TO Target object

G[11] Gate

R[11] R-gate



G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO




119889= 097015



119889= 096000








119883 [m] 119884 [m]

TO 0410 0400 26 25 207 114

1 1 G 0350 0340 29 27 213 71

1 2 G 0350 0340 29 27 209 148

1 3 R 0280 0300 33 31 185 101

1 4 R 0360 0340 31 31 194 100

1 5 R 0390 0370 41 43 186 148

1 6 R 0410 0450 35 38 180 135

1 7 R 0360 0340 36 38 181 114

2 1 G 0340 0360 35 32 145 155

2 2 G 0350 0360 31 34 161 72

2 3 G 0340 0360 35 32 159 188

2 4 R 0340 0360 29 27 186 71

2 5 R 0420 0390 31 32 220 71

2 6 R 0380 0340 28 24 213 155

2 7 R 0390 0340 40 38 179 155

3 1 G 0410 0380 35 34 32 115

3 2 G 0380 0350 29 28 165 31

3 3 G 0410 0380 35 34 159 188

3 4 R 0400 0390 34 36 123 32

3 5 R 0370 0360 28 26 233 36

3 6 R 0390 0340 28 23 217 179

3 7 R 0280 0290 26 28 101 182






1 0100 0080 5600 55002 0130 0110 5400 55003 0150 0120 5400 5500

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO





Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect



gt 0PathsCount

geGateProbabDetect














TargetProbabDetect

TargetProbabDetect

















TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R[36]

G[31]

R[37]

G[21]

G[23] = G[33]

G[22]

R[35]R[34] G[32]

R[25]G[11]

R[24]

R[15]

R[16]

R[17]

R[14]R[13]

R[26]R[27]

G[12]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



119889119863 = 1 minus (1 minus

119875

+


119875

minus

119889119863 = 1 minus (1 minus 119875

minus

119889Z)119879minus119863

6 Algorithms












G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

R[24]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



TO Target object

G[11] Gate

R[11] R-gate



G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO




119889= 097015



119889= 096000








119883 [m] 119884 [m]

TO 0410 0400 26 25 207 114

1 1 G 0350 0340 29 27 213 71

1 2 G 0350 0340 29 27 209 148

1 3 R 0280 0300 33 31 185 101

1 4 R 0360 0340 31 31 194 100

1 5 R 0390 0370 41 43 186 148

1 6 R 0410 0450 35 38 180 135

1 7 R 0360 0340 36 38 181 114

2 1 G 0340 0360 35 32 145 155

2 2 G 0350 0360 31 34 161 72

2 3 G 0340 0360 35 32 159 188

2 4 R 0340 0360 29 27 186 71

2 5 R 0420 0390 31 32 220 71

2 6 R 0380 0340 28 24 213 155

2 7 R 0390 0340 40 38 179 155

3 1 G 0410 0380 35 34 32 115

3 2 G 0380 0350 29 28 165 31

3 3 G 0410 0380 35 34 159 188

3 4 R 0400 0390 34 36 123 32

3 5 R 0370 0360 28 26 233 36

3 6 R 0390 0340 28 23 217 179

3 7 R 0280 0290 26 28 101 182






1 0100 0080 5600 55002 0130 0110 5400 55003 0150 0120 5400 5500

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO





Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect



gt 0PathsCount

geGateProbabDetect














TargetProbabDetect

TargetProbabDetect

















TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

R[24]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO



TO Target object

G[11] Gate

R[11] R-gate



G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO




119889= 097015



119889= 096000








119883 [m] 119884 [m]

TO 0410 0400 26 25 207 114

1 1 G 0350 0340 29 27 213 71

1 2 G 0350 0340 29 27 209 148

1 3 R 0280 0300 33 31 185 101

1 4 R 0360 0340 31 31 194 100

1 5 R 0390 0370 41 43 186 148

1 6 R 0410 0450 35 38 180 135

1 7 R 0360 0340 36 38 181 114

2 1 G 0340 0360 35 32 145 155

2 2 G 0350 0360 31 34 161 72

2 3 G 0340 0360 35 32 159 188

2 4 R 0340 0360 29 27 186 71

2 5 R 0420 0390 31 32 220 71

2 6 R 0380 0340 28 24 213 155

2 7 R 0390 0340 40 38 179 155

3 1 G 0410 0380 35 34 32 115

3 2 G 0380 0350 29 28 165 31

3 3 G 0410 0380 35 34 159 188

3 4 R 0400 0390 34 36 123 32

3 5 R 0370 0360 28 26 233 36

3 6 R 0390 0340 28 23 217 179

3 7 R 0280 0290 26 28 101 182






1 0100 0080 5600 55002 0130 0110 5400 55003 0150 0120 5400 5500

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO





Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect



gt 0PathsCount

geGateProbabDetect














TargetProbabDetect

TargetProbabDetect

















TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119883 [m] 119884 [m]

TO 0410 0400 26 25 207 114

1 1 G 0350 0340 29 27 213 71

1 2 G 0350 0340 29 27 209 148

1 3 R 0280 0300 33 31 185 101

1 4 R 0360 0340 31 31 194 100

1 5 R 0390 0370 41 43 186 148

1 6 R 0410 0450 35 38 180 135

1 7 R 0360 0340 36 38 181 114

2 1 G 0340 0360 35 32 145 155

2 2 G 0350 0360 31 34 161 72

2 3 G 0340 0360 35 32 159 188

2 4 R 0340 0360 29 27 186 71

2 5 R 0420 0390 31 32 220 71

2 6 R 0380 0340 28 24 213 155

2 7 R 0390 0340 40 38 179 155

3 1 G 0410 0380 35 34 32 115

3 2 G 0380 0350 29 28 165 31

3 3 G 0410 0380 35 34 159 188

3 4 R 0400 0390 34 36 123 32

3 5 R 0370 0360 28 26 233 36

3 6 R 0390 0340 28 23 217 179

3 7 R 0280 0290 26 28 101 182






1 0100 0080 5600 55002 0130 0110 5400 55003 0150 0120 5400 5500

G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO


G[31] G[32] R[34]G[33]

G[21] G[22] G[23]

G[11] G[12]

R[35] R[36] R[37]

R[24] R[25] R[26] R[27]

R[13] R[14] R[15] R[16] R[17]

TO





Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect



gt 0PathsCount

geGateProbabDetect














TargetProbabDetect

TargetProbabDetect

















TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Start

Gate

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes

TargetProbabDetect



gt 0PathsCount

geGateProbabDetect














TargetProbabDetect

TargetProbabDetect

















TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










TargetTime

GateTimeIn

Time le TargetTime








PathPenetTime +=


Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

YesNoYes

No

Yes


GateTargetTime



gt 0PathsCount























Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

Path = null


Path = null

Yes

No

No

Yes

No



Path = null



Finish

No

No

Yes

Yes

Yes

No

YesNo

Yes

NoYes

= 0PathsCount

gt 0PathsCount














TargetProbabDetect

TargetProbabDetect


TargetProbabDetect

geGateProbabDetect


=Gate2Type









TO







Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

Path = null

Yes

No

Yes

No

Path = null

Finish

No

No

Yes

Yes

No

Yes

No

Yes

No

Yes








gt 0PathsCount





TargetProbabDetect




TargetProbabDetect
































8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















8 Conclusions







References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Mathematical Approach to Security Risk … · 2019. 7. 31. · Research Article Mathematical Approach to Security Risk Assessment RobertVrabel,MarcelAbas,PavolTanuska,PavelVazan,MichalKebisek,