research article caf: cluster algorithm and a-star with...

Research ArticleCAF Cluster Algorithm and A-Star with Fuzzy Approach forLifetime Enhancement in Wireless Sensor Networks

Yali Yuan1 Caihong Li1 Yi Yang1 Xiangliang Zhang2 and Lian Li1

1 School of Information Science and Engineering Lanzhou University Lanzhou Gansu 730000 China2 Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science andTechnology (KAUST) PO Box 2925 Thuwal 23955-6900 Saudi Arabia

Correspondence should be addressed to Caihong Li licaihonggmailcom

Received 7 January 2014 Revised 13 February 2014 Accepted 9 March 2014 Published 28 April 2014

Academic Editor Haiyan Lu

Copyright copy 2014 Yali Yuan et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Energy is amajor factor in designingwireless sensor networks (WSNs) In particular in the real world battery energy is limited thusthe effective improvement of the energy becomes the key of the routing protocols Besides the sensor nodes are always deployed faraway from the base station and the transmission energy consumption is index times increasing with the increase of distance as wellThis paper proposes a new routing method forWSNs to extend the network lifetime using a combination of a clustering algorithma fuzzy approach and an A-star method The proposal is divided into two steps Firstly WSNs are separated into clusters usingthe Stable Election Protocol (SEP) method Secondly the combined methods of fuzzy inference and A-star algorithm are adoptedtaking into account the factors such as the remaining power theminimumhops and the trafficnumbers of nodes Simulation resultsdemonstrate that the proposedmethodhas significant effectiveness in terms of balancing energy consumption aswell asmaximizingthe network lifetime by comparing the performance of the A-star and fuzzy (AF) approach cluster and fuzzy (CF)method clusterand A-star (CA)method A-star method and SEP algorithm under the same routing criteria

1 Introduction

In recent years wireless sensor networks have gained theworldwide attention In particular with the development ofmicroelectromechanical system (MEMS) technology moreand more intelligent sensors have come into being Thesesensors are small with limited processor and computingresources and are cheaper than traditional sensors Smartsensor nodes can be used to detect measure and gatherinformation from the surrounding environment They canpass the collected data to the user through a number of localdecision-making systems

As a new mode of information acquisition and process-ing wireless sensor networks (WSNs) [1] have now becomea hot research of concern at home and abroad WSN collectsthe information of perceived objects in the network coveragearea through large numbers of sensor nodes deployed in themonitoring region and periodically provides the end userswith information which has been collected and processedby means of the multihop wireless communication method

WSNs which do not acquire the support of the fixed networkhave features of fast expansion and strong survivability andalso have a wide field of applications for military targettracking and monitoring [2 3] natural disasters rescue[4] biological medical health monitoring [5 6] dangerousenvironmental exploration and remote earthquake sensing[7] environmental monitoring [8 9] and special fields likethese In large-scale data acquisition networks the energyof sensor nodes is provided by a small inexpensive batterywhich can run a long time [10] Figure 1 shows the schematicof WSN which consists of sensor nodes base station taskmanagement centers and so on

WSNs are always deployed at a remote location wherepeople are unable to get close and energy cannot be replacedWireless sensor nodes are small and battery-powered whichleads to many problems such as limited capacity of singlenode processing and storage limited communication rangeand limited energy The energy issue is the most criticalUneven energy consumption is an inherent problem inmultihop WSNs Imbalance in energy consumption reduces

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 936376 17 pageshttpdxdoiorg1011552014936376

2 Abstract and Applied Analysis

Server

User access management

Sensor StorageProcessor Network MAC Transceiver

Power support module

Sensing module Processing module Wireless communication module

Gateway

Base station

ACDC

Internet satelliteWLAN and so on

Non-CH

CH

Figure 1 Components of a wireless sensor network

the network lifetime to a large extentThe whole network canbe split because of the death of somenodes ultimately leadingto the termination of the entire network The lifetime of awireless sensor network is a critical parameter for evaluatingthe performance of routing protocols [11 12] Thereforedesigning an effective policy to control energy loss is the coreissue for prolonging the network lifetime [13 14]

In order to solve this problem in WSNs researchershave proposed a variety of methods In [15] the authorsproposed to shorten the hop count of the route to reduce theenergy consumption of end-to-end transmission and usedthe ratio of currently selected path hops and the shortestpath as metrics The authors of [16] presented the optimizedforwarding by fuzzy inference systems (OFFIS) for flat sensornetworksOFFIS protocol favors four descriptors (small hopsshortest path maximum remaining battery power and linkusage) to select the best node from candidate nodes in theforwarding paths A link-aware clusteringmechanism (LCM)for energy-efficient routing in WSNs is proposed to deter-mine an energy-efficient and reliable routing path [17] TheLCM algorithm evaluated the qualification of cluster head(CH) nodes and gateways to construct clusters Liao et al[18] used a balanced clustering algorithm with distributedself-organization for WSNs on the basis of their distanceand density distribution taking into account the optimalconfiguration of clusters This makes it essentially differentfrom the previous clustering algorithms A Stable Election

Protocol (SEP) for clustered heterogeneous WSNs is used toprolong the time interval before the death of the first nodewhich is crucial for many applications where the feedbackfrom the sensor network must be reliable SEP is based onweighted election probabilities of each node to become aCH according to the remaining energy in each node [19]Some of the researchers adopted the combination of fuzzyand clustering method In [20] Gupta et al proposed touse three fuzzy descriptors (residual energy concentrationand centrality) during the cluster-head selection In [21] afuzzy-logic-based clustering approach with an extension tothe energy predication (LEACH-ERE) has been proposedto prolong the lifetime of WSNs using the fuzzy method toevenly distribute the workload It selects the CHs consideringexpected residual energy of the sensor nodes Rana andZaveri [22] exploited the A-star method to search the optimalroute from the source to destination If the sensor nodersquosresidual energy level is below the predefined minimumenergy level it would not participate in routing The workin [23] presented a combination of a fuzzy approach andan A-star algorithm (AF) in terms of balancing energyconsumption and maximization of network lifetime Theauthors used three fuzzy descriptors (the highest remainingbattery power minimum number of hops and minimumtraffic loads) and compared the proposed approach with theA-star search algorithm and fuzzy approach using the samerouting criteria in two different topographical areas

Abstract and Applied Analysis 3

From the aforementioned literatures we note that mostof the present routing algorithms are routing methods basedon flat multipath routing protocols or routing protocols withsimple clustering As for the WSNs with large sensor nodesuneven nodes distribution density and real-time changingnetwork topology the flat multipath routing methods maycause network partition because some nodes that are partof the efficient path are drained from their battery energyquicker In many cases the lifetime of a sensor network isover as soon as the battery power in critical nodes is depletedThe simple clustering routing protocols send the mergingdata directly to the sink node However most of the clusterheads are far away from the sink node and the energy costby transmission is index times increasing with respect todistance As to the above problems this paper presents a newrouting method called CAF which means a cluster algorithmand A-star with fuzzy approach for lifetime enhancement inWSNs The CAF routing method which takes into accountthe advantage of SEP algorithm A-star method and fuzzyinference approach by favoring the remaining power theminimum hops and the traffic numbers of nodes provides agood solution to the problem of uneven energy consumptionand prolongs the lifetime of the network effectively Theproposal adopts a method of single hop within clusters andmultihop among clusters to communicate

The rest of this paper is organized as follows Section 2describes the concept of routing metrics and presents someassumptions In Section 3 the SEP method is put forwardfuzzy approach and A-star algorithm by describing each partof the algorithem in detail In Section 4 the CAF algorithmis proposed Then a radio energy dissipation model andthe simulation results based on simulation environmentparameters and metrics are presented in Section 5 FinallySection 6 concludes this paper

2 Preliminaries

This section describes the concept of routing metrics whichhave been used to prolong the lifetime of WSNs and presentssome assumptions on which all works have been done in thispaper

21 Assumptions

(a) The sensors are randomly (uniformly) distributed andare not mobile the coordinates of the sink and thedimensions of the sensor field are known and the sinkis not energy limited (at least in comparison with theenergy of other sensor nodes) [19] The base stationrsquosposition is fixed and far from sensor nodes in thenetwork

(b) A percentage of the node population is equippedwith more energy than the rest of the nodes in thesame network this is the case of heterogeneous sensornetworks and they are all uniformly distributed inspace [19]

(c) Each node can be informed of the total energy inthe network and set the probability of selected CHsaccording to their residual energy [24]

(d) All sensor nodes are randomly distributed in the areaand each sensor node is assumed to recognize its ownposition as well as that of its neighbors and the sink[23]

