Research Article A Probabilistic Analysis of Path Duration Using …downloads.hindawi.com/archive/2014/495036.pdf · 2019-07-31 · Research Article A Probabilistic Analysis of Path

Research ArticleA Probabilistic Analysis of Path Duration Using RoutingProtocol in VANETs

Ram Shringar Rao1 Sanjay Kumar Soni2 Nanhay Singh1 and Omprakash Kaiwartya3

1 Ambedkar Institute of Advanced Communication Technologies amp Research Delhi 110031 India2Delhi Technological University Delhi 110042 India3 Jawaharlal Nehru University New Delhi 110067 India

Correspondence should be addressed to Ram Shringar Rao rsrao08yahooin

Received 28 February 2014 Revised 20 June 2014 Accepted 5 July 2014 Published 17 July 2014

Academic Editor Rakesh Mishra

Copyright copy 2014 Ram Shringar Rao 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

In recent years various routingmetrics such as throughput end-to-end delay packet delivery ratio path duration and so forth havebeen used to evaluate the performance of routing protocols in VANETs Among these routing metrics path duration is one of themost influential metrics Highly mobile vehicles cause frequent topology change in vehicular network environment that ultimatelyaffects the path duration In this paper we have derived a mathematical model to estimate path duration using border node-basedmost forward progress within radius (B-MFR) a position based routing protocol The mathematical model for estimation of pathduration consists of probability of finding next-hop node in forwarding region estimation of expected number of hops probabilitydistribution of velocity of nodes and link duration between each intermediate pair of nodes The analytical results for the pathduration estimation model have been obtained using MATLAB The model for path duration estimation has been simulated inNS2 Each of the analytical results has been verified through respective simulation results The result analysis clearly reveals thatpath duration increases with the increase in transmission range and node density and decreases with the increase in the number ofhops in the path and velocity of the nodes

1 Introduction

The intelligent transport system (ITS) has been working tomake the road safer and efficient to cope up with increasingnumber of on-road vehicles day by day The number ofaccidents on the roads is continuously increasing due to highgrowth in on-road vehicle population The increasing num-ber of accidents has become an issue of concern worldwideIt has made the roads vulnerable and threatening as everyyear millions of people are dying in accidents throughout theworld Amodern network concept VANETs has become thehope for providing safer andwell-organized transportation innear future [1]

In VANETs routing is the process of finding optimal pathfor information forwarding between source and destinationnode Position-based routing protocols have become oneof the most investigated choices among researches due togeographic region sharing of on-road vehicles [2ndash7] In

position-based routing forwarding of information is per-formed either through direct communication or throughintermediate nodes between source and destination nodeThe intermediate nodes are high speed moving vehicles inVANETs that act as router during forwarding of informationfrom source to destination Due to ad hoc network architec-ture no fixed infrastructure is considered during informationforwarding in VANETs [8] The mobility of nodes is one ofthe most critical factors in design analysis and performanceevaluation of information forwarding techniques inVANETsHigh mobility of nodes causes dynamic changes in thenetwork topology that ultimately results in infrequent pathfailure The communication between source and destinationnode is frequently interrupted due to path failure Oncea path failure occurs a new path has to be set up forfurther communication The frequent path failures degradethe performance of routing protocol and add overhead interms of establishing a new path [9]

Hindawi Publishing CorporationInternational Journal of Vehicular TechnologyVolume 2014 Article ID 495036 10 pageshttpdxdoiorg1011552014495036

2 International Journal of Vehicular Technology

In VANETs any routing path is made up of one or morelinks between pair of intermediate nodes Therefore lifetimeof a link is one of the most important contributors in pathduration Path duration can be defined as the duration oftime till every link of the route is active In VANETs thelifetime of a link is a random variable whose probabilitydistribution depends on mobility node density transmissionrange different traffic scenarios and various impairmentsof radio communications The links between intermediatenodes frequently break due to the high mobile nodes movingout of each otherrsquos transmission range Therefore estimationof path duration between source and destination decreasesthe chances of path breakage The effective use of knowledgeof path duration improves the performance and efficiency ofrouting protocols in VANETs [10]

In this paper our main contribution is in terms of prob-abilistic and mathematical analysis of path duration usingborder-node based most forward progress within radius (B-MFR) routing protocol For this analysis a mathematicalmodel to estimate path duration has been developed B-MFR is a position-based routing protocol which selects next-hop node from the nodes belonging to the border area ofthe transmission range The mathematical model for pathduration estimation consists of the following modules (1)The probability of finding at least one node in the consideredforwarding region has been mathematically derived (2)Average number of hops between source and destination hasbeen estimated (3)The probability distribution function forrelative velocity between any two nodes has been derived(4) Link duration between any two nodes has been obtainedAnalytical results for the path duration estimation modelhave been obtained using MATLAB The model for pathduration estimation has been simulated in NS2 also Eachof the analytical results has been verified through respectivesimulation results

The rest of the paper is organized as follows In Section 2the two communication modes in VANETs have beendescribed Section 3 presents related works in detail InSection 4 all themathematical formulations for the proposedpath duration estimation have been presented In Section 5the simulation results and analysis have been discussedFinally we concluded the work presented in the paper inSection 6

2 Communication Modes in VANETs

Vehicular communications in on-road traffic environmentshave been realized throughVANETs In this network vehiclestermed as nodes share information through wireless commu-nicationDedicated short range communications (DSRC) [11]technology is used for wireless communication DSRC is anenhanced version of Wi-Fi technology specially designed forVANETs environment and this is known as wireless access invehicular environment (WAVE) During the communicationwhen a sender node does not find any neighboring nodes itforwards the information using road side units (RSUs) avail-able along the road However the availability of RSUs is notstrictly considered in VANETsThus the communications in

VANETs can be categorized in the following two modes (1)vehicle-to-vehicle (V2V) communication and (2) vehicle-to-roadside (V2R) communication [12]

21 V2V Communication V2V communication is the basicand primary aim in VANETs It is pure ad hoc communica-tion between two vehicles [13] V2V communication can bethrough direct link or through multihop links (see Figure 1)If the destination node is present within the transmissionrange of the source node then the direct link is establishedfor communication and this type of communication is knownas single-hop communication If the destination node ispresent outside the transmission range of the source nodethen the intermediate nodes are used to deliver the messageup to the destination and this type of communication isknown asmultihop communications V2V communication ismainly used for the safety applications such as road blockadealarm electronic brake warning incoming traffic warningvehicle stability warning lane change warning and collisionwarning This type of communication is also used for thedifferent types of the protocol operations To set up RSUssuch as fixed infrastructure access points internet gatewaysand base station on the road side is expensive ThereforeVANETs should useV2V communication asmuch as possiblefor communication purpose

22 V2RCommunication V2R communication is the combi-nation of ad hoc network and fixed infrastructure networks[14] This mode of communication (as shown in Figure 1)involves on-road vehicles as well as RSUs Only single-hopcommunication between a vehicle and RSU is used in V2Rcommunication Further vehicle sends the message to theroad side unit which broadcasts themessage to all the vehiclesin the neighborhood Generally RSUs use links of higherbandwidth for communication and broadcasting RSUs maybe placed at every one kilometer or less to enable andmaintain high data rate in highly dense traffic environment

