Capacity of Shared-Short Lanes at Unsignalised Intersections

Transcription

1 Nig Wu itle ad Abstract Author's Address apacity of Shared-Short aes at Usigalised Itersectios Nig Wu ABSA he calculatio procedures i recet highway capacity mauals do ot eactly treat ared/ort laes at usigalized itersectios i a eact maer he capacity of idividual streams left tur through ad right tur are calculated separately If the streams are a commo traffic lae the capacity of the ared lae is the calculated accordig to the ared lae procedure from Harders 968 ie the legths of the ort laes are cosidered either as ifiite or as zero he eact legths of the separate ort laes caot be tae ito accout herefore the capacity computed from covetioal methods is overestimated whereas that from the ared laes formula as i hapter 0 of the 994 Highway apacity Maual is uderestimated his paper presets a aalytical procedure based o probability theory for estimatig the capacity of this combiatio of ared ad ort laes his procedure combies the eistig procedures for estimatig the capacity of ared ad ort laes It was tested by simulatios i the style of the KNOSIMO simulatio ad it ca be used for arbitrary lae cofiguratios or simple ared/ort lae cofiguratios eplicit euatios are derived for estimatig the capacity or complicated ared/ort lae cofiguratios iteratio procedures are give As a special case the so-called flared mior approaches are treated accordig to the theory derived eywords: capacity usigalised itersectio ort laes ared laes flared laes Author's address: Dr Nig Wu Istitute for raffic Egieerig uhr-uiversity Bochum el: 49/34/ a: 49/34/ Bochum Germay NigWu@ruhr-ui-bochumde

2 apacity of Shared-Short aes at Usigalised Itersectios INODUION Usigalized Itersectios cross-roads ad -uctios where traffic is regulated by traffic sigs are the most commoly used itersectios i traffic maagemet he right-of-way regulated by traffic sigs presupposes that a driver maes the decisio to pass through if he is at the first waitig positio directly at the stop lie or if o other vehicle is waitig i frot of him he calculatio procedures developed for this situatio which are also used i umerous mauals 3 are stadard for calculatig the capacity of usigalised itersectios he two best ow ad simplest procedures are these from Harders 4 ad Siegloch 5 he calculatio procedures i recet mauals 3 assume that traffic streams that must give way have their ow traffic laes at the itersectio he capacity of the idividual streams left tur through ad right tur are calculated separately If the streams are a commo traffic lae the capacity of the ared lae is the calculated accordig to the ared lae procedure from Harders 4 he procedures for cosiderig the lae distributio at usigalized itersectios are based o the assumptio that left-tur ad/or right-tur streams have either ifiitely log eclusive laes or o eclusive laes at all I reality however this is ot the case If a approach with ort traffic laes igure a for the left-tur ad/or right-tur streams is calculated the capacity is either overestimated legth of the eclusive laes as ifiite or uderestimated legth of the eclusive laes as zero I this paper a procedure is derived with which the legth of the tur laes ca be cosidered eactly for calculatig the capacity of the ared lae he precisio of this calculatio procedure is tested through simulatios he followig symbols ad idices are used: Symbols: m umber of sub - streams [-] capacity [vph]

3 Nig Wu 3 traffic flow [vph] degree of saturatio [-] legth of ueue space i umber of vehicles [veh] P s probability that a poit o the street is occupied by traffic [-] factor for estimatig the capacity of ared lae / real [-] real real degree of saturatio of ared lae / [-] capacity of ared lae [vph] traffic flow of ared lae [vph] apparet degree of saturatio of ared lae [-] Idices for systems with arbitrarily may sub-streams: i ide for the i-th sub-stream i ide for the i-th sub-stream of the level i ide for the i-th sub-stream of the level ide for the -th step of iteratios ide for ared lae ide for ared lae of the level ide for ared lae of the level Idices for systems with three sub-streams: ide for left-tur streams ad their traffic laes ide for through streams ad their traffic laes ide for right-tur streams ad their traffic laes ide for ared streams cosistig of a left-tur ad a through stream ad their traffic laes ide for ared streams cosistig of a through ad a right-tur stream ad their traffic laes

