2-MCM2004B-M-美賽國賽論文

ControlNumber501PAGEPagePAGE1ofNUMPAGES27SummaryTheQuickPasssystemallowsthetouriststodoubleusetheoriginalwaitingtimeandoffersthemmoretimeforotheractivitiesinthepark.However,therestillexitsomeproblemsinthecurrentQPsystem.Inthispaper,weanalyzetheexistingQPsystem,proposebetterQPschemes,andtesttheirvalidity.AfterthecarefulanalysisofthecommonpursuitoftouristsandCEOs,weconcludethattheprincipleobjectiveofaQPsystemistoincreasetheaveragefreetimeoftourists.Thisincrementisdirectlyreflectedinthedecreaseofthelengthofawaitingline.Empirically,weassumethatthelengthofaregularlinehasanoptimalvalue.WealsosetalimittothelengthofthelineforQPholders.WhenformulatingQPschemes,themostimportantfactoristhereturntimedesignatedtoeachtourist.First,weproposeScheme1,asimplifiedversionoftherealQPsystem,inwhichthereturntimeisdeterminedbythecurrentlengthoftheregularline.Weusetheresultofthisschemeasthebaselineofanalyzingthelaterschemes.ThenwetakeintoaccountthenumberofQPholdersinthevirtuallineandcreateScheme2.ThisisaFIFOsystemandthuscanavoidtheproblemoftheinstabilityoftheassignedreturntime.InScheme3,wepredictthearrivalpatternofthedayunderdiscussionusingthehistoricaldataandcomeupwithanalgorithmthatguaranteesthereturningQPholderstoavoidthepeaktimeoftourists’arrival.Scheme3cansuccessfullyimprovetheperformanceofthesystem.BasedonScheme3,wefurtherrecordthereturntimeofeachQPholderinthevirtuallineandintroduceScheme4.WestaggerthereturntimeofQPholdersandthuscontrolthelengthofthelineforreturningQPholderswithinadesirablelevel.Finally,weformulateScheme5fromanewperspective.Thisnewschemewillmakesurethateachdesignatedreturntimemaximizesthefreetimeincrementofthesystem!Astheextensionofthismodel,wealsoproposeasystemwithpartialpriorityforQPholders.WebrieflycomparethenewsystemwiththeoriginalQPsystemwithabsolutepriorityandrecommendthesituationsofintroducingthispartialprioritystrategy.Totesttheaboveschemes,weestablishaconsideratesimulationprogram,takingintoaccountmanycomplexfactorsintherealworldsituation.Thenwehavethepowerfulweapontotestourproposedschemes.Basedontheresultsofsimulation,wecometosomeusefulconclusionsandstrongrecommendationsfortheCEOs:Thetwocriteriaofevaluatingaschemearethefreetimeincrementandtheregularlinelength.Thetwoquantitiesfollowalinearrelationshipapproximately.AftertheimplementationofQPsystems,thelengthofregularlinereducesbyabout20%.Tomaintainthelengthofregularlinewithinaproperrange,anappropriatenumberofQPticketsshouldbeissued.Itisimportanttoaccumulateandupdatethehistoricaldata.

