Research Article Fortifying Intrusion Detection Systems in ...downloads.hindawi.com/journals/ijdsn/2014/608162.pdf · Multihop ad hoc wireless networks are a set of nodes equipped

Research ArticleFortifying Intrusion Detection Systems in Dynamic Ad Hoc andWireless Sensor Networks

Abdelouahid Derhab1 Abdelghani Bouras2

Mustapha Reda Senouci3 and Muhammad Imran1

1King Saud University PO Box 92144 Riyadh 11543 Saudi Arabia2Industrial Engineering Department College of Engineering King Saud University PO Box 800 Riyadh 11421 Saudi Arabia3Laboratory of Research in Artificial Intelligence Ecole Militaire Polytechnique PO Box 17 Bordj-El-Bahri 16111 Algiers Algeria

Correspondence should be addressed to Abdelouahid Derhab abderhabksuedusa

Received 24 September 2014 Revised 1 December 2014 Accepted 4 December 2014 Published 29 December 2014

Academic Editor Jinsung Cho

Copyright copy 2014 Abdelouahid Derhab et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We investigate three aspects of dynamicity in ad hoc and wireless sensor networks and their impact on the efficiency of intrusiondetection systems (IDSs) The first aspect is magnitude dynamicity in which the IDS has to efficiently determine whether thechanges occurring in the network are due to malicious behaviors or or due to normal changing of user requirements The secondaspect is nature dynamicity that occurs when a malicious node is continuously switching its behavior between normal andanomalous to cause maximum network disruption without being detected by the IDS The third aspect named spatiotemporaldynamicity happenswhen amalicious nodemoves out of the IDS range before the latter canmake an observation about its behaviorThe first aspect is solved by defining a normal profile based on the invariants derived from the normal node behavior The secondaspect is handled by proposing an adaptive reputation fading strategy that allows fast redemption and fast capture ofmalicious nodeThe third aspect is solved by estimating the link duration between two nodes in dynamic network topology which allows choosingthe appropriate monitoring period We provide analytical studies and simulation experiments to demonstrate the efficiency of theproposed solutions

1 Introduction

Multihop ad hoc wireless networks are a set of nodesequipped with wireless interfaces and data are forwardedthrough multiple nodes to reach the intended destinationsThey include many types of networks such as mobile ad hocnetworks (MANETs) [1] wireless sensor networks (WSNs)[2] and vehicular ad hoc networks (VANETs) [3]

In the last decade there has been a substantial research inthe area of security in ad hoc and wireless sensor networks[4 5] The security solutions have been designed with thegoal of protecting the networks against some attacks suchas selective forwarding black hole wormhole sinkhole andenergy exhausting attack Prevention mechanisms like keymanagement and authentication which represent the firstline of defense are not sufficient to provide an efficientsecurity solutionTherefore there is a need to deploy a secondline of defence named intrusion detection system (IDS)

In general intrusion detection systems are divided intotwo major approaches misuse detection and anomaly detec-tion [6] Misuse detection performs signature analysis bycomparing on-going activities with patterns representingknown attacks and those matched are labeled as intrusiveattacksThemisuse approach is showing its limits as it cannotdetect new attacks Anomaly detection on the other handbuilds profile of normal behavior and attempts to identifythe patterns or activities that deviate from the normal profileThemain advantage of anomaly detection is that it can detectunknown attacks

The detection model that we consider in the ad hoc andwireless sensor network is as followsThe IDS is implementedin a distributed manner each node can act as a monitoringnode that observes the behavior of its neighbors Eachobservation lasts for amonitoring time interval of durationΔcalled the monitoring periodThe IDS can judge whether the

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2014 Article ID 608162 15 pageshttpdxdoiorg1011552014608162

2 International Journal of Distributed Sensor Networks

90

90

9090

90 180360

720

90 180 270

L

K

A

J

C

DF

E

B

G H I

(a)

90

90

9090

90 100270

550

90 120 210

L

K

A

J

C

DF

E

B

G H I

(b)

L

K

A

J

C

DF

E

B

90

90

9090

90 270450

900

GH I

90 270 360

(c)

Figure 1 Impact of feature choice on false positive rate

monitored node is normal or anomalous after one ormultipleconsecutive observations

Although intrusion detection systems have received con-siderable attention in ad hoc and wireless sensor networks[7 8] to the best of our knowledge there are no studies onthe impact of network dynamicity on IDS efficiency and howthe IDS can react or adapt to these changes

In this paper we investigate the following three aspectsof behavioral dynamicity that occur in the network and cannegatively affect the IDS performance and efficiency

(i) Magnitude Dynamicity Due to change of user re-quirements a node changes the rate at which itgenerates data For instance a legitimate user wantsto change (ie increasedecrease) the data collectionrate received at the sink node The challenge facingthe IDS here is to be efficient at detecting attacks anddistinguishing between the changes due to normalbehaviors and the changes due to malicious attacks

(ii) Nature Dynamicity In some detection models amonitoring node has to observe the behavior of themonitored node during a set of consecutive moni-toring periods before judging whether the monitorednode is malicious or not A monitored node mightevade from IDS detection and confuses it by switch-ing continuously its behavior between normal andanomalous In this case the malicious node strives tocause network disruption without being detected bythe IDS

(iii) Spatiotemporal Dynamicity The IDS detection mech-anism is based on collecting a set of consecutiveobservations about the monitored node An IDSis able to observe the behavior of the monitorednode if the latter stays within the monitoring nodersquostransmission range for a duration exceeding Δ Byknowing this fact a malicious node can evade IDSdetection bymoving around in the network at a speedwhich prevents it from being within the monitoringnodersquos transmission range for a duration higher thanΔ

In this paper we propose a solution for each aspect ofdynamicity mentioned above The contributions of the paperare threefold Firstly the magnitude dynamicity aspect issolved by defining a normal profile based on the invariantsderived from the normal node behavior This is achieved

by generating a dependency graph consisting of stronglycorrelated features and then derives the high-level featuresfrom the graph The high-level features are obtained byapplying the divide-and-conquer strategy on the maximalcliques algorithm and the maximum weighted spanning treealgorithm Secondly to handle nature dynamicity aspect weadopt the carrot and stick strategy (ie reward generouslyand punish severely) to prevent a malicious node fromevading the IDS To do so we propose an adaptive reputationfading strategy to allow fast redemption and fast capture ofmalicious nodeThirdly we use statistical analysis to estimatethe link duration between two nodes in dynamic networktopology Based on this estimation the monitoring nodechooses the appropriate monitoring period which allows itto observe the monitored nodersquos behavior

The rest of the paper is organized as follows In Section 2we describe the normal profile construction and the featureselection method Section 3 presents the adaptive reputationfading strategy In Section 4 we analyze link-node durationin amobile wireless network and explain how themonitoringtime period is estimated Finally Section 5 concludes thepaper

2 Magnitude Dynamicity

21 Background

211 One-Feature Profile In the one-feature profile we use asingle feature to describe and detect anomalous behavior Todetect the network malicious behavior a node can measurethe following features as shown in Table 1 [9] The disadvan-tage of this profile structure is that there is a need to assignone feature for each known attack In this case the IDS hasto measure each feature and check whether it has anomalousvalue When the number of attacks increases the detectionspeed of the IDS becomes slow It also becomes slower whenthe size of rule set increases

The one-feature profile might fail at distinguishingbetween normal and anomalous behaviors Figure 1 showsthat using some features individually to describe normalbehavior is misleading and might make the detection systemfalsely accuse a legitimate node of beingmalicious Figure 1(a)depicts a tree-based wireless network rooted at the sink 119861

and it shows the normal traffic rates of the network Thevalue above each link indicates the flow rate traversing thislink Each node measures the flow rate coming from its

International Journal of Distributed Sensor Networks 3

Table 1 Relation between attacks and features

Attack FeaturePacket sending rate Energy exhausting attack

Packet dropping rate Selective forwarding and black holeattacks

Packet receiving rate Sinkhole attackPacket sending power Hello attack wormhole attack

upstream neighbors Figure 1(b) (resp Figure 1(c)) showsthe state of the network when nodes 119863 119867 and 119870 becomecompromised and start behaving maliciously by droppingsome packets (resp generating more packets) As 119863 119867 and119870 reduce (resp increase) their sending rate their respectivedownstream neighbors 119868 and 119871 have also to reduce (respincrease) their sending rate accordingly As a result node119861 will falsely accuse nodes 119868 and 119871 of performing selectiveforwarding attack (resp energy exhaustion attack) andhence a high false positive rate will be observed

212 Multifeature Profile In the multifeature profile wedescribe the normal behavior by a 119889-feature vector and eachelement of the vector represents a feature In this way theIDS can determine whether some features together showan anomalous behavior Experiments have shown that wecan obtain better detection accuracy by combining relatedfeatures rather than individually [10] If node119861 in the exampleof Figure 1 considers two features (a) the flow entering themonitored node and (b) the flow leaving themonitored nodeit will conclude that nodes 119868 and 119871 are just forwarding whatthey received from their upstream neighbors and hence theyare not malicious

Loo et al [11] group the observed data into clusters anduse a profile of 12 features to describe normal profile Tocheck whether a test instance belongs to a given cluster theymeasure the Euclidian distance between the test point andthe centroid of the cluster If such a distance is higher thana threshold distance the test point is considered anomalousThe following example shows that the Euclidian distancebetween two 119889-feature profiles reduces the detection accu-racy Let (119891119905

1 1198911199052) be a vector profile such that each feature

of the vector is used to detect one attack 1198911199051and 119891119905

2take

values in [0 10] Let (10 10) be the centroid vector The firstand the second attacks are detected when 119891119905

1le 7 and

1198911199052

le 6 respectively We take the distance between (10 10)

and (7 6) which is 5 as the threshold distance Let a testvector be (6 10) the distance between the two vectors is 4which is lower than the distance threshold In this case thetest point will be considered normal whereas the value of 119891119905

