CHAPTER 5SPECIFICNETWORK

QUEUEMODELSUNDER

FUZZYENVIRONMENT 

 

                          CHAPTER‐5 

 SPECIFICQUEUEMODELSUNDER

FUZZYENVIORNMENT

5.1INTRODUCTION

Networkofqueues canbedescribedas groupofnodeswhere eachnode

represents a service facility of some kind i.e customer may arrive from

outside the system at any node andmay depart from the system at any

node.Thesenetworkmodelshavetakenonspecialimportancebecauseof

increase capability to modelling manufacturing facility and computer /

communication nets Disney 1996, Kelly 1976, Walrand 1988, Jackson

1957,1963 ,Vandijk 1993 etc.LiandLee 1989 havemadeananalysis

fortwofuzzyqueuingsystemsbasedonZadeh’sextensionsprinciples,one

is the possibility concept and the other fuzzy Markov chains, namely

M F 1,andFM 1whereFdenotes fuzzytimeandFMdenotes fuzzified

exponential time. Negi & Lee’s 1992 claimed that their approach can

utilizetheadvantagesofboththefuzzyandprobabilityapproachtomake

themodelmorerealisticandlessrestrictive.

Inorder toovercometheuncertaintyand imprecision, fuzzy logic theory

helpstorestoretheintegrityofreliability.Fuzzylogicconsistsofthetheory

offuzzysetsandpossibilitytheory.ItwasintroducedbyZadeh 1965 asa

means of representing and manipulating data that was not precise but

rather fuzzy.Fuzzysetsare functions thatmapavalue,whichmightbea

memberofaset, toanumberbetweenzeroandone, indicating itsactual

degreeofmembership.Adegreeofzeromeansthatvalueisnotintheset

anddegreeofonemeansthatthevalueiscompletelyrepresentativeofthe

membershipofeachelement in theset.Possibility theoryworkswith the

possibilitythatavariablemayhaveaparticularvalue.

The model finds its applications in manufacturing or assembling line

process in which units proceed through a series of work stations, each

performing a given task. Practical situation can be observed in a

registration process such as registration, vehicles RC where the

registrantshavetovisitaseriesofdesks advisor,departmentchairperson,

cashier etc. , or on a clinical physical test procedure where the patients

havetopassthroughaseriesofstagesalabtests,electrocardiogram,chest

Xrayetc. .Thecustomersoritemsaftergettingserviceatstage forman

inputforstage2i.etothesecondphaseat .Inclassicalqueuingtheory

inter arrival times and service times are required to follow certain

probabilitydistribution.

In thischapterwestudied twospecific typesof fuzzyqueuemodels.The

chapterisdividedintotwosections5.2&5.3.

Section5.2:Fuzzynetworkqueuewithblocking.

Section5.3:Machinerepairingqueuemodelunderfuzzyenvironment.

SECTION5.2

Inthissection,weproposeageneralproceduretoconstructanetworkof

fuzzy queue model with blocking by considering two servers in service

each with its own queue. In some of these kind of system it generally

happensthatthearrivalrateorservicerateareuncertain.Soweintroduce

fuzzyarrivalrate andfuzzyservicerate usingthefuzzyconceptwhich

results in deriving the state probability as a fuzzy number. The

objectiveofthesectionistoderivethevariousperformancemeasuressuch

asexpectednumberofcustomerinsystemi.e ,expectedwaitingtimeof

customer in system , fractional part of potential customer lost etc,which

areextractedfromthefuzzystateprobabilityfunctiongoverningbehavior

of the system. The queue discipline is considered FCFS. A numerical

illustration has also beenmentioned to support the derived result under

trapezoidal fuzzy arithmetic. The basic idea is to apply the ‐ cut and

Zadeh’s extension principle to transform the fuzzy queuing system to a

familyofcrispqueue.