3 Related Work

In spite of the fundamental importance of estimating thepath duration of communication links in VANET therehave been some mathematical and experimental studies inMANETs The estimation of path duration in MANETs isproposed using several theoretical and analytical modelsMany research work and models are also proposed for theimplementation and improvement of the VANET Some ofthe related research works about the path duration have beencarried out in the recent past decade

The path duration is an important design parameter forthe better performance and routing decision in VANETsAuthors in [15] show the analysis of path duration andprovide different parameters related and dependent on thepath duration in MANETs Authors also present the pathduration impact on the reactive routing protocols The resultshows that the path duration probability density function(pdf ) for large number of hops can be estimated with the helpof exponential distribution The path duration depends on

International Journal of Vehicular Technology 3

V2Rcommunication

V2Rcommunication

Figure 1 VANETs and its communication modes

different parameters like relative speed of the moving nodestransmission range and number of hops This result helps toenhance the performance of the reactive routing protocolsused in MANETs It also shows that the inverse of the pathduration is directly proportional to the throughput of the adhoc networks for dynamic source routing (DSR) protocol

Tomaximize the path duration of the route link durationof that path must be maximum If there is any link breaksin the route it means whole route will be expired The linkstability and route lifetime are directly proportional to eachother In [16] link stability and route lifetime are analyzed forthe ad hoc networks ldquoEdge Effectrdquo phenomenon occurringin the highly dense networks is also discussed ldquoEdge Effectrdquophenomenon is an adverse effect on the network performancewhen the greedy routing approach is used in dense networksIn greedy routing approach sender node selects the bordernode or node cover maximum distance towards the destina-tion In dense network nodes are easily available at borderof the transmission range and are selected as the next-hopnode for further transmission Small movement of the bordernodes outside the transmission range breaks the path anddegrades the performance of the network Therefore bordernode must lie within the transmission range or on the borderline of the senderrsquos transmission range to improve the routingas well as overall network performance

Path selection is important to decide the path durationof the route in VANETs The shortest path is not always thebest path in terms of the path duration To maximize thepath duration the shorter average link duration of nodesshould be avoided over the longer average link duration ofnodes In [17] a scheme is proposed using ad hoc on demanddistance vector (AODV) routing protocol to maximize thepath duration and also provide a local path recovery in caseof path failure with the help of the cached alternative pathcomputing In this schema path information is recorded

in a table with five fields (i) destination sequence number(ii) next-hop to the destination (iii) hop count (iv) inversepath duration (IPD) and (v) time stamp The path forthe transmission is chosen first on the basis of destinationsequence number and then on the basis of the IPD valuesIf two paths tie on the basis of above two fields then the pathwill be selected on the basis of hop count The path on firstrank is selected as primary path and other paths are cachedas backed recovery path

In [17] result shows that expected link duration ofthe path is the parameter of the exponential distributionThis exponential distribution can be used to approximatethe distribution where hop count is large In MANETs orVANETs the greedy routing and least remaining distance(LRD) approach are used In LRD approach the next-hopnode is the node which attempts to minimize the remainingdistance between source and destination in every hop Theaverage progress per hop towards destination also helps tofind the number of hops from source-to-destination Authorsin [18] show that the progress per hop and number of hopsare related to the node density and distance of the path ingreedy routing approach

To estimate the path duration of the route in VANETsthe use of the suitable routing protocol is also a critical factorThe position-based routing protocols may be the suitablerouting protocols inVANETsThe routing protocols using theposition information of the node in the networks are knownas the position-based routing protocols In these protocolsthe next-hop node will be selected on the basis of maximumdistance covered towards the destination within the senderrsquostransmission range Some position-based routing protocolssuch as border-node based most forward progress withinradius routing (B-MFR) [19] and edge-node based directionalrouting (E-DIR) [20] have been proposed for VANET toselect the best node for further transmission These routing


S

AB

DA

998400

B998400

90∘

90∘

Source vehicleNeighbor vehicle

Destination vehicleBorder vehicle

Figure 2 B-MFR forwarding method

protocols can be used in estimation of path duration forVANET We are introducing B-MFR for our proposed work

B-MFR is a modified version of MFR (most forwardprogress within radius) routing protocol It avoids usingthe interior nodes within the transmission range for packetforwardingTherefore in B-MFR a packet is sent to the next-hop node which is positioned nearer to border or on theborder of the transmission range towards the destination (seeFigure 2) It means the border node with the greatest progresson the line joining source and destination is chosen as a next-hop node Thus in B-MFR only the border nodes presentnearer to border or on the border of the senderrsquos transmissionrange are selected as the next-hop nodes for further packettransmission

In contrast B-MFR exploits the dynamics of VANETs bychoosing the node farthest from the source node to makethemultihop forwardingmore efficientThis routing protocolis especially important in VANETs when the node densityis high and gives better performance in the dense networkslike city traffic scenario In the dense network nodes onthe border of the transmission range are easily available toforward the data Therefore the border-node based protocolhelps to reduce the number of hops between source and des-tination Moreover B-MFR is a very useful routing conceptto estimate expected distance and expected number of hopsbetween source and destination node mathematically Thesemathematical expressions can be used in estimation of pathduration in VANETs

Themain drawback of the B-MFR is that the node farthestfrom the source cannot be sufficient selection criteria ina network with frequent topological changes Further B-MFR is not more suitable for sparse VANETs where smallernumber of vehicles moving on the road and no or fewerroad side units are present along the road In such networksit is important to guarantee high packet delivery rate withminimum delay and therefore robustness of the routingprotocol is of main concern To achieve this we need tofind the path duration by taking into consideration both thedirection and speed of the vehicular nodes

4 Proposed Work

The randomness in vehicular traffic environment motivatesthe need for a suitable stochastic model to determine theability of a routing protocol in terms of successful packetdelivery and to analyze the network performance Poissondistribution has been used in the model to realize dynamictraffic environment in VANETs We followed Poisson distri-bution since we are interested in finding number of nodespresent in specified forwarding area given the mean densityof nodes in the network area Further the arrival of each nodeis independent We assume that the nodes in the networkarea are deployed in a two-dimensional space according toa spatial Poisson process The mathematical model attemptsto estimate the average path duration using a position-basedrouting concept Once path duration of a path is estimatedwell before the path breakage then the performance of arouting protocol can be enhanced significantly

41 The Routing Protocol Used The role of routing protocolis very important in the estimation of the path duration inVANETs where the network topology frequently changesFor the proposed model the B-MFR position-based routingprotocol is used which we have explained in Section 3 B-MFR is based on greedy routing approach and typically usedfor long distance multihop communications It minimizesthe number of hops that the message has to travel inbetween source and destination In greedy routing approachsender node first finds the position information of its directneighbors and then selects a node that is closest to thedestination as a next-hop node for further transmission TheB-MFR method is useful to estimate the path duration as thenumber of hops can be decreased significantly by selectingthe border node as a next-hop node