1

individually indicates the occurrence of an attack The aboveexample shows that aggregating features through the use ofEuclidian distance result in loss of detection accuracy

In [9 12] the normal profile of a monitored node 119894

is defined by a 119902-feature vector 119891119894

= (1198911199051198941 119891119905

119894119902) If a

node monitors a set of 119899 nodes it forms a matrix 119865 =

(1198911 119891

119899)119879 Both schemes assume that all feature vectors 119891

119894

follow the same multivariate normal distribution with mean120583 and variance-covariance matrix M Node 119894 is considered

suspicious if the Mahalanobis distance between 119891119894and the

center of the set 119865 is greater than a predefined threshold Theauthors of both works use the orthogonalized Gnanadesikan-Kettenring estimation to find the center of the set 119865 Let 120583

and M denote the simple mean and the simple variance-covariance of 119865 such that 120583 = (1119899)sum

119899

119894=1119891119894and M = (1(119899 minus

1))sum119899

119894=1(119891119894minus 120583)(119891

119894minus 120583)119879 The Mahalanobis distance between

119891119894and the vector 120583 is given by radic(119891

119894minus 120583)119879Mminus1(119891

119894minus 120583) The

Mahalanobis distance differs from the Euclidian distance inthat it takes into account the correlations between featuresIn [12] nodes are evaluated in terms of packet droppingrate packet sending rate forwarding delay time and nodereadings In [9] the attacks are detected by monitoringpacket sending rate packet dropping rate packet mismatchrate packet receiving rate and received signal strength Asstated in [13] the works of [9 12] suffer from two majorcriticisms (1) the circumstances underwhich the assumptionof multivariate normal distribution holds are not explainedand (2) the network features such as packet sending packetdropping and packet receiving rates do not follow the normaldistribution for tree-based routing protocol

22 Profile Construction Based on Strongly Correlated Fea-tures When it comes to comparing distances we find thatthe Mahalanobis distance is a powerful technique as it takesthe covariances into account which leads to elliptic decisionboundaries in the 2D space While the Euclidean distancebuilds circular boundaries and considers equal variances ofthe features it appears that the Mahalanobis distance is moreappropriate for multivariate data

In our paper we take a novel approach to select relevantfeatures and construct the normal profile vector We do notassume multivariate normal distribution and we feed onlystrongly correlated features to the distancemeasure unlike theMahalanobis distance which considers correlation betweenall features

In the training phase we investigate the significant associ-ations between features We are interested in identifying thelevel of correlation between those features called Pearsonrsquoscorrelation coefficient which measures the strength of thelinear association between features Pearsonrsquos correlationcoefficient between two feature vectors 119883 and 119884 is definedby

120588 (119883 119884) =COV (119883 119884)

120590119883120590119884

=119864 [(119909 minus 120583

119883) (119910 minus 120583

119884)]

120590119883120590119884

(1)

where 120583119883(resp 120583

119884) and 120590

119883(resp 120590

119884) are the mean and

standard deviation values of feature 119883 (resp feature 119884) If120588(119883 119884) = 1 then 119883 and 119884 have a linear correlation If 07 le

120588(119883 119884) lt 1 then 119883 and 119884 have a strong linear correlationif 05 le 120588(119883 119884) lt 07 then 119883 and 119884 have a modest linearcorrelation and if 0 le 120588(119883 119884) lt 05 then 119883 and 119884 are saidto have a weak linear correlation

The Pearson correlation indicates to what extent variablesshow a linear relationship (correlation) among them Thecorrelation takes its values in the range from minus1 to +1 Theextreme value +1 (resp minus1) informs about a perfect


directincreasing linear relationship (resp inversedecreasing) Indeed strong relationship between variablesis reflected by values close to the limits (minus1 le 120588 le minus09

or 09 le 120588 le +1) [14] Pearsonrsquos correlation coefficienttakes value 0 if we are in presence of independent variablesHowever the reverse is not true since this coefficient dealsonly with figuring out linear dependencies between variables

In our approach we first use the training dataset 119865

represented by 119899 times 119889 119865 consists of 119899 profile instances 119891119894such

that 119894 = 1 119899 and each 119891119894= (119891119905

1198941 119891119905

119894119889) From 119865 we

construct a correlation matrix Ω The latter is a 119889 times 119889 matrixwhere Ω

119894119895isin R and minus1 le Ω

119894119895le +1

Ω = (

Ω11

Ω12

sdot sdot sdot Ω1119889

Ω21

Ω22


d

Ω1198891

Ω1198892


) (2)

We consider the set of 119889 feature vectors 1198651 119865

119889 such

that 119865119894

= (

1198911199051119894

119891119905119899119894

) For each pair of features (119865119894 119865119895) we

compute Ω119894119895

= 120588(119865119894 119865119895) Then we derive a weighted graph

119866 = (119881 119864 119908) from matrix Ω defined as follows

(i) 119881 = V1sdot sdot sdot V119889 the set of vertices (features) where

|119881| = 119889(ii) 119864 = (V

119894 V119895) where Ω

119894119895= 0 and |119864| = 119898

(iii) 119908(V119894 V119895) = 119908119894119895

= Ω119894119895

A subgraph 119866[Th]

= (119881[Th]

119864[Th]

119908[Th]

) is then inducedfrom the graph 119866 where 0 lt Th le 1 by removing all theedges (V

119894 V119895) whose 119908

119894119895lt Th 119866[Th] is defined as follows

(i) 119864[Th]

= (V119894 V119895) where 119908

119894119895ge Th

(ii) 119881[Th]

= 119909 isin 119881 exist119910 isin 119881 and (119909 119910) isin 119864[Th]

|119881[Th]| le119889

(iii) 119908[Th]119894119895

= 119908119894119895

The induced graph 119866[Th] from 119866 might be composed of a

set of disjoint connected partitions The more the Th is closeto 1 the stronger the correlations exist in 119866

[Th]We aim at finding the set of features that increase and

decrease altogether in order to avoid the missed detectionproblem as in [11] The best way to do so is to extract from119866 the set of cliques composed of strongly correlated featuresOne of the widely adopted solutions [15] to computemaximalcliques in an arbitrary graph of 119889 vertices runs in time119874(31198893

) = 119874(144119889) Instead of applying the maximal cliques

algorithm on graph 119866 we propose to adopt the divide andconquer strategy by applying this algorithm on each con-nected component of the subgraph 119866

[Th] A clique CL[Th]119894

=

(119881[Th]119894

119864[Th]119894

) (119894 ge 1) of a graph 119866[Th] is a set of vertices

119881[Th]119894

sube 119881[Th] such that all the pairs of 119862[Th]

119894are adjacent This

strategy significantly reduces the computational complexityto find maximal strongly correlated cliques Let us considerthat 119866[Th] is composed of 119889 vertices belonging to a set of 119872

connected components Each connected component 119875119894119894 =

1 sdot sdot sdot119872 is composed of 119878119894vertices There are 120572 singleton

vertices and 120573 partitions with two vertices and the restof connected components are composed of more than twovertices The computational complexity incurred by applyingthe maximal cliques algorithm on graph 119866 is

144119889= 144

(120572+2120573+sum119895119878119895gt2119878119895)

= 1441205721442120573

prod

119895119878119895gt2

144119878119895 (3)

By applying the same algorithm on each connectedpartition of 119866

[Th] we notice that there is no need to applyit on isolated vertices and the partitions of two verticesare cliques by definition and hence we get the followingcomputational complexity sum

119895119878119895gt2

144119878119895 It is obvious that

applying the divide and conquer strategy can significantlyreduce the running time of the algorithm andmake it suitablefor resource-constrained nodes

Let 120601 be the set of edges belonging to all cliques in 119866[Th]

and |120601| = 1198891015840 For each edge (119865

119897 119865119896) which is the 119905th element

of 120601 (119905 = 1 1198891015840) we define a high-level feature 119867

119905= 119865119897119865119896

From the training dataset 119865 we derive its high-level trainingdataset119867119865defined as follows for each119889-profile vector119891

119894isin 119865

we derive its 1198891015840-profile high-level vector 119892119894= (1198921199051198941 119892119905

1198941198891015840)

such that 119892119905119894119905

= 119891119905119894119897119891119905119894119896and 119891119905

119894119896= 0 If 119891119905

119894119896= 0 the

high-level vector 119892119894is then removed from the training dataset

119867119865 This choice is justified by the fact that the stronger thecorrelation between 119865

119897and 119865

119896is the more the data instances

of (119865119897 119865119896) fall on the same straight line 119865

119897= 119886119865119896+ 119887 where 119886

is the slope and 119887 is the interceptThe high-level features belonging to the same clique

CL[Th]119894

are grouped into a single vector 120585119894 We consider that

119870 cliques are obtained from119866[Th] Thus the normal profile is

then defined as the set of vectors 120585119894(119894 = 1 119870) To further

reduce the number of features in each vector 120585119894 we apply the

maximum weighted spanning tree algorithm on each cliqueTo do so we apply Kruskalrsquos algorithm originally used toobtain the minimum spanning tree by negating the weightof each edge [16]The high-level features whose edges do notbelong to the tree are removed from the normal profile Theresulted profile is called the minimum normal profile Thetime complexity of the maximum weighted spanning tree is119874(|119864CL| log |119864CL|) where 119864CL is the number of edges in theclique As |119864CL| = |119881CL|(|119881CL| minus 1)2 the time complexitybecomes proportional to 119874(|119864CL| log |119881CL|) As the maximalcliques algorithm the maximum weighted spanning tree isonly applied on cliques with more than two vertices The useof maximum weighted spanning tree is justified by the factthat all the low-level features of each clique in 119866

[Th] havestrong correlation between them In each clique if 119883 and 119884

are strongly correlated and 119884 and 119885 are strongly correlatedthen 119883 and 119885 are strongly correlated Hence we can removethe redundant (119883 119885) edge from the clique

To illustrate further the above method we consider anexample of seven network features namely 119865