Recently,Gani&Ritha 2006 &SinghT.Petal 2009 extendedtheworkof

other researchersby introducing the fuzzy set theory into little’s formula

bytakingTriangular fuzzynumbers. Thisworkis furtheranextensionof

the work made by Gani, Ritha & Singh T.P in the sense that we have

considered a trapezoidal fuzzy number .The performance measure for

fuzzy tandem queue network has been worked out under the situation

wherearrivalandserviceratesarefuzzyvariables.Thestateprobabilities

as a fuzzy numbers have been calculated followed by numerical

illustration.

Thischapterisbasedupontheresearchpaper“Trapezoidalfuzzynetwork

queuemodelwithblocking”publishedinAryabhattajournalofmathematics

andinformatics,Vol‐3.issue‐1,2011,pp.185‐192.

Ourcontributionisstructuredasfollows:The5.2.2describesthenotation

and trapezoidal fuzzy number dealing with their operations. The 5.2.5

shows tandem fuzzy queue model with blocking. After that a simple

solution showing the performance measures is outlined. The 5.2.8

describesthenumericalproblemthroughfuzzyarithmeticsolutionaswell

asα‐cutapproach.Finallysomeconclusionsareoutlined.

5.2.1FUZZYSET

IntheuniverseofdiscourseX,afuzzysubsetAonXisdefinedbythe

membershipfunction X WhichmapseachelementxintoXtoareal

numberintheinterval 0,1 . X denotesthegradeordegreeof

membershipanditisusuallydenotedas X :X→ 0,1 .

5.2.2Notations:

λ:Fuzzyarrivalrate

μ:Fuzzyservicerate

ρ:Fuzzybusytimeoftheserver

P t :Possibilitythatthesystemisinstate I,j atanytimet.

L :Expectednumberofcustomersinthesystem.

W :Expectedwaitingtimeinthesystem.

A&B:TrapezoidalFuzzyNumber TFN .

5.2.3FUZZYNUMBER

ThefuzzynumberAissaidtobetrapezoidalfuzzynumberifitisfully

determinedby , , , ofcrispnumberssuchthat

withmembershipfunction,representingatrapezoidoftheform

X , x ,

1, x ,

, x ,

0,otherwise

Where , , are the lower limit, lowermode, uppermode and

upper limit respectively of the fuzzy numberA. When , the

trapezoidalfuzzynumberbecomesatriangularfuzzynumber.

5.2.4TAPEZOIDALFUZZYNUMBEROPERATION:

LetA , , , andB , , , be two trapezoidal fuzzy

numbers,thenthearithmeticoperationonAandBaregivenasfollows:

i AdditionofAandB

A B , , ,

ii SubtractionofAandB

A‐B , , ,

iii MultiplicationofAandB

AxB , , ,

iv DivisionofAandB

A/B , , ,

v Scalarmultiplication

Let beanyrealnumber.Then

For 0, xA , , ,

For 0, xA , , ,

5.2.5TANDEMQUEUESWITHBLOCKING:

Weconsiderasimplesequentialtwostations,singleserverateachstation

where no queue is allowed to form at either station. If a customer is in

station 2 and service is completed at service station 1, the station 1

customer must wait there until the station 2 customer completes his

service,i.ethesystemisblocked.

ServiceStation1 ServiceStation2

Input Output

Inordertoconstructthemodel,firstidentifythestateofthesystematany

pointintimet.

Eachstationmaybe free,busyor inblockedstaterepresentedbysymbol

0,1,b respectively. Station 1 is said to be blocked if the customer in this

stationcompleteshisservicebeforestation2isfree.Let i,j representthe

statesofstation1andstation2thenthestatesofthesystemaregivenby,

i,j 0,0 , 0,1 , 1,0 , 1,1 , b,1

Possiblesystemstates:

i,jDescriptionofsystem

0,0Systemempty

1,0Customerinprocessat1only

0,1Customerinprocessat2only

1,1Customerinprocessat1&2

b,1Customerinprocessat2&acustomerfinishedat1butwaitingfor

2tobecomeavailable,i.e.systemisblocked

Assumingarrivalsat the system Station1 arePoissonwithperimeterλ

andserviceisexponentialwithparameters , .

Considering , is the state probability of customers in first station

and insecondstation.