(e) Each node has a certain amount of traffic pendingin the nodersquos queue The nodersquos queue includes theapplication traffic and also the traffic that a node hasalready committed to forward [23]

(f) The base station prepares the routing schedule andbroadcasts it to each node The A-star algorithmwhich is used to find the optimal route from the nodeto the base station is applied to each node [23]

22 Definition of Terminology

(a) Remaining power (RP) in WSNs the most criticalaspect of routing is energy efficiency Under thisstandard the size of the battery capacity in the currentnode is in focus Routing protocols adopted by thismeasure tend to select a path which has maximumtotal energy from the source node to the destinationnode That is the node which has more residualenergy is more likely to be selected as the node in thebest path [25 26]

(b) Small multiple hops (SMH) the SMH are a commonstandard in routing protocols which passes througha minimum number of relay nodes from the sourcenode to the destination node The basic idea of thismeasure is that it takes the shortest path to get end-to-end relay nodes and uses less resource consumptionbecause of a minimum of forwarding nodes involved[15 25ndash27]

(c) Traffic numbers (TN) the node traffic numbers canbe defined as the pending amount of traffic in a nodersquosqueue These traffic numbers include the amount ofthe transfer being sent and previously being trans-mitted In the case of the concentrated events of aparticular subregion using the shortest path wouldcause an implosion phenomenon on this path Highloading will cause sensor node data queue overflowsresulting in the loss of important information Inaddition the entire sensor network lifetime would beshortened due to the quick energy consumption ofbatteries in the sensor nodes [26ndash28]

(d) Stable phase it is the time interval between theoperation of the network and the death of the firstsensor node in the network

(e) Period of instability it is the time interval between thedeath of the first node and the death of the last one inthe network

(f) Network lifetime it is the time interval between theoperation of the network and the death of the lastsensor node in the network


3 SEP Method Fuzzy Approach andA-Star Algorithm

31 SEP Method Organizing WSNs into clusters enables theefficient utilization of the limited energy resources of thedeployed sensor nodes Low energy adaptive clustering hier-archy (LEACH) protocol is an application-specific protocolarchitecture [24 29] LEACH protocol is the first one toraise the hierarchical routing protocol of data aggregationwhich is to balance the energy consumption of each node inthe network Periodically CH is elected in each round andeach round of elections is as follows Each node generates arandom number in the range (0 1] if a number is less than athreshold 119879(119899) then the node would be seen as a CH in thecurrent round The threshold is set as

119879 (119899) =

119901opt

1 minus 119901opt sdot (119903 mod (1119901opt))if 119899 isin 119866

0 otherwise(1)

where 119901opt is the probability of becoming a CH 119899 is the tokenof the node 119903 is the number of the current round of electionsand 119866 is the most recent 1119901opt round of the set of the CH

The clustering method in this paper uses the SEP algo-rithm in [19] which is an improved algorithm based on theprotocol of LEACH SEP improves the stable region of theclustering hierarchy process by the characteristic parametersof heterogeneity in which 119898 is the advanced nodes fractionand 120572 is the additional energy factor between advanced andnormal nodes

To achieve the goal of extending the time of stable phaseadvanced nodes have more chance to become cluster headsthan the normal nodes which is equivalent to a fairnessconstraint on energy consumption The new heterogeneoussetting which consists of advanced and normal nodes has noeffect on the spatial density of the network so the priori settingof 119875opt has the same value as the one from (1) SEP methodis based on the weighted election probabilities of each nodein order to become a CH according to the residual energy ineach node

Themain objective of the SEPmethod is to avoid the nodewith low energy becoming a CH and control the number oftheCHs to the optimal number aswell as reducing the unevendistribution of CHs during every round In this way it canreduce the energy cost and extend the life of the network

32 Fuzzy Approach Fuzzy inference is an approximateinference that simulates peoplersquos everyday inference which isbooming in the mathematical core of fuzzy control technol-ogy Zadeh introduced fuzzy logic in the 1960s to representand use fuzzy and uncertain knowledge [30] From the early1980s there have been many scholars engaged in research onartificial intelligence and committed to the study of the fuzzylogic method application in the field of artificial intelligenceThese provide the basis for further improvement of thefuzzy inference or incorporation into the appropriate logicalframework [31ndash33] In recent years many academicians havegradually made various improved methods of the fuzzyinference [34ndash36] At present in the research of the fuzzy

theory and its applications fuzzy logic and fuzzy inferencehave achieved certain results and are moving in the directionof the development of axioms intelligence and informationFuzzy logic has been successfully used in a variety of areasincluding communication systems [37] and prediction ofwater level in reservoirs [38] and as a classifier in faultdiagnosis of roller bearing [39] All of these studies haveachieved good results

The fuzzy logic system a nonlinear input-outputmappingas a whole processes information by using fuzzy set Thefuzzy set 119865 in universe 119880 is expressed as a membershipfunction 120583

119865 the value of which is in the interval [0 1] 119865 can

be expressed as 119865 = (119906 120583119865(119906))119906 isin 119880

The two most important concepts in fuzzy logic are fuzzylanguage variables and fuzzy inference rules

Fuzzy language variables refer to words or sentences in astatement Their value is not a usual number but a fuzzy setis represented in fuzzy languageThis value is a description tosome extent For example the language variable ldquoagerdquo can bereferred to as ldquoyoungrdquo ldquomiddlerdquo or ldquooldrdquo Language variableis an important concept which provides a formal way toquantize language description

The fuzzy inference rule which is also named as the ldquoif-thenrdquo rule is the core of fuzzy logic For example if 119909 is 119860then 119910 is 119861 In this example 119860 and 119861 are language valuesdefined respectively in the 119883119884 range by a fuzzy set Theif part ldquo119909 is 119860rdquo in the fuzzy rule is known as the premise orassumption of the rule and then the part ldquo119910 is 119861rdquo is known asthe result or conclusion In essence this expression describesthe relationship between variables 119909 and 119910 Therefore the if-then rule can be defined as binary fuzzy relations

There is a fuzzy control rule ldquoif119909 is119860119894and119910 is119861

119894 then 119911 is

119862119894rdquo which is achieved by a fuzzy implication (fuzzy relation)

119877119894and is defined as

120583119877119894≜ 120583(119860119894 119861119894rarr119862119894)

(119906 V 119908)

= [120583119860119894(119906) 120583119861119894 (V)] 997888rarr 120583

119862119894(119908)

(2)

where 119860119894and 119861

119894is a fuzzy set 119860

119894times 119861119894in 119880 times 119881 119877

119894≜

(119860119894and 119861

119894) rarr 119862

119894is a fuzzy implication (relation) in 119880 times

119881 times119882 and rarr represents a fuzzy implication functionThemost widely used application in engineering is a rule-

based fuzzy inference system (FIS) which has the followingfour basic elements as shown in Figure 2

The fuzzy system contains all of the applicable fuzzyalgorithms and the necessary ingredients resolving all relatedambiguities It consists of the following four basic elements

(1) Knowledge base it includes the definition of fuzzysets and fuzzy operators

(2) Inference mechanisms they perform all of the outputcalculation

(3) Fuzzifier it represents the true input value as a fuzzyset

(4) Defuzzifier it transfers the output fuzzy sets into realvalue

Knowledge base contains the definitions for each of thefuzzy sets and maintains a system operator to achieve the


Fuzzy systems

Fuzzy setsfuzzyoperatorsKnowledge

base Fuzzy rulesalgorithms

Real input valuesFuzzification Inference computing Defuzzification

Real output values

Figure 2 Fuzzy system structure

basic logic (AND OR etc) as well as expressing fuzzy rulesmapping with a rule reliability matrix Inference units withfuzzifiers and antifuzzies calculate real output values fromthe real input values The fuzzifier represents the input as afuzzy set making the inference unit match with it in the rulesstored in the knowledge base Then inference units calculatethe strength of each rule and output a fuzzy distribution

33 A-Star Algorithm Peter Hart Nils Nilsson and BertramRaphael of Stanford Research Institute (now SRI Interna-tional) first described the A-star algorithm in 1968 [40] It is acombination of heuristic methods such as BFS and methodslike Dijkstra algorithm [41]

A-star search method is a highly efficient heuristic algo-rithm used in the graph searching algorithm and it is alsoused in finding a variable or low cost path Noted for itsperformance and accuracy it is widely used in WSNs Inpractical systems it can preprocess the graph to attain betterperformance [42]

A-star method uses a best-first search and finds a least-cost path from a given initial node to a one goal node (out ofone ormore possible goals) As the A-star traverses the graphit follows a path of the lowest expected total cost or distancekeeping a sorted priority queue of alternate path segmentsalong the way