1 1198652 1198653 1198654 1198655


(1) Let 119885 be the high-level test profile composed of 119885119897vectors (119897 = 1 119870)

(2) for All vectors 119862119897such that 119897 = 1 119870 do

(3) if (119863119894119904(119885119897 119862119897) notin [119871119900119908

119897 119880119901119897]) then

(4) return 119885 is anomalous(5) end if(6) end for(7) return 119885 is normal

Algorithm 1 Intrusion detection algorithm

097

093

099

098

098

095

094F1

F2

F3

F4 F5

F6

F7

(a) Normal profile

097

099

098

098F1

F2

F3

F4 F5

F6

F7

(b) Minimum normal profile

Figure 2 Graph-based normal behavioral model

1198656 and 119865

7 The correlation coefficient matrices Ω between

these features are

Ω =

1198651

1198652

1198653

1198654

1198655

1198656

1198657

1198651

1198652

1198653

1198654

1198655

1198656

1198657

(((

(

1 093 097 025 073 082 098

093 1 099 081 054 062 094

097 099 1 073 087 043 095

025 081 073 1 098 052 071

073 054 087 098 1 078 060

082 062 043 052 078 1 053

098 094 095 071 060 053 1

)))

)

(4)

According to the correlation matrix we generate thegraph 119866

[Th] where Th gt 09 as shown in Figure 2(a) In thegraph there are two cliques 119865

1 1198652 1198653 1198657 and 119865

4 1198655

The network normal profile is defined as (11986511198652 11986511198653

1198651119865711986521198653 1198652119865711986531198657) (11986541198655) After applying themax-

imum weighted spanning tree algorithm the edges (1198651 1198652)

(1198652 1198657) and (119865

3 1198657) are removed and the minimum normal

profile becomes (11986511198653 11986521198653 11986511198657) (11986541198655)

Proposition 1 For any data set of 119889 low-level features thenumber of high-level features induced by the graph-basedgeneration method is upper-bounded by 119889 minus 119870 such that 119870

is the number of cliques in 119866[Th]

Proof Consider 119881[Th]

sube 119881 that is in the worst case eachlow-level feature belongs to a given clique CL[Th]

119894(119894 ge 1) As a

result sum119870119894=1

|119881[Th]119894

| le 119889 It is known that the number of edgesinduced by executing the maximum weighted spanning treeon the clique CL[Th]

119894is ℎ119894

= |119881[Th]119894

| minus 1 As sum119870

119894=1(ℎ119894+ 1) le

119889 sum119870119894=1

ℎ119894

le 119889 minus 119870 Thus the number of edges (ie high-level features) induced by executing the maximum weighted

spanning tree on all the cliques of 119866[Th] is upper-bounded by119889 minus 119870

23 Detection Process Each node constructs its local datasetrepresented by 119899 times 119889 matrix (ie 119899 vector instances and119889 features) It then extracts 119870 cliques from this dataset asshown above as well as its minimum profile composed of 119870vectors 120585

119897of size 119898

119897 where 119897 = 1 119870 The node computes

the centroid vector 119862119897for all the 119899 instances of 120585

119897

To check whether a profile 119885 is normal or anomalous wederive from 119885 its corresponding high-level profile 119867119885 andwe execute the pseudocode depicted in Algorithm 1 In thealgorithm Dis denotes the Euclidian distance between twovectors Low119897 and Up119897 denote the lowest and highest valuesobtained from estimating Dis(120585

119897 119862119897) for all the 119899 instances of

120585119897

24 Simulation Results Westudy the performance of the pro-posed IDS using GloMoSim simulator [17] Each node sendsone packetsec toward the sink A watchdog is implementedat each node and its role is to monitor the network activitiesof all the nodersquos neighbors At every 10 seconds (ie onetime period) amonitoring node 119894measures the feature vectorof its monitored node 119895 After a training phase of 119879 timeperiods testing phase lasts for 1800 seconds The role of IDSwhich is implemented at a node 119894 is not just to detect if 119894rsquosneighbor (node 119895) is malicious or not but also to detect ifnode 119895 is malicious during a given time period We evaluatethe performance of the IDS using two metrics detection rateand false positive rateWe select the following five quantitativefeatures

(i) number of generated packets (GEN)(ii) number of received packets (RCV)(iii) number of forwarded packets (FWD)


1

09289

09727

09289

09727

09828

RCVFWD

LOSS

SENT

1

09727

09828

RCVFWD

LOSS

SENT

Figure 3 Normal profile and minimum normal profile

60

65

70

75

80

85

90

95

100

0 01 02 03 04 05 06 07 08 09 1

Det

ectio

n ra

te (

)

Dropping probability

T = 3T = 5T = 10T = 20

T = 30T = 40T = 50

Figure 4 Detection rate versus dropping probability

(iv) number of sent packets (SENT)(v) number of lost packets (LOSS)

We generate then the correlation matrix Ω as well asthe minimum normal profile after performing the maximalcliques algorithm and the maximum weighted spanning treealgorithm as shown in Figure 3

Ω =

GEN RCV FWD SENT LOSSGENRCVFWDSENTLOSS

(

1 04205 04205 07263 06032

04205 1 1 09289 09727

04205 1 1 09289 09727

07263 09289 09289 1 09828

06032 09727 09727 09828 1

)

(5)

Figure 4 shows the detection rate of the proposed IDSas a function of dropping probability The first observationthat we can draw from the figure is that the detectionrate is 100 when the dropping probability is higher than005 and it is under 100 when the dropping probabilityis le002 This can be explained as follows under very lowdropping probabilities the malicious nodes drop packets at

60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50

Det

ectio

n ra

te (

)

Training period

P = 1P = 05P = 01

P = 005P = 001

Figure 5 Detection rate versus training time

0

05

1

15

2

25

3

35

4

5 10 15 20 25 30 35 40 45 50

False

pos

itive

rate

()

Training period

P = 08P = 05P = 02

P = 005P = 003P = 001

Figure 6 False positive rate

low intensities and their activities become unnoticeable Thishappens when the dropping probability becomes very closeto or less than the normal packet loss which is at most 2during each time period Figure 5 shows the detection rateof the IDS as a function of training period The results arepresented under the following levels of dropping probability119875 = 1 05 01 005 001 The results show that the detectionrate does not depend on the training period but on thedropping probability Under high dropping probabilities thedetection rate is 100 for all the training periods Under lowdropping probabilities the detection rate decreases as themalicious behavior becomes very close to the normal one

Figure 6 shows the false positive rate of IDS as a functionof training period under the following levels of droppingprobability 119875 = 08 05 01 005 003 001 We can notice


that the false positive becomes 0 when the training period119879 = 30 for all 119875 gt 002 At 119879 = 30 the IDS has learned all thepossible instances of the normal profile and can accuratelydistinguish between normal and anomalous traffic When119879 lt 30 the IDS still has not learned all the instances of thenormal profile In other words the normal profiles which arenot observed during the training phase will be consideredanomalous during the testing phase Thus the false positiverate depends in this case on the number of times unlearnednormal profiles are observed during the testing phase whichitself depends on the number of lost packets that are due to (1)

normal packet loss and (2) dropping activities As packet lossis an event that occurs randomly the false positive curves arealso random when 119879 lt 30 For 119875 = 001 the false positivebecomes 0 only when 119879 = 40 Given that the behavior ofthemalicious node becomes very close to the legitimate nodethe IDS needs more time to learn about new instances of thenormal profile

3 Nature Dynamicity

31 Background Constant Fading Reputation Strategy Repu-tation is defined as the general opinion of a society of nodestowards a certain node in a specific domain of interest and itis the global perception on the future behavior of this nodeIn the IDS based on multiple observations the IDS collectsa series of consecutive observations each of which occursduring a separate monitoring period

Since reputation aggregates past experiences and dynam-ically evolves it is similar to Bayesian analysis which is a sta-tistical procedure that estimates parameters of an underlyingdistribution based on observations Starting with prior dis-tribution which is the initial state before any observation ismade Bayesian analysis continuously takes into account newexperiences and derives posterior probability [18] One of theused distributions in Bayesian analysis is Beta distribution

Beta distribution has been recognized as a useful formaltool to model reputation [18ndash20] A reputation value assumesa tuple of (120572 120573 ge 1) such that 120572 and 120573 represent positive andnegative observations respectively

The Beta distribution and its probability density function(PDF) are defined as follows

119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)

The reputation denoted by 119877 is defined as the expecta-tion (denoted by E) of the Beta distribution and it takes thefollowing simple form

119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)

We model the reputation of a node with a Beta distribu-tion (120572 120573) Initially 120572 = 1 and 120573 = 1

The standard Bayesian procedure is as follows Initiallythe prior is Beta(1 1) the uniform distribution on [0 1]Then when a new observation is made say with 119899 observedmisbehaviors and 119901 observed correct behaviors the prior isupdated according to120572 = 120572+119901 and120573 = 120573+119899The reputationrelies on the nodersquos direct observation When the monitoringnode makes one individual observation about the monitorednode it updates 120572 and 120573 as follows

(i) If the observation is qualified as misbehavior 120573 is setto 120573 + 1

(ii) If the observation is qualified as correct behavior 120572 isset to 120572 + 1

The standard Bayesian method is modified in [19] togive less weight to the observations received in the past soas to allow reputation fading and prevent any node fromcapitalizing on its previous good behavior forever To achievethis aim a discount factor for past observations is usedWhena new observation (119901 119899) is made 120572 and 120573 are updated asfollows

120572 = 120596120572 + 119901

120573 = 120596120573 + 119899

where 0 le 120596 le 1

(8)

The weight 120596 is a constant discount factor for pastobservations which serves as the fadingmechanismWe referhereafter to the reputation system described above as theconstant fading reputation strategy