TheQuickPassStrategy1AnalysisoftheProblemNowadays,theemergenceof"QuickPass"systemhasreducedthepeople’stimewaitinginlineandthushasbeenwidelyusedinmanyareas.Especiallyinamusementparks,thesystemhelpthetouristsgetridofthehorriblylongwaitinglineandofferthemmoretimeforotherrecreationalactivities.Itseemstobeawin-winstrategy.WehaveaccumulatedlotsofcommentspraisingtheintroducingofthissystembothfromtouristsandCEOsoftheparks.However,doestheQuickPassreallyhavethegreatpowertoenhancethesatisfactionoftouristsononehandandontheother,tobringmoreprofitstothebusinessmen?Actually,wehavefoundsome“necessaryevils”fromtheQuickPasssystem(wewillcallitQPforshort),whichweshouldpayspecialattentionto:AlongreturntimedesignatedbyQuickPass(say,6-8hours)willblockatouristfromgettinganyfurtherQPfortherestoftheday!AtouristmaynotgetaQPfromabigattractionlateinaday.Thereturntimemaybecomeunstablesometimes.Forinstance,atouristmayfinditunfairtobeassignedtoreturn4hourslaterwhenheobservesthatjustashortlateronthesameride,theQuickPasswasgivenforanhourorsolater.PerhapsthemostdissatisfactorysituationisthattheQPholdersreturntotheridehappilyonlytofindthattheystillhavetowaitinalinenearlyaslongastheregularlines.Accordingtothosedrawbacks(maybemore),itisclearthatthere’sstillgreatneedtoimprovethecurrentsystem.Ourtaskisanalyzingtheproblemsintheoldsystem,proposingourbetterstrategiesandthentestingthevalidityofourschemes.Moreover,wewillcomeupwithsomekeycriteriatoevaluatetheQPschemesbasedonourmodel.Beforeestablishingourmathmodel,itisimportanttogetadeeperunderstandingofthe“QuickPass”System.ThekeywordinaQuickPassSystemis“virtualline”.Thisconceptderivesfromaverysimpleidea,thatis,tousesomecertainformtorecordthepositionofapeopleinaline,toestimatethetimefromnowonuntilhegetstheserviceandtoassignareturntimeforthepeople.Thesystemthenallowsthepeopleto“idle”duringtheperiodofhisoriginalwaitingtime,doingwhateverhewants,solongashereturnstoaccepttheserviceatthedesignatedtime.Disneylandisoneofthefirstamusementparkswhoadoptthiskindofsystem,namely,FastPass,andthecompanyhasachievedsomesuccess.Inthepast,touristsvisitingDisneylandcomplainironicallythatthe“mostfrequentactivity”issimplytowaitinaline!Althoughitmaybealittleexaggerating,thelongwaitinglineisindeedtheNO.1headacheofbothtouristsandtheCEO.Obviously,itiscausedbythecontradictionbetweentheever-increasingtouristsandthelimitedcapacityofsomehotattractions.Longwaitinglineswillnotonlycauselargeamountsofcomplaintsfromtouristsbutalsothelackoftimeforthemtogoshoppingordininginthepark,whichisagainstthehopeoftheCEO?(Sincetouristsdonotbuyextraticketsaftertheyenterthepark,theprofitgainfromshoppingcentersorrestaurantsistheonlypossiblesourceforprofitincrement.)Therefore,DisneylandintroducestheFastPasssystemtoskillfullyallocatetourists’timeandmakespossiblethedoubleuseoftheirtime.NowwehavegotsomeideaofthereasonforintroducingtheQuickPasssystem.ButwestillwouldliketofurtherexploretheexpectedbenefitsoftheQPsystemfromtwodifferentperspectives:theenjoymentrelatedconsiderationoftouristsandtheprofitrelatedconsiderationsofCEOs.TouristsToplayinasmanyattractionsaspossibleOfcourseitisthesincerehopeofeverytouristsandtoday’sQPsystemfacilitatetherealizationofthishope.Ourschemeswillfurtherimprovetheenjoymentoftourists.ToreducethewaitingtimeQPsystemwillgivetouristsmoretimeforotheractivities,includingplayinginmoreattractions.Therefore,thisfactorincorporatesthefirstone.CEOs1.TostretchtheaveragefreetimeforeachtouristThe“averagefreetimeofatourist”isdefinedastheaveragevalueofatourist’stotaltimeusedintheparkminusthetimeusedtowaitinlinesandactuallyplayinattractions.TheincreaseofthistimemeansthatthelikelihoodforthetouriststogoshoppingordiningintheparkmaybecomelargerandsodoesthelikelihoodfortheCEOstoearnmoreprofitgains(althoughwecannotexactlycontrolthebehavioroftourists).Ontheotherhand,touristsalsoprefertohavemorefreetimeforplayingorshoppingratherthanpassivelywaitinanendlessline.Weputemphasisontheincreaseofaveragefreetimeforallthetourists,includingQPholdersandtouristsintheregularline.Obviously,QPsystemcanincreasethefreetimeofthoseQPholders,butatthesametimesomeoftheQPstrategiesmayalsostretchthewaitingtimeofthoseintheregularlines.SincetotheCEOsthetwogroupsoftouristshaveequalstatus,weshouldtakeintoaccounttheinterestsofbothgroups.SomeofourQPschemescanincreasethefreetimeforbothQPholdersandregulartourists.2.TomaintainawaitinglinewithappropriatelengthinfrontofeachattractionThisstatementmayseemalittleconfusingatthefirstsight,butaccordingtothecommentsfrommanyCEOSofamusementparks,theyhopethatthereisawaitinglineinfrontofeachattractionbecauseitisasymbolofattractivenessofthisattraction.Obviously,theQPrulestrytoreducethewaitingline.ButthegoalofourQPsystemistomaintainawaitinglinewithappropriatelengthinsteadofeliminatingtheline.Fromtheaboveanalysis,wefindthatthepursuitsofCEOandtouristsarequitesimilar:morefreetimefortourists.WithregardtothesecondhopeofCEO,althoughitisobviouslynotthehopeoftourists,mosttouristsarewillingtoacceptalinewithanappropriatelength,especiallyinathemepark.2AssumptionsAtouristonlybuysonetickettoenteranamusementparkandnoextramoneyforplayinginanyattraction.Butgoingshoppingordiningintheparkisextracost.TouristscanfreelychoosewhetherornottorequestaQPticket.EverytouristisallowedtohaveonlyoneQPticketatatime.Thelengthofreturntimewindowisaconstantvalue.Acertainpercentage(1-m%)ofQPholderswillreturntotheattractionatthedesignatedtime.OverdueQPticketisinvalid.Thereisnobreakdownofanattractionandstaffsoftheattractionarealwaysworkingduringbusinesshours.Therefore,thecapacityofanattractionremainsconstantduringaday.3TermsandDefinitionNameDefinitionrithearrivetimeoftouristiqithelengthofregularlinewhentouristiarrivesvithelengthofvirtualline(thenumberofthepreviousvalidQPholders)whentouristiarrivesTithereturntimeoftheithtouristifhe/sheiswillingtohaveaQPWithereturntimewindowoftheithtouristifhe/sheiswillingtohaveaQPCjthecapacityofattractionj(thenumberoftouriststhatitcanserveatatime)m%theno-showrateofQPholders