A-starmethoduses a knowledge-plus-heuristic cost func-tion of node 119899 (usually denoted by 119891(119899)) to determine theorder in which the search visits the nodes in the treeThe costfunction is a sum of two functions Consider the following

(a) The past path-cost function is the known distancefrom the starting node 119904 to the current node 119899 (usuallydenoted by 119892(119899))

(b) A future path-cost function is an admissible ldquoheuristicestimaterdquo of the distance from 119899 to the goal (usuallydenoted by ℎ(119899)) ℎ(119899) depends on the heuristicinformation In the heuristic function the more theheuristic information (the problem of knowledge) isthe less the extending node will be so the destinationnode can be searched quickly

The evaluation function used in the paper is given as

119891 (119899) = 119892 (119899) + ℎ (119899) (3)

The ℎ(119899) part of the 119891(119899) function must be an admissibleheuristic that is it must not overestimate the distance tothe goal Thus for an application-like routing ℎ(119899) mightrepresent the straight-line distance to the goal since that isphysically the smallest possible distance between any twonodes

In short A-star algorithm firstly creates two lists an openlist and a closed list The open list corresponds to a containerof nodes that will be checkedThe closed list corresponds to acontainer of nodes that have already been examined Initiallyput the starting node into the open list Then extending thestarting nodemeans to calculate the evaluation function of alladjacent nodes Assuming that the adjacent nodes are not inthe closed list if the evaluation function values of the nodesare larger than the values of nodes in the open list replacethe nodes in the open list andif not add them to the openlist When it iterates the optimal node will take the top of thepriority list Then check whether it is the destination nodeIf so the algorithm is finished Otherwise repeat the aboveoperations If it does not find a solution then it can guaranteethat no such solution exists A-star algorithm will find a pathwith the lowest possible cost It looks for the shortest pathwith minimum cost which mainly depends on the choice ofthe evaluation function [43]

A-star method can be accomplished as follows

(1) Put the initial node 119878 into the open list let 119892(119878) = 0and 119891(119878) = 0

(2) If the open list is empty then exit the algorithm andreturn false

(3) Move out the first node 119899 from the open list which hasthe smallest 119891(119899) value let it be best node and put itinto the closed-list

(4) If best node is the target node then exit the algorithmand return to the target node

(5) Otherwise extend the best node If the node does nothave the successor node go to (2)


(a) Otherwise generate its entire legal successornodes For each successor node 119899suc calculate119892 = 119892(Best-node) + 119888(Best-node 119899suc)

(b) If it neither belongs to the open list nor to theclosed list then let 119892(119899suc) = 119892 and 119891(119899suc) =119892 + ℎ(119899suc) and put 119899suc into the open list

(c) If 119899suc belongs to either the open list or the closedlist and 119892(119899suc) gt 119892 then let 119892(119899suc) = 119892 and119891(119899suc) = 119892 + ℎ(119899suc) and if 119899suc belongs to theopen list then it is retained If it belongs to theclosed list then it will be moved out from theclosed list into the open list

(d) Reorder the 119891(119899) values of the nodes in ascend-ing order in the open list

(e) Go to (2)

4 Proposed CAF Algorithm

The lifetime of a wireless sensor network is a critical parame-ter for assessing the performance of routing protocols Imbal-ance in energy consumption reduces the network lifetime to alarge extentTherefore designing an effective routingmethodto control energy loss is the core issue for prolonging thenetwork lifetime

The primary goal of this paper is to design a protocolwhich can extend the network lifetime by limiting energyconsumption and balance the distribution of energy costThispaper proposes the CAF routing method which makes use ofthe SEP method fuzzy approach and A-star algorithm Theproposal selects the optimal node to construct the multihoptree by considering the three routing criteria mentioned inSection 22 (ie highest remaining power minimum numberof hops and lowest traffic numbers) The implementation ofthe proposal includes four parts

41 Implementation of the SEP Method As with LEACHSEP in each round consists of two phases set-up phase andstability phase

411 Set-Up Phase The set-up phase is essentially the prepa-ration stage of data transmission This stage is primarily forthe transmission of the control information the formationof the clusters and the random generation of the CHsThis stage without data transmission but with a series ofpreparatory work prior to data transmission fully preparesfor the data transfer

This paper optimizes the CH selection algorithm andaims to let the nodes withmore energy rather than less energybecome the first cluster head in the clusters thus avoidingthe lesser energy nodersquos premature death because of thefast energy consumption while being elected as the CH andsubsequently affecting the network lifetime Therefore theenergy factor can be added during the process of the choosingCHs which ensures that the node with more residual energycan be elected to be the CH and all nodes run out of energyalmost at the same time

The critical aspect of the cluster set-up phase is to set thethreshold as 119879(119899)The selection of the CH in the SEPmethodis carried out from the advanced node and the normal nodeThe ratio of the advanced node in the total number of nodesis119898 Set the total nodes as 119899The energy of the advanced nodeis 120572 times larger than the normal node

SEP provides constraints and limits for balancing energyconsumption in order to extend the lifetime of the steadystateThe advanced node is more likely to become a CH nodethan a normal node which is fairly a constraint for the energyconsumption If the energy of each node is initialized to 119864

0

the energy of each advanced nodes is 1198640sdot (1 + 120572) The total

energy of the initial nodes in the network is 119899 sdot (1 minus 119898) sdot 1198640+

119899 sdot 119898 sdot 1198640sdot (1 + 120572) = 119899 sdot 119864

0sdot (1 + 120572 sdot 119898)

Total energy of the SEP is (1 +120572 sdot119898) times larger than thetotal energy of the LEACH

Let the weighted selection probabilities of the normalnode be 119901nrm and let the weighted selection probabilities ofthe advanced node be 119901adv They are defined as

119901nrm =119901opt

1 + 120572 sdot 119898

119901adv =119901opt

1 + 120572 sdot 119898times (1 + 120572)

(4)

where the ratio of the advanced node in the total number ofnodes is 119898 and the energy of the advanced node is 120572 timeslarger than the normal node

119879(119904nrm) the threshold of the normal node is the averageprobability of the normal node which has not yet becomethe CH in the 119903 round to be the CH 119879(119904adv) the thresholdof advanced node is the average probability of the advancednode which has not yet become the CH in the 119903 round to bethe CH They could be obtained by (5) and (6) Consider

119879 (119904nrm) =

119901nrm

1 minus 119901nrm sdot (119903 mod (1119901nrm))if 119904nrm isin 119866

1015840

0 otherwise(5)

where 119903 is the current round and1198661015840 is the set of normal nodesthat have not been CHs within the last 1119901nrm round at eachstep 119879(119904nrm) is to ensure that each normal node can have theopportunity to become the CH node in the (1119901opt) sdot (1 + 120572 sdot119898) round at each step

119879 (119904adv) =

119901adv

1 minus 119901adv sdot (119903 mod (1119901adv))if 119904adv isin 119866

10158401015840

0 otherwise(6)

where 119903 is the current round and 11986610158401015840 is the set of advancednodes that have not been CHs in the last 1119901adv round at eachstep 119879(119904adv) is to ensure that each advanced node can havethe opportunity to become the CH in the (1119901opt) sdot (1 + 120572 sdot119898)(1 + 120572) round at each step

Figures 3 and 4 have shown the different situation of thecluster figures using the CAF algorithm In these two figures


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100

Figure 3 A clustering wireless sensor network with all nodes beingalive

0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100

Figure 4 A clustering wireless sensor network with some nodesbeing dead

the Voronoi diagram was used to cluster the nodes in WSNConsider a set of coplanar points P For each point Px in theset P you can draw a boundary enclosing all the intermediatepoints lying closer to Px than to other points in the set PSuch a boundary is called a Voronoi polygon and the set ofall Voronoi polygons for a given point set is called a Voronoidiagram which is composed of many Voronoi cells Each cellhas a CH

In these two figures ldquo∘rdquo is a normal node ldquolowastrdquo is a CH ldquo+rdquois an advanced node and ldquo∙rdquo is a dead node If the nodes arealive the nodes which are in the sameVoronoi cell will reportmessages to the CH of this cell

412 Stability Phase The stability phase is the cluster com-munication phase In the LEACHprotocol between the nodein the cluster and the CH and between the CH and the basestation both use direct single hop communication The nodein the cluster is close to the CH so the easy direct com-munication can be used However the CH node is often farfrom the base station and the transport energy consumptionincreases exponentially as the distance increases Thereforedirect communication will consume a relatively large amountof energy of the CH To solve this issue the CAF protocolin this paper uses multihop communication to transmit datafrom the CH to the base station which creates a multihoppath tree using A-star method and fuzzy approach Only theroot node of the path or the independent node which has noparent node needs a long distance communication with thebase station while other CHs only need close communicationwithin the sensing area