32 Adaptive Fading Reputation Strategy Theconstant fadingreputation mechanism uses the same discount factor for alltypes of observations and during all the time The higher(resp lower) the value of 120596 is the slower (resp quicker)the histories are forgotten By knowing the value of 120596 amalicious node can evade from IDSdetection bymisbehavingfor a given time and goes back to normal behavior Underhigh discount factor the change of node behavior (fromwell-behaved to misbehaved and vice versa) will be detectedafter a long time During this time well-behaved nodescan count on their good histories and act maliciously Inaddition misbehaved nodes will have to wait a longer timeto redeem themselves On the other hand a low discountfactor permits a quicker detection redemption of nodesHowever it might raise false alarms especially when networkfaults and attacks both share the same failure symptoms Forinstance amisbehavior is detected if the observed node is notforwarding a packet This rule is set to detect black hole andselective forwarding attacks In addition this rule is appliedwhen packets are not forwarded due to collisions whichmeans that a well-behaved observed node might be falselyconsidered malicious

To deal with this issue we propose an adaptive fadingreputation mechanism This mechanism uses the carrot andstick strategy that is reward the well-behaved node and pun-ish the misbehaved node The adaptive mechanism uses twotypes of discount factors one for past positive observations


Positive discount factor Negative discount factor

R0 1

1

Reward strategyPunishment strategy

NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th

Figure 7 Positive and negative discount factors

and the second one for past negative observations The valueof the discount factors is adjusted as function of reputation 119877

as shown in Figure 7In the adaptive fading reputationmechanismwhen a new

observation (119901 119899) is made 120572 and 120573 are updated as follows

120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)

120595(119877) and 120593(119877) denote the discount factors for past posi-tive and negative histories respectively whose values fall intothe range of [0 1] According to the value of 119877 a reputationsystem executes the following two fading strategies

(i) Reward Strategy It is applied when the reputation119877 ge th such that th isin [0 1] The IDS forgets thenegative history more quickly than the positive one(ie 120595(119877) gt 120593(119877)) this strategy is used when a nodeis well-behaved

(ii) Punishment Strategy It is applied when the reputation119877 lt th The IDS forgets the positive history morequickly than the negative one (ie 120595(119877) lt 120593(119877)) thisstrategy is used when a node is misbehaved

Formally 120595(119877) and 120593(119877) are written as follows

120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +

PRmin minus PRmax times 119905

1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905

120593 (119877) = (PRmax + NRmin) minus 120595 (119877) when 119877 ge 119905

(NPmax + PPmin) minus 120595 (119877) when 119877 lt 119905

(10)

where PRmax and PRmin are the upper and the lower boundsof the positive discount factor respectively under rewardstrategy NRmax and NRmin are the upper and the lowerbounds of the negative discount factor respectively underreward strategy PPmax and PPmin are the upper and the lower

N M

PN

PM

PN PM

Figure 8 Probabilistic evasion model

bounds of the positive discount factor respectively underpunishment strategy NPmax and NPmin are the upper andthe lower bounds of the negative discount factor respectivelyunder punishment strategy

For new nodes positive and negative histories are keptwith a discount factor equal to 1 when the number ofobservations is less than a given value named experiencethreshold

From the above upper and lower bounds we define thefollowing two distance metrics

(i) Punish-to-Reward (PTR) Distance It is defined byPRmin minus PPmax and it shows to what extent the nodeis rewarded by the IDS when it transits from themisbehaved state to the well-behaved state that is thehigher the PTR is the slower the positive histories areforgotten

(ii) Reward-to-Punish (RTP) Distance It is defined byNPmin minusNRmax and it shows to what extent the nodeis punished by the IDS when it transits from the well-behaved state to the misbehaved state that is thehigher the RTP is the slower the negative histories areforgotten

33 Performance of Adaptive Discount Factor Strategy Weevaluate the performance of the constant and adaptive dis-count factor strategies in terms of detection time To do sowe implement three behavioral models

(i) Deterministic redemption model in this model anode with reputation 119877 = 0 behaves correctly in thenetwork

(ii) Deterministic evasion model in this model a nodewith reputation 119877 = 1 behaves maliciously in thenetwork

(iii) Probabilistic evasion model the nodersquos behavior ismodeled with a two-state Markov chain as depictedin Figure 8 In state 119873 the node is well-behavedand in state 119872 the node is misbehaved Initially thenodersquos reputation 119877 = 1 The node transits towardsstate119873 with probability 119875

119873and towards state119872 with

probability 119875119872 such that 119875

119873+ 119875119872

= 1 119875119872

is calledthe evasion probabilityThe time spent in state119873 andstate 119872 is the monitoring time period

The parameters for the experiment are shown in Table 2We define three settings for the adaptive fading reputation

(i) Setting 1 PTR and RTP are high for example theyequal 07


0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n

Time (number of observations)

120596 = 02

120596 = 05

120596 = 08

(a) Constant discount factor strategy

0

02

04

06

08

1

12

Repu

tatio

n

0 1 2 3 4 5 6 7 8 9Time (number of observations)

Setting 1Setting 2Setting 3

(b) Adaptive discount factor strategy

Figure 9 Deterministic redemption model

Table 2 Experiment parameters

Parameter Setting 1 Setting 2 Setting 3NPmax PRmax 1 1 1

PRmin NPmin 09 09 09

PPmax NRmax 02 06 08

NRmin PPmin 01 05 07

120596 02 05 08

119905ℎ 05

(ii) Setting 2 PTR and RTP are medium for examplethey equal 03

(iii) Setting 3 PTR and RTP are low for example theyequal 01

As for constant fading reputation we define three levelsof discount factor 120596 = 02 05 08

We study the evolution of reputation over time whenapplying constant and adaptive discount factor In Fig-ure 9(a) the convergence time increases as 120596 increases Thisis because higher (resp lower) values of 120596 mean that thenegative histories are forgotten at slower (resp faster) ratewhich leads to longer (resp shorter) time to converge to119877 = 1 In Figure 9(b) we observe that the deterministicredemption model under adaptive discount factor strategyrequires less converge time than the constant one It rangesbetween 3 and 9 observations under setting 1 and setting 3respectively The reason for this is that a node under setting1 is rewarded more generously as long as it is well-behavingthat is its positive histories are forgotten slower than those ofsetting 2 and setting 3

In Figure 10 we also notice that the malicious node thatfollows the deterministic evasion is detected more quicklywhen the adaptive discount factor strategy is applied The

time to converge to 119877 = 0 is between 3 and 9 observationsunder the adaptive discount factor strategy and between4 and 14 observations under the constant discount factorstrategy For instance let 119877 = 01 be the boundary betweenmalicious behavior and normal behavior the malicious nodecan evade IDS detection for a time required to collect only3 observations if the IDS adopts the adaptive discount factorstrategy under setting 3 Under the constant discount factorstrategy and if 120596 = 08 IDS can detect the malicious after atime period of 5 observations

By knowing the required number of observations todetect a malicious node the latter can adopt the probabilisticevasion model which do discontinuous harm to the networkto confuse the IDS and hence evade detection Figures 1112 and 13 show that the adaptive discount factor strategycan quickly detect this type of behavior In the figures weconsider that a node is malicious when 119877 = 01 When theevasion probability 119875

119872= 05 the adaptive strategy succeeds

at detecting the malicious node after a time between 2 and37 observations On the other hand the malicious node canevade the IDS adopting the constant strategy for a time of751 observations when 120596 = 08 This value decreases to 10and 2 when 120596 = 05 and 120596 = 02 respectively When119875119872

= 06 the detection time decreases to 40 and 27 under120596 = 08 and setting 3 respectively When 119875

119872is between

07 and 09 the adaptive strategy (resp constant strategy)achieves a detection time between 2 and 4 (resp between 2and 5) observations

4 Spatiotemporal Dynamicity

Amonitoring node 119894 can make at least one observation abouta monitored node 119895 if the wireless link lasts for a durationhigher than the monitoring period Δ The malicious node 119895


0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n

Time (number of observations)0 1 2 3 4 5 6 7 8 9



Figure 10 Deterministic evasion model

0

02

04

06

08

1

Repu

tatio

n

Time (number of observations)1 10 100 1000

120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n

Time (number of observations)0 5 10 15 20 25 30 35 40



Figure 11 Probabilistic evasion model (119875119872

= 05)

which knows this fact can move around in the network tocreate links with its neighbors of duration less than Δ

As shown in Figure 14 the nodes start operating at time 1199050

Awireless link between themonitoring node 119894 andmonitorednode 119895 is created at time 119905

1when node 119895 comes within the

transmission range of node 119894 Node 119894 loses its link with node119895 either (1)when node 119895moves out of the transmission rangeof node 119894 at time 119905

2or (2) when node 119895 runs out of its battery

power at time 1199053 Therefore node 119894 estimates the link-node

lifetime by the following equation min(1199052minus1199051 1199053minus1199051) (1199052minus1199051)

is the estimation of the link lifetime and (1199053minus1199051) is the residual

node lifetime after node 119895 has been in existence for (1199051minus 1199050)

time unitsIn this section we statistically analyze the link-node

distribution Based on this analysis we choose appropri-ate values for the monitoring period so that the mobilemonitored node cannot evade IDS detection We use therandomwaypointmobilitymodel inwhich eachmobile noderandomly selects a location within an area of 100m times 100mwith a random speed uniformly distributed between 0 and acertain maximum speed 119881max then it stays stationary duringa pause time of 1 second before moving to a new random


0 5 10 15 20 25 30 35 40Time (number of observations)

0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)

location In our analysis we consider two different numbersof nodes (NN) that is 10 and 20 nodes

41 Link Lifetime Distribution We obtain from our simu-lation the frequency of link durations and plot them intoa histogram as shown in Figures 15 and 16 The EasyFitsoftware [21 22] is used to measure the compatibility of arandom sample with the theoretical probability distributionfunctions As shown in the figures the software approximatesthe simulation data to a Weibull distribution [23] with twoparameters 120572 = 1031 and 120573 = 2874 (resp 120572 = 1029 and120573 = 3285) when 119881max = 20 and NN = 10 (resp NN = 20)