42 Mathematical Models Since we are interested in the pathduration therefore the main goal of this section is to derivea mathematical expression for path duration between twovehicles by deriving other useful mathematical expressions


S

R1

Shaded area A2

D

R2

45∘ 1205722

Shaded area A1

Source vehicleNeighbor vehicleDestination vehicle

Figure 3 Area of shaded region (119860119904= 1198601+ 1198602)

such as average number of hops and link duration In thiswork we use traditional traffic flow principle to describethe vehicular environment which will be more accurate forour path duration estimation Vehicles are assumed to followPoisson distributed arrivals for obtaining the probabilitydistribution function (pdf )

421 Modeling Assumptions In our proposed model someassumptions are made which are given as follows

(1) Nodes are equipped with GPS receiver digital mapand sensors

(2) No other fixed infrastructure for communication ispresent

(3) Transmission range for every node is same(4) Average speed of the nodes in the network is constant(5) Link duration for the nodes moving away from the

source node

422 Mathematical Notations The notations are used in ouranalysis See Notations section

423 Area of Border Region for Finding the Next-Hop NodeSince the node closer to the border or on the border linecovers maximum distance it may reduce the number of hopcounts between source and destination Itmay not be possibleto find even a single node at the extreme end of the senderrsquostransmission range Therefore we have considered a regionaround the extreme endof the transmission range towards thedestination That region is shown by shaded area in Figure 3

In Figure 3 circle shows the transmission range of sourcenode 119878 Shaded region of the circle (119860

1+ 1198602) can be used as

the border region Further the shaded region can be dividedinto two parts showing areas119860

1and119860

2 Border nodes can be

chosen from the shaded area 1198602 because area 119860

2lies closest

to border line of the senderrsquos transmission range Greedyrouting is the appropriate method to select a next-hop nodefor the given shaded region

Area of the shaded region can also be known as the areaof interaction of the two circles one with the radius 119877

1and

the other with the radius1198772 Area119860

119904of the shaded region can

be calculated as

119860119904= 1198601+ 1198602 (1)

where

1198601= 1198772

1sdot 1205721minus11987721sdot sin (2120572

1)

2

1198602= 1198772


2)

2

(2)

As we can see in Figure 3 the line from source to destination(SD) is the bisector of the angle 90∘ (120572

1= 45∘) therefore area

of shaded region 119860119904 is

119860119904= 1198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2] (3)

Thus we can say that the shaded region is the combination oftwo arcs one with radius119877

1and the other with radius119877

2The

value of 1205722depends on transmission range 119877

1and distance

between source and destination

424 Probability of Finding Nodes in the Shaded Region Inthis section our aim is to find the probability of at least onenode in the border region to improve the performance ofthe network We assume that nodes are two-dimensionallyPoisson distributed over the network with node density 120596The number of nodes present in selected region can becalculated as

Number of nodes = 120596 times Area of the region (4)

If119883 is the random variable representing the number of nodesin the shaded region 119860

119904 then the probability that a number

of nodes 119909 are located in the region 119860119904can be calculated as

119875 (119883 = 119909) =(120596119860)119909

sdot119890minus120596119860

119909 119909 = 0 1 2 3 (5)

The probability of selecting 119899 nodes out of 119909 nodes is given by

119875 (119884 = 119899) = (119909

119899) (119901)119899

(1 minus 119901)119909minus119899

(6)

If a node is present in the selected region only two possibil-ities are there one is selecting the node 119901

119904 and the other is

not selecting the node 119902 (= 1 minus 119901119904)Therefore probability of

both the cases occurring is equal that is 119901119904= 119902 = 12 Now

probability of selecting exactly 119899 nodes in the given shadedregion is [21]

119875 (119899) =

infin

sum119909=119899

(119909

119899) (119901119904)119899

(1 minus 119901119904)119909minus119899

sdot(120596119860119904)119909

119909sdot 119890minus120596119860119904

=(119901119904120596119860119904)119899

119899sdot 119890minus119901119904120596119860119904

(7)


Put the value of 119901119904= 12 and 119860

119904in the above equation then

119875 (119899)

= (120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119899

times (119899)minus1

sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(8)

Similarly the probability for selecting at least 119899 nodes in theshaded region is

119875119899= 1

minus

119899minus1

sum119894=0

(120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119894


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(9)

From (9) we can easily obtain the probability 119875 of having atleast one node within the border region as

119875 = 1 minus 119875 (119883 = 0)

= 1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(10)

425 Average Number of Hops between Source and Destina-tion Node Number of hops can be defined as the number ofintermediate nodes in the route (source to destination) Themain assumption is that each hop results in the same progresstowards the destination equal to the average distance coveredby a node in one hop Number of hops should be as low aspossible It will decrease the chances of link breakage andimprove the path duration between nodes [10]

To determine the average number of hop counts nodeswithin the transmission range 119877 follow the Poisson dis-tributed model If destination node is present in the senderrsquostransmission range then the probability of finding destinationnode is the same as the probability of finding next-hop nodeWe assume that 119885

1is the distance between the source and

next-hop node The probability density function (pdf ) ofthe link distance 119885

1 between source and next-hop node is

defined as [22]

119891 (1198851) = 2120587120596119885

1sdot 119890minus120587120596119885

2

1 (11)

Distance between two nodes which provide a link to aroute can be defined as the link distance Link distancecan be increased by increasing the distance between sourcenode and next-hop node towards border line within thetransmission range [23] The probability of one-hop countcan be calculated as

119875 (1) = int1198771

0

119891 (1198851) 1198891198851= 1 minus 119890

minus1205871205961198772

1 (12)

However the destination node can be far away from thesource node which may be two three or more hop counts

If the destination node is out of transmission range 1198771 and

less than 21198771 then at least one intermediate node is required

between source and destination to transmit the packet furtherin the network The probability of two-hop counts can becalculated as follows

119875 (2) = int21198771

1198771

21205871205961198851sdot119890minus120587120596119885

2

11198891198851times [1 minus 119890

minus(1205962)sdot119860119904]

119875 (2) = [119890minus120587120596119877

2

1 minus 119890minus4120587120596119877

2

1] times [1 minus 119890minus(1205962)sdot119860

119904]

(13)

Similarly the probability for three-hop count is

119875 (3) = [119890minus4120587120596119877

2

1 minus 119890minus9120587120596119877

2


119904]2

(14)

Consequently the 119896-hop counts probability can be defined as

119875 (119896) = [119890minus(119896minus1)

2

1205871205961198772

1 minus 119890minus1198962

1205871205961198772


119904]119896minus1

(15)

Now by using (12) (13) (14) and (15) we can calculatethe expected number of hops 119864

119867 between source and

destination as follows

119864119867=

119896

sum119867=1

119867119875 (119867) = 119875 (1) + 2119875 (2) + 3119875 (3) + sdot sdot sdot + 119896119875 (119896)

119864119867=

119896

sum119867=1

119867[119890minus(119867minus1)

