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ThemeParkQueueingSystems301Theme-ParkQueueingSystemsAlexanderV.FrolkinFrederickD.W.vanderWyckStephenBurgessMertonCollege,OxfordUniversityOxford,EnglandAdvisor:UlrikeTillmannSummaryWedetermineanoptimalsystemforallocatingQuickPasses(QPs)totheme-parkguests,subjecttokeycriteriathatweidentify.Werecommendaspecificsystem-awayofdecidingwhenaguestasksforaQPwhethertheyshouldgetone,andifsoforwhattime.Ontheotherhand,wewarnagainstsomeplausiblesystemsthatwouldactuallyworsenthequeueingsituation.WealsoexplainwhysomethemeparksuseanunfairwayofallocatingQPs,wherelateguestscanfarebetterthanearlyarrivals.Thekeystoourapproacharetwoverydifferentsimulationswiththesameparameters.TheExcelsimulationbreaksthedayinto10-minintervals,workswithgroupsofpeople,andisnonrandom.ItisfastandallowsustotestquicklymanydifferentQPallocationsystems.ThePerlsimulationbreaksthedayinto1-minintervals,modelsindividualpeople,andincludesrandomness.Itismorerealisticandflexible.Thus,thesimulationsareusefulindifferentcontexts.Thefactthattheirresultsareconsistentprovidesastrongsafeguardagainstincorrectresultscausedbycodingerrors,ariskinlargesimulations.Inaddition,wecarryoutextensivetestsofthestabilityofourmodelandtherobustnessofourrecommendation.WeconcludethatitisbesttoallocatelotsofQPsforslotsearlyandlateintheday,andfewerforthepeakdemandinthemiddle.WealsoexploremodificationstotheQPconcept,includingcharging.TheUMAPJournal25(3)(2004)301-318.(CCopyright2004byCOMAP,Inc.Allrightsreserved.Permissiontomakedigitalorhardcopiesofpartorallofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthisnotice.Abstractingwithcreditispermitted,butcopyrightsforcomponentsofthisworkownedbyothersthanCOMAPmustbehonored.Tocopyothenvise,torepublish,topostonservers,ortoredistributetolistsrequirespriorpermissionfromCOMAP.302TheUMAPJournal25.3(2004)IntroductionTheProblemThemeparkshaveintroducedtwobasictypesofnewqueueingsystems:*Invirtualqueuesystems,guestsuseapagertoregisterinaqueue;itpageswhentheyareneartheheadofthequeueandshouldcometotheride.Forexample,Lo-QPlc.hasdevelopedsuchasystem,usedinSixFlagsthemeparksintheU.S.[Sixflags...2002].*InQuickPasssystems,ridescanissueguestsaQuickPass,allowingthemtoreturntotherideataspecifiedtime,whentheycanridewithminimalqueueing.ExamplesofthistypeofsystemareDisney'sFastPass®[DisneyTickets20041andUniversalStudiosExpressPass[UniversalStudiosn.d.].Eithertypeofsystemmayormaynotbefree;Disney'sFastPass®isfreebutguestspayforLo-Q'spagers.WefocusontheQuickPasssystemandanalysehowtoimplementiteffec-tively.Weconcludewithabriefcomparisonwithvirtualqueuesystems.CriteriaforaGoodQuickPasssystemWetakethefollowingasgeneralguidelinesinchoosingaQuickPasssystem.*Atnotimeshouldmorethan50%ofaride'scapacitybeQPusers.*Norideshouldhaveaqueuelongerthan45min.*Theaveragewaitingtimeshouldbeasshortaspossible.*Waitingtimesshouldbeevenlydistributed.(Itisbetterthat100peoplewait20minthanthat50peoplehavenowaitbutanother50wait40min.)*Thesystemshouldseemfair.PeoplearrivinglatershouldnotgetQPswhenpreviousguestshavebeenrefusedthem.Similarly,peoplearrivinglatershouldnotbeallocatedearlierslotsthanpreviousguests.*QPsshouldnotbeallocatedformorethan4hinthefuture.(Weassume,basedonpersonalexperience,thatpeoplestayforonlyabout5h.)SummaryofOurApproach*Wecollectdataandperformcalculationstoobtainreasonableinitialmod-ellingassumptions.ThemeParkQueueingSystems303*Weconstructtwocomputersimulationsandmodifythemwithfurtheras-sumptions.Eventuallywefindthebehaviouragreeablewithcommonsenseandconsistentbetweenthetwosimulations.*Weusetwoverydifferentsimulationswiththesameparameters.TheExcelsimulationbreaksthedayinto10-minintervals,workswithgroupsofpeople,andisnonrandom.ThePerlsimulationbreaksthedayinto1-minintervals,modelsindividualpeople,andincludesrandomness.Eachapproachhasitsadvantages.Thefactthatthetwosimulationsgivesimilarconclusionsprovidesverypowerfulevidenceforthevalidityoftheconclusions.iWelistsystemsforallocatingQPsthatseemlikelytoworkwellandtestthemusingthesimulations.iWeanalysetheresultswithgraphicalinterpretationsandsummarystatistics.*Weassessthestabilityofourmodelundervariationsininputparametersandtherobustnessofourrecommendationunderdifferingconditions.TheSimulationProcessInitialmodellingassumptionsvWehaveinmindasanexampleaparticularthemepark,ThorpePark,Surrey,UK,whichweregardasatypicalmedium-sizedpark.*Wemodeladayrunningfrom8A.M.-6P.M.Thenumberofpeopleintheparkvariesovefthedayandhasakeyimpactonqueuelengths.*ThorpeParkhas2millionvisitorsperyear,andtheparkisopenforaround200daysperyear[ThorpeParkGuiden.d.].Soweassumethat10,000peoplevisittheparkonatypicalday.Mostarrivelatemorningorearlyafternoonandadmissionsstopwellbeforeclosingtimesothatqueuescansubside.*Peoplewhoarriveearlytypicallystayforabout5h;laterarrivalsstayuntilashorttimebeforeclosing.*Thereisoneoverwhelminglypopularride,theBigRide,theonlyrideforwhichweissueQPs.*Weestimatefrompersonalexperiencethatpopularridestake40peopleandleaveevery5min,sotheBigRidehasacap