View of Analysis of M/G/1 Feedback Queue under Steady State When Catastrophes Occur

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640 Analysis of M/G/1 Feedback Queue under Steady State When Catastrophes Occur

S.Shanmugasundaram1,_s_G.Sivaram2,

1_Assistant_s_Professor,_s_Department_s_of_s_{Mathematics, Government}_s_Arts_s_College,_s_Salem–_s₆₃₆_007,_s_India

Email:ssundaramsss@rediffmail.com

2_Associate_s_Professor,_s_Department_s_of_s_{Mathematics,Government}_s_Arts_s_College,_s_Salem–_s₆₃₆_007,_s_India

Email: sgsivaram1965@gmail.com

Article History: Received: 11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract: In this paper we analyse the M/G/1 feedback queue under steady states conditions when catastrophe occur. The stationary probability of ‘n’ and zero customers sins the system are derived. The

asymptotic behaviour of the model and the averages queues length

are also obtained. The numerical example are provided to test the feasibility of the model.

Keywords: Bernoullisprocess, scustomer,s feedback,scatastrophes,sstationarysdistribution,s asymptotic sbehaviour

1. INTRODUCTION

S Asqueue issa waitings line swhich sdemands service from asserver.The squeues does not include a customers beings serviced.s Queueings mathematician sA.K. Erlang.s The sErlang

work[1]sonsqueueing sstimulated smany sauthors

tosdevelop sa svarietys ofs queueings models. Many squeueing ssituations shave sfeatures sthat the scustomer smay sbe sserviceds once sagain. Ifsascustomersis not ssatisfied sby shis sservice or hesexpectss more servicesthen she sjoinss the queue stos gets additional sservice sisscalled feedback. Thescustomer smay s(or)s may snotsopt for sa sfeedback.s In sthe syear s 1963s Takacs [8] first

sintroducedsthe sconcept sofs feedback

mechanism sinsqueuess. In s1996 s sGautams Choudharysands Madhus Chandapaul [3] have proposed s a stwos phases queueings systemswiths Bernoullis feedback.

Inscertainsqueueingsmodelssbeforesstarting a service,sthes servers may shavestosdossomes preparatorys worksor ssomes alignments musts be done in sthe scase sofs certain snecessities.sThis sort sof spreparatory swork sfors customers occur sin shospitals, sproduction sprocess, bank etc. sSanthakumaran sands Thangaraj [7]shave proposeds as singles servers queues with impatient sand sfeedback s customers. Santhakumaran sand sShanmugasundaram [6s] have preparatory swork sons arriving scustomerss withs as single sserver sfeedbacks queue. Santhakumaran, Ramasamy and Shanmuga sundaram[13] havesalsosstudiedsas single queue with sinstantaneous sBernoulli feedback sand setup stime. Thangaraj sands Vanitha [12] shave focussed son sas continued fractions approach sto a sM/M/1 squeues with feedback.Chandrasekaran and Saravanarajan[14] made a sstudy sonstransient and sreliability analysiss of M/M/1s feedback queue ssubjectsto catastrophes, servers failures and srepairs.Insqueueingssystemscatastrophessmeansssuddens calamitysthats occurs ins queues ors service facility.Whenscatastrophessoccursinsthessystem,allthesavailable customers aresdestroyedsimmediately and the server sgets sinactivated.s Catastrophes modelling sand sanalysis shass been splaying asvitals role sinsvarious sareas sofs science sand technology.

Chao s[2] s shass modelled sa

queueing snetwork s smodesl swiths catastrophes and sproducts forms solution. sShanmugasundaram ands Chitra [9] shaves made sasstudy son time dependent solution sof sa ssingle server feedback queue scustomer has sas serviceswithandwithout spreparatorysworkwhen catastrophes occur. Krishnakumar, Krishnamoorthy, Pavai Madheswariands Sadiqs Basha s[15]s studied a transient s analysis sof sa ssingle sserver squeue withs catastrophes, sfailuress ands repairs.

Krishnakumars,s Arivudainambi [10]s focussed on transientsstate ssolutionstosasM/M/1squeue withscatastrophes. Parthasarathy s[11s] smade sa study sonsas transients solution sto sa sM/M/1 queue. sJain sandskumars[4s] shavesstudied son M/G/1 squeueswithscatastrophes.sKrishnakumar andsPavaisMadheswaris[5] shaves mades asstudy on stransientsanalysissof san sM/M/1 squeue subject stoscatastrophes sands server sfailures. S

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Insthissmodel,externalscustomerssarrivesaccordingtosaspoissonsprocess swiths rate sλ.sThe sservice follows generalsdisciplineswithsservicesrate sµ and thes servicesfor sans arriving scustomers beginssinstantaneously sifsthesserver sis sidle supons an sarrival.

s s ss

After getting s service,s the scustomers makesa decision sdepending sons the slevels of sservice whether s stos

depart s(or) sfeedback. s If the

customer sdoess feedback, she sjoins s the feedback stream swith sprobability sq sand sjoins the end of the queue. If a customer sdoes not feedback,he joinss the sdeparture sprocesss withsprobability pssosthatsps+sqs= s1. sThe squeue sdiscipline sis FIFO andsthe capacity sof sthe squeue is infinite.