After a period of the stability of the data transmissionphase the network reenters the set-up phase of the clusterand goes to the next cycle of the cluster of refactoring Theflowchart of the SEP method is shown in Figure 5

42 Implementation of Fuzzy Approach Find the best nodeby integrating several factors in the problem such as residualenergy traffic numbers and minimum number of hops Thisis equivalent to finding the optimal path from the start nodeto the sink node For the fuzzy method the fuzzy value isprocessed by FIS which includes a rule base and variousways to infer these rules There are a series of if-then rulesin rule base and fuzzy sets are combined with the inputand output fuzzy language variables which are describedby the fuzzy implication operator to construct the FISjointly

To handle uncertainties this paper has used the FISfor the probability computation of each node as shownin Figure 6 The two input variables for the FIS are theresidual energy and the traffic numbers and the single outputparameter is the probability of a node to be selected as thebest node in the optimal path The greater the probabilityvalue will mean that the node has more chance to be abest node The value range of RE is [0 10] the value rangeof TN is [0 10] and the probability is [0 1] Accordingto the current simulation environment the fields of fuzzyvariables are selected for inputs as well as outputs Whenthe environment changes the value ranges can be modifiedas well

The values of the residual energy for fuzzy input variablesare very low (VL) low (L) medium (M) high (H) andvery high (VH) The values of the traffic number for fuzzyinput variables are very low (VL) low (L) medium (M) high(H) and very high (VH) The values of the node probabilityfor fuzzy output variables are very small (VS) small (S)medium (M) large (G) and very large (VG) Membershipfunction (MF) is a curve and it defines how each pointof the input space is mapped to a membership between 0and 1 This paper selects the Gaussian membership functionas the membership function for input and output variablesEquation (7) shows the Gaussian membership function asfollows

119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)

where 119888 and 120590 are the center and standard deviation ofmembership function respectively

The membership functions of the input-output fuzzyvariables and surface viewer of the FIS are given inFigures 7 and 8 respectively Nodes with small probabilitywould not be selected as the next hop

The probability calculation is accomplished using prede-fined fuzzy if-then mapping rules to handle the uncertaintyBased on the two fuzzy input variables 25 fuzzy mappingrules are defined in Table 1 From the fuzzy rules thefuzzy variable probability can be obtained which has to be


Start

Initialize the network environment

End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax

Generate a random number R1

R1 lt KCH

Mark node as Nadv Mark node as Nnrm

Calculate the threshold of nodes ascluster head nodes T(Snrm ) and T(Sadv )


S(i) node is Nnor

S(i) is nor node S(i) is adv nodeMark S(i) node as NCH and

store it in C[i]

i le n

Calculate the distance d1 d2 middot middot middot dkfrom each node to the cluster head in C[k]

Select the nearest cluster head and mark thenode as the member of it

R2 lt T(Snrm ) R2 lt T(Sadv )

maximum number of rounds Rmax

network node n best cluster head KCH and

Figure 5 Flowchart of the SEP method

transformed to a single crisp number that is a form whichcan be used in practice In the CAF approach the processof output in fuzzy inference is actually a defuzzifier processGenerally fuzzy rules can be generated either from heuristicsor from experimental data Fuzzy rules are used with theprinciple a node which holds more remaining power andlower traffic numbers has a higher probability The greaterthe probability of nodes as well as the lower the node hopsis the easier the node will be selected to be the next hop inthe optimal path

The defuzzification method adopted in this paper is themost commonly used formula of gravity calculation The

value of the probability obtained using the gravity method ofthe defuzzifier could be represented as

pro value =sum119899

119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)

where 119880119896is the output of the rule base 119896 and 119888

119896is the center

of the output membership functionsThe flowchart of the AFalgorithm is shown in Figure 9

43 Implementation of A-Star Algorithm A-star algorithmwhich is used to find the optimal route from the CHs to the


Table 1 If-then rules

Number Antecedent ConsequentRP(119899) TN(119899) Probability(119899)

1 VL VL S2 VL L VS3 VL M VS4 VL H VS5 VL VH VS6 L VL M7 L L M8 L M S9 L H S10 L VH VS11 M VL G12 M L M13 M M M14 M H S15 M VH S16 H VL VG17 H L G18 H M G19 H H M20 H VH M21 VH VL VG22 VH L VG23 VH M VG24 VH H G25 VH VH G

Residual energy

Traffic numberFuzzifier Inference

engine Defuzzifier

Fuzzy rule base

Input Output

Probability

Figure 6 Fuzzy structure with two inputs and one output

base station is applied to each CH and builds a multihoptransmission path among CHs using a weight function 119891(119894)Evaluation function 119891(119894) can be obtained via

119891 (119894) = 119908 sdot (cos 119905 (119894)

max cos 119905 (119894)) + (1 minus 119908) sdot (

119863max minus ℎ (119894)

119863max)

(9)

where cos 119905(119894) = 119908 sdot (119873119888(119894)max cos 119905(119894)) + (1 minus 119908) sdot ((119863max minus119892(119894))119863max) and cos 119905(119894) is cost of the CH node of the 119894 Thevalue of 119873119888(119894) is obtained by the fuzzy inference algorithmwhich considers the residual energy theminimumnumber of

0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1

Figure 7 Membership functions of the input-output fuzzy vari-ables

0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN

Figure 8 Surface viewer of the FIS with two inputs and one output

hops and the traffic numbers of each node Its range is [0 1]The maximum cost of the node 119894 is max cos 119905(119894)

119863max is the maximum distance from the base station inthe networkThe actual cost from the initial node (start node)to the next node 119894 is 119892(119894) The estimated cost of the optimalpath from node 119894 to the target node (destination node) is ℎ(119894)which depends on the heuristic information of the problemarea [44] In the above formula 119908 is a constant 0 lt 119908 lt

1 which is used to balance the energy cost of the node anddistance parameters Clearly the node with a larger cost willbe closer to the base station

44 Implementation of CAF Algorithm CAF algorithm isdivided into two steps Firstly apply the Stable ElectionProtocol (SEP) algorithm to the wireless sensor network inorder to cluster The nodes are divided into advanced nodesand normal nodes The advanced nodersquos energy is two timesthe normal node It is proposed to let the candidate nodesrun for the CH based on the residual energy within a certainrange to make the even distribution of the CH Secondlyamong the CH nodes the combined algorithms of the fuzzyinference and A-star method are used taking into accountthe residual energy minimum hops count and traffic loadsof each CH and looking for the best path from the sourcenode to the destination node Figures 10 and 11 present theresults using the CAF routing method to find the optimal


Start

using gravity method to make the defuzzification

Use fuzzy method and fuzzy rule to inference calculation

OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure

Remove the first node N from open listand add it to closed list

N is the target nodeExit successful

routing path is constructed bynodes in closed list

Use the Voronoi algorithm to divideexperimental zone based on CHs

Obtain the neighboring CH of CH N

according to the divided zone

Extend N and put the neighbor of Nin the expanded list

i from expanded listin open list

Put all nodes in expanded list to open list

in open list to neighboring CH

W(i j) = factorlowast (pro(i j)max cost) +(1 minus factor) lowast ((max distance minusdistance)max distance)

Arrange nodes in the open list inascending order by probability value

Calculate pro(m) accordingto the fuzzy inference phase

Calculate RP(i j) and TN(i j)

Get the fuzzification of RP(i j) andTN(i j) respectively and map them tothe corresponding fuzzy sets as fuzzy

inputs

Get node cost pro(i j)

Calculate the distance(i j) from CH C(i)

the value of pro(m) is [0 1]

and closed list = emptyPut start node N into open list Let f = h

of node C(i) in the open list (i)

Figure 9 Flowchart of the AF algorithm where distance (119894 119895) is the distance between the node 119894 and one of its neighboring nodes 119895 RP(119894 119895)and TN(119894 119895) are the remaining energy and traffic load of the neighboring node 119895 of node 119894 respectively Pro(119894 119895) is the node cost of theneighboring node 119895 of node 119894

route The pseudocode of the proposed CAF routing methodis described as Algorithm 1

5 Performance Evaluations

The simulation environment parameters and metrics arepresented in this section followed by the simulation resultsThis paper adopts the combinedmethods of the SEPmethodthe A-star algorithm and the fuzzy approach and experi-mental simulation results are given to evaluate the merits

of the algorithm This paper selects the SEP AF and A-staralgorithms as a comparison of the CAF method Simulationresults show that the CAF algorithm performs better thanothers