4 apacity of Shared-Short aes at Usigalised Itersectios 4 MAHEMAIA DEIVAIONS I igure a the possible combiatios of ort traffic laes are preseted he ort traffic laes at usigalized itersectios have usually two basic forms: All three directio streams divide at a poit igure b type he streams divide oe after aother at two poits igure b type ad 3 Mathematical derivatios are give i this paper for both basic forms of ort traffic laes igure a Possible ueues at the approaches of usigalised itersectios ype ype 3 ype igure b ombiatio forms of ort traffic laes

5 Nig Wu 5 irst a geeralized system with m sub-streams which all develop at the poit A from oe ared lae igure is cosidered he sub-stream i is described by the parameters i traffic flow i capacity ad i degree of saturatio he capacity i ad the degree of saturatio i i / i are cosidered uder the assumptio that there are ifiite ueue places for the subect stream i Accordigly the ared lae has the parameters ad A i m i i i m m m igure elatioip betwee a ared lae ad its sub - streams or poit A the followig fudametal state coditio holds: Poit A is eually occupied from the left ared lae ad from the right all substreams by waitig vehicles hat is the probability that poit A is occupied o the side of the ared lae is eual to the probability that poit A is occupied o the side of the sub-streams It follows that P P P P P P s s s s i s m s i i m he probability that poit A is occupied by a sub-stream is eual to the probability that the ueue legth i this sub-stream is larger tha the legth of the ueue space sectio from the stop lie to poit A ie for the sub-stream i Ps i Pr N > i

6 apacity of Shared-Short aes at Usigalised Itersectios 6 he distributio fuctio of ueue legths i a waitig stream at usigalized itersectios ca be represeted approimately by the followig euatio see also Wu 6 : Pr N a b i i i i 3 with i i i a b parameters Accordigly oe obtais a bi s i Pr N > i i i P 4 he M/M/-ueuig system also offers good approimatio for the ueuig system at usigalized itersectios see also Wu 6 I this case oe has a ad b hus i s i i i P Pr N > 5 or further derivatios the ueuig system at itersectios without traffic sigals is cosidered as a M/M/-ueuig system he resultig deviatio ca be cosidered egligible see also Wu 6 If oe cosiders poit A as a couter i the sese of ueuig system the the probability that poit A is occupied o the side of the ared lae is eual to the degree of saturatio of the ared lae ie 0 s P Pr N > 0 6 Isertig Euatios 6 ad 5 ito Euatio oe obtais m i i Ps i 7 Here however is oly the apparet degree of saturatio of the ared lae hat meas

7 Nig Wu 7 ad accordigly oe has also m i m i i i i he establimet of these ieualities lies i the fact that o liear relatioip eits betwee the traffic flow ad the degree of saturatio i the ared lae as a result of the epoets of i he capacity of the ared lae ca be determied oly i other ways or estimatig the capacity of the ared lae the followig defiitio is made: he capacity of the ared lae is the traffic flow at which the merge poit A o both sides is occupied 00% hat is Ps ma ma i As a rule the traffic flows i eistig or predicted do ot describe the complete saturatio of the ared lae he capacity of the ared lae is geerally greater tha the sum of i i case of udersaturatio by eistig i I this case the traffic flows at the subect traffic stream would approach the limit of the capacity if the i values icrease I geeral each i value could have differet icreases It is assumed however that a eual icrease factor ca be applied for these fictioal icreases of eistig traffic flows hus is that factor by which all traffic flows o the subect approach has to icrease to reach the maimal possible traffic flow: the capacity By multiplyig the degree of saturatio of all sub-streams by this factor ad postulatig i P s ma ma i i m! 8 oe obtais the capacity of the subect ared lae m i i 9