2

1205871205961198792

minus 119890minus1198672

1205871205961198792

]

times [1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

]119867minus1

(16)

426 Velocity of Nodes Direction of movement and speed ofa node are very essential parameters for the calculation of thepath duration in case of VANETs Link duration depends onthe relative velocity of the nodes as it can increase the linkdistance between nodes The relative velocity between nodesis inversely proportional to the link duration The relativevelocity of the source node and next-hop node should beknown to determine the expected link duration Let 119881

1and

1198812be the velocity of source and next-hop nodes then relative

velocity 119881119877of the nodes can be calculated as

119881119877= radic11988121+ 11988122minus 2 sdot 119881

1sdot 1198812cos 120579 (17)

In this work we assume that all the nodesmovewith constantvelocity in the network that is

119881 = 1198811= 1198812 (18)

Therefore relative velocity is

119881119877= 119881 sdot radic2 (1 minus cos 120579) (19)

In the above equation 120579 can vary from 0 to 1205872 as the next-hop node can move in the direction of destination only tomaintain the communication link (link duration) betweennodesWe assume that angle 120579 is uniformly distributedwithin


(0 1205872) and pdf of 119891120579(120579) is 2120587 Then the pdf of 119881

119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1

radic1 minus sin2 (1205792)sdot2

120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)

427 Link Duration Link duration is the time for which thedirect link between two nodes within the transmission rangeis active and it is a part of the route It is necessary that next-hop node must be present within the transmission range ofthe source node tomaintain the communication link betweensource and next-hop node In this work as we assumedborder node will be the next-hop node for each hop betweenthe source and destination Since the velocity of each nodein the network is constant it means that the links betweensource and next-hop node will always be maintained As wehave assumed119885

1to be the distance between source and next-

hop node within radius1198771 then the expected value of119885

1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)

Therefore link duration 119879 can be expressed as

119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)

The pdf of 119879 119891119879(119879) is given by

119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)

428 Path Duration The path duration is one of the keyparameters which could be useful to improve the perfor-mance and throughput of a highly dynamic network such asVANET The path duration will be helpful in the process ofpath selection during the transmission of packet from sourceto destination [25 26] Path duration can be derived from thepdf of the link duration Let119879

1 1198792 1198793 119879

119864119867

denote the linkduration of 1 2 3 119864

119867hops respectively119864

119867is the average

number of hops required to reach the destination as estimatedin (16) Therefore the path duration can be expressed as

119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)

By using Bayesrsquo theorem [21] the pdf of 119879path is

119891 (119879path) = 119864119867 sdot 119864119885 sdot 119862119864119867minus1

119879 (25)

Here 119879 represents the link duration and 119862119879= 1 minus 119865

119879is

the complementary cumulative density function (cdf) of 119879Therefore

119891 (119879path) = 119864119867sdot 119891119879(119879) sdot [ 1 minus int

infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)

Finally the average path duration can be estimated as

119864119879path

= int120579

0

119879path sdot 119891 (119879path) sdot 119889119879path

119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)

sdot [1 minus intinfin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)

5 Simulations and Results Analysis

In this section extensive simulations have been performedto analyze the mathematical model for estimation of pathduration presented in Section 4 The impact of four param-eters namely transmission range number of hops velocityof nodes and density of nodes on path duration have beenanalyzedThe simulation results of themodel have been com-pared with analytical results obtained for the mathematicalformulation of path duration estimation

51 Simulation Environment Themathematical estimation ofpath duration has been simulated using network simulator(NS-234) MOVE (mobility model generator for vehicularnetworks) [27] has been used to generate realistic vehiculartraffic environment along with open-source microtraffic sim-ulator SUMO (simulation of urban mobility) The vehiculartraffic scenario consists of roads traffic lanes on roads junc-tions traffic lights at junctions vehicles speed probability ofturning left or right of a vehicle at junctions and so forth hasbeen set up using road map editor and movement editor ofMOVE The trace file used in ns2 is produced following thesetup procedure of MOVE

A set of five horizontal and five vertical roads cross-ing each other and thus making twenty five junctions isused as simulation area The lane width used is 5m Thevelocity range 0ndash60Kmh is used for node movement andtransmission range varies from 100m to 600m The otherbasic parameters used for the simulation are packet size of512 bytes traffic type as CBR wireless channel omnidirec-tional antenna 80211p as MAC wireless standard and 300 ssimulation time The position-based routing protocol usedfor the simulation is B-MFR After setting the network andtraffic flow with above discussed parameters we conductedthe simulation The average of ten different simulation runsis taken for data record where different source and differentdestination are selectedMATLAB is used to obtain analyticalresults for the mathematical formulation of the model

52 Result Analysis The results obtained for the modelhave been analyzed in the following subsections In eachsubsection impact of a specific parameter on path durationhas been analyzed In each analysis the simulation andanalytical results have been discussed comparatively

521 The Impact of Transmission Range Figure 4 shows theimpact of transmission range on path duration Path duration


100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11

Transmission range (m)

Aver

age p

ath

dura

tion

(s)

Hops = 10 [analytical results]Hops = 10 [simulation results]Hops = 15 [analytical results]Hops = 15 [simulation results]

Figure 4 Average path duration versus transmission range

of the routes is dependent on the transmission range ofthe nodes in the network It can be clearly observed thatthe average path duration increases with the increase intransmission range This can be attributed to the fact that thelarger transmission range increases the probability of findingnext-hop node in the border area of senderrsquos transmissionrange It means selection of border node gives better resultas compared to interior node within the transmission rangeAdditionally increasing the number of hops in the consideredpath decreases the average path duration The simulationresults are close to the analytical results that validate themodel

522 The Impact of Number of Hops The plots in Figure 5show the impact of number of hops on average path durationwhich depends on each hop of the route and it varieswith number of hops for fixed transmission range It clearlyreveals that increasing the number of hops in the pathdecreases average path duration The reason behind this isthat increasing the number of hops in the path also increasesthe probability of link failure Figure 5 also shows that thehigh velocity of nodes (eg 119881 = 55ms) has a relativelyreduced average path duration as compared to low velocityof nodes (eg 119881 = 50ms) This is due to the increasingprobability of link failure with higher velocity of nodes Thesimulation results are close to the analytical results that verifythe model

523 The Impact of Node Density The node density is also acritical factor for the path duration as the possibility of find-ing suitable next-hop node is increased with the increasednumber of nodes The results depicted in Figure 6 show

Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)

V = 50kmh [analytical results]V = 50kmh [simulation results]V = 55kmh [analytical results]V = 55kmh [simulation results]

Figure 5 Average path duration versus number of hops

the impact of node density on average path duration It clearlyconveys that increasing the node density in a given areaincreases path duration This is due to the increasing proba-bility of availability of suitable next-hop node with increasingnode density in the network Moreover the results also con-firm that for a given node density increasing the number ofhops in the path decreases average path duration The close-ness between analytical and simulation results for hops = 15is smaller as compared to hops = 10 due to the increment inthe network dynamics with increasing number of hops