Catastrophes occur sfrom s the sarrival sands the

service process with rateΩ.When scatastrophes occur, all sthe savailablescustomerssaresdestroyedimmediately sands thes servers becomes inactive.s The sserver swill sbes ready sfors service sat sthe times ofs as news arrival.s The smotivation sfor this smodels comes sfrom sbank, hospital, production ssystems, restaurant setc. Ss LetsPn(t)=sP[x(t)=n],n=0,1,2,…..sdenotesthesprobabilitiessthatstheresares‘n’customers sin sthe system at

stime st sand slet sP(x,t)=∑∞𝑃(𝑡)𝑥𝑛

0 sbe sits

probabilitys generatingsfunction. Assume sthat

there sare sno scustomers sin sthe ssystem s at time t= s0.si.e., s s sP0(0)=1

Thessystemsofsdifferentialsdifferencesequations forsthesprobabilitysPn sis

s s-λP0+ sµP1s+Ω(1-P0) s= s0 s s s(1)

and sforsn s= s1,2,3,……….

s-λPn-1-(λ+ sµ+Ω)Pn s+µPn+1 s= s0 s s s(2) Theorem sd:1

Thes Stationarysd Probabilitys distribution

{𝜋𝑛 , nsd≥ 0}sfor thesM/G/1 queues when

catastrophessd occur dissd 𝜋0 sd= sd1-ρsd

𝜋𝑛=sd(1-ρ)ρn,sdnsd= sd1,2,…………

dWhere 𝜌 = [(𝜆+µ+𝛺)−√𝜆2+µ2+𝛺2+2(𝜆𝛺+µ𝛺−𝜆µ)

2µ ]

Proofs:ThesLaplacestransformsdofsdthesdsteady statesdprobabilitysdforsdnosdcustomerssdinsdthe systems is 𝑃0′(𝑥) = 1 +𝛺_𝑥 (𝑥 + 𝜆 + 𝛺) − (𝑤 − √𝑤₂2− 4𝜆µ) Where𝑤 = (𝑥 + 𝜆 + µ + 𝛺) 𝜋0 = lim 𝑥→0𝑥𝑃0 ∗_(𝑥) Queue Service Evaluation Arrival Departure Feedback

P

q

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=𝑑lim 𝑥→0 𝑥 + Ω (𝑥 + 𝜆 + 𝛺) − (𝑤 − √𝑤 2_{− 4𝜆µ} 2 ) = 2Ω 2(𝜆 + Ω) − [(𝜆 + 𝜇 + Ω) + √(𝜆 + 𝜇 + Ω)2_{− 4𝜆𝜇]} = 2Ω (𝜆−𝜇+Ω)+√(𝜆+𝜇+Ω)2_−4𝜆𝜇 𝜋0 = 1 − [ (𝜆 + 𝜇 + Ω) − √𝜆2_{+ 𝜇}2_{+ Ω}2_{+ 2𝜆Ω + 2𝜇Ω − 2𝜆𝜇} 2𝜇 ]

Also sdtaking sdLaplace sdtransform sdof sdthe steady sdstate sdprobability sdfor sdn sdcustomers in sdthe sdsystem, sdwe sdobtain sd lim 𝑥→0𝑥𝑃𝑛 ′_{(𝑥) = (}𝑤 − √𝑤2− 4𝜆µ 2𝜇 ) 𝑛 lim 𝑥→0𝑥𝑃0 ∗_(𝑥) 𝑠 𝑠 𝑠𝑑𝜋𝑛= lim 𝑥→0( 𝑤 − √𝑤2_{− 4𝜆µ} 2𝜇 ) 𝑛 𝜋0 𝑠 𝑠 𝑠𝑑𝜋𝑛= 𝜋0[ (𝜆 + 𝜇 + Ω) − √𝜆2_{+ 𝜇}2_{+ Ω}2_{+ 2𝜆Ω + 2𝜇Ω − 2𝜆𝜇} 2𝜇 ] 𝑛