The steady state differential difference equationswithmulti dimensional

Markovchainaregivenby

‐λ ,  +    ,  = 0                                              (1) 

,  +    , +λ , = 0                                (2) 

λ ,  +    , , = 0                (3)   

  ,  + λ , = 0                              (4) 

,  +    , = 0                                           (5) 

Withboundarycondition∑∑ , 1 6

Wehave 6 equationsinfiveunknowns.Theequation 3 canbeusedto

getallprobabilities intermsof , andthecondition 6 isusedtofind

, .

If say theresulthasbeenderivedbyJackson’s 1957,1963

asfollows

  ,  =  ,  ,                 ,  =  ,   ,              ,  =  ,   ,            

      ,  =  ,  

,  =   

5.2.6 FUZZY QUEUE MODEL WITH BLOCKING IN TRANSIENT STATE: 

Define astheprobabilitythatthesystemisinstate i,j atthetimet.

the transition probabilities between t and t ∆t in fuzzy environment by

taking . Following equations in transient form can be

establishedas

, (t+∆t) =  , (t) (1‐  ∆t) +  ∆t  , (t)  

, (t+∆t) = (1‐ ∆t‐ ∆t)  , (t) +  ∆t  , (t) +  ∆t  , (t) 

, (t+∆t) = (1‐ ∆t)  , (t) +  ∆t  , (t) + ∆t  , (t) 

, (t+∆t) = (1‐2 ∆t)  , (t) + ∆t  , (t) 

, (t+∆t) = (1‐ ∆t)  , (t) + ∆t  , (t) 

5.2.7 MATHEMATICAL MODELLING IN STEADY STATE: 

Byrearrangingthetermsandtakinglimitt→∞thesteadystateequations

canbewrittenas:

,  ‐ ,  = 0 

  ,  +   ,  ‐ ( 1+  )  ,  = 0 

,  +   , ‐   ,  = 0 

,  ‐ 2  ,  = 0 

,  ‐  ,  = 0                                                                      where   =   

Oneoftheseequationsisredundant,hence,withtheboundaryconditions,

  ,  +  , ,  +  , ,   = 1               or         ∑∑ , =1        

Thesolutionfor isgivenby

,  =   =              where   = 3 +4  +2                   ,  =             

   ,  =  ,  =                             ,  =                          where   =   

1. Expectednumberofcustomersinthesystem canbeobtained

as

0 , 1 , , 2 , ,

2. Expectedwaitingtimeofcustomerinthesystemorfuzzyaverage

delayperproductisgivenbyfuzzyLittle’sformulaas

Effectivefuzzyarrivalrate

3. Theprobabilitythatanarrivingitem/customerwillenterat

station1is , ,

4. Fractionofpotentialcustomerlost i.efractionoftimesystemis

full

5. Effectivearrivalrate 1‐

Effectivetrafficintensity

WhereNisthecapacityofthesystemi.enomorethanNcustomercanbe

accommodatedinthesystemduetocertainreason.Thuscustomerarriving

whenthesystemalreadycontainsNcustomersdoesnotenterthe

customerinthesystemandislost.

5.2.8Numericalillustration: 

Consideratwostation‐subassemblylinewhichoperatedbyabeltsystem

thesizeofassembledproductdoesnotallowstoringmorethanoneunitin

eachstation.Theproductarrivetothislinefromanotherservicefacilities

asperPoissondistributiontheTFNaveragearrivalrateis 2,3,4,5 per

hours.Assembledtimesofstation1,station2andstation3isexponentially

distributedwithaveragefuzzydepartureservicerateis 4,6,8,10 per

hour.Ourobjectiveistocalculatetheexpectedwaitingtimeofcustomers

insystem,potentialcustomerlost,andeffectivearrivalrate.