524 The Impact of Velocity of Nodes VANETs are knownfor their mobility The high velocity of nodes makes vehiclesin and out from the transmission range of the source nodewhich causes most of the link or path breakage in thenetwork The results in Figure 7 show the impact of velocityof nodes on average path duration It can be observed thatincreasing the velocity of nodes decreases the path durationThis can be attributed to the fact that increasing the velocity ofnodes in the network increases the probability of link failurewhich ultimately decreases average path duration Moreoverit also reveals that increasing the number of hops for a givenvelocity decreases average path duration due to the incrementin the number of links in the path that again increases theprobability of link failure In the figure simulation results arevery close to the analytical results

6 Conclusion

In this paper we have derived a mathematical model forestimation of path duration between source and destinationnodes using position-based routing concept This model has


25 50 75 100 125 15025

3

35

4

45

5

55

Node density in area 1000 times 1000 (m2)

Aver

age p

ath

dura

tion

(s)


Figure 6 Average path duration versus node density

0 10 20 30 40 50 600123456789

101112131415

Velocity of nodes (kmh)

Aver

age p

ath

dura

tion

(s)


Figure 7 Average path durations versus velocity of nodes

been verified by means of simulations and analytical resultsBoth the results are well approximated by the mathematicalmodel The mathematical model that we present is able todescribe the effects of various road traffic parameters includ-ing the transmission range number of hops node densityand velocity of nodes on path duration The use of the B-MFR routing protocol in this work is to find the routing path

with maximum path duration It demonstrates the selectionof next-hop nodes positioned in a region around the extremeend of the transmission range towards the destination

The mathematical analysis and simulation results revealthat high transmission range and smaller number of hopsincrease the path duration but it decreases as the velocityof the nodes increases From the above observations it mayappear that in highly dynamic networks such as in VANETit is very necessary to maintain the path duration betweensource and destination nodes Therefore message can beforwarded timely to reduce the large number of accidents onthe roadThus we can say that the work in this paper helps usto improve the routing performance and decrease the numberof path failures generally occurring in VANETs

Notations and Their Descriptions

1198771 Transmission range of nodes(omnidirectional)

119860 Region covered by transmission range119860119904 Selected region of the transmission range

120596 Node density119909 Number of nodes in the shaded region119899 Number of nodes selected out of 119909 nodes119875119904 Probability of successfully selecting a node

119902 Probability of not selecting a node119875119899 Probability for selecting at least 119899 nodes inthe shaded region

119864119867 Expected number of hops between sourceand destination nodes

1198772 Distance between next-hop node anddestination node

1198851 Distance between source and next-hopnode

119881119877 Relative velocity between source node andnext-hop node

1205721 Angle between 119877

1and SD


2and SD

120579 Relative angle between source andnext-hop node

Conflict of Interests

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

References

[1] WHO Status Report on Road Safety World Health Organiza-tion (WHO) Geneva Switzerland 2013

[2] T Taleb E Sakhaee A Jamalipour K Hashimoto N Kato andY Nemoto ldquoA stable routing protocol to support ITS services inVANET networksrdquo IEEE Transactions on Vehicular Technologyvol 56 no 6 pp 3337ndash3347 2007

[3] O Kaiwartya and S Kumar ldquoGeocast routing Recent advancesand future challenges in vehicular adhoc networksrdquo in Proceed-ings of Signal Processing and Integrated Networks (SPIN rsquo14) pp291ndash296 2014


[4] O Kaiwartya and S Kumar ldquoCache agent based geocasting(CAG) in VANETsrdquo International Journal of Information andCommunication Technology In press

[5] O Kaiwartya S Kumar and R Kasana ldquoTraffic light basedtime stable geocast (T-TSG) routing for urban VANETsrdquo inProceedings of the 6th International Conference onContemporaryComputing (IC3 rsquo13) pp 113ndash117 2013

[6] O Kaiwartya and S Kumar ldquoGeoPSO geocasting throughparticle swarm optimization in vehicular adhoc networksrdquo inProceedings of Information Systems and Design of Communica-tion (ISDOC rsquo14) ACM May 2014

[7] O Kaiwartya and S Kumar ldquoEnhanced caching for geocastrouting in vehicular Ad-Hoc networks (ECGR)rdquo in Proceedingsof the International Conference on Advanced Computing Net-working and Informatics (ICACNI rsquo13) vol 243 pp 213ndash220Springer 2013

[8] J J Blum A Eskandarian and L J Huffman ldquoChallenges ofintervehicle Ad Hoc networksrdquo IEEE Transactions on IntelligentTransportation Systems vol 5 no 4 pp 347ndash351 2004

[9] W Alasmary andW Zhuang ldquoMobility impact in IEEE 80211pinfrastructure less vehicular networksrdquo Ad Hoc Networks vol10 no 2 pp 222ndash230 2012

[10] R J La and Y Han ldquoDistribution of path durations in mobilead hoc networks and path selectionrdquo IEEEACM Transactionson Networking vol 15 no 5 pp 993ndash1006 2007

[11] ldquoDSRC Standards Whatrsquos Newrdquo ITS Standards Advisory num-ber 3 US Department of Transportation httpwwwitsdotgovDSRC

[12] R Popescu-Zeletin I Radusch and M A Rigani Vehicular-2-X Communication State-Of-The-Art and Research in MobileVehicular AdHocNetworks Springer NewYork NYUSA 2010

[13] A F Molisch F Tufvesson J Karedal and C F Meck-lenbrauker ldquoA survey on vehicle-to-vehicle propagation chan-nelsrdquo IEEE Wireless Communications vol 16 no 6 pp 12ndash222009

[14] J Gozalvez M Sepulcre and R Bauza ldquoIEEE 80211p vehicle toinfrastructure communications in urban environmentsrdquo IEEECommunications Magazine vol 50 no 5 pp 176ndash183 2012

[15] N Sadagopan F Bai B Krishnamachari and A HelmyldquoPATHS analysis of PATH duration statistics and their impacton reactiveMANET routing protocolsrdquo in Proceedings of the 4thACM International Symposium on Mobile Ad Hoc Networkingand Computing (MOBIHOC rsquo03) pp 245ndash256 June 2003

[16] G Lim K Shin S Lee H Yoon and J S Ma ldquoLink stabilityand route lifetime in ad-hoc networksrdquo EURASIP Journal onWireless Communications and Networking pp 1ndash6 2007

[17] Y Han and R J La ldquoMaximizing path durations in mobile ad-hoc networksrdquo in Proceedings of the 40th Annual Conferenceon Information Sciences and Systems (CISS rsquo06) pp 26ndash31Princeton NJ USA March 2006

[18] S De A Caruso T Chaira and S Chessa ldquoBounds on hop dis-tance in greedy routing approach in wireless ad hoc networksrdquoInternational Journal of Wireless and Mobile Computing vol 1no 2 pp 131ndash140 2006

[19] R S Raw and D K Lobiyal ldquoB-MFR routing protocol forvehicular ad hoc networksrdquo in Proceedings of the InternationalConference on Networking and Information Technology (ICNIT10) pp 420ndash423 Manila Philippines June 2010