Weibull distribution has a PDF as shown in the followingequation

119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)

Based on the properties of the Weibull distribution themean (expected value) is

Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime

Residual node lifetime

t0

t1 t2

t3

t0 t3

t0 t1 t3

Figure 14 Link-node lifetime

016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e

Distribution of link durations

Figure 15 Link lifetime distribution under NN = 10 and119881max = 20

Table 3 Comparison between theoretical and approximative 120573

Number ofnodes (NN)

Node velocity(ms) Approximative 120573 Theoretical 120573

10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386

On the other hand Samar and Wicker [24 25] havedescribed the expected link lifetime as a function of nodevelocity say V

1 with the following equation

119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601

119886 minus radic1198862 minus V21sin2120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

PDF

of li

nk li

fetim

e

HistogramWeibull



where 119877 is the radius of the circle centered at the nodeV1is uniformly distributed between 119886 and 119887 expressed in

meterssecond 120601 is the direction of motion 1206010

= 120587 minus

sinminus1(119886V1)

Since (12) and (13) are both describing the expected valueof the link lifetime we can write

120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)

We derive then 120573 as a function of velocity V1as follows

120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)

Simulations have been conducted to compare betweenthe theoretical 120573 obtained from (15) and the Weibull approx-imative one obtained from simulations as shown in Table 3The results show that the Weibull distribution fits wellsimulation data

42 Residual Node Lifetime Distribution We assume thatthe node lifetime follows an exponential distribution with a


parameter 120582 This distribution is similar to the one used tomodel ldquotime to failurerdquo in reliability engineeringWe considerthat 120582 is the rate at which nodersquos battery is discharged Theprobability density function is then

119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)

The probability density function of the residual nodelifetime for a node of age 119886 is given by the following equation[26]

119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)

where 119865 is the cumulative density function (CDF) of theexponential distributionThus the residual node lifetime alsofollows an exponential distribution The expected value forthe random variable 119883 following an exponential distributionis

E (119883) =1

120582 (18)

43 Link-Node Lifetime Distribution Consider a randomvariable 119885 where 119885 = min(119883 119884) 119883 (resp 119884) is arandom variable related to link lifetime (resp residual nodelifetime) following a Weibull distribution (resp exponentialdistribution) with a joint cumulative distribution function119868119883119884

(119909 119910) Then since 119883 and 119884 are independent we have

119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore

119875 (119885 gt 119905) = 1 minus 119875 (119883 le 119905) minus 119875 (119884 le 119905) + 119875 (119883 le 119905 119884 le 119905)

(20)

Consequently the cumulative distribution function(CDF) of 119885 is

119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)

The approximated density function for the combinedvariables 119883 and 119884 is a Phased Bi-Weibull distribution [27]which has a PDF as shown in

119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)

EasyFit software [22] approximates the simulation datato the Phased Bi-Weibull distribution as shown in Figure 17(resp Figure 18) with parameters 120572

1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)

Distribution of link-node durations

032

028

024

02

016

012

008

004

0

0 10 20 30 40 50 60 70 80 90 100 110 120 130

PDF

of li

nk-n

ode l

ifetim

e

Link-node lifetime (s)

HistogramPhased Bi-Weibull

Figure 17 Link-node lifetime distribution under NN = 10 and119881max = 20


032

036

028

024

02

016

012

008

004

0

0 10 20 30 40 50 60 70 80 90 100 110 120

PDF

of li

nk-n

ode l

ifetim

e




Remark 2 (see [28]) For real values 119909 119910 isin R min(119909 119910) =

119909 + 119910 minus max(119909 119910)

The result of this remark is extended to random variablesby the following theorem

Theorem 3 (see [28]) Given two real-valued continuousrandom variables X Y isin Ω rarr R then the expected value ofthe minimum of the two variables is E(min(119883 119884)) = E(119883) +

E(119884) minus E(max(119883 119884))

Lemma 4 (see [28]) Given two real-valued continuous ran-dom variables X Y isin Ω rarr R then the expected valueof the maximum of the two variables is E(max(119883 119884)) =

intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910

Based on Theorem 3 and Lemma 4 the expected link-node lifetime is given by

E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20

Figure 19 Expected link-node lifetime

where E(119883) is given in (12) and E(119884) in (18) Figure 19shows that the expected link-node lifetime resulted fromsimulation as a function of node velocity The results showthat the expected link-node lifetime decreases rapidly as itsvelocity is increased and it shows a significant decrease when119881max isin [1 5]The results also show that under higher networkdensity the expected link-node lifetime becomes longer Thereason for this is that a node in this case shares links withlarger number of neighbors and consequently links withlonger durations will be observed

44 Monitoring Period Estimation Based on the above statis-tical analysis we propose a method to choose the appropriatevalue for the monitoring period This method is low-costand more appropriate for resource-constrained networkslike sensor networks We also propose another method thatrequires some communication cost and can be implementedon nodes with higher capabilities such as mobile sinks ormobile ad hoc networks and vehicular ad hoc networks

441 Low-Cost Method We assume that the monitoringnode has no information about themonitored nodersquos velocityposition or residual battery and it wants to ensure that 119897 ofits links are observable that is they exist for a duration gt

Δ As the link-node lifetime follows a Phased Bi-Weibulldistribution the minimum value of Δ which ensures thisrequirement is 119905 such that 119875(119885 le 119905) = 119897100

442 High-Cost Method We assume that each node 119894 canestimate its remaining battery power 119864

119894and its rate of energy

dissipation EDisip119894for every time periodΔ an ultraconserva-

tive estimate of the residual node lifetime is derived as shownin the following equation

120599119894=

119864119894

max (EDisip119894)(119904) (25)

Each node 119894 periodically broadcasts a beacon messagecontaining its residual node lifetime 120599

119894and its position

obtained from GPS Upon receiving such a message fromnode 119894 node 119895 first calculates 119889

119894119895 that is the distance

separating it from its neighbor 119894 The relative velocity of node119894with respect to node 119895 isradicV2

119894+ V2119895minus 2V119894V119895cos 120579 where V

119894and

V119895are node 119894rsquos and node 119895rsquos velocity respectively 120579 denotes the

angle between vectors 997888rarrV119894and 997888rarrV119895in the Cartesian coordinate

system The relative velocity is maximum when V119894

= V119895

=

119881max and 120579 = 180∘ and it equals then to 2119881max Node 119895 then

calculates a conservative estimate of the residual link lifetimethat is the minimum time for node 119894 to move out of thetransmission range of node 119895 The residual link lifetime 120585

119894119895 is

given by the following equation where TR is the transmissionrange

120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)

After that each node 119895 estimates the residual link-nodelifetime given by

120594119894119895

= min (120599119894 120585119894119895) (27)

Therefore the monitoring period required to observe themonitored node 119894 must be less than 120594

119894119895

5 Conclusion

In this paper we have proposed IDS solutions for threeaspects of dynamicity in ad hoc andwireless sensor networksThe magnitude dynamicity aspect is solved by defining anormal profile based on the invariants derived from thenormal node behavior We have generated a dependencygraph consisting of strongly correlated features and we havederived the high-level features from the graphThe high-levelfeatures are obtained by applying the divide-and-conquerstrategy on themaximal cliques algorithm and themaximumweighted spanning tree algorithm Simulation results showthat the IDS can achieve a detection rate of 100 whenthe malicious behavior is not similar to the normal oneIn addition it can also achieve a false positive rate of 0when the duration of the training time exceeds a givenvalue To handle nature dynamicity aspect we have adoptedthe carrot and stick strategy to prevent a malicious nodefrom evading the IDS To do so we have proposed anadaptive reputation fading strategy to allow fast redemptionand fast capture of malicious node We have analyticallystudied link-node lifetime distribution and have shown thatit can be approximated to the Phased Bi-Weibull distributionBased on this analysis we have proposed a low-cost methodto estimate the minimum monitoring period required toobserve the monitored nodersquos behavior In addition based onsome topology information we have proposed a high-costmethod designed for network having nodes less constrainedwith resource limitations


Conflict of Interests

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

Acknowledgment

The authors would like to extend their sincere appreciation tothe Deanship of Scientific Research at King Saud Universityfor funding this research through Research Group Project(RG no 1435-051)

References

[1] C E PerkinsAd hoc Networking Addison-Wesley ProfessionalReading Mass USA 2008

[2] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[3] S Al-Sultan M M Al-Doori A H Al-Bayatti and H ZedanldquoA comprehensive survey on vehicular Ad Hoc networkrdquoJournal of Network and Computer Applications vol 37 no 1 pp380ndash392 2014

[4] D Djenouri L Khelladi and N Badache ldquoA survey of securityissues in mobile ad hoc and sensor networksrdquo IEEE Communi-cations Surveys and Tutorials vol 7 no 4 pp 2ndash28 2005

[5] S Gillani F Shahzad A Qayyum and R Mehmood ldquoA surveyon security in vehicular ad hoc networksrdquo in CommunicationTechnologies for Vehicles pp 59ndash74 Springer New York NYUSA 2013

[6] P Garcıa-Teodoroa J Dıaz-Verdejoa G Macia-Fernandezaand E Vazquezb ldquoAnomaly-based network intrusion detectiontechniques systems and challengesrdquo Computers amp Security vol28 no 1-2 pp 18ndash28 2009

[7] M Xie S Han B Tian and S Parvin ldquoAnomaly detectionin wireless sensor networks a surveyrdquo Journal of Network andComputer Applications vol 34 no 4 pp 1302ndash1325 2011

[8] B Sun L Osborne Y Xiao and S Guizani ldquoIntrusion detectiontechniques in mobile ad hoc and wireless sensor networksrdquoIEEE Wireless Communications vol 14 no 5 pp 56ndash63 2007

[9] G Li J He and Y Fu ldquoGroup-based intrusion detection systemin wireless sensor networksrdquo Computer Communications vol31 no 18 pp 4324ndash4332 2008