Given =(2,3,4,5) =(4,6,8,10)

Utilizationofsubassemblylineis

= =(.2,.375,.666,.25)using(2.3)

0,0=(.6849,.510,.336,.627)

0,1=(.1369,.1912,.2222,.1568)

Theprobabilitythatanarrivingitem/customerwillenteratstation1is

0,0+ 0,1=(.8218,.7012,.558,.7838)

1,0=(.1506,.2271,.2955,.1764)

1,1= ,1=(.0136,.0358,.0733,.0196)

Expectedwaitingtimeofcustomerinthesystem

=(.3424,.35618,.811,.4117)

Fractionofpotentialcustomerlost = 0=(.1369,.1912,.2237,.1567)

Effectivearrivalrate = (1‐ )=(1.726,2.426,3.105,4.216)

Expectedwaitingtimeofcustomerinthesystem

= =Effectivefuzzyarrivalrate

==(.1983,.1467,.2611,.0976)

5.2.9TheFuzzyQueueModelThrough cut:

Thenusing cut

Given =(2,3,4,5) =(4,6,8,10)

( )= 2, 5 ( )= 2 4, 10 2

= =2

10 2,5

2 4

2 =2 4 4

4 2 40 100,

2 10 25

4 2 16 16

=3 2+4 +2=3 3 21 2 512 1460

4 3 60 2 300 500,3 3 192 2 79 374

4 3 24 2 48 32

=5 2 4

=3 3 79 2 120 500

3 3 21 2 512 1460,

3 3 32 2 253 470

3 3 192 2 79 374

When =0

Then =(.3472,1.256)

When =1

Then =(.5887,1.0617)

=[.3472,.5887,1.0617,1.256]

Ingraphverticallineshowsthemembershipfunctionwhichliesin[0,1]

andhorizontallineshowstheexpectedwaitingtimeofcustomersinthe

systemi.e .

5.2.10GRAPH:

Expectedwaitingtimeofcustomerinthesystem

5.2.11Conclusion:

In this model we have made analysis regarding the probability that an

arrivalitemwillenterthestationat , , .Theeffectivefuzzyarrival

rate,andtheexpectedwaitingtimeofthecustomerhasalsobeendeduced.

The fuzzy queue length has been calculated. The technique is able to

provide fuzzy number performance measures for service queues with

fuzzifiedexponentialarrivalrateandfuzzyservicerate.Ifmorestationsare

considered then theproblemalso expands, if one allows limit other than

zero on queue length, as we have finite set of numerical technique for

solving simultaneous equations. If then trapezoidal fuzzy number

converts intotriangular fuzzynumberandthemodelresemblesthework

madebyGani&Ritha 2006 andT.PSingh 2009 .The showsasa

bridgebywhich fuzzysetandcrispsetsareconnected.The fuzzysetcan

uniquelyberepresentedbythefamilyofallitsof . 

0

0.2

0.4

0.6

0.8

1

0.3472 0.5887 1.0617 1.256

SECTION‐5.3

MACHINEREPAIRINGQUEUEMODELUNDERFUZZYENVIORNMENT

ConsideraqueuingsystemconsistingofRrepairmen’s technicians who

supportsMmachinessubjecttostochasticfailurewhereM R.Whenever,a

machinebreakdownitwillresultinalossofproductionunit,itisrepaired

by the repairman. Each repair keeps a repairman busy for the period of

timeduringwhichtheycannotgiveservicetootherbrokenmachines.This

type of queue model can be referred as equivalent to the finite calling

sourcewithmaximumlimitofMpotentialcustomers Gross,Harris 1998

andGrossJ.F.INC 1981 .

