Custo mers eat in the restaurant with probability 05 and carry out their orde r without eating with probability 05 A meal requires an average of 20 minute s What is the average number of customers in the restaurant Question 2 IITK sports facility ha ID: 57849
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PracticeProblems:Question1:Customersarriveatafastfoodrestaurantatarateof5perminuteandwaittoreceivetheirorderforanaverageof5minutes.Customerseatintherestaurantwithprobability0.5andcarryouttheirorderwithouteatingwithprobability0.5.Amealrequiresanaverageof20minutes.Whatistheaveragenumberofcustomersintherestaurant?Question2:IITKsportsfacilityhas4tenniscourts.PlayersarriveatthecourtsataPoissonrateofonepairper10minanduseacourtforanexponentiallydistributedtimewithmean40min.Supposethatapairofplayersarrivesand ndsallcourtsbusyandkotherpairswaitinginqueue.Howlongwilltheyhavetowaittogetacourtontheaverage?Question3:ConsideranM/D/1queuewithservicetimeequaltobtimeunits.Supposefurtherthatoneisabletodeterminethesystemsizewhentimeisamultipleofb.DeterminethetransitionmatrixoftheMarkovchainoffXn;n=0;1;2::::jXnequaltothesystemsizeattimet=nbg.Question4:Considerathreestationqueueingsystem(singleserverateachstation)withPoissoninput(parameter)andexponentialservice(parameters1,2and3).Thereisnocapacitylimitonthequeueinfrontofthe rsttwostations,butatthethird,thereisalimitofKallowed(includingservice).IfKareinthethirdstation,thenanysubsequentarrivalsareshuntedoutofthethirdstation.Findtheexpectedtimespentinthesystembyacustomerwhocompletesallthreestagesofservice.1 Question5:Consideraringnetworkwithnodes1,2,...,K.Inthisnetwork,acustomerthatcompletesserviceatnodeiexitsthenetworkwithprobabilityp,oritisroutedtonodei+1withprobability1-p,fori=1,2,...,K-1.CustomersthatcompleteserviceatnodeK,eitherexitthenetwork,orareroutedtonode1,withrespectiveprobabilitiespand1-p.Ateachnode,externalcustomersarriveaccordingtoaPoissonprocesswithrate\r.Theservicetimesateachnodeareexponentiallydistributedwithrate.Thearrivalprocessesandtheservicetimesatthevariousnodesareindependent.1.Findtheaggregatearrivalratesi,i=1;2;:::;K2.Findthestationarydistributionofthenetwork.Underwhatconditionsdoesthisstationarydistributionexist?3.Findtheaveragetimethatacustomerspendsinthenetwork.4.Isthe\rowonthelinkbetweennodes1and2Poisson?Justifyyouranswer.Question6:Customersarriveaccordingtopoissonprocesswithratetoatwoserversystem.TheservicetimeoftheserversAandBareexponentiallydistributedwithratesAandB,respectively.Acustomerwho ndsthesystememptyonarrivalisallocatedtotheserverthathasbeenidleforthelongesttime.Otherwise,thecustomerattheheadofthequeuegoestothe rstfreeserver.1.De nethestatesdescribingthesystemasaMarkovchainanddrawthestate-transitiondiagram.2.Findthestationarydistribution.3.Isthesystemreversible?Justifyyouranswer.4.FindthetransitionratesofthereversedMarkovchainanddrawitsstate-transitiondiagram.2 Question7:CustomersarriveataservicestationaccordingtoaPoissonprocesswithrate.Theservicetimesareexponentiallydistributedandindependentofeachotherandthearrivalprocess.Queuebu ersarein niteandcustomersareservedona rst-come- rst-servedbasis.Considerthefollowingcases:Case1:Therearetwoserverssharingacommonqueue.Eachserverprovidesserviceatrate.Acustomerthatuponarrival ndsthesystememptyisroutedrandomlytooneoftheservers.Otherwise,itentersthequeueandwaitsforservice.Thecustomerattheheadofthequeuegoestothe rstfreeserver.Case2:Thereisasingleserver(withasinglequeue)thatprovidesserviceatrate2.Case3:Therearetwoservers,eachwithitsowndedicatedqueue.Theservicerateofeachserveris.Uponarrivaltothecombinedservicesystem,acustomerisroutedtothe rstqueuewithprobability0.5,ortothesecondqueuewiththesameprobability.Answerthefollowingquestions:1.ForaCase1station:Drawthestatetransitiondiagram,andderivethestationarydistribution.Findtheaveragenumberofcustomersinthesystemandtheaveragetimeacustomerspendsinthesystem.Also ndtheaveragenumberofcustomersthatarequeued2.ForaCase2station, ndtheaveragenumberofcustomersinthesystemandtheaveragetimeacustomerspendsinthesystem.3.ForaCase3station, ndtheaveragenumberofcustomersandtheaveragetimeacustomerspendsinthesystem.4.Comparetheaveragetimedelayofthesethreesystems.Isitbettertousetwoidenticalserversorasingleserverwithdoubleprocessingpowertoserveasinglequeue?Ifwehavetwoservers,isitbettertousededicatedqueues,oracommonqueueforwaiting.Explaintheresultsintuitively.3 Question8:IntheM/G/1systemwithparametersandX,deter-minethefollowing:Whatistheprobabilitythattheserverisidle?Whatistheaveragelengthofthetimebetweenbusyperiods?Whatistheaveragelengthofabusyperiod?Whataretheaveragenumberofcustomersservedinabusyperiod?Question9:Consideranopennetworkofsingle-serverM/M/1queuesasshowninFigure.1.ThemeanservicetimesatthequeuesQ1,Q2,Q3andQ4are1=;2=;2=and2=secondsrespectively.Letbetheaverageexternalarrivalrate(Poisson)toQ1.Whatisthemaximumvalueofunderwhichthenetworkwilloperateinastablemanner? Figure1:Question10:Twotypesofpacketsaretransmittedoveradatanetwork.48-bitlongcontrolpacketsaredesignatedforoperationssuchassignaling,congestionnoti cationandroutingchangeinformation.Thesepacketshaveahigherpri-oritytoanydatapacketpassingthroughthenetworkelements.Letassumethatthedatapacketsare960bitslongontheaverage,withanexponentialdistributedpacketlength.Thetransmissionlinksallhaveacapacityof96004 bps.Controlpacketsconstitute20%ofthetotaltrac.Assumethattheoveralltracutilizationoveratransmissionlinkis0.5.Ifnon-priorityserviceisused,showthattheaveragewaitingtimeforeithertypeoftrac(controlordatapackets)is99ms.Ifnon-preemptivepriorityisgiventothecontrolpackets,calculatethewaitingtimesforthetwotypesofpackets.Question11:Studentsenterthedininghallforbreakfastinequallylikelygroupsofeitheroneortwowithagrouparrivalrateof.The rstmemberofthegroupisservedinanexponentiallydistributedtime.Thesecondmemberifanyordersanextraside-dishwhichrequiressecondsmore,whereis xed.Themessoperatesasasingleserverqueue.Findthemeandelaythatanarrivingstudentwillencounterbeforebeingserved.5