[10] Y Zhang N Meratnia and P Havinga ldquoOutlier detectiontechniques for wireless sensor networks a surveyrdquo IEEE Com-munications Surveys and Tutorials vol 12 no 2 pp 159ndash1702010

[11] C E Loo M Y Ng C Leckie and M Palaniswami ldquoIntrusiondetection for routing attacks in sensor networksrdquo InternationalJournal of Distributed Sensor Networks vol 2 no 4 pp 313ndash3322006

[12] F Liu X Cheng and D Chen ldquoInsider attacker detection inwireless sensor networksrdquo in Proceedings of the 26th IEEE Inter-national Conference on Computer Communications (INFOCOMrsquo07) pp 1937ndash1945 May 2007

[13] A Stetsko L Folkman and V Matyas ldquoNeighbor-based intru-sion detection for wireless sensor networksrdquo in Proceedingsof the 6th International Conference on Wireless and MobileCommunications (ICWMC rsquo10) pp 420ndash425 IEEE September2010

[14] S Dowdy S Wearden and D Chilko Statistics for ResearchJohn Wiley amp Sons New York NY USA 3rd edition 2004

[15] E Tomita A Tanaka and H Takahashi ldquoThe worst-case timecomplexity for generating all maximal cliques and computa-tional experimentsrdquoTheoretical Computer Science vol 363 no1 pp 28ndash42 2006

[16] P Sriram and S Skiena ldquoComputational discrete mathematicscombinatorics and graph theory withmathematicardquoComputingReviews vol 45 no 12 p 775 2004

[17] X Zeng R Bagrodia and M Gerla ldquoGloMoSim a libraryfor parallel simulation of large-scale wireless networksrdquo inProceedings of the 12th Workshop on Parallel and DistributedSimulation (PADS rsquo98) pp 154ndash161 May 1998

[18] J Liu and V Issarny ldquoEnhanced reputation mechanism formobile ad hoc networksrdquo in Proceedings of 2nd InternationalConference on Trust Management pp 48ndash62 Springer NewYork NY USA 2004

[19] S Buchegger and J-Y L Boudec ldquoA robust reputation systemfor peer-to-peer and mobile ad-hoc networksrdquo in Proceedingsof the 2nd Workshop on the Economics of Peer-to-Peer Systems(P2PEcon rsquo04) Cambridge Mass USA 2004

[20] P Michiardi and R Molva ldquoCore a collaborative reputationmechanism to enforce node cooperation in mobile ad hoc net-worksrdquo in Advanced Communications and Multimedia Securitypp 107ndash121 Springer New York NY USA 2002

[21] ldquoMathwave data analysis amp simulationrdquo httpwwwmathwavecomproductseasyfithtml

[22] K Schittkowski ldquoEASY-FIT a software system for data fitting indynamical systemsrdquo Structural and Multidisciplinary Optimiza-tion vol 23 no 2 pp 153ndash169 2002

[23] C Forbes M Evans N Hastings and B Peacock StatisticalDistributions John Wiley amp Sons 2011

[24] P Samar and S B Wicker ldquoOn the behavior of communicationlinks of a node in amulti-hopmobile environmentrdquo in Proceed-ings of the 5th ACM International Symposium onMobile Ad HocNetworking and Computing (MoBiHoc rsquo04) pp 145ndash156 ACMMay 2004

[25] P Samar and S B Wicker ldquoLink dynamics and protocol designin a multihop mobile environmentrdquo IEEE Transactions onMobile Computing vol 5 no 9 pp 1156ndash1172 2006

[26] MGerharz C deWaalM Frank and PMartini ldquoLink stabilityin mobile wireless ad hoc networksrdquo in Proceedingsof the 27thAnnual IEEE Conference on Local Computer Networks (LCNrsquo02) pp 30ndash39 IEEE 2002

[27] F Louzada-Neto andA C Davison A note on bayesian analysisof the poly-weibull model 1998

[28] G Lewellen Expected maximum and minimum of real-valuedcontinuous random variables 2013 httpsantimatroidword-presscom201301

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



90

90

9090

90 180360

720

90 180 270

L

K

A

J

C

DF

E

B

G H I

(a)

90

90

9090

90 100270

550

90 120 210

L

K

A

J

C

DF

E

B

G H I

(b)

L

K

A

J

C

DF

E

B

90

90

9090

90 270450

900

GH I

90 270 360

(c)

Figure 1 Impact of feature choice on false positive rate

monitored node is normal or anomalous after one ormultipleconsecutive observations

Although intrusion detection systems have received con-siderable attention in ad hoc and wireless sensor networks[7 8] to the best of our knowledge there are no studies onthe impact of network dynamicity on IDS efficiency and howthe IDS can react or adapt to these changes

In this paper we investigate the following three aspectsof behavioral dynamicity that occur in the network and cannegatively affect the IDS performance and efficiency

(i) Magnitude Dynamicity Due to change of user re-quirements a node changes the rate at which itgenerates data For instance a legitimate user wantsto change (ie increasedecrease) the data collectionrate received at the sink node The challenge facingthe IDS here is to be efficient at detecting attacks anddistinguishing between the changes due to normalbehaviors and the changes due to malicious attacks

(ii) Nature Dynamicity In some detection models amonitoring node has to observe the behavior of themonitored node during a set of consecutive moni-toring periods before judging whether the monitorednode is malicious or not A monitored node mightevade from IDS detection and confuses it by switch-ing continuously its behavior between normal andanomalous In this case the malicious node strives tocause network disruption without being detected bythe IDS

(iii) Spatiotemporal Dynamicity The IDS detection mech-anism is based on collecting a set of consecutiveobservations about the monitored node An IDSis able to observe the behavior of the monitorednode if the latter stays within the monitoring nodersquostransmission range for a duration exceeding Δ Byknowing this fact a malicious node can evade IDSdetection bymoving around in the network at a speedwhich prevents it from being within the monitoringnodersquos transmission range for a duration higher thanΔ

In this paper we propose a solution for each aspect ofdynamicity mentioned above The contributions of the paperare threefold Firstly the magnitude dynamicity aspect issolved by defining a normal profile based on the invariantsderived from the normal node behavior This is achieved

by generating a dependency graph consisting of stronglycorrelated features and then derives the high-level featuresfrom the graph The high-level features are obtained byapplying the divide-and-conquer strategy on the maximalcliques algorithm and the maximum weighted spanning treealgorithm Secondly to handle nature dynamicity aspect weadopt the carrot and stick strategy (ie reward generouslyand punish severely) to prevent a malicious node fromevading the IDS To do so we propose an adaptive reputationfading strategy to allow fast redemption and fast capture ofmalicious nodeThirdly we use statistical analysis to estimatethe link duration between two nodes in dynamic networktopology Based on this estimation the monitoring nodechooses the appropriate monitoring period which allows itto observe the monitored nodersquos behavior

The rest of the paper is organized as follows In Section 2we describe the normal profile construction and the featureselection method Section 3 presents the adaptive reputationfading strategy In Section 4 we analyze link-node durationin amobile wireless network and explain how themonitoringtime period is estimated Finally Section 5 concludes thepaper

2 Magnitude Dynamicity

21 Background

211 One-Feature Profile In the one-feature profile we use asingle feature to describe and detect anomalous behavior Todetect the network malicious behavior a node can measurethe following features as shown in Table 1 [9] The disadvan-tage of this profile structure is that there is a need to assignone feature for each known attack In this case the IDS hasto measure each feature and check whether it has anomalousvalue When the number of attacks increases the detectionspeed of the IDS becomes slow It also becomes slower whenthe size of rule set increases

The one-feature profile might fail at distinguishingbetween normal and anomalous behaviors Figure 1 showsthat using some features individually to describe normalbehavior is misleading and might make the detection systemfalsely accuse a legitimate node of beingmalicious Figure 1(a)depicts a tree-based wireless network rooted at the sink 119861

and it shows the normal traffic rates of the network Thevalue above each link indicates the flow rate traversing thislink Each node measures the flow rate coming from its











2take


1le 7 and

1198911199052



1




= (1198911199051198941 119891119905

119894119902) If a


(1198911 119891


119894





119899

119894=1119891119894and M = (1(119899 minus

1))sum119899

119894=1(119891119894minus 120583)(119891



119894minus 120583)119879Mminus1(119891

119894minus 120583) The





120588 (119883 119884) =COV (119883 119884)

120590119883120590119884

=119864 [(119909 minus 120583

119883) (119910 minus 120583

119884)]

120590119883120590119884

(1)

where 120583119883(resp 120583

119884) and 120590

119883(resp 120590










that 119894 = 1 119899 and each 119891119894= (119891119905

1198941 119891119905

119894119889) From 119865 we



119894119895le +1

Ω = (

Ω11

Ω12


Ω21

Ω22


d

Ω1198891

Ω1198892


) (2)


119889 such

that 119865119894

= (

1198911199051119894

119891119905119899119894


compute Ω119894119895




|119881| = 119889(ii) 119864 = (V

119894 V119895) where Ω

119894119895= 0 and |119864| = 119898

(iii) 119908(V119894 V119895) = 119908119894119895

= Ω119894119895


= (119881[Th]

119864[Th]

119908[Th]


119894 V119895) whose 119908


(i) 119864[Th]

= (V119894 V119895) where 119908

119894119895ge Th

(ii) 119881[Th]


|119881[Th]| le119889

(iii) 119908[Th]119894119895

= 119908119894119895








=

(119881[Th]119894

119864[Th]119894


119881[Th]119894







144119889= 144

(120572+2120573+sum119895119878119895gt2119878119895)

= 1441205721442120573

prod

119895119878119895gt2

144119878119895 (3)