4Model1QPSystemwithAbsolutePriority4.1ACoarseQuantitativeAnalysisofQPSystemWefirstdosomecoarsemathematicalestimateofaQPsystem.WehopetofindthepossibleinteractionbetweenQPholdersandtheregulartourists.WesimplifytheproblembynotconsideringtheconcreteQPrules.AnattractionthatusesQPrulesisreducedtoasingle-serversystemwithtwokindsofclients:QPtouristsandnon-QPtourists.Onaverage,λ1QPtouristsandλ2non-QPtouristsarriveattheattractioneachtimeunit.QPtouristshavepriorityandthusnon-QPtouristscannotgettheservicesolongasthereareQPtouristsinthesystem.Thenwetrytofindsomeconclusionsinthesenseofstatisticalaverage.WedescribethestateofaQPsystematthetimeinstanttasN(t)=(i,j)whichmeansthatthereareiQPtouristsandjnon-QPtouristsinthesystematthismoment.SetPi,j(t)equaltoP{N(t)=(i,j)},denotingtheprobabilityoftheappearanceofstateN(t).AfterallthesimplificationsitbecomesaMarkovprocess.Thestatetransitionequationsareasfollows:Aftertransposing,wedividethetwosidesoftheequationsbyΔtandsetΔtgotozero,yielding:AccordingtotheMarkovlimittheorem(K.LChung(1960)[42],Part2),weknowthatthelimitofastgoestoinfinityexists(denotedbyPi,j).Intheequations(2),weset.Thenthederivativesontheleftsidegotozeroandwegetlinearfunctionsasfollows:Thegeneralsolutionofequations(3)seemstobedifficulttofind.Fortunately,welearnthe“MotherFunction”methodofsolvingforsuchequationsinamonographofrandomservicesystem[Ref1].Usingthismethod,weobtaintheexpressionoftheaveragenumberofnon-QPtouristsinthesystemthatreflecttheinfluenceofQPholdersinthesystem.([Ref1]showsthebasicmethodweusedtosolvefortheequations.)Theaveragenumberofnon-QPtouristsis(4)whereλ1,λ2arerespectivelytheaveragenumberofQPtouristsandnon-QPtouristsarrivinginthesystemeachtimeunit,anduistheaveragenumberoftouriststheattractioncanserveeachtimeunit.Inourcase,uequalstothecapacityoftheattractiondividedbythetimeofplayingintheattractionforonetime,whichisaconstantvalue.Wecanseefromtheaboveexpressionthattheaveragenumberofnon-QPtouristsremaininginthesystemwillgoupwiththeriseofthearrivingdensityofQPtourists.Therefore,themoretouristsholdQPtickets,themorenon-QPtouristsmayhavetowaitintheline.Itseemstobeapessimisticestimate,sincetheimplementationofQPrulesmaycausethelengtheningofwaitinglineformanynon-QPtourists.However,sincethisisaverycoarseestimate,withoutconsideringtheconcreterulesofaQPsystem,theconclusionsabovecanprovidelimitedguidance.Actually,laterwewillfindthatwiththeimplementationofappropriateQPrules,wecanavoidsuchexacerbatingsituation.4.2QPSchemesAQPsystemhasseveralkeyfactors,say,whetherornotanattractionshoulduseQPrules,thenumberofQPticketsanattractioncanissueduringaday,etc.Butthemostimportanttaskistodeterminethereturntimeandthereturntimewindowforeverytouristuponarrivalbasedonthecurrentdata.Inourmodel,wewillalsoputemphasisontheschemetofindoutthevalueofthesetwokeyquantities.Firstofall,weofferasimpleruletodeterminewhetherornotanattractionshouldintroduceQPrules.AccordingtothecommentsfrommanyCEOs,tomaintainawaitinglineofappropriatelengthisdesirable.Ofcoursethelineshouldnotbetoolong.Therefore,weempiricallysetanoptimallengthofawaitingline.Basedontheinformationwehaveaccumulated,thewaitingtimeoftenminutesisacceptabletomosttouristsinalargeamusementpark.Therefore,weassigntheoptimallengthofregularlineofanattractionasthenumberoftouriststheattractioncanserveintenminutes.Iftheaveragelengthofthewaitinglineisshorterthanthisoptimalvalue,thereisnoneedfortheattractiontouseQPrules.NowwefocusononespecificattractionusingQPrules.Weonlyconsideroneattractionsincewebelievethattheinfluenceofotherattractionsisonlyreflectedinthehistoricaldata(say,theaveragenumberofpeoplewhoarriveattheattractionduringacertainperiodoftime),whichwewillalsotakeintoaccountinourschemes.