Most of the researchers and authors derived the system performance

measure of machine interference problems and its variants when their

parametersarenotexact.Butinrealworldsituationtheseparametersmay

not be presentedprecisely due to uncontrollable factors. The breakdown

and the service patterns aremore suitably described by linguistic terms

such as slow,medium and fast based on possibility theory. To dealwith

uncertain information indecisionmakingZadeh 1965,1978 introduced

the concept of fuzziness. The concept of fuzzy logic has been applied by

various researchers in different situations.W.Ritha etal 2011 proposed

an inventory model with partial backordering in a fuzzy situation by

employing trapezoidal fuzzy numbers. Li & lee 1989 investigated the

analyticalresultfortwospecialqueuesM/F/1/∞andFM/FM/1/∞where

F denotes the fuzzy time and FM denotes the fuzzified exponential

distribution.NegiandLi 1992 proposedaprocedureusingα‐cut.Chen’s

1996 introducedafunctionalprincipleinfuzzyinventorymodel.Robert

& Ritha 2010 studied the machine interference model in trapezoidal

fuzzyenvironment.OurworkdiffersfromRobert&Rithainthesensethat

we have applied Triangular Fuzzy Numbers instead of trapezoidal fuzzy

arithmetic. In thework of Robert & Ritha the fuzzy system performance

measure has been calculated on chain function principle while in our

proposed model Yager’s formula for defuzzification of Triangular Fuzzy

systemhasbeenapplied.

5.3.1FUZZYSET:

1.IntheuniverseofdiscourseX,afuzzysubsetAonXisdefinedbythe

membershipfunction X WhichmapseachelementxintoXtoareal

numberintheinterval 0,1 . X denotesthegradeordegreeof

membershipanditisusuallydenotedas X :X→ 0,1 .

2.IfafuzzysetAisdefinedonX,forany ∈ 0,1 ,the ‐cuts is

representedbythefollowingcrispset,

Strong ‐cuts: x∈ ; ∈ 0,1

Weak ‐cuts: x∈ ; ∈ 0,1

Hence,thefuzzysetAcanbetreatedascrispset inwhichallthe

membershavetheirmembershipvaluesgreaterthanoratleastequalto .

Itisoneofthemostimportantconceptsinfuzzysettheory.

5.3.2Notations:

λ:Fuzzyarrivalrate

μ:Fuzzyservicerate

ρ:Fuzzybusytimeoftheserver

P t :Possibilitythatthesystemisinstate I,j atanytimet.

L :Expectednumberofcustomersinthesystem.

W :Expectedwaitingtimeinthesystem.

A&B:TriangularFuzzyNumber TFN .

5.3.3TRIANGULARFUZZYNUMBER:

A fuzzy number is simply an ordinary number whose precise value is

somewhat uncertain. Taking a1, a2 supporting interval and the point

am,1 asthepeak,wedefineatriangularfuzzynumber withmembership

function definedonRbyconfidence,

, x ,

, x ,

0,otherwise

Ingeneralpractice,thepointsam∈ a1,a2 islocatedatthemidofthe

supportingintervali.eam

Puttingthesevalues,weget

, x ,

, x ,

0,otherwise

i.ethreevaluesa1,am,anda2constructatriangularnumberdenotedby

a1,am,a2

5.3.4TRIANGULARFUZZYNUMBEROPERATION:

Let a1,a2,a3 and b1,b2,b3 betwoTFN,thenthearithmetic

operationon and aregivenasfollows:

Addition a1 b1,a2 b2,a3 b3

Subtraction a1‐b3,a2‐b2,a3‐b1

Multiplication ∗ a1b2 a2b1‐a2b2,a2b2,a2b3 a3b2‐a2b2

Division / 2a1/b1 b3,a2/b2,2a3/b1 b3

Provided areallnon‐zeropositivenumbers.

5.3.5DEFUZZIFICATIONOFTFN:

If a1, , isaTFNthenitsassociatedcrispnumberisgivenby

Yager’sformulaasfollows:

A

5.3.6PERFORMANCEMEASURES:

In this chapter, we have made an attempt to find the following fuzzy

performance measures that are commonly used in traditional queuing

theory

1. Operatorutilizationi.e. .

2. Averagenumberofmachinewaitingforservicei.e. .

3. Averagenumberofmachineoutofactioni.e.breakdowni.e

4. Effectivearrivalratei.e.

5.3.7FUZZYQUEUEMODEL:

Inthismodel,thereareMmachineswhichareservicedbyRrepairmen.

Wheneveramachinebreaksdown,resultingalossofproductionunititis

repaired.Consequently,abrokenmachinecannotgeneratenewcallswhile

inservice.Thisisequivalenttothefinitecallingsourcewithmaximumlimit

ofMpotentialcustomers.