119895119878119895gt2




and |120601| = 1198891015840 For each edge (119865



119905= 119865119897119865119896


119894isin 119865


1198941198891015840)

such that 119892119905119894119905

= 119891119905119894119897119891119905119894119896and 119891119905

119894119896= 0 If 119891119905

119894119896= 0 the



119897and 119865



119897= 119886119865119896+ 119887 where 119886


CL[Th]119894









1 1198652 1198653 1198654 1198655




(3) if (119863119894119904(119885119897 119862119897) notin [119871119900119908

119897 119880119901119897]) then



097

093

099

098

098

095

094F1

F2

F3

F4 F5

F6

F7

(a) Normal profile

097

099

098

098F1

F2

F3

F4 F5

F6

F7



1198656 and 119865


these features are

Ω =

1198651

1198652

1198653

1198654

1198655

1198656

1198657

1198651

1198652

1198653

1198654

1198655

1198656

1198657

(((

(

1 093 097 025 073 082 098

093 1 099 081 054 062 094

097 099 1 073 087 043 095

025 081 073 1 098 052 071

073 054 087 098 1 078 060

082 062 043 052 078 1 053

098 094 095 071 060 053 1

)))

)

(4)



1 1198652 1198653 1198657 and 119865

4 1198655




(1198652 1198657) and (119865


profile becomes (11986511198653 11986521198653 11986511198657) (11986541198655)





119894(119894 ge 1) As a

result sum119870119894=1

|119881[Th]119894


119894is ℎ119894

= |119881[Th]119894


119894=1(ℎ119894+ 1) le

119889 sum119870119894=1

ℎ119894




119897of size 119898



119897



120585119897




1

09289

09727

09289

09727

09828

RCVFWD

LOSS

SENT

1

09727

09828

RCVFWD

LOSS

SENT


60

65

70

75

80

85

90

95

100

0 01 02 03 04 05 06 07 08 09 1

Det

ectio

n ra

te (

)


T = 3T = 5T = 10T = 20

T = 30T = 40T = 50




Ω =


(

1 04205 04205 07263 06032

04205 1 1 09289 09727

04205 1 1 09289 09727

07263 09289 09289 1 09828

06032 09727 09727 09828 1

)

(5)


60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50

Det

ectio

n ra

te (

)

Training period

P = 1P = 05P = 01

P = 005P = 001


0

05

1

15

2

25

3

35

4

5 10 15 20 25 30 35 40 45 50

False

pos

itive

rate

()

Training period

P = 08P = 05P = 02

P = 005P = 003P = 001







3 Nature Dynamicity





119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)


119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)






120572 = 120596120572 + 119901

120573 = 120596120573 + 119899


(8)






R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

PDF

of li

nk li

fetim

e

HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


032

028

024

02

016

012

008

004

0

0 10 20 30 40 50 60 70 80 90 100 110 120 130

PDF

of li

nk-n

ode l

ifetim

e





032

036

028

024

02

016

012

008

004

0

0 10 20 30 40 50 60 70 80 90 100 110 120

PDF

of li

nk-n

ode l

ifetim

e





119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















2take


1le 7 and

1198911199052



1




= (1198911199051198941 119891119905

119894119902) If a


(1198911 119891


119894





119899

119894=1119891119894and M = (1(119899 minus

1))sum119899

119894=1(119891119894minus 120583)(119891



119894minus 120583)119879Mminus1(119891

119894minus 120583) The





120588 (119883 119884) =COV (119883 119884)

120590119883120590119884

=119864 [(119909 minus 120583

119883) (119910 minus 120583

119884)]

120590119883120590119884

(1)

where 120583119883(resp 120583

119884) and 120590

119883(resp 120590










that 119894 = 1 119899 and each 119891119894= (119891119905

1198941 119891119905

119894119889) From 119865 we



119894119895le +1

Ω = (

Ω11

Ω12


Ω21

Ω22


d

Ω1198891

Ω1198892


) (2)


119889 such

that 119865119894

= (

1198911199051119894

119891119905119899119894


compute Ω119894119895




|119881| = 119889(ii) 119864 = (V

119894 V119895) where Ω

119894119895= 0 and |119864| = 119898

(iii) 119908(V119894 V119895) = 119908119894119895

= Ω119894119895


= (119881[Th]

119864[Th]

119908[Th]


119894 V119895) whose 119908


(i) 119864[Th]

= (V119894 V119895) where 119908

119894119895ge Th

(ii) 119881[Th]


|119881[Th]| le119889

(iii) 119908[Th]119894119895

= 119908119894119895








=

(119881[Th]119894

119864[Th]119894


119881[Th]119894







144119889= 144

(120572+2120573+sum119895119878119895gt2119878119895)

= 1441205721442120573

prod

119895119878119895gt2

144119878119895 (3)



119895119878119895gt2




and |120601| = 1198891015840 For each edge (119865



119905= 119865119897119865119896


119894isin 119865


1198941198891015840)

such that 119892119905119894119905

= 119891119905119894119897119891119905119894119896and 119891119905

119894119896= 0 If 119891119905

119894119896= 0 the



119897and 119865



119897= 119886119865119896+ 119887 where 119886


CL[Th]119894









1 1198652 1198653 1198654 1198655




(3) if (119863119894119904(119885119897 119862119897) notin [119871119900119908

119897 119880119901119897]) then



097

093

099

098

098

095

094F1

F2

F3

F4 F5

F6

F7

(a) Normal profile

097

099

098

098F1

F2

F3

F4 F5

F6

F7



1198656 and 119865


these features are

Ω =

1198651

1198652

1198653

1198654

1198655

1198656

1198657

1198651

1198652

1198653

1198654

1198655

1198656

1198657

(((

(

1 093 097 025 073 082 098

093 1 099 081 054 062 094

097 099 1 073 087 043 095

025 081 073 1 098 052 071

073 054 087 098 1 078 060

082 062 043 052 078 1 053

098 094 095 071 060 053 1

)))

)

(4)



1 1198652 1198653 1198657 and 119865

4 1198655




(1198652 1198657) and (119865


profile becomes (11986511198653 11986521198653 11986511198657) (11986541198655)





119894(119894 ge 1) As a

result sum119870119894=1

|119881[Th]119894


119894is ℎ119894

= |119881[Th]119894


119894=1(ℎ119894+ 1) le

119889 sum119870119894=1

ℎ119894




119897of size 119898



119897



120585119897




1

09289

09727

09289

09727

09828

RCVFWD

LOSS

SENT

1

09727

09828

RCVFWD

LOSS

SENT


60

65

70

75

80

85

90

95

100

0 01 02 03 04 05 06 07 08 09 1

Det

ectio

n ra

te (

)


T = 3T = 5T = 10T = 20

T = 30T = 40T = 50




Ω =


(

1 04205 04205 07263 06032

04205 1 1 09289 09727

04205 1 1 09289 09727

07263 09289 09289 1 09828

06032 09727 09727 09828 1

)

(5)


60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50

Det

ectio

n ra

te (

)

Training period

P = 1P = 05P = 01

P = 005P = 001


0

05

1

15

2

25

3

35

4

5 10 15 20 25 30 35 40 45 50

False

pos

itive

rate

()

Training period

P = 08P = 05P = 02

P = 005P = 003P = 001







3 Nature Dynamicity





119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)


119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)






120572 = 120596120572 + 119901

120573 = 120596120573 + 119899


(8)






R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

PDF

of li

nk li

fetim

e

HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


032

028

024

02

016

012

008

004

0

0 10 20 30 40 50 60 70 80 90 100 110 120 130

PDF

of li

nk-n

ode l

ifetim

e





032

036

028

024

02

016

012

008

004

0

0 10 20 30 40 50 60 70 80 90 100 110 120

PDF

of li

nk-n

ode l

ifetim

e





119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














that 119894 = 1 119899 and each 119891119894= (119891119905

1198941 119891119905

119894119889) From 119865 we



119894119895le +1

Ω = (

Ω11

Ω12


Ω21

Ω22


d

Ω1198891

Ω1198892


) (2)


119889 such

that 119865119894

= (

1198911199051119894

119891119905119899119894


compute Ω119894119895




|119881| = 119889(ii) 119864 = (V

119894 V119895) where Ω

119894119895= 0 and |119864| = 119898

(iii) 119908(V119894 V119895) = 119908119894119895

= Ω119894119895


= (119881[Th]

119864[Th]

119908[Th]


119894 V119895) whose 119908


(i) 119864[Th]

= (V119894 V119895) where 119908

119894119895ge Th

(ii) 119881[Th]


|119881[Th]| le119889

(iii) 119908[Th]119894119895

= 119908119894119895








=

(119881[Th]119894

119864[Th]119894


119881[Th]119894







144119889= 144

(120572+2120573+sum119895119878119895gt2119878119895)

= 1441205721442120573

prod

119895119878119895gt2

144119878119895 (3)



119895119878119895gt2




and |120601| = 1198891015840 For each edge (119865



119905= 119865119897119865119896


119894isin 119865


1198941198891015840)

such that 119892119905119894119905

= 119891119905119894119897119891119905119894119896and 119891119905

119894119896= 0 If 119891119905

119894119896= 0 the



119897and 119865



119897= 119886119865119896+ 119887 where 119886


CL[Th]119894









1 1198652 1198653 1198654 1198655




(3) if (119863119894119904(119885119897 119862119897) notin [119871119900119908

119897 119880119901119897]) then



097

093

099

098

098

095

094F1

F2

F3

F4 F5

F6

F7

(a) Normal profile

097

099

098

098F1

F2

F3

F4 F5

F6

F7



1198656 and 119865


these features are

Ω =

1198651

1198652

1198653

1198654

1198655

1198656

1198657

1198651

1198652

1198653

1198654

1198655

1198656

1198657

(((

(

1 093 097 025 073 082 098

093 1 099 081 054 062 094

097 099 1 073 087 043 095

025 081 073 1 098 052 071

073 054 087 098 1 078 060

082 062 043 052 078 1 053

098 094 095 071 060 053 1

)))

)

(4)



1 1198652 1198653 1198657 and 119865

4 1198655




(1198652 1198657) and (119865


profile becomes (11986511198653 11986521198653 11986511198657) (11986541198655)