51 Simulation Environment Construct a simulation envi-ronment of the algorithm test using theMatlab2010a In orderto verify the adaptive capacity of the CAF algorithm underdifferent network topologies this paper sets two topologicalregions A and BThe region size of A is 100mtimes100m while


Input119873 a network119901 optimal election probability of a node to become aCH119899 number of nodes in the field119898 percentage of nodes that are advanced119886 multiple of the advanced node is 120572 times greaterthan the normal node119903 the number of rounds

outputNA(119903) the number of alive nodes in each roundRP(119903) the remaining power of each node ineach round119877 FND the round of first node are dead119877 HNA the round in which half of the nodes are alive

FunctionFuzzyLogic(Remain Power Traffic Number)Insert Open(119909val 119910val parent 119909val parent 119910valℎ119899 cost 119909119899)Expand Array(node 119909 node 119910 hn 119909Target119910Target (LOSED MAX 119883 MAX 119884 exp neibor)

Max Fn(OPEN OPEN COUNT 119909Target 119910Target)Seek Node Id(C xval 119910val)Node Index(OPEN 119909val 119910val)Arrow(119875start 119875end)

Initialization(1) 119903 larr 0(2) 119870 larr 119899 lowast 119901(3) RPlarr (119878(119862 (119894) sdot 119894119889) sdot 119864) lowast 10(4) TNlarr rand(1) lowast10(5) Pro(119894)larr FuzzyLogic(RP TN)(6) 119882 cost2dislarr 07Main(7) For 119894 = 1 1 n(8) If (temp rnd gt=119898 lowast 119899 + 1)(9) 119878(119894)typelarr 119873nor(10) else 119878(119894)typelarr 119873adv(11) end if(12) end for(13) lowast for every clustering round lowast(14) For 119894 = 1 1 119899(15) if((119878(119894)type==119873nor ampamp (temp rand lt= (119901nrm

(1 ndash 119901nrm lowastmod(119903 round(1119901nrm)))))))(16) 119878(119894)typelarr 119873ch Election CHs from normal nodes(17) end if(18) if((119878(119894)type ==119873adv ampamp (temp rand lt= (119901adv

(1 minus 119901advlowastmod(119903 round(1119901adv)))))))(19) 119878(119894)typelarr 119860ch Election CHs from advanced nodes(20) end if(21) end for(22) Voronoi(119883119884)(23) [V 119888] larr voronoin([119883 119884]1015840)(24) For 119896 = 1 1 cluster minus1(25) lowastFind the optimal route using the A-star methodlowast(26) MH(119894) larr sqrt((119878 (119862 (119894) sdot 119894119889) sdot 119909119889 minus (119878 (119899 + 1) sdot 119909119889))and2 + (119878(119862 (119894) sdot 119894119889)) sdot 119910119889

minus (119878(119899 + 1)119910119889))and2)

(27) NC(119894)larr119882 cost2dislowast(Pro(119894)Max cost) + (1 minus119882 cost2dis)lowast ((Max distance-MH(119894))Max distance)

(28) 119862(119894)verlarr [V(119888119894 1) V(119888119894 2)](29) if the polygon 119894 and 119895 have two same vertexes or more(30) 119862(119894)neighbor(119899Nei)larr 119895 Find the neighbor of each CH

Algorithm 1 Continued


(31) end if(32) while((119909Node sim= 119909Target ||119910 Node sim= 119910Target) ampamp

NoPath == 1)(33) if node 119894 is in Exp Array table and not in OPEN table(34) OPEN(OPEN COUNT)larr Insert Open(

Exp Array(119894 1) Exp Array(119894 2) 119909Node119910Node Exp Array(119894 3) Exp Array(119894 4) Exp Array(119894 5))

(35) end if(36) index max nodelarrMax Fn(OPEN OPEN COUNT

119909Target 119910Target)(37) if (index max node sim= minus1)(38) CLOSED(CLOSED COUNT1)larr 119909Node(39) CLOSED(CLOSED COUNT2)larr 119910Node(40) else NoPathlarr 0(41) end if(42) end while(43) while(parent 119909 sim= 119909Start || parent 119910 sim= 119910Start)(44) parent 119909 larr OPEN(Node Index(OPEN 119909val 119910val)4)(45) parent 119910 larr OPEN(node index(OPEN 119909val 119910val)5)(46) 119894nodelarr node index(OPEN parent 119909parent 119910)(47) 119901Startlarr [parent 119909 parent 119910](48) 119901Endlarr [119909val 119910val](49) Arrow(119901Start119901End)(50) end while(51) end for

Algorithm 1 Proposed CAF method

0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100

Figure 10 The CAF routing method result with all nodes beingalive

the region size of B is 200m times 50m According to the actualsituation coordinates of the base station are fixed and are farfrom the sensor nodes in the network so this paper sets thebase station coordinates as follows A is (50 300) and B is(100 150) In the two different topological regions A and B100 nodes are placed randomly as a test network

For nonclustering routing algorithms such as AF andA-star each node has the same initial energy of 055 J Forclustering routing algorithms such as SEP and CAF theinitial energy of the advanced node is 1 J and normal nodersquosinitial energy is 05 J The size of the message that nodes sendto their CHs as well as the size of the (aggregate) message thata CH send to the sink is 2000 bits The packet size is 2000 bitand the initial ratio of CH is 01

0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100

Figure 11 The CAF routing method result with some nodes beingdead

Simulations are done using the values 50 nJbit and100 pJbitm2 for 119864elec and 119864amp respectively Table 2 presentsthe systems parameters in detail

52 Energy Model Once the node is placed it is no longermoved and the node died when the energy is zero Theexperiment utilized a first-order radio model [45] which iscommonly used in theWSNsThe radio energy consumptionmodel is shown in Figure 12

The energy consumed by the sensor nodes in this paper isdivided into two categories First is the energy consumed bythe normal or advanced node (non-CHandnonparent node)According to Figure 12 when the distance is 119889 and transmit


Bit data Bit data

Transmitting Transmittingelectronics electronicsamplifier

Receiving

ETx (l d)d

lEelec lEelecl120576fsd

2

Figure 12 Radio energy dissipation model

119871-bit message the energy required by common sensor nodeswhile transmitting electronics can be defined as

119864NTx (119871 119889) = 119871 sdot 119864elec + 119871 sdot 120576fs sdot 119889

2 if 119889 le 1198890

119871 sdot 119864elec + 119871 sdot 120576mp sdot 1198894 if 119889 gt 119889

0

(10)

where 119871 is the packet size of transmitted data 119889 is the distancebetween the sender and the receiver 119864elec is power circuitsfor sending or receiving determined by the digital codingmodulation filtering and spread signal and other factors and1198890is the distance threshold When the transmission distance

is less than the threshold the free space channel model isused On the contrary when the transmission distance ismore than the threshold themultipath fading channel modelis used [46] Amplifier power 120576fs and 120576mp is determined bytransmission distance and the accepting error rate When119889 = 119889

0 let 1198890= radic120576fs120576mp

The received electronics energy consumption when thenormal node receives the 119871-bit message is shown as

119864NRx = 119871 sdot 119864elec (11)

The next is the energy consumed by the CH or parentnode (PN) As a CH or PN except that the sending andreceiving messages consume energy the data processing alsoconsumes the energyThe energy consumed by the CH or PNwhile transferring 119871-bit data can be obtained via

119864CHPTx (119871 119889) = 119871 sdot 119864elec + 119871 sdot 119864DA + 119871 sdot 120576fs119889

2 if 119889 le 1198890

119871 sdot 119864elec + 119871 sdot 119864DA + 119871 sdot 120576mp1198894 if 119889 gt 119889

0

(12)

where 119864DA is the energy consumed by CH or PN whileprocessing 1 bit message and 119889 is the distance betweenthe sender and the receiver The received energy circuitconsumption for CH or PN when they receive 119871-bit messagecan be defined as

119864CHPRx = 119871 sdot (119864elec + 119864DA) (13)

The values used in the first-order radio model aredescribed in Table 2

Table 2 Simulation parameters

Parameter ValueTopographical area (meter)

A 100m times 100mB 200m times 50m

Sink locationA (50 300)B (100 150)

Initial energy of nodeAF A-star 055 JSEP CF CA CAF

Normal nodes 05 JAdvanced nodes 1 J

Radio model119864DA 5 nJbit119864Txelec or 119864

Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4

Number of nodes 100Limit of transmission distance (meters) 30mPacket data size 2 k bit119875opt 01119898 01120572 1