𝑠 𝑠 𝑠𝑑𝜋𝑛= (1 − 𝜌)𝜌𝑛,𝑠𝑑𝑛 = 1,2,3 …. sd sdand sdthe sdstationary sdprobability sddistribution sdexists sdif sdand sdonly sdif sd𝜌 < 1.𝑠𝑑

Theorem:2

Thes dasymptotic sdbehaviours dof sdaveragesd queue sdlength sd𝐻(𝑡) sdwhen sdΩ > 0 sdis 𝐻(𝑡) = (𝜆−𝜇

Ω ) +𝑑𝑑

2𝜇

2(𝜆+Ω)−[(𝜆−𝜇+Ω)−√(𝜆+𝜇+Ω)2_{−4𝜆𝜇]} 𝑠 assd 𝑠𝑑𝑡 → ∞.

Proof:sd Consider sdthe sd probability generating

functionsd𝑃(𝑥, 𝑡) = ∑∞𝑛=0𝑃𝑛(𝑡)𝑥𝑛together s dwith

initials dconditions sdand sdusings dthe dequations (1) and (2), the probabilitysgenerating s function 𝑃(𝑥, 𝑡) becomessd 𝜕𝑃(𝑥, 𝑡) 𝜕𝑡 = [𝜆 + 𝜇 𝑥− (𝜆 + 𝜇 + Ω)] 𝑃(𝑥, 𝑡) + 𝜇 (1 − 1 𝑥) 𝑃0+ Ω Thesd average sdqueuesd lengthsdissd

𝐻(𝑡) = ∑ 𝑛𝑃𝑛(𝑡) = 𝜕𝑃(𝑥,𝑡) 𝜕𝑡 ∞ 𝑛=1 𝑠𝑑at 𝑠𝑑𝑥 = 1 𝑑𝐻(𝑡) 𝑑𝑡 + Ωℎ(𝑡) = 𝜆 − 𝜇(1 − 𝑃0)

Thissd differential sdequation sisslinearsdinsd𝐻(𝑡)andsd solving sdfor sdH(t)sdwes dgetsd

𝐻(𝑡)𝑒∫ Ω𝑑𝑡_{= ∫ [𝜆 − 𝜇(1 − 𝑃} 0)]𝑒∫ Ω𝑑𝑡𝑑𝑡 + 𝑐 𝑡 0 𝐻(𝑡) = 𝜆 Ω(1 − 𝑒 Ω𝑡_{) −}𝜇 Ω(1 − 𝑒 Ω𝑡_{) + 𝜇 ∫ 𝑃} 0(𝑢)𝑒−Ω(𝑡−𝑢) 𝑡 0 𝑑𝑢 TakingsdLaplacesdtransformsdforsdthesdaboves

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𝐻∗_{(𝑥) =} 𝜆 x(x + Ω)− 𝜇 𝑥(𝑥 + Ω)+ 𝑃 (𝑥 + Ω)𝑃0∗(𝑥) lim 𝑡→∞𝐻(𝑡) = lim𝑥→0𝑥𝐻 ∗_{(𝑥) = lim} 𝑥→0 𝜆 − 𝜇(𝑡) 𝑥 + Ω + 𝜇(𝑡) (𝑥 + 𝜆 + Ω) − [(𝑥 + 𝜆 + 𝜇(𝑡) + Ω) − √(𝑥 + 𝜆 + 𝜇(𝑡) + Ω) 2_{− 4𝜆𝜇(𝑡)} 2 ] =𝜆 − 𝜇(𝑡) Ω + 2µ(t) 2(𝜆 + Ω) − [(𝜆 + 𝜇(𝑡) + Ω) − √(𝜆 + 𝜇(𝑡) + Ω)2_{− 4𝜆𝜇(𝑡)]} assdsd𝑡 → ∞ d𝐻(𝑡) = (𝜆−𝜇 Ω ) + 2𝜇 2(𝜆+Ω)−[(𝜆−𝜇+Ω)−√(𝜆+𝜇+Ω)2_{−4𝜆𝜇]}

If sthesservicesiss carrieds outs withdandswithout

preparatorysworksand stherespectivesservice srates are taken das 𝜇1𝑑, d𝜇2 and if p is the

probabilitysforspreparatoryswork and q is sdthe

probabilitywithoutspreparatoryswork suchdthat

pd+sqs= 1,sthen the asymptotic behaviour of average queue length 𝐻 = (𝜆−(𝑝𝜇1+𝑞𝜇2)

Ω )

+ 2(𝑝𝜇1+ 𝑞𝜇2)