119894(119894 ge 1) As a

result sum119870119894=1

|119881[Th]119894


119894is ℎ119894

= |119881[Th]119894


119894=1(ℎ119894+ 1) le

119889 sum119870119894=1

ℎ119894




119897of size 119898



119897



120585119897




1

09289

09727

09289

09727

09828

RCVFWD

LOSS

SENT

1

09727

09828

RCVFWD

LOSS

SENT


60

65

70

75

80

85

90

95

100

0 01 02 03 04 05 06 07 08 09 1

Det

ectio

n ra

te (

)


T = 3T = 5T = 10T = 20

T = 30T = 40T = 50




Ω =


(

1 04205 04205 07263 06032

04205 1 1 09289 09727

04205 1 1 09289 09727

07263 09289 09289 1 09828

06032 09727 09727 09828 1

)

(5)


60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50

Det

ectio

n ra

te (

)

Training period

P = 1P = 05P = 01

P = 005P = 001


0

05

1

15

2

25

3

35

4

5 10 15 20 25 30 35 40 45 50

False

pos

itive

rate

()

Training period

P = 08P = 05P = 02

P = 005P = 003P = 001







3 Nature Dynamicity





119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)


119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)






120572 = 120596120572 + 119901

120573 = 120596120573 + 119899


(8)






R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

PDF

of li

nk li

fetim

e

HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


032

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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(3) if (119863119894119904(119885119897 119862119897) notin [119871119900119908

119897 119880119901119897]) then



097

093

099

098

098

095

094F1

F2

F3

F4 F5

F6

F7

(a) Normal profile

097

099

098

098F1

F2

F3

F4 F5

F6

F7



1198656 and 119865


these features are

Ω =

1198651

1198652

1198653

1198654

1198655

1198656

1198657

1198651

1198652

1198653

1198654

1198655

1198656

1198657

(((

(

1 093 097 025 073 082 098

093 1 099 081 054 062 094

097 099 1 073 087 043 095

025 081 073 1 098 052 071

073 054 087 098 1 078 060

082 062 043 052 078 1 053

098 094 095 071 060 053 1

)))

)

(4)



1 1198652 1198653 1198657 and 119865

4 1198655




(1198652 1198657) and (119865


profile becomes (11986511198653 11986521198653 11986511198657) (11986541198655)





119894(119894 ge 1) As a

result sum119870119894=1

|119881[Th]119894


119894is ℎ119894

= |119881[Th]119894


119894=1(ℎ119894+ 1) le

119889 sum119870119894=1

ℎ119894




119897of size 119898



119897



120585119897




1

09289

09727

09289

09727

09828

RCVFWD

LOSS

SENT

1

09727

09828

RCVFWD

LOSS

SENT


60

65

70

75

80

85

90

95

100

0 01 02 03 04 05 06 07 08 09 1

Det

ectio

n ra

te (

)


T = 3T = 5T = 10T = 20

T = 30T = 40T = 50




Ω =


(

1 04205 04205 07263 06032

04205 1 1 09289 09727

04205 1 1 09289 09727

07263 09289 09289 1 09828

06032 09727 09727 09828 1

)

(5)


60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50

Det

ectio

n ra

te (

)

Training period

P = 1P = 05P = 01

P = 005P = 001


0

05

1

15

2

25

3

35

4

5 10 15 20 25 30 35 40 45 50

False

pos

itive

rate

()

Training period

P = 08P = 05P = 02

P = 005P = 003P = 001







3 Nature Dynamicity





119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)


119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)






120572 = 120596120572 + 119901

120573 = 120596120573 + 119899


(8)






R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

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Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

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002

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

09289

09727

09289

09727

09828

RCVFWD

LOSS

SENT

1

09727

09828

RCVFWD

LOSS

SENT


60

65

70

75

80

85

90

95

100

0 01 02 03 04 05 06 07 08 09 1

Det

ectio

n ra

te (

)


T = 3T = 5T = 10T = 20

T = 30T = 40T = 50




Ω =


(

1 04205 04205 07263 06032

04205 1 1 09289 09727

04205 1 1 09289 09727

07263 09289 09289 1 09828

06032 09727 09727 09828 1

)

(5)


60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40 45 50

Det

ectio

n ra

te (

)

Training period

P = 1P = 05P = 01

P = 005P = 001


0

05

1

15

2

25

3

35

4

5 10 15 20 25 30 35 40 45 50

False

pos

itive

rate

()

Training period

P = 08P = 05P = 02

P = 005P = 003P = 001







3 Nature Dynamicity





119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)


119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)






120572 = 120596120572 + 119901

120573 = 120596120573 + 119899


(8)






R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

PDF

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












3 Nature Dynamicity





119861 (120572 120573) = int

1

0

119905120572minus1

(1 minus 119905)120573minus1

119889119905

119891 (119901 | 120572 120573) =1

119861 (120572 120573)119901120572minus1

(1 minus 119901)120573minus1

where 0 le 119901 le 1 120572 120573 ge 0

(6)


119877 = E (119861 (120572 120573)) =120572

120572 + 120573 (7)






120572 = 120596120572 + 119901

120573 = 120596120573 + 119899


(8)






R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

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fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

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012

01

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











R0 1

1


NPmaxNPmin

PPmaxPPmin

NR maxNR min

PR maxPR min

th





120572 = 120595 (119877) 120572 + 119901

120573 = 120593 (119877) 120573 + 119899

where 0 le 120595 (119877) 120593 (119877) le 1

(9)





120595 (119877) =

(PRmax minus PRmin

1 minus 119905)119877 +


1 minus 119905

when 119877 ge 119905

(PPmax minus PPmin

119905) 119877 + PPmin

when 119877 lt 119905



(10)


N M

PN

PM

PN PM













119873+ 119875119872

= 1 119875119872





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

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012

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008

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n










120596 02 05 08

119905ℎ 05











119872is between





0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

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008

006

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0

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08

1

12

0 2 4 6 8 10 12 14

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





0

02

04

06

08

1

Repu

tatio

n


120596 = 02

120596 = 05

120596 = 08


0

02

04

06

08

1

12

Repu

tatio

n





= 05)















0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

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fetim

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Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

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012

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

02

04

06

08

1

Repu

tatio

n

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

0 5 10 15 20 25 30

Repu

tatio

n




= 06)


0

02

04

06

08

1

Repu

tatio

n

0 1 2 3 4 5

120596 = 02

120596 = 05

120596 = 08



0

02

04

06

08

1

12

Repu

tatio

n

0 05 1 15 2 25 3 35 4




= 07 08 09)




119891 (119909 120572 120573) =120572

120573(

119909

120573)

120572minus1

119890minus(119909120573)

120572

(11)


Mean = 120573 times Γ (120572 + 1

120572) (12)


Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

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0

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


032

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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Time

Time

Time

Link lifetime


t0

t1 t2

t3

t0 t3

t0 t1 t3


016

014

012

01

008

006

004

002

0

0 20 40 60 80 100 120 140 160 180 200 220

Link lifetime (s)

HistogramWeibull

PDF

of li

nk li

fetim

e




Number ofnodes (NN)


10

20 2874 283615 3553 358310 5363 50175 8820 8855

20

20 3457 328515 4004 394410 5607 52295 8450 80386



119865V1

link =119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(13)

018

016

014

012

01

008

006

004

002

0

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Link lifetime (s)

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HistogramWeibull





= 120587 minus

sinminus1(119886V1)


120573Γ (120572 + 1

120572) =

119877

2 (119887 minus 119886)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(14)


120573 =119877

2 (119887 minus 119886) Γ ((120572 + 1) 120572)

times (int

0

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119887 + radic1198872 minus V21sin2120601

V1+ V1cos120601

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601

minusint

1206010

120587

log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119886 + radic1198862 minus V21sin2120601


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889120601)

(15)





119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


032

028

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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119891 (119905) = 0 if 119905 lt 0

120582119890minus120582119905

if 119905 ge 0(16)


119903119886(119905) =

119891 (119905 + 119886)

1 minus 119865 (119886)= 120582119890minus120582119905

(17)


E (119883) =1

120582 (18)



119875 (119885 gt 119905) = 119875 (min (119883 119884) gt 119905) = 119875 (119883 gt 119905 119884 gt 119905) (19)

Therefore


(20)


119867119885(119905) = 1 minus 119875 (119885 gt 119905)

= 119875 (119883 le 119905) + 119875 (119884 le 119905) minus 119875 (119883 le 119905 119884 le 119905)

(21)

Thus

119867119885(119905) = 119865

119883(119905) + 119866

119884(119905) minus 119868

119883119884(119905 119905) (22)


119892 (119905) =

1205721

1205731

(119905 minus 1205741

1205731

)

1205721minus1

119890minus((119905minus120574

1)1205731)1205721 if 120574

1le 119905 le 120574

2

1205722

1205732

(119905 minus 1205742

1205732

)

1205722minus1

119890minus((119905minus120574

2)1205731)1205722 if 120574

2lt 119905 lt infin

(23)


1= 087118 120573

1= 19482

1205741

= 0 1205722

= 068969 1205732

= 31875 and 1205742

= 3 (resp1205721= 090481 120573

1= 22976 120574

1= 0 120572

2= 071509 120573

2= 14819

and 1205742= 4)


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119909 + 119910 minus max(119909 119910)



E(119884) minus E(max(119883 119884))


intinfin

minusinfin119909119891119883(119909)119865119884(119909)119889119909 + int

infin

minusinfin119910119891119884(119910)119865119883(119910)119889119910


E (119885) = E (119883) + E (119884) minus E (max (119883 119884)) (24)


20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Expe

cted

link

-nod

e life

time (

s)

Node velocity (ms)

NN = 10NN = 20










120599119894=

119864119894

max (EDisip119894)(119904) (25)







119894and




= V119895

=



119894119895 is


120585119894119895

=

TR minus 119889119894119895

2119881max(119904) (26)


120594119894119895

= min (120599119894 120585119894119895) (27)


119894119895

5 Conclusion





Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Acknowledgment


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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