53 Simulation Results Handy et al [47] proposed themetricfirst node dies (FND) and half of the nodes alive (HNA)FND metric is useful in these scenarios where it is necessarythat all nodes stay alive as long as possible since networkquality decreases considerably as soon as one node dies suchas intrusion or fire detection In these cases it is importantto know when the first node dies HNA metric is useful inthese scenarios where sensors are placed in proximity to eachother Adjacent sensors could record related or identical dataIn these cases the loss of a single or few nodes does notautomatically diminish the quality of service of the networkHence HNA metric is useful in densely deployed sensornetworks

The rounds of FND and HNA are simulated using A-starAF CF CA SEP and CAF the six different approaches for


54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500

A-star AF CF CA SEP CAF

FNDHNA

Tim

e (ro

unds

)

Figure 13 Simulation results of FND and HNA in sample networkin Area A

FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283

A-star AF CF CA SEP CAF0

500

1000

1500

2000

2500

Tim

e (ro

unds

)

Figure 14 Simulation results of FND and HNA in sample networkin Area B

both areas A and B The results are shown in Figures 13 and14 respectively Clearly the CAFmethod outperforms the A-star AF CFCA and SEP algorithms in both areas A andB InFigure 13 the FND improvements 3240 1216 121 30and 104 are accomplished by comparing the algorithms ofA-star AF CF CA and SEP respectively HNA improves by282 270 51 39 and 57 while in Figure 14 theFND improvements 3660 766 151 37 and 42 areaccomplished by comparing the algorithms of A-star AF CFCA and SEP respectively HNA improves by 275 26441 43 and 33 It can be seen that the times for the FNDand HNA in the proposed method are much longer than thetimes for the FND andHNA for all the threemethods in bothareas A and B

Moreover Figures 15 and 16 show the simulation resultsfor this configuration It can be seen that the number of nodesalive in the proposed method is always higher than that of allA-star AF CF CA and SEP algorithms The two figures canalso show that the performance of the SEP and CF algorithmis obviously affected by the topographic regions The SEPand CF algorithm performance varies greatly in different

0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)

Cluster and A-star and fuzzyA-star and fuzzyCluster and fuzzy

Cluster and A-starA-starSEP

Figure 15 Distribution of the nodes being alive according to thenumber of rounds in Area A

topographic regions In Figure 15 with the position of thesink node (50 300) the survival rate of nodes for SEP and CFalgorithm is lower than those in Figure 16 with the position ofthe sink node (200 50) while the CAF algorithm this paperproposed has achieved good performance both in areasA andB The CAF algorithm has a strong adaptive capacity underdifferent network topologies

Figures 17 and 18 show the total remaining energyof a WSN with the round number increasing for the sixapproaches in areas A and B respectively This validates thatthe proposed CAF method actually consumes less energythan the A-star AF CF CA and SEP algorithms A betterenergy balance in aWSN is achieved by the proposedmethodin both areas A and BThe simulation results indicate that theproposed CAF is superior to the other mechanisms in termsof FNDHNA residual total energy and the number of sensornodes alive with different topographical areas

From the aforementioned simulation results it is clearthat the proposed CAF method outperforms the AF algo-rithm CF approach CA algorithm SEP method and A-star approach in terms of balancing energy consumption andmaximization of network lifetime This is because the CAFalgorithm selects a CH first The CH is selected randomlyfrom the advanced nodes as well as normal nodes and italways selects the node with the highest residual energy asa CH The sensor nodes are clustered in the zone using theVoronoi cell and themember nodes sendmessages directly tothe CH instead of the base station which reduces themessagecommunication distance of themember nodes greatlyMean-time the AF routing algorithm is used to find the optimalpath among the CHs The multiple factors including theresidual energy traffic load andminimumhop are taken intoconsideration to propose a metric of finding the next hop inthe optimal path In this way the energy cost of CHs canbe controlled in minimum in message communication TheCAF algorithm guarantees that the discovered routing pathcan remain consistent


0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



Figure 16 Distribution of the nodes being alive according to thenumber of rounds in Area B

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10

Figure 17 Distribution of the total remaining energy according tothe number of rounds in Area A

6 Conclusions

Energy is amajor factor for designingWSNs in the real worldUnfortunately the battery energy is inherently limitedThere-fore efficient utilization of the energy is most important Toimprove the energy efficiencymany algorithms are proposedOne of themain characteristics of these algorithms shows thatthe network lifetime is highly related to the routing selectionThis paper has proposed a cluster algorithm and the A-starwith fuzzy method called CAF providing an energy-efficientrouting in WSNs The main goal of the CAF algorithm isto prolong the lifetime of the WSN by evenly distributingthe workload retaining the remaining energy and reducingthe number of the hops To achieve this goal this paper hasmostly focused on selecting proper CHs from existent sensornodes and the optimum next hop from the CHs The SEPalgorithm has been used to choose the best CHs while the A-star method and fuzzy approach is used to select the propernext hop of the optimal path

Simulation results confirming the CAF method producethe following advantages Firstly this algorithm is stable and

0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)



Figure 18 Distribution of the total remaining energy according tothe number of rounds in Area B

adaptable to different topologies secondly the algorithmhas low energy consumption and effectively balances theenergy consumption of each sensor node and the algorithmsignificantly extends the survival time of the network

In conclusion the proposed CAF achieves a better FNDHNA residual total energy and the number of sensor nodesalive than the original SEP A-star CF CA andAF techniqueswith different topographical areasHowever in the real worldthere may be certain situations that one or even more ofthe sensors in the critical pathway becomes intermittent inthe ability to function normally Such behavior may addperformance noise (fluctuations) into the WSNs Ongoingresearch is investigating the practicality of using the CAFrouting protocol for data gathering in WSNs

Conflict of Interests

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

Acknowledgments

The authors would like to thank to the Natural ScienceFoundation of China (61073193 and 61300230) the KeyScience and Technology Foundation of Gansu Province(1102FKDA010) Natural Science Foundation of GansuProvince (1107RJZA188) and the Science and TechnologySupport Program of Gansu Province (1104GKCA037) forsupporting this research

References

[1] I F Akyildiz W Su Y Sankarasubramaniam and E Cayirci ldquoAsurvey on sensor networksrdquo IEEE Communications Magazinevol 40 no 8 pp 102ndash105 2002

[2] X Wang J Ma S Wang and D Bi ldquoDistributed energyoptimization for target tracking in wireless sensor networksrdquoIEEE Transactions on Mobile Computing vol 9 no 1 pp 73ndash86 2010


[3] H Mahboubi A Momeni A G Aghdam K Sayrafian-Pourand V Marbukh ldquoAn efficient target monitoring scheme withcontrolled node mobility for sensor networksrdquo IEEE Transac-tions onControl SystemsTechnology vol 20 no 6 pp 1522ndash15322011

[4] M Castillo-Effen D H Quintela R Jordan W Westhoff andWMoreno ldquoWireless sensor networks for flash-flood alertingrdquoin Proceedings of the 5th IEEE International Caracas Conferenceon Devices Circuits and Systems (ICCDCS rsquo04) pp 142ndash146November 2004

[5] Q Ling Z Tian Y Yin and Y Li ldquoLocalized structural healthmonitoring using energy-efficient wireless sensor networksrdquoIEEE Sensors Journal vol 9 no 11 pp 1596ndash1604 2009

[6] L Wang G-Z Yang J Huang et al ldquoA wireless biomedicalsignal interface system-on-chip for body sensor networksrdquoIEEE Transactions on Biomedical Circuits and Systems vol 4no 2 pp 112ndash117 2010

[7] G Werner-Allen K Lorincz M Welsh et al ldquoDeploying awireless sensor network on an active volcanordquo IEEE InternetComputing vol 10 no 2 pp 18ndash25 2006

[8] H-C Lin Y-C Kan and Y-M Hong ldquoThe comprehensivegateway model for diverse environmental monitoring uponwireless sensor networkrdquo IEEE Sensors Journal vol 11 no 5 pp1293ndash1303 2011

[9] D Dardari A Conti C Buratti and R Verdone ldquoMathe-matical evaluation of environmental monitoring estimationerror through energy-efficient wireless sensor networksrdquo IEEETransactions on Mobile Computing vol 6 no 7 pp 790ndash8022007

[10] S Cui A J Goldsmith and A Bahai ldquoEnergy-constrainedmodulation optimizationrdquo IEEE Transactions onWireless Com-munications vol 4 no 5 pp 2349ndash2360 2005

[11] J N Al-Karaki andA E Kamal ldquoRouting techniques inwirelesssensor networks a surveyrdquo IEEEWireless Communications vol11 no 6 pp 6ndash27 2004