[20] R S Raw and D K Lobiyal ldquoE-DIR a directional routing pro-tocol for VANETs in a city traffic environmentrdquo InternationalJournal of Information and Communication Technology vol 3no 3 pp 242ndash257 2011

[21] S Ukkusuri and L Du ldquoGeometric connectivity of vehicularad hoc networks analytical characterizationrdquo TransportationResearch Part C Emerging Technologies vol 16 no 5 pp 615ndash634 2008

[22] S M Harb and J McNair ldquoAnalytical study of the expectednumber of hops in wireless Ad Hoc networkrdquo in WirelessAlgorithms Systems and Applications vol 5258 of Lecture Notesin Computer Science pp 63ndash71 2008

[23] K Namuduri and R Pendse ldquoAnalytical estimation of pathduration in mobile ad hoc networksrdquo IEEE Sensors Journal vol12 no 6 pp 1828ndash1835 2012

[24] R S Raw and S Das ldquoPerformance analysis of P-GEDIR proto-col for vehicular ad hoc network in urban traffic environmentsrdquoWireless Personal Communications vol 68 no 1 pp 65ndash782013

[25] H Ghafoor K Insoo and G Nasirud-Din ldquoNeighboring andconnectivity-aware routing in VANETsrdquo The Scientific WorldJournal vol 2014 Article ID 789247 10 pages 2014

[26] C Chen Y Jin Q Pei and N Zhang ldquoA connectivity-awareintersection-based routing in VANETsrdquo EURASIP Journal onWireless Communications and Networking vol 2014 article 422014

[27] F K Karnadi Z H Mo and K Lan ldquoRapid generation ofrealisticmobilitymodels for VANETrdquo in Proceedings of the IEEEWireless Communications and Networking Conference (WCNCrsquo07) pp 2508ndash2513 IEEE March 2007

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



In VANETs any routing path is made up of one or morelinks between pair of intermediate nodes Therefore lifetimeof a link is one of the most important contributors in pathduration Path duration can be defined as the duration oftime till every link of the route is active In VANETs thelifetime of a link is a random variable whose probabilitydistribution depends on mobility node density transmissionrange different traffic scenarios and various impairmentsof radio communications The links between intermediatenodes frequently break due to the high mobile nodes movingout of each otherrsquos transmission range Therefore estimationof path duration between source and destination decreasesthe chances of path breakage The effective use of knowledgeof path duration improves the performance and efficiency ofrouting protocols in VANETs [10]

In this paper our main contribution is in terms of prob-abilistic and mathematical analysis of path duration usingborder-node based most forward progress within radius (B-MFR) routing protocol For this analysis a mathematicalmodel to estimate path duration has been developed B-MFR is a position-based routing protocol which selects next-hop node from the nodes belonging to the border area ofthe transmission range The mathematical model for pathduration estimation consists of the following modules (1)The probability of finding at least one node in the consideredforwarding region has been mathematically derived (2)Average number of hops between source and destination hasbeen estimated (3)The probability distribution function forrelative velocity between any two nodes has been derived(4) Link duration between any two nodes has been obtainedAnalytical results for the path duration estimation modelhave been obtained using MATLAB The model for pathduration estimation has been simulated in NS2 also Eachof the analytical results has been verified through respectivesimulation results

The rest of the paper is organized as follows In Section 2the two communication modes in VANETs have beendescribed Section 3 presents related works in detail InSection 4 all themathematical formulations for the proposedpath duration estimation have been presented In Section 5the simulation results and analysis have been discussedFinally we concluded the work presented in the paper inSection 6

2 Communication Modes in VANETs

Vehicular communications in on-road traffic environmentshave been realized throughVANETs In this network vehiclestermed as nodes share information through wireless commu-nicationDedicated short range communications (DSRC) [11]technology is used for wireless communication DSRC is anenhanced version of Wi-Fi technology specially designed forVANETs environment and this is known as wireless access invehicular environment (WAVE) During the communicationwhen a sender node does not find any neighboring nodes itforwards the information using road side units (RSUs) avail-able along the road However the availability of RSUs is notstrictly considered in VANETsThus the communications in

VANETs can be categorized in the following two modes (1)vehicle-to-vehicle (V2V) communication and (2) vehicle-to-roadside (V2R) communication [12]

21 V2V Communication V2V communication is the basicand primary aim in VANETs It is pure ad hoc communica-tion between two vehicles [13] V2V communication can bethrough direct link or through multihop links (see Figure 1)If the destination node is present within the transmissionrange of the source node then the direct link is establishedfor communication and this type of communication is knownas single-hop communication If the destination node ispresent outside the transmission range of the source nodethen the intermediate nodes are used to deliver the messageup to the destination and this type of communication isknown asmultihop communications V2V communication ismainly used for the safety applications such as road blockadealarm electronic brake warning incoming traffic warningvehicle stability warning lane change warning and collisionwarning This type of communication is also used for thedifferent types of the protocol operations To set up RSUssuch as fixed infrastructure access points internet gatewaysand base station on the road side is expensive ThereforeVANETs should useV2V communication asmuch as possiblefor communication purpose

22 V2RCommunication V2R communication is the combi-nation of ad hoc network and fixed infrastructure networks[14] This mode of communication (as shown in Figure 1)involves on-road vehicles as well as RSUs Only single-hopcommunication between a vehicle and RSU is used in V2Rcommunication Further vehicle sends the message to theroad side unit which broadcasts themessage to all the vehiclesin the neighborhood Generally RSUs use links of higherbandwidth for communication and broadcasting RSUs maybe placed at every one kilometer or less to enable andmaintain high data rate in highly dense traffic environment

3 Related Work

In spite of the fundamental importance of estimating thepath duration of communication links in VANET therehave been some mathematical and experimental studies inMANETs The estimation of path duration in MANETs isproposed using several theoretical and analytical modelsMany research work and models are also proposed for theimplementation and improvement of the VANET Some ofthe related research works about the path duration have beencarried out in the recent past decade

The path duration is an important design parameter forthe better performance and routing decision in VANETsAuthors in [15] show the analysis of path duration andprovide different parameters related and dependent on thepath duration in MANETs Authors also present the pathduration impact on the reactive routing protocols The resultshows that the path duration probability density function(pdf ) for large number of hops can be estimated with the helpof exponential distribution The path duration depends on


V2Rcommunication

V2Rcommunication









S

AB

DA

998400

B998400

90∘

90∘








4 Proposed Work





S

R1

Shaded area A2

D

R2

45∘ 1205722

Shaded area A1








source node






1and119860



2lies closest



1and



be calculated as

119860119904= 1198601+ 1198602 (1)

where

1198601= 1198772


1)

2

1198602= 1198772


2)

2

(2)




119860119904= 1198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2] (3)



2The


1and distance







119875 (119883 = 119909) =(120596119860)119909

sdot119890minus120596119860

119909 119909 = 0 1 2 3 (5)


119875 (119884 = 119899) = (119909

119899) (119901)119899

(1 minus 119901)119909minus119899

(6)