In the model, the approximate probability of a single service during an

instant∆ is n ∆ for n , and ∆ for . On the other hand, the

probability of a single arrival during∆ is approximately M‐n ∆

for ,whereλisdefinedasthefuzzybreakdownratepermachinein

fuzzynumber.

Once a machine is to be repaired it turns to good condition and again

susceptible to breakdown. The length of time that machine is in good

condition follows an exponential rate with breakdown rate and repair

rate whichareintriangularfuzzynumber.

5.3.8Intransientstate:

Define be thepossibilitymeasureof ‘n’ customers at any time‘t’ the

differentialdifferenceequationcanbemodelledas:

∆ 1 λ∆ ∆ 1 1 λ∆ , 

0, 1  

∆ 1 ∆ 1 ∆

1 ∆ 1 1 ∆

1 1 ∆ 1 ∆ , 0

, 2  

∆ 1 ∆ 1 ∆

1 ∆ 1 1 ∆

1 1 ∆ ∆ , , 3  

∆ 1 ∆ 1 ∆

1 ∆ 1 ∆

1 1 ∆ ∆ ,

1, 4  

∆ 1 ∆ ∆ 1 ∆ , , 

                                                                                                                             (5) 

5.3.9Insteadystate:

The steady state condition is reachedwhen the behaviour of the system

becomes independent of the time. Since inmachine repairingmodel the

initial start‐ups and ending stages do not change the arrival rate and

servicerateofthequeueandqueueultimatelysettledownwithvaluesof

the parameters oscillating around their predictable averages hence the

queuesystem is takenas thesteadystate.Thesteadystateequationsare

t→ ∞and∆ → 0wehave

M = , for n=0                                                                                                    (6) 

{(M‐n) +n} = (M‐n+1) +(n+1) , for 0 ,                               (7) 

{(M‐n) +R} = (M‐n+1) +R  , for R 1,                            (8) 

R = ,   for n=K,                                                                                            (9)               

Tosolvethesteadystateequations:

  From(6), 1=M 0

Substitutingn 1in 7 ,2 M‐1

Byinduction,itcanbeshownthat, n 1 M‐n ,0

Similarly,fromequation 8 and 9 wegetR M‐n ,for

R

Aftersolvingtheequationsweget,

 =        ,               0 , 

          !

! ,        R .      

!!

 

Theothermeasuresareobtainedasfollows:

 

where expectednumberofidlerepairmen ∑

Suppose definetheeffectivearrivalrate,valueof canbe

convenientlydeterminedfrom

 = ‐                             or    =  [  ‐  ] 

=    ,                               =  

 

5.3.10NUMERICAL:

Considertherearefivemachinesonrunning,eachsuffersbreakdownatthe

fuzzyrateof 1,2,3 therearetwoservicemanandonlyonemancanwork

onmachineatatime.Ifnmachinesareoutoforderwhenn 2then n‐2

of them wait until a service man free the time to complete the repair

followstheexponentialdistributionwithfuzzydeparturerate 2,3,6 .Our

objectiveistofindoutthefirstdistributionofnumberofmachinesoutof

actionatagiventime,averagetimeanoutofactionmachinehastospend

waitingfortherepairstostarti.eWq.

Take 1,2,3 2,3,6

M totalnumberofmachinesinthesystem 5,

R numberofservicemen 2

Letn numberofmachinesoutoforder

 = ( ,  ,  )         

 ,                0 2     

 =       ,                0 2   

              ,                0 2                                                       

                       !

!  ,              2 5      

     =   !

!   ,          2 5      

                        !

!     ,      2 5                         (A)                  

                 

5 5 !2! 2

 

Usingthisweget

0=2048

6503,81

1311,256

5491=(.31,.06,.04)

Puttingthisvaluein A weget

.31  ,                0 2     

 =      .06  ,                0 2                    

              .04 ,                0 2                          

              !

!.31   ,          2 5                              (B) 

 =      !

!.06   ,          2 5      

              !