8 apacity of Shared-Short aes at Usigalised Itersectios 8 Accordigly the real degree of saturatio of the ared lae becomes 0 real hereby is determied implicitly by Euatio 8 or i m that is all sub-streams have the same legth of ueue space resultig i all i m i i ad alli m i m i i i or i with geeral values the Euatio 8 caot be solved eplicitly for he solutio for ca be foud however accordig to the Newto method iteractively ad umerically he iteratio procedure is f f 0 ; 0 with 3 m i f i i he iteratios are coverget for all > 0

9 Nig Wu 9 A i i i i B i B B i i ii mi i i i i i i ii ii ii mi mi mi m m m m B m igure 3 elatioip betwee ared laes ad their sub- ad sub-sub-streams If a sub-stream agai cosists of several sub-sub-streams this sub-stream must be cosidered as a ared stream itself igure 3 Accordigly aalogously to Euatio for the merge poit A oe obtais P m Ps i 4 s i Ad for the sub merge poits B i oe obtais P m i Ps i i 5 s i i If oe cosiders the ueuig systems i all sub-streams ad sub-sub-streams as M/M/-ueuig systems respectively the for the sub-sub-stream with ide i i oe obtais s i i i i i i P 6 for the sub merge poit with ide B i sectio betwee poit A ad B i mi mi i i s i i s i i i i i i P P 7

10 apacity of Shared-Short aes at Usigalised Itersectios 0 for the sub-stream with ide i m i i mi i i i i Ps i i Ps i Ps i i i i i i i 8 ad for the merge poit A m m m i i m m i i i i Ps Ps i Ps i i i i 9 i i i i i Multiplyig the degree of saturatio of all sub-sub-streams by a factor ad postulatig m m i i! i i Ps ma ma i i 0 i i the capacity of the total ared stream becomes m m i i i i i he Euatios 0 ad are the geeralized forms of Euatios 7 8 ad 9 Settig m i or m eual to oe obtais here the Euatio 7 8 ad 9 agai or i i ad i with geeral values the iteratio procedure for solvig becomes f f 0 ; 0 with m mi i f i i i i i i Aalogously oe ca also treat systems with a arbitrarily umber of levels of sub-sub-streams

11 Nig Wu PAIA APPIAIONS O HE HEOY ype lae combiatios see igure b ad 4 A Stop lie igure 4 Parameters for ype of ort traffic laes Settig m ad i i Euatios 9 0 ad oe obtais the followig euatios for estimatig the capacity of the ared lae for ype ort laes at usigalized itersectios: P i s type Ps i P N > i i 3 i Ps ma type ma i! 4 5 type I geeral the three streams ad must stop ad wait at the same stop lie his meas that the umbers of available ueue places are eual for all three streams I this case settig oe obtais type i 6 ad type type 7 At 0 oe gets

12 apacity of Shared-Short aes at Usigalised Itersectios type 0 8 hat is eactly the well-ow ared-lae formula from Harders 4 or ad with geeral values the iteratio procedure for solvig see also Euatio 3 yields type 0 ; 0 9 With this euatio the Newto iteratio method is to be used for determiig the subect he capacity of the whole approach ca the be obtaied accordig to Euatio 9 ype lae combiatios see igure b ad 5 A B Stop lie igure 5 Parameters for ype of ort traffic laes Settig i ad i i Euatios 9 0 ad oe obtais the followig euatio for estimatig the capacity of the ared lae for ype ort laes at usigalized itersectios: type s P 30 [ ]! ma ma type s P 3 type 3

13 Nig Wu 3 Also i this case the three streams must geerally stop ad wait at the same stop lie he followig relatioips eist betwee the available ueue places: where is the commo umber of ueue spaces for left-tur ad through streams Accordig to these relatioips oe obtais type s P 33 [ ]! ma ma type s P 34 Settig II II I I 35 oe obtais for the postulate Euatio 34! ma ma II I II I type s P 36

14 apacity of Shared-Short aes at Usigalised Itersectios 4 his meas that uder the margial coditio that all three sub-streams stop ad wait at the same stop lie the ared-lae system with three sub-streams ca be simplified i a ared-lae system with oly two sub-