119875 (119899) =

infin

sum119909=119899

(119909

119899) (119901119904)119899

(1 minus 119901119904)119909minus119899

sdot(120596119860119904)119909

119909sdot 119890minus120596119860119904

=(119901119904120596119860119904)119899

119899sdot 119890minus119901119904120596119860119904

(7)




119875 (119899)

= (120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119899


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(8)


119875119899= 1

minus

119899minus1

sum119894=0

(120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119894


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(9)


119875 = 1 minus 119875 (119883 = 0)

= 1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(10)






defined as [22]

119891 (1198851) = 2120587120596119885

1sdot 119890minus120587120596119885

2

1 (11)


119875 (1) = int1198771

0

119891 (1198851) 1198891198851= 1 minus 119890

minus1205871205961198772

1 (12)





119875 (2) = int21198771

1198771

21205871205961198851sdot119890minus120587120596119885

2

11198891198851times [1 minus 119890

minus(1205962)sdot119860119904]

119875 (2) = [119890minus120587120596119877

2

1 minus 119890minus4120587120596119877

2


119904]

(13)


119875 (3) = [119890minus4120587120596119877

2

1 minus 119890minus9120587120596119877

2


119904]2

(14)


119875 (119896) = [119890minus(119896minus1)

2

1205871205961198772

1 minus 119890minus1198962

1205871205961198772


119904]119896minus1

(15)




119864119867=

119896

sum119867=1

119867119875 (119867) = 119875 (1) + 2119875 (2) + 3119875 (3) + sdot sdot sdot + 119896119875 (119896)

119864119867=

119896

sum119867=1

119867[119890minus(119867minus1)

2

1205871205961198792


1205871205961198792

]


2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

]119867minus1

(16)


1and



119881119877= radic11988121+ 11988122minus 2 sdot 119881

1sdot 1198812cos 120579 (17)


119881 = 1198811= 1198812 (18)






119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1


120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)




1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)


119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)


119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)


1 1198792 1198793 119879

119864119867





119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)



119879 (25)


119879is



infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)


119864119879path

= int120579

0


119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)


119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)








100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










V2Rcommunication

V2Rcommunication









S

AB

DA

998400

B998400

90∘

90∘








4 Proposed Work





S

R1

Shaded area A2

D

R2

45∘ 1205722

Shaded area A1








source node






1and119860



2lies closest



1and



be calculated as

119860119904= 1198601+ 1198602 (1)

where

1198601= 1198772


1)

2

1198602= 1198772


2)

2

(2)




119860119904= 1198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2] (3)



2The


1and distance







119875 (119883 = 119909) =(120596119860)119909

sdot119890minus120596119860

119909 119909 = 0 1 2 3 (5)


119875 (119884 = 119899) = (119909

119899) (119901)119899

(1 minus 119901)119909minus119899

(6)






119875 (119899) =

infin

sum119909=119899

(119909

119899) (119901119904)119899

(1 minus 119901119904)119909minus119899

sdot(120596119860119904)119909

119909sdot 119890minus120596119860119904

=(119901119904120596119860119904)119899

119899sdot 119890minus119901119904120596119860119904

(7)




119875 (119899)

= (120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119899


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(8)


119875119899= 1

minus

119899minus1

sum119894=0

(120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119894


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(9)


119875 = 1 minus 119875 (119883 = 0)

= 1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(10)






defined as [22]

119891 (1198851) = 2120587120596119885

1sdot 119890minus120587120596119885

2

1 (11)


119875 (1) = int1198771

0

119891 (1198851) 1198891198851= 1 minus 119890

minus1205871205961198772

1 (12)





119875 (2) = int21198771

1198771

21205871205961198851sdot119890minus120587120596119885

2

11198891198851times [1 minus 119890

minus(1205962)sdot119860119904]

119875 (2) = [119890minus120587120596119877

2

1 minus 119890minus4120587120596119877

2


119904]

(13)


119875 (3) = [119890minus4120587120596119877

2

1 minus 119890minus9120587120596119877

2


119904]2

(14)


119875 (119896) = [119890minus(119896minus1)

2

1205871205961198772

1 minus 119890minus1198962

1205871205961198772


119904]119896minus1

(15)




119864119867=

119896

sum119867=1

119867119875 (119867) = 119875 (1) + 2119875 (2) + 3119875 (3) + sdot sdot sdot + 119896119875 (119896)

119864119867=

119896

sum119867=1

119867[119890minus(119867minus1)

2

1205871205961198792


1205871205961198792

]


2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

]119867minus1

(16)


1and



119881119877= radic11988121+ 11988122minus 2 sdot 119881

1sdot 1198812cos 120579 (17)


119881 = 1198811= 1198812 (18)






119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1


120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)




1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)


119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)


119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)


1 1198792 1198793 119879

119864119867





119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)



119879 (25)


119879is



infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)


119864119879path

= int120579

0


119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)


119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)








100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










S

AB

DA

998400

B998400

90∘

90∘








4 Proposed Work





S

R1

Shaded area A2

D

R2

45∘ 1205722

Shaded area A1








source node






1and119860



2lies closest



1and



be calculated as

119860119904= 1198601+ 1198602 (1)

where

1198601= 1198772


1)

2

1198602= 1198772


2)

2

(2)




119860119904= 1198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2] (3)



2The


1and distance







119875 (119883 = 119909) =(120596119860)119909

sdot119890minus120596119860

119909 119909 = 0 1 2 3 (5)


119875 (119884 = 119899) = (119909

119899) (119901)119899

(1 minus 119901)119909minus119899

(6)






119875 (119899) =

infin

sum119909=119899

(119909

119899) (119901119904)119899

(1 minus 119901119904)119909minus119899

sdot(120596119860119904)119909

119909sdot 119890minus120596119860119904

=(119901119904120596119860119904)119899

119899sdot 119890minus119901119904120596119860119904

(7)




119875 (119899)

= (120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119899


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(8)


119875119899= 1

minus

119899minus1

sum119894=0

(120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119894


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(9)


119875 = 1 minus 119875 (119883 = 0)

= 1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(10)






defined as [22]

119891 (1198851) = 2120587120596119885

1sdot 119890minus120587120596119885

2

1 (11)


119875 (1) = int1198771

0

119891 (1198851) 1198891198851= 1 minus 119890

minus1205871205961198772

1 (12)





119875 (2) = int21198771

1198771

21205871205961198851sdot119890minus120587120596119885

2

11198891198851times [1 minus 119890

minus(1205962)sdot119860119904]

119875 (2) = [119890minus120587120596119877

2

1 minus 119890minus4120587120596119877

2


119904]

(13)


119875 (3) = [119890minus4120587120596119877

2

1 minus 119890minus9120587120596119877

2


119904]2

(14)


119875 (119896) = [119890minus(119896minus1)

2

1205871205961198772

1 minus 119890minus1198962

1205871205961198772


119904]119896minus1

(15)




119864119867=

119896

sum119867=1

119867119875 (119867) = 119875 (1) + 2119875 (2) + 3119875 (3) + sdot sdot sdot + 119896119875 (119896)