4.2.1Scheme1Atfirst,wetrytofindtheapproachusedbytheFastPasssystemofDisneylandfromtheinformationontheInternet,butlater,webelieveitismorelikelytobeacommercialsecret.Fortunately,fromthewordofmouthofmanytouristsandunofficialguidetoDisneyland,wecanstillgettoknowtheapproximaterulesofdeterminingthereturntimeinaQPsystem.ThebiggestvariableinthesystemthatwillaffectthereturntimeofatouristishowlongtheregularlinefortheattractionisatthetimehegetstheQP.Therefore,weestimatethereturntimeoftheithtouristaccordingtothefollowingsimplefunction:whereTiisthereturntime,riisthearrivetime,Ciisthecapacityoftheattraction,qiisthecurrentlengthoftheregularline,andεisamodifiedfactorspecifiedbythesystemcontroller.Forconvenience,inthisbasicmodelwesetthereturntimewindowWiequaltoanhour,whichisthetimewindowusedbyDisneyland.Weusethisvaluefortheconvenienceofthetourists.AnalysisofScheme1Thisschemefordeterminingreturntimeisverysimpleandeasytoimplement.Andusingscheme1willencouragepeopletogetQPticketssincethereturntimeisshortandQPholderscanalsoenjoythebenefitsofgettingtheservicebeforesomenon-QPtouristswhoarrivedearlier.Butbasedonthisscheme,thefreetimeincrementofQPholdersissmallandontheotherhand,thoseintheregularlinemayhavetowaitforevenmoretimebeforetheimplementationofQPrules.Thisresultisagainstouroriginalobjective.Moreover,sincethiskindofQPsystemisnotFIFO,itwillcauseaseriousproblemwehavepreviouslydiscussed:Thereturntimemaybecomeunstable.Thereasonisthatweonlyconsiderthelengthoftheregularlineandthusthesuddenchangeofthisline(say,alargegroupoftouristsarriveattheattractionduringaveryshortperiodoftime)willcausetheaboveproblem.Fig1Boththeregularlineandvirtuallineincreaseasthereturntimegoesup.Theappearanceofthepeakisbecausethatwehavenottakenintoaccountofthevirtualline.Thereforelaterwewillconsiderthelengthofvirtualline.4.2.2Scheme2Scheme1isaverysimpleapproach.NowweaddinconsiderationthenumberofpeopleintheQPvirtualline.InthiswaythesystembecomesFIFOandthuscanavoidtheprobleminscheme1.Tobeconcrete,wereservethepositionofaQPholderinthelineandallowthetouristtodoanythinghewantsduringhisoriginalwaitingtime.Therefore,weestimatethereturntimeoftheithtouristaccordingtothefollowingsimplefunction:whereTi,ri,Ci,qi,andεhavethesamemeaningasthoseinscheme1,andviisthecurrentlengthoftheQPvirtualline.AnalysisofScheme2AfterconsideringtheQPvirtualline,scheme2solvesthemainproblemofscheme1.Moreover,itfurtherincreasesthetotalfreetime,sinceaccordingtoFIFO,thewaitingtimeforpeopleintheregularlineremainsthesameastheircounterpartsinhistoryandthushasnotbeenstretchedbecauseofQPrules.4.2.3Scheme3AgoodQPstrategyshouldnotonlyincreasethefreetimebutalsomakethelengthofthewaitinglineinanappropriaterange.Therefore,wehopethattheQPholderswillreturntotheattractionduringthe“valley”ofthetourists’arrival.Toachievethis,wenowtakeintoaccountthefluxoftouristsofanattractionshowedinhistoryandassumethatthesituationoftourists’arrivalduringthedayunderdiscussionfollowsthesamepatternasthehistoricaldata.Foraspecificattraction,weneedfourtypesofbasichistoricaldatawhenformulatingourscheme.thenumberoftouristsanattractioncanserveinatimeunittheaveragenumberoftouristsarrivingattheattractionduringeveryperiodoftimeofanordinarydaythelengthoftheregularlinewhenatouristarrivesattheattractionthelengthoftheQPvirtuallinewhenatouristarrivesattheattractionBasedonthefourtypesofdataabove,wewillformulatearuletodeterminethereturntime(denotedbyTi)oftheithtourist.Ourruleisbasedonthefollowingcriteria:tomovethepositionoftheQPtouristbackwardsinordertoavoidthe“peaktime”manifestedinthehistoricaldataandfillhispositioninthe“valley”ofthedatagraph.Inthisway,wetrytomaintainthenumberofthetourists’arrivalinastablelevel.Wefirstfigureoutthelengthofwaitinglineduringeveryperiodoftimeinadayusingthenumberoftouristsarrivedattheattraction,includingthenumberoftouristswhorequestedaQPticket,andthecapacityoftheattraction.ThelengthofwaitinglineisdenotedbyA.Itcanbeaminusvalue,whichmeansthattheattractionisnotinuseatthattime.Nowwedescribetheconcreteprocedurestosearchforthereturntime.ForaQPtourist,thereisalowerboundforhisreturntime.WesetthislowerboundbethereturntimederivedfromScheme2.Thislowerboundtimeisthestarttimepointofoursearchingalgorithm.SearchbackwardsandfindthefirsttimepointthatAisaminusvalue.Theattractionhasalowworkloadatthistimepoint.Therefore,weassignthistimepointasthereturntime.Consideringthetourists’interests,wealsointroduceanupperboundofthereturntime.Oursearchingprocesswillstopifitreachestheupperboundandwillreturnthisvalue.Fig2TheredcurverepresentsthedataofanattractionwithoutaFPsystem,whiletheblueonerepresentsthedataofthesameattractionwithaFPsystemBycomparingtwocurves,wecanobservethattheFPsystemassigntheQPholdersalaterreturntimethatavoidthepeaktimeoftourists’arrival.4.2.4Scheme4InScheme4,wesetthereturntimeofQPholderstoavoidthepeaktimeoftourists’arrival(itisalsothetimethattheregularlineisverycongested).However,thisstrategycannotstaggerthereturntimeofQPholders.ItmeansthattheremayalsoexistpeaktimeforQPholderstoreturn,sothatthoseQPtouristsmaystillhavetowaitinaquitelongQPline.Therefore,inScheme4,wewanttoformulateaQPrulewhichcanavoidsuchsituation.Inotherwords,wehopethatthelengthofQPlinecanalsobecontrolledinashortandstablelevel.Inordertoachievethisgoal,wemustrecordthereturntimeofeachQPholderofthisattractioninourprogramsothatwecanadjustthelengthofQPvirtualline.WehopethattheintroducingofQPsystemwilllimitthewaitingtimeforareturningQPholderwithinfiveminutes,whichisthestandardexpectedvalueusedbyDisneyland.Therefore,wesettheupperboundoftheQPlineequaltothenumberoftouriststheattractioncanserveinfiveminutes.Thefollowingistheconcretesearchingprocedure.First,wefindthereturntimepointderivedfromScheme3(denotedasRt).Then,basedontherecordedreturntimeofeachQPholderinthesystem,wefigureoutthelengthoftheQPlinewhenthisQPholderreturnstotheattractionatthetimepointRt.IfthelengthoftheQPlineatRtexceedstheupperboundwesetabove,wefurthermovethereturntimepointbackwardsforfiveminutes.ThisstrategywillguaranteethattheQPholders’waitingtimeisnomorethanfiveminutes.TheeffectofScheme4isshowedinthefollowingfigure.Fig3Theeffectivenessofscheme4ontheadjustmentoftheQPlineTheblueonerepresentsthedatebeforetheimplementationofScheme4andtheredonerepresentsthedatebeforetheimplementationofScheme44.2.5Scheme5InScheme5,westillfocusonthemaximizationofthefreetimeofalltourists.Butwewilluseanewapproachtoformulateourscheme.Beforedescribingtheconcreterules,wewanttofurtheranalyzethemechanismofstretchingthefreetimeinaQPsystem.Forthesakeofconvenience,wedonottakeintoaccountthelengthofthereturntimewindowandsimplyassumethereturntimeasadesignatedtimepoint.InScheme2,theQPsystemmaintainsthepositioninthelineforaQPholder.Whenthetimefor“serving”thispositioncomes,theQPholderreturnstoaccepttheservice.Therefore,afterintroducingScheme2,theonlyincreasedfreetimederivesfromtheoriginalwaitingtimeoftheQPholder.Asforthetouristsintheregularline,theirfreetimehasneitherincreasednordecreased,comparedtothesituationbeforeintroducingQPrules.Actually,thisisaninterestinghiddenfact.Manynon-QPtouristscomplainthattheirwaitingtimeislengthenedbecauseofthecontinuousarrivalofreturningQPholders.ButifthesystemfollowstheFIFOpatternasinScheme2,theysimplywaitthesamelongtimeastheircounterpartsinthehistorydid!Inaword,Scheme2bringsnobenefittothetouristsintheregularline.TheonlygroupofpeoplewhoobtainmorefreetimeareQPholders.Seethefollowingfigure.QPholderonlyreservestheposition.QPholderonlyreservestheposition.Fig4SinceScheme1maystretchthewaitingtimeoftouristsintheregularline,thefreetimeincrementofScheme1isevensmallerthanthatofScheme2.ButinmostofthetimeScheme1canstillincreasethetotalfreetimeoftourists.InScheme3,wemovethepositionofaQPholderbackwardssothatsometouristsarrivinglatercanenjoythebenefitsofgettingtheserviceearlier.Ontheotherhand,theQPholderssacrificetheircurrentpositioninthelinetogetmorefreetimeforotheractivitiesinreturn.Thisstrategycanfurtherimprovetheaveragefreetimeoftouristsinthepark.Theeffectisillustratedinthefollowingfigure.QPholdersQPholderssacrificetheircurrentposition. Fig5Sincetheregular-linetouristswhoarebetweenthetwoarrowscangettheservicebeforetheearlierarrivingQPholders,thetotalfreetimenowisnt+Rt,wherenisthenumberofregular-linetouristsinthedashedframe,tistheaverageservicetimeforeachtouristandRtisthereturntimeoftheQPholder.Fromtheaboveanalysis,wecanseethatdifferentQPrulescanbringdifferentfreetimeincrement.Butinmostofthesituations,theintroducingofQPruleswilldefinitelyincreasethefreetimeofthesystem.Therefore,inordertomaximizethetotalfreetimeofasystem,itseemsthatthenumberofQPholdersshouldbeaslargeaspossibleandthereturntimeofQPholdersshouldbeassignedaslongaspossible.However,itisnotthecase.First,weconsiderthelimitsituation.IfallthetouristsholdQPticketsandreturntotheattractionatthedesignatedtime,thelineinfrontoftheattractionwillnearlydisappear.ItisagainstthepreviousdiscussedhopeofCEO—tomaintainawaitinglinewithappropriatelength.Therefore,thesystemshouldsetalimittothenumberofQPticketsavailable.Second,wewouldliketotakeintoaccountthepsychologicalfactorofthetouristsandmakethefollowingassumption.Westillusethelowerboundofreturntimespecifiedinscheme3andassumethattheprobabilityoftouriststorequestaQPtickethasthehighestvaluecorrespondingtothisreturntime.Withtheincreasingofthereturntimemanifestedontheboardoftheattraction,fewerandfewerpeoplearewillingtogetaQPticket.(Sincetheyhavetoidletoomuchtimebeforeplayinginthisattraction,itwillsuretocausetheirinconvenienceanddissatisfactionthatleadtotheabandonmentoftheideatogetaQPticketatthattime.)Weusethesimplecurvetodescribethisassumption:Fig6TheprobabilityoftouriststorequestaQPtickethasthehighestvaluecorrespondingtothisreturntime.Withtheincreasingofthereturntimemanifestedontheboardoftheattraction,fewerandfewerpeoplearewillingtogetaQPticket.ThenwemultiplythenumberofthepeoplewhoarewillingtogetaQPticketbythecorrespondingreturntime,yieldinganewcurveasfollows:Fig7ItshowsthetendencyofthechangesinfreetimeofQPholderswithrespondtothechangesinreturntime.Thereexistsapeakvalueofthefreetime.Further,weaddthefreetimeincrementofregular-linetourists(asshowedbytheredlineinthefollowinggraph).Fig8Thegreenlineshowsthetotalfreetimeincrement.Wecanseethatwiththeincreasingofthereturntime,thetotalfreetimeofth