DESIGN THE SWITCHING PERIOD FOR TRAFFIC LIGHT.docx

DESIGN THE SWITCHING PERIOD FOR TRAFFIC LIGHT.docx

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DESIGNTHESWITCHINGPERIODFORTRAFFICLIGHT

DESIGNTHESWITCHINGPERIODFORTRAFFICLIGHT

Abstract

Thisisaproblemaboutthecapacityofthetrafficlightsintheintersection. Wefirstlydividedonehourinto 360 periodsoflength 10s. thenumberofthecarsarrivedattheintersectionisassumedtobeinPoissondistribution,andthevaluewithinagivenrange.WecangeneratearandomnumberinPoissondistributiontosimulatethenumberofthecarsarrived. Inaddition,weusetheconceptofquantityofflowtoanalogyarrivalandpassingcars,introductingtheconceptofstrandedcarsatthestartofeveryperiod.Itisthenumberofcarsstrandedonthestartofthelastperiodplusthenumberofcarsarrivedanddeductingthenumberofcarspassedinthelastperiod. Ineachperiod,whenthesumofthenumberofvehiclesstrandedandthenumberofvehiclesarrivingarefewerthanthecapacity,theyallcanpass.Whenmorethanthecapacity,only36carsper10seccanpasswhiletherestwillhavetowaitforthenextperiod. Thuswemakethemodeltosimulatethenumberofcarsstuckinacycle,thenumberofcarsarrived,aswellasthenumberofcarspassedineachperiod,thentheproblemcanbesolved.

Fortheproblemone,bythecumulativenumberofcarspassedduringeachperiod,wecanget thenumberofthepassedcarinonehouris3364inthedirection1.

 Astothesecondprobleminvolvingthewaitingtime.Basedontheoriginalmodel,weintroducetheconceptofthetimeofthecarsarrivedineveryperiodneedtowait.Thatisalsothetimefortheretentedcarstopassthroughtheintersection.Wecanobtaintheaveragewaitingtimewhenintheredlightis 43sbyaveragingthesumofwaitingtimeineachperiod.Comparethewaitingtimeineachperiod,wecangetthemaximumwaitingtime70s. 

Forthethirdissues,weassumethatallthecarsareofthesamelength,thevehicleisalsoequallyspacedwhenstopped. Soweconvertedthequeuelengthforthesakeofthenumberofcarsstranded. Accordingtothepreviousmodelestablished,wehavealreadygottheaveragenumberofcarsstrandedandthemaximumnumberofstrandedvehiclesbycomparingthenumberineachperiod.Theresultsisthattheaveragequeuelengthis 198mwhilethemaximumqueuelengthis 480m. 

Tothefourthquestion,becausethecarcanonlythroughtheintersectionwheninthegreenlight,theaveragenumberofcarspassingthroughtheintersectionduringthetimewhenthetrafficlightisgreenisequaltothetotalnumberofcarspassingthroughtheintersectiondividedbythenumberofgreenlightcyclewithinonehourperiod,approximately 93. 

Asforthefifthproblem,substantially,thedirectionforthecarsinthedirection2isthesamewiththatinthedirection1, thedifferenceisthetimeofgreenlightis30secinthedirection1withthe 70s redlight.Bycontrastthetimeofgreenlightinthedirection2is70s,whiletheredlighttakes30s . Inaddition,thedirection2has anumberofthecarsneedtopassthrough,whilethegreenlightownsasmallercapacity.Soweneedtaketheconditonthattherewillhavethecarsstrandedintheintersectionafteragreenlightcycleintotheconsideration.Weshouldanalysethewaitingtimeandthequeuelengthofthestrandedcarsinanotherway.Theresultsoftheprogrammingisthatthereare 5039 carsinthedirection2passthroughtheintersectionwithinonehour.Whenstoppedintheredlight,theaveragewaitingtimeforacaris 108s, themaximumwaitingtimeis141s. Theaveragequeuelengthis282mwhilethemaximumqueuelengthis834m. Wheningreenlight,theaveragenumberofthecarspassedthroughtheintersectionis 140,upto 140 ,themostcarscancross.

Thefinalproblem,weallowthetimeofthegreenlightinthedirection1changingbythestepina certainrange,togetthedifferenttotalwaitingtime.Andthenwefindtheapproximategreentimemakingtheshortestwaitingtime.Whentakingintoaccountthedirection2,weusethesameway.Thenweshouldaccumulatethetotalwaitingtimeinbothdirections,andchoosethetimeofgreenlightmakingthewaitingtimeshortestis28s. 

Keywords:

 trafficlightcycle,vehicle,waitingtime

1.Introduction

 Nowadaysthetrafficjamstravelproblemsisveryserious,howtodesignatrafficlightconversioncycletomakethecartheshortestwaitingtimeandqueuelengthisaveryimportantissue.

Consideranintersectionoftwoone-waystreetscontrolledbyatrafficlight.Assumethatbetween5and15cars（varyingprobabilistically）arriveattheintersectionevery10secindirection1,andthatbetween6and24carsarriveevery10secgoingindirection2.Supposethat36carsper10seccancrosstheintersectionindirection1andthat20carsper10seccancrosstheintersectionindirection2ifthetrafficlightisgreen.Noturningisallowed.Initially,assumethatthetrafficlightisgreenfor30secandredfor70secindirection1.Writeasimulationalgorithmtoanswerthefollowingquestionsfora60-mintimeperiod:

Howmanycarspassthroughtheintersectionindirection1duringthehour?