[12] S C Ergen and P Varaiya ldquoPEDAMACS power efficient anddelay awaremedium access protocol for sensor networksrdquo IEEETransactions on Mobile Computing vol 5 no 7 pp 920ndash9302006

[13] J C F Li S Dey and J Evans ldquoMaximal lifetime power andrate allocation for wireless sensor systems with data distortionconstraintsrdquo IEEE Transactions on Signal Processing vol 56 no5 pp 2076ndash2090 2008

[14] C Karakus A Gurbuz and B Tavli ldquoAnalysis of energyeffficiency of compressive sensing in wireless sensor networksrdquoIEEE Sensors Journal vol 13 no 5 pp 1999ndash2008 2013

[15] M-J Tsai H-Y Yang and W-Q Huang ldquoAxis-based virtualcoordinate assignment protocol and delivery-guaranteed rout-ing protocol in wireless sensor networksrdquo in Proceedings of the26th IEEE International Conference on Computer Communica-tions (INFOCOM rsquo07) pp 2234ndash2242 May 2007

[16] M A Azim and A Jamalipour ldquoPerformance evaluation ofoptimized forwarding strategy for flat sensor networksrdquo inProceedings of the 50th Annual IEEEGlobal TelecommunicationsConference (GLOBECOM rsquo07) pp 710ndash714 November 2007

[17] S-S Wang and Z-P Chen ldquoLCM a link-aware clusteringmechanism for energy-efficient routing in wireless sensor net-worksrdquo IEEE Sensors Journal vol 13 no 2 pp 728ndash736 2013

[18] Y Liao H Qi and W Li ldquoLoad-balanced clustering algorithmwith distributed self-organization for wireless sensor networksrdquoIEEE Sensors Journal vol 13 no 5 2013

[19] G Smaragdakis I Matta and A Bestavros ldquoSEP a stableelection protocol for clustered heterogeneous wireless sensornetworksrdquo in Proceedings of the 2nd International Workshop onSensor and Actor Network Protocols and Applications (SANPArsquo04) Boston Mass USA 2004

[20] I Gupta D Riordan and S Sampalli ldquoCluster-head electionusing fuzzy logic for wireless sensor networksrdquo in Proceedings ofthe 3rd Annual Communication Networks and Services ResearchConference (CNSR rsquo05) pp 255ndash260 May 2005

[21] J S Lee and W L Cheng ldquoFuzzy-logic-based clusteringapproach for wireless sensor networks using energy predica-tionrdquo IEEE Sensors Journal vol 11 no 9 pp 2891ndash2897 2012

[22] K M Rana and M A Zaveri ldquoASSER a routing method toextend life of two-tiered wireless sensor networkrdquo InternationalJournal of Advanced Smart Sensor Network Systems vol 11 no2 pp 1ndash16 2011

[23] I S Alshawi L YanW Pan and B Luo ldquoLifetime enhancementin wireless sensor networks using fuzzy approach and A-staralgorithmrdquo IEEE Sensors Journal vol 12 no 10 pp 3010ndash30182012

[24] W Heinzelman Application-specific protocol architectures forwireless networks [PhD thesis] Massachusetts Institute ofTechnology 2000

[25] Y M Lu and V W S Wong ldquoAn energy-efficient multipathrouting protocol for wireless sensor networksrdquo InternationalJournal of Communication Systems vol 20 no 7 pp 747ndash7662007

[26] W Dargie and C Pollabauer ldquoNetwork layerrdquo in Fundametalof Wireless Sensor Networks Theory and Practice pp 163ndash204Wiley 2010

[27] S-Y Chiang and J-L Wang ldquoRouting analysis using fuzzylogic systems in wireless sensor networksrdquo in Knowledge-BasedIntelligent Information and Engineering Systems vol 5178 ofLecture Notes in Computer Science no 2 pp 966ndash973 2008

[28] C Park and I Jung ldquoTraffic-aware routing protocol for wirelesssensor networksrdquo in Proceedings of the International Conferenceon Information Science and Applications pp 1ndash8 April 2010

[29] W B Heinzelman A P Chandrakasan and H Balakrish-nan ldquoAn application-specific protocol architecture for wirelessmicrosensor networksrdquo IEEE Transactions onWireless Commu-nications vol 1 no 4 pp 660ndash670 2002

[30] L A Zadeh ldquoSoft computing and fuzzy logicrdquo IEEE Softwarevol 11 no 6 pp 48ndash56 1994

[31] K L Tang and R J Mulholland ldquoComparing fuzzy logic withclassical controller designsrdquo IEEE Transactions on SystemsManand Cybernetics vol 17 no 6 pp 1085ndash1087 1987

[32] C-W Xu and Y-Z Lu ldquoFuzzy model identification and self-learning for dynamic systemsrdquo IEEE Transactions on SystemsMan and Cybernetics vol 17 no 4 pp 683ndash689 1987

[33] C C Lee ldquoFuzzy logic in control systems fuzzy logiccontrollermdashpart Irdquo IEEE Transactions on Systems Man andCybernetics vol 20 no 2 pp 404ndash418 1990

[34] J J Simpson and R H Keller ldquoAn improved fuzzy logicsegmentation of sea ice clouds and ocean in remotely sensedarctic imageryrdquo Remote Sensing of Environment vol 54 no 3pp 290ndash312 1995

[35] J Schneider N Bitterlich andG Schulze ldquoImproved sensitivityin the diagnosis of gastro-intestinal tumors by fuzzy logic-basedtumor marker profiles including the tumorM2-PKrdquo AnticancerResearch vol 25 no 3A pp 1507ndash1515 2005


[36] M N Uddin and M A Rahman ldquoHigh-speed control ofIPMSM drives using improved fuzzy logic algorithmsrdquo IEEETransactions on Industrial Electronics vol 54 no 1 pp 190ndash1992007

[37] Q Deng D-L Xie and S-Z Chen ldquoA fuzzy logic based QoSevaluationmodel for wireless sensor networkrdquo in Proceedings ofthe 5th International Conference on Wireless CommunicationsNetworking and Mobile Computing (WiCOM rsquo09) pp 1ndash4September 2009

[38] F-J Chang and Y-T Chang ldquoAdaptive neuro-fuzzy inferencesystem for prediction of water level in reservoirrdquo Advances inWater Resources vol 29 no 1 pp 1ndash10 2006

[39] V Sugumaran and K I Ramachandran ldquoAutomatic rule learn-ing using decision tree for fuzzy classifier in fault diagnosis ofroller bearingrdquo Mechanical Systems and Signal Processing vol21 no 5 pp 2237ndash2247 2007

[40] P E Hart N J Nilsson and B Raphael ldquoA formal basisfor the heuristic determination of minimum-cost pathsrdquo IEEETransactions on Systems Science and Cybernetics vol 4 no 2pp 100ndash107 1968

[41] S Bandi and D Thalmann ldquoPath finding for human motionin virtual environmentsrdquo Computational Geometry Theory andApplications vol 15 no 1ndash3 pp 103ndash127 2000

[42] DDelling P Sanders D Schultes andDWagner ldquoEngineeringroute planning algorithmsrdquo Algorithmics of Large and ComplexNetworks Springer vol 5515 pp 117ndash139 2009

[43] X H Li S H Hong and K L Fang ldquoWSNHA-GAHR a greedyandAlowast heuristic routing algorithm for wireless sensor networksin home automationrdquo IET Communications vol 5 no 13 pp1797ndash1805 2011

[44] K M Passino and P J Antsaklis ldquoMetric space approach to thespecification of the Heuristic function for the Alowast algorithmrdquoIEEE Transactions on Systems Man and Cybernetics vol 24 no1 pp 159ndash166 1994

[45] W R Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)pp 1ndash10 January 2000

[46] T S RappaportWireless Communications Principles and Prac-tice Prentice-Hall Englewood Cliffs NJ USA 2002

[47] M J Handy M Haase and D Timmermann ldquoLow energyadaptive clustering hierarchy with deterministic cluster-headselectionrdquo in Proceedings of the International Workshop Mobileand Wireless Communication Networks pp 368ndash372 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Server

User access management

Sensor StorageProcessor Network MAC Transceiver

Power support module

Sensing module Processing module Wireless communication module

Gateway

Base station

ACDC

Internet satelliteWLAN and so on

Non-CH

CH

Figure 1 Components of a wireless sensor network

the network lifetime to a large extentThe whole network canbe split because of the death of somenodes ultimately leadingto the termination of the entire network The lifetime of awireless sensor network is a critical parameter for evaluatingthe performance of routing protocols [11 12] Thereforedesigning an effective policy to control energy loss is the coreissue for prolonging the network lifetime [13 14]