!.04      ,          2 5 

Averagenumberofmachinesoutofactionisgivenby

2 2

2 3  

Puttingthevaluein B forfinding , thenweget

 =      . 04, .17, .16   ,                0 2  ,      

               . 07, .26, .25 ,                 2 5 

2  =      . 012, .1, .12   ,              0 2  ,      

               . 036, .1, .12 ,                 2 5 

 

3  =     . 0009, .021, .027 ,       0 2  ,      

               . 006, .15, .21 ,               2 5 

  2 3  

      =        . 052, .29, .30   ,           0 2  ,      

             . 11, .51, .58 ,                 2 5 

Averagetimeanoutofactionmachinehastospendwaitingfortherepairstostartis

 =   

5 5 4 3 2  

5  =   1.55, .3, .2   ,              0 2  ,      

             3.1, .6, .4 ,                 2 5 

4  =   1.52, .8, .6   ,                0 2  ,      

             1.55, .8, .6 ,                  2 5 

3  =   . 57, .78,1.35   ,             0 2  ,      

             . 57, .78,1.35 ,               2 5 

2  =       . 08, .34, .32   ,               0 2  ,      

                  . 14, .52, .5 ,                   2 5 

 =          . 006, .05, .06   ,            0 2  ,      

                  . 018, .05, .06 ,              2 5 

Puttingallvaluesin C weget

  =          3.72,2.27,2.53   ,      0 2  ,      

                      5.34,2.75,2.91 ,        2 5 

         = (1,2,3)    3.72,2.27,2.53 ,     0 2  ,      

                      5.34,2.75,2.91 ,     2 5                                 

Solvetheaboveexpressionwiththehelpoffuzzyoperation,weget

          =     5.17,4.54,7.33   ,       0 2  ,      

                       7.93,5.5,8.57 ,           2 5 

Similarlywecanfindout

 =    .

.,.

.,.

. ,                   0 2  ,                  

               .

.,.

.,.

.,                    2 5         

   =  . 008, .063, .048 ,            0 2   

                  . 013, .092, .070 ,           2   5 

5.3.11FUZZYQUEUEMODELTHROUGHα–CUT:

Thenusingα‐cut

Given 1,2,3 2,3,6

1,3 , 2,6 3

,

5 5 !2! 2

 

 =      ,          

                     ,               

 =        ,               0 , 

          !

! ,        R .        

Forfinding ,

2 3

Where

 =           ,           

                          ,                 

                                                                                          For 0 2         

    =        !

!,     .        

                        !

!   

                                                                                      For 2 5 

   =           ,                ,                 

                                                          

                                                                                       For 0 2          

    =        !

!, 

                         !

!      

        .                                                                           For 2 5 

   =           ,  

                                                                                

                                                                                    for  0 2         

 

    =        !

!,                   

                        !

!   

                                                                                   for 2 5 

Now  

  2 3  

  =        (63 324 423  

,                                                              

                     3 32 127  

                                                      For 0 2 

495 810 1485  

21 975 8040 28140 46995 34107

2 6 3, 

=                            30 180 645  

3 45 520 3200 7965 13729

2 2 

                                                       For 2 5      

For  0, we get 

     =   (.11, 22986.6)            0 2 

                    (.20, 58371.7)            2 5   

Similarly,fordifferentvaluesof ,thefuzzyqueuelengthofthesystemcanbedepictedbelow:

For .2,weget

     =   (.374, 6609.20)            0 2 

                    (1.39, 30204.74)          2 5      

For .4,weget

     =   (9.31, 1937.79)            0 2 

                    (38.07, 7981.61)          2 5    

For .6,weget

     =   (4.29, 552.85)            0 2 

                    (20.14, 2056.30)        2 5      

For .8,weget

     =   (17.47, 138.55)           0 2 

                    (98.80, 466.46)           2 5      

For 1,weget

     =   (89.6, 28.02)                0 2 

                    (323.6, 42.81)              2 5 

. 11, 9.6, 22986.6    0 2 

              (.20, 323.6, 58371.7)         2 5   

 

                      For i.efuzzyaveragequeuelengthininterval 2,

0

0.2

0.4

0.6

0.8

1

0.11 89.6 22986.6

   

For . ininterval2

Inabovegraphs,verticallineshowsthemembershipfunctioninfuzzythat

liesintheinterval 0,1 andhorizontallinesshowsthe i.efuzzyaverage

queuelengthintheinterval 0,2 and 2,5 .