Whatistheaveragewaitingtimeofacarstoppedwhenthetrafficsignalisredindirection1?

Themaximumwaitingtime?

Whatistheaveragelengthofthequeueofcarsstoppedforaredlightindirection1?

Themaximumlength?

Whatistheaveragenumberofcarspassingthroughtheintersectionindirection1duringthetimewhenthetrafficlightisgreen?

Whatisthemaximumnumber?

AnswerProblemsa-dfordirection2.

Howwouldyouuseyoursimulationtodeterminetheswitchingperiodforwhichthetotalwaitingtimeinbothdirectionsisassmallaspossible?

（Youwillhavetomodifyittoaccountforthewaitingtimesindirection2.）

2.TheDescriptionoftheProblem

 Accordingtotheexistingtrafficflowtheory,webelievethatvehiclesqueuedservicetimeandthenumberofvehiclesinlinewiththePoissondistribution.

Forthequestionasked,wecalculatethe 10s asaminimumperiod.Assumingthenumberofcarsreachedtheintersection10s ineveryonehourareinPoissondistribution,andinthegivenrangeofonehour,thenwegeneratedatotalof 360 Poissonrandomnumbers,asthenumberofcarstoreach.Andwegivetheconceptofthenumberofcarsstranded,meansthatthenumberofcarsarrivedattheintersectionbutdidnotcrossitboforeamoment.Comparethenumberof thestrandedcarsand36,whichisthenumberofcarscrosstheintersectionindirection1per10sec.Wechoosethesmalleroneasthenumberofthecarscrosstheintersectioninthe10sec. Accumulatethenumberofeach10seccanobtainthenumberofthecarspassingthroughtheintersectionindirection1duringonehour.

Thesecondproblemconcerningtheaveragewaitingtimeofacarstoppedwhenthetrafficsignalisred.Tosolvethisproblems,weneedtoknowthetotalwaitingtimeofallcars,wedeterminedthewaitingtimeofthecarstrandedineach10secandaccumulatedtoobtainthetotalwaitingtime.Afterthetotalwaitingtimegot,wecanacheivetheaveragewaitingtimebydividingitintothenumberofallthecars. Then,comparetheretentiontimeofeachcarwaitingperiodtoobtainthemaximumwaitingtime.

Aboutthethirdproblem,weassumethatthelengthofallcarareinthesameandallcarsmaintainthesamesapcingwheninthequeue. Similarly,wefindthenumberofstrandedcarsineach10secperiod,multipliedbythevehiclelengthandspacing,andgetthelengthofthequeueofcarsstoppedforeachredlight,Dividedbythetotalnumberofperiodsafterthecumulativeresults,wecanobtaintheaveragelengthofthequeue. Then,bycomparingeachtimethelengthofthecarsqueuinginlonglines,wecangetthemaximumqueuelength.

Tothefourthproblem,becausethecarscanonlypasstheintersectionwheninthegreenlight,theaveragenumberofcarspassingthroughtheintersectionduringthetimewhenthetrafficlightisgreenisequaltothetotalnumberofcarspassingthroughtheintersectiondivided36thenumberofgreenlightcyclewithinonehourperiod.

Asforthefifthproblem,substantially,thedirectionforthecarsinthedirection2isthesamewiththeabovesolutionforthatinthedirection1, thedifferenceisthetimeofgreenlightis30secinthedirection1withthe 70s redlight.Bycontrastthetimeofgreenlightinthedirection2is70s,whiletheredlighttakes30s . Inaddition,thedirection2has anumberofthecarsneedtopassthrough,whilethegreenlightownsasmallercapacity.Soweneedtaketheconditonthattherewillhavethecarsstrandedintheintersectionafteragreenlightcycleintotheconsideration.Weshouldanalysethewaitingtimeandthequeuelengthofthestrandedcarsinanotherway.

Asforthelastextendedquestion,allowingthetimeoftheredlightinthedirection1changingbythestepof1sina certainrange,togetthetotalwaitingtimeineachperiod.Andthenwefindtheshortestwaitingtimeinthecertainrangeandgetthecertaintimeintheredlight.Whentakingintoaccountthedirection2,weusethesamewayinthedirection1.Thenweshouldaccumulatethetotalwaitingtimeinbothdirections,andchoosethetimeofredlightmakingthewaitingtimeshortest.

3.Models

3.1BasicModel

3.1.1Terms,DefinitionsandSymbols

Thesignsanddefinitionsaremostlygeneratedfromqueuingtheory.

●Q（t）:

  thenumberofthecarsarrivedatthetheintersectioninthedirection1inthe‘t’timeperiod.

●I（t）:

 thenumberofstrandedcarsreachtheintersectioninthedriection1 atthebeginningofthe‘t’period .

●q（t）:

  thenumberofthecarsinthedirection1passingthroughtheintersectioninthe‘t’timeperiod.

●D（t）:

 thetimeofthearrivedcarsnee