In order to solve this problem in WSNs researchershave proposed a variety of methods In [15] the authorsproposed to shorten the hop count of the route to reduce theenergy consumption of end-to-end transmission and usedthe ratio of currently selected path hops and the shortestpath as metrics The authors of [16] presented the optimizedforwarding by fuzzy inference systems (OFFIS) for flat sensornetworksOFFIS protocol favors four descriptors (small hopsshortest path maximum remaining battery power and linkusage) to select the best node from candidate nodes in theforwarding paths A link-aware clusteringmechanism (LCM)for energy-efficient routing in WSNs is proposed to deter-mine an energy-efficient and reliable routing path [17] TheLCM algorithm evaluated the qualification of cluster head(CH) nodes and gateways to construct clusters Liao et al[18] used a balanced clustering algorithm with distributedself-organization for WSNs on the basis of their distanceand density distribution taking into account the optimalconfiguration of clusters This makes it essentially differentfrom the previous clustering algorithms A Stable Election

Protocol (SEP) for clustered heterogeneous WSNs is used toprolong the time interval before the death of the first nodewhich is crucial for many applications where the feedbackfrom the sensor network must be reliable SEP is based onweighted election probabilities of each node to become aCH according to the remaining energy in each node [19]Some of the researchers adopted the combination of fuzzyand clustering method In [20] Gupta et al proposed touse three fuzzy descriptors (residual energy concentrationand centrality) during the cluster-head selection In [21] afuzzy-logic-based clustering approach with an extension tothe energy predication (LEACH-ERE) has been proposedto prolong the lifetime of WSNs using the fuzzy method toevenly distribute the workload It selects the CHs consideringexpected residual energy of the sensor nodes Rana andZaveri [22] exploited the A-star method to search the optimalroute from the source to destination If the sensor nodersquosresidual energy level is below the predefined minimumenergy level it would not participate in routing The workin [23] presented a combination of a fuzzy approach andan A-star algorithm (AF) in terms of balancing energyconsumption and maximization of network lifetime Theauthors used three fuzzy descriptors (the highest remainingbattery power minimum number of hops and minimumtraffic loads) and compared the proposed approach with theA-star search algorithm and fuzzy approach using the samerouting criteria in two different topographical areas




2 Preliminaries


21 Assumptions

















119879 (119899) =

119901opt


0 otherwise(1)














119894 then 119911 is



120583119877119894≜ 120583(119860119894 119861119894rarr119862119894)

(119906 V 119908)

= [120583119860119894(119906) 120583119861119894 (V)] 997888rarr 120583

119862119894(119908)

(2)

where 119860119894and 119861


119894times 119861119894in 119880 times 119881 119877

119894≜

(119860119894and 119861

119894) rarr 119862











Fuzzy systems




Real output values










119891 (119899) = 119892 (119899) + ℎ (119899) (3)














(e) Go to (2)









0




0sdot (1 + 120572 sdot 119898)



119901nrm =119901opt

1 + 120572 sdot 119898

119901adv =119901opt

1 + 120572 sdot 119898times (1 + 120572)

(4)



119879 (119904nrm) =

119901nrm


1015840

0 otherwise(5)


119879 (119904adv) =

119901adv


10158401015840

0 otherwise(6)




0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100









119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)





Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2 Preliminaries


21 Assumptions

















119879 (119899) =

119901opt


0 otherwise(1)














119894 then 119911 is



120583119877119894≜ 120583(119860119894 119861119894rarr119862119894)

(119906 V 119908)

= [120583119860119894(119906) 120583119861119894 (V)] 997888rarr 120583

119862119894(119908)

(2)

where 119860119894and 119861


119894times 119861119894in 119880 times 119881 119877

119894≜

(119860119894and 119861

119894) rarr 119862











Fuzzy systems




Real output values










119891 (119899) = 119892 (119899) + ℎ (119899) (3)














(e) Go to (2)









0




0sdot (1 + 120572 sdot 119898)



119901nrm =119901opt

1 + 120572 sdot 119898

119901adv =119901opt

1 + 120572 sdot 119898times (1 + 120572)

(4)



119879 (119904nrm) =

119901nrm


1015840

0 otherwise(5)


119879 (119904adv) =

119901adv


10158401015840

0 otherwise(6)




0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100









119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)





Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119879 (119899) =

119901opt


0 otherwise(1)














119894 then 119911 is



120583119877119894≜ 120583(119860119894 119861119894rarr119862119894)

(119906 V 119908)

= [120583119860119894(119906) 120583119861119894 (V)] 997888rarr 120583

119862119894(119908)

(2)

where 119860119894and 119861


119894times 119861119894in 119880 times 119881 119877

119894≜

(119860119894and 119861

119894) rarr 119862











Fuzzy systems




Real output values










119891 (119899) = 119892 (119899) + ℎ (119899) (3)














(e) Go to (2)









0




0sdot (1 + 120572 sdot 119898)



119901nrm =119901opt

1 + 120572 sdot 119898

119901adv =119901opt

1 + 120572 sdot 119898times (1 + 120572)

(4)



119879 (119904nrm) =

119901nrm


1015840

0 otherwise(5)


119879 (119904adv) =

119901adv


10158401015840

0 otherwise(6)




0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100









119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)





Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fuzzy systems




Real output values










119891 (119899) = 119892 (119899) + ℎ (119899) (3)














(e) Go to (2)









0




0sdot (1 + 120572 sdot 119898)



119901nrm =119901opt

1 + 120572 sdot 119898

119901adv =119901opt

1 + 120572 sdot 119898times (1 + 120572)

(4)



119879 (119904nrm) =

119901nrm


1015840

0 otherwise(5)


119879 (119904adv) =

119901adv


10158401015840

0 otherwise(6)




0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100









119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)





Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(e) Go to (2)









0




0sdot (1 + 120572 sdot 119898)



119901nrm =119901opt

1 + 120572 sdot 119898

119901adv =119901opt

1 + 120572 sdot 119898times (1 + 120572)

(4)



119879 (119904nrm) =

119901nrm


1015840

0 otherwise(5)


119879 (119904adv) =

119901adv


10158401015840

0 otherwise(6)




0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100









119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)





Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100


0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100









119906 (119909 120590 119888) = exp(minus[(119909 minus 119888)120590

]2

) (7)





Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start


End

Yes

Yes

No

No

No

No

No

NoYes

Yes

Yes

Yes

r lt rmax


R1 lt KCH




S(i) node is Nnor


store it in C[i]

i le n











119896=1119880119896sdot 119888119896

sum119899

119896=1119880119896

(8)


119896is the center







Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Residual energy


engine Defuzzifier

Fuzzy rule base

Input Output

Probability



119891 (119894) = 119908 sdot (cos 119905 (119894)


119863max minus ℎ (119894)

119863max)

(9)


0 1 2 3 4 5 6 7 8 9 10

002040608

1

RP

TN0 1 2 3 4 5 6 7 8 9 10

0Probability

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

002040608

1

Deg

ree o

fm

embe

rshi

p

VS S M G VG

VL L M H VH

VL L M H VH

Input 2

Input 1

Output

01 02 03 04 05 06 07 08 09 1


0 1 2 3 4 5 6 7 8 9 10

01234567891002030405060708

Prob

abili

ty

RPTN







Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start



OPEN is empty Yes

Yes

Yes

No

No

No

Exit failure
















inputs




















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















minus (119878(119899 + 1)119910119889))and2)











0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100




0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100






Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Bit data Bit data


Receiving

ETx (l d)d


2




2 if 119889 le 1198890


0

(10)



0 let 1198890= radic120576fs120576mp


119864NRx = 119871 sdot 119864elec (11)



2 if 119889 le 1198890


0

(12)











Rxelec 50 nJbit

120576fs 10 pJbitm2

120576mp 00013 pJbitm4





54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










54 137

815

1348

885

1804

554 572

1403 15211351

2118

0

500

1000

1500

2000

2500


FNDHNA

Tim

e (ro

unds

)


FNDHNA

50 217

749

1367 1326

1880

609 627

1615 1593 1711

2283


500

1000

1500

2000

2500

Tim

e (ro

unds

)




0 200 400 600 800 1000 1200 1400 1600 1800 20000

102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

Time (round)








0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0102030405060708090

100

Num

ber o

f nod

es st

ill al

ive

0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)




0 200 400 600 800 1000 1200 1400 1600 1800 2000Time (round)



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus10


6 Conclusions



0

10

20

30

40

50

60

Tota

l rem

aini

ng en

ergy

minus100 200 400 600 800 1000 1200 1400 1600 1800 2000

Time (round)








Acknowledgments


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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