5.3.12FUZZYQUEUEMODELTHROUGHα–CUTINWAITINGTIMEINAQUEUE:

Forfinding thenwehave:

5.17,4.54,7.33 ,0 2,

7.93,5.5,8.57 ,2 5

Now

5.17 .63 , 7.33 2.79 ,0 2,

2.43 7.93, 8.57 3.07 ,2 5    

. .

63 324

423 ,                                               

=   . .

 

3 32 127  

                                                      For 0 2 

0

0.2

0.4

0.6

0.8

1

0.2 323.6 58371.7

 

. .

495 810

1485 ,                     

          . .

30 180

645  

                                                       For 2 5      

For 0,weget

.023,3135.96 0 2

.052,13622.33 2 5

Similarly,fordifferentvaluesof ,theAveragefuzzywaitingtimeforthe

repairstostartinthesystemcanbedepictedbelow:

For .2,weget

     =   (.070, 975.38)              0 2 

                    (.165, 3796.47)             2 5      

For .4,weget

     =   (1.71, 311.84)              0 2 

                     (4.27, 1087.11)            2 5    

For .6,weget

     =   (.773, 97.74)                0 2 

                    (2.14, 305.63)              2 5      

  For .8,weget

     =   (3.07, 27.17)                0 2 

                     (10.006, 76.29)           2 5      

For 1,weget

     =    (15.45, 17.01)             0 2 

                      (62.47, 42.92)             2 5 

.023,15.45,3135.96 0 2

.052,62.47,13622.33 2 5

 

For i.eAveragefuzzywaitingtimeininterval 2,

 

For . ininterval2

Inabovegraphs,verticallineshowsthemembershipfunctioninfuzzythat

liesintheinterval 0,1 andhorizontallinesshowsthe i.eaveragefuzzy

waitingtimeintheinterval 0,2 and 2,5 .

0

0.2

0.4

0.6

0.8

1

0.023 15.45 3135.76

0

0.2

0.4

0.6

0.8

1

0.052 62.47 13622.33

5.3.13CONCLUSION:

It has been observed thatwhen the breakdown rate and service rate

are in fuzzy numbers, the performance measures in machine repairing

system are expressed by fuzzy number that completely conserve the

fuzzinessofinputinformationwhilesomeparametersinthismodelareina

fuzzy number. By applying the fuzzy arithmetic operators and yager’s

defuzzification, we derive the system performance measures as clear by

numericalillustration.Fuzzyaveragenumberofsystembreakdownis

.023,15.45,3135.96 0 2

.052,62.47,13622.33 2 5

Consideringthefuzzyaveragequeuemodelfortherepairthesystemgives

the

.11, 89.6, 22986.6 0 2

.20,323.6,58371.7 2 5

From the graph, it is clear that expected fuzzy waiting time can not be

below.023orneverexceedabove3135.96in 0,2 .Similarly,theexpected

fuzzywaitingtimecannotbebelow.052orneverexceedabove13622.33

in 2,5 this give the lowerboundandupperboundsof the fuzzywaiting

timeforthesaidproblem.

Similarly,itisclearthatexpectedfuzzyqueuelengthcannotbebelow.11

orneverexceedabove22986.6in 0,2 .Similarly,theexpectedfuzzyqueue

length can not be below .20 or never exceed above 58371.7 in 2,5 this

givethe lowerboundandupperboundsofthefuzzyqueuelengthforthe

saidproblem.

The result obtained as Average length queue, Average waiting time in

queue obtained from the proposed approach maintain the fuzziness of

input data describe the machine repairing model more appropriate and

practicallybetter.Theproposedmodelcanbeusedindesigningmachines,

repair system under fuzzy environment, since it deals with incomplete

information. Approach and fuzzy arithmetic operator are used to

construct system characteristic membership functions which are more

suitableforpractioners.