2(𝜆 + Ω) − [(𝜆 − (𝑝𝜇1+ 𝑞𝜇2) + Ω) − √(𝜆 + (𝑝𝜇1+ 𝑞𝜇2) + Ω)2− 4𝜆(𝑝𝜇1+ 𝑞𝜇2)]

Ifsthescustomerssdwithoutsdpreparatorysdworksareonlysallowedsto feedbacks dwith probability q but thescustomersswithspreparatorysworksare not allowedsto feedback and depart from the system withsdprobabilityspssuchdthatdps+qd= 1 with servicesdratessd𝜇1𝑑ands𝜇2 respectively then also the symptotic behaviour ofs dthe average queue lengthsdcoincidessdwithsdthesdresultsdofsd ShanmugasundaramsdandsdChitra.sd

NumericalsdStudy

Insthisdsectiondadnumericalsstudysis made basedsonsthesaveragesqueueslength of thes model. Forsthisspurposestwostablessarescomputedsbysvaryingsthesvaluessof λ and Ω by keeping µ fixedsand then varying µ and Ω by keeping λ fixed with λ<µ.

Table sd:1

sdThesdaveragesdlengthsdofsdthesdsystemsdH(t)sdissdcomputedsdforsdcatastrophicsdeffectsdofs Ωsd= s0.3,0.6, 0.9s dwith sdµsd= sd10.s λ Ω(0.3) Ω(0.6) Ω(0.9) 1 0.107156478 0.103497187 0.100099205 2 0.238900952 0.228902577 0.219832183 3 0.404243315 0.383147227 0.364602356 4 0.616804097 0.575908711 0.541468249 5 0.897773718 0.820692544 0.759471622 6 1.280883932 1.136017172 1.029732001 7 1.820063752 1.546060566 1.364936683 8 2.597927127 2.079246401 1.777777778 9 3.723532349 2.762735242 2.278240709 10 5.295112884 3.61298756 2.870624736 0 1 2 3 4 5 6 A ve rag e Qu e u e Len gth Ω(0.3) Ω(0.6) Ω(0.9)

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Fig 1

Tablesd:2sd

ThesdaveragesdlengthsdofsdthesdsystemsdH(t)s dis computedsdforsdcatastrophicseffectsdof sΩ = 0.3, 0.6, 0.9swithsλsd= sd5.s µ Ω(0.3) Ω(0.6) Ω(0.9) 6 2.455141541 1.846464005 1.527030955 7 1.766765872 1.442734638 1.243925591 8 1.349574196 1.163331999 1.034895452 9 1.081228688 0.96539335 0.878617872 10 0.897773718 0.820692544 0.759471622 11 0.765729744 0.711548385 0.666666667 12 0.666666667 0.626871006 0.592867953 13 0.589834758 0.559551324 0.533062874 14 0.528620236 0.504900548 0.483774357 15 0.478760818 0.459734573 0.442544832 Fig 2 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 A ve rag e q u e u e le n gth Service rate µ Ω(0.6) Ω(0.3) Ω(0.9)

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Tablesd: sd3

Thesstationarydprobabilitysdistributiond𝜋0 for fixedsvalued ofdµs= 10sandsdΩsd= s0.3, 0.6, 0.9 for svarioussvalues of λ are computed as follows.

λ Ω(0.3) Ω(0.6) Ω(0.9) 1 0.903214694 0.906209831 0.909008928 2 0.807167029 0.813734155 0.819784896 3 0.71212729 0.722988834 0.732814212 4 0.618504123 0.634554523 0.648732142 5 0.526933212 0.549241553 0.568352446 6 0.438426518 0.6816103 0.49267588 7 0.354601913 0.392763634 0.422844302 8 0.277937814 0.324754784 0.36 9 0.21170597 0.265764115 0.305041664 10 0.158853387 0.216779254 0.258356226 Fig 3 Conclusion:

sssHereswe derive the probability sof n’ number of customers in the system and no customer ind thesdsystems. The snumerical exampless shows whens dthesd arrivals rate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10

π

0 Series1 Series2 Series3

λ

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increases thesaverage squeue sdlength

increasess (as din Fig-1).The increases din service rate decreases sthesaverages thesd queues length (as in Fig-2). Ass thes arrivals drate dincreases the stationarys probabilitysofd nos customerss in sthe system sdecreases (assdin sdFig-3) . Itsshowsdthe correctness sof sdthesmodel.

References:

[1]sA.K.Erlang,sdThesdTheorysdofsdprobabilitiessandsdTelephonesdconversations,sNyts Jindsskriff for Mathematic,sdB sd20, sd33-39sd(1909).

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Bernoulli s feedback, International Journal of Informationsdand dManagements Sciences, Vol 16, pp.35-52, sd2005.

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