119864119867=

119896

sum119867=1

119867[119890minus(119867minus1)

2

1205871205961198792


1205871205961198792

]


2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

]119867minus1

(16)


1and



119881119877= radic11988121+ 11988122minus 2 sdot 119881

1sdot 1198812cos 120579 (17)


119881 = 1198811= 1198812 (18)






119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1


120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)




1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)


119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)


119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)


1 1198792 1198793 119879

119864119867





119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)



119879 (25)


119879is



infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)


119864119879path

= int120579

0


119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)


119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)








100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










S

R1

Shaded area A2

D

R2

45∘ 1205722

Shaded area A1








source node






1and119860



2lies closest



1and



be calculated as

119860119904= 1198601+ 1198602 (1)

where

1198601= 1198772


1)

2

1198602= 1198772


2)

2

(2)




119860119904= 1198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2] (3)



2The


1and distance







119875 (119883 = 119909) =(120596119860)119909

sdot119890minus120596119860

119909 119909 = 0 1 2 3 (5)


119875 (119884 = 119899) = (119909

119899) (119901)119899

(1 minus 119901)119909minus119899

(6)






119875 (119899) =

infin

sum119909=119899

(119909

119899) (119901119904)119899

(1 minus 119901119904)119909minus119899

sdot(120596119860119904)119909

119909sdot 119890minus120596119860119904

=(119901119904120596119860119904)119899

119899sdot 119890minus119901119904120596119860119904

(7)




119875 (119899)

= (120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119899


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(8)


119875119899= 1

minus

119899minus1

sum119894=0

(120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119894


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(9)


119875 = 1 minus 119875 (119883 = 0)

= 1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(10)






defined as [22]

119891 (1198851) = 2120587120596119885

1sdot 119890minus120587120596119885

2

1 (11)


119875 (1) = int1198771

0

119891 (1198851) 1198891198851= 1 minus 119890

minus1205871205961198772

1 (12)





119875 (2) = int21198771

1198771

21205871205961198851sdot119890minus120587120596119885

2

11198891198851times [1 minus 119890

minus(1205962)sdot119860119904]

119875 (2) = [119890minus120587120596119877

2

1 minus 119890minus4120587120596119877

2


119904]

(13)


119875 (3) = [119890minus4120587120596119877

2

1 minus 119890minus9120587120596119877

2


119904]2

(14)


119875 (119896) = [119890minus(119896minus1)

2

1205871205961198772

1 minus 119890minus1198962

1205871205961198772


119904]119896minus1

(15)




119864119867=

119896

sum119867=1

119867119875 (119867) = 119875 (1) + 2119875 (2) + 3119875 (3) + sdot sdot sdot + 119896119875 (119896)

119864119867=

119896

sum119867=1

119867[119890minus(119867minus1)

2

1205871205961198792


1205871205961198792

]


2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

]119867minus1

(16)


1and



119881119877= radic11988121+ 11988122minus 2 sdot 119881

1sdot 1198812cos 120579 (17)


119881 = 1198811= 1198812 (18)






119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1


120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)




1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)


119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)


119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)


1 1198792 1198793 119879

119864119867





119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)



119879 (25)


119879is



infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)


119864119879path

= int120579

0


119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)


119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)








100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119875 (119899)

= (120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119899


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(8)


119875119899= 1

minus

119899minus1

sum119894=0

(120596

21198772

1[120587 minus 2

4] + 1198772

2[1205722minussin (2120572

2)

2])

119894


sdot 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(9)


119875 = 1 minus 119875 (119883 = 0)

= 1 minus 119890minus(1205962)119877

2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

(10)






defined as [22]

119891 (1198851) = 2120587120596119885

1sdot 119890minus120587120596119885

2

1 (11)


119875 (1) = int1198771

0

119891 (1198851) 1198891198851= 1 minus 119890

minus1205871205961198772

1 (12)





119875 (2) = int21198771

1198771

21205871205961198851sdot119890minus120587120596119885

2

11198891198851times [1 minus 119890

minus(1205962)sdot119860119904]

119875 (2) = [119890minus120587120596119877

2

1 minus 119890minus4120587120596119877

2


119904]

(13)


119875 (3) = [119890minus4120587120596119877

2

1 minus 119890minus9120587120596119877

2


119904]2

(14)


119875 (119896) = [119890minus(119896minus1)

2

1205871205961198772

1 minus 119890minus1198962

1205871205961198772


119904]119896minus1

(15)




119864119867=

119896

sum119867=1

119867119875 (119867) = 119875 (1) + 2119875 (2) + 3119875 (3) + sdot sdot sdot + 119896119875 (119896)

119864119867=

119896

sum119867=1

119867[119890minus(119867minus1)

2

1205871205961198792


1205871205961198792

]


2

1[(120587minus2)4]+119877

2

2[1205722minus(sin(2120572

2)2)]

]119867minus1

(16)


1and



119881119877= radic11988121+ 11988122minus 2 sdot 119881

1sdot 1198812cos 120579 (17)


119881 = 1198811= 1198812 (18)






119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1


120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)




1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)


119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)


119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)


1 1198792 1198793 119879

119864119867





119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)



119879 (25)


119879is



infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)


119864119879path

= int120579

0


119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)


119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)








100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119877 119891119881119877

(119881119877)

can be expressed as

119891119881119877

(119881119877) =

1


120587= radic4119881

2 minus 119881119903

119881sdot2

120587

(20)




1[24]

can be computed as

1198641198851

=1198991198771

(119899 + 1) (21)


119879 =1198641198851

119881119877

=1198991198771

119881119877(119899 + 1)

(22)


119891119879(119879) = int

119881

0

119881119877sdot 119891119889119881119877

(119881119877119879119881) 119889V

= int119881

0

[1198641198851

] sdot[[

[

2

radic41198812 minus 1198812119877

sdot1

120587

]]

]

119889119881119903

(23)


1 1198792 1198793 119879

119864119867





119879path = MIN (1198791 1198792 1198793 119879

119864119867

) (24)



119879 (25)


119879is



infin

119879=0

119891119879(119879) 119889119879]

119864119867minus1

(26)


119864119879path

= int120579

0


119864119879path

= int120579

0

119879path sdot 119864119867 sdot 119891119879 (119879)


119879=0

119891119879(119879) 119889119879]

119864119867minus1

sdot 119889119879path

(27)








100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100 150 200 250 300 350 400 450 500 550 6001

2

3

4

5

6

7

8

9

10

11


Aver

age p

ath

dura

tion

(s)






Aver

age p

ath

dura

tion

(s)

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

12

Number of hops (N)





6 Conclusion



25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










25 50 75 100 125 15025

3

35

4

45

5

55


Aver

age p

ath

dura

tion

(s)



0 10 20 30 40 50 600123456789

101112131415


Aver

age p

ath

dura

tion

(s)
















1and SD


2and SD




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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article A Probabilistic Analysis of Path Duration Using …downloads.hindawi.com/archive/2014/495036.pdf · 2019-07-31 · Research Article A Probabilistic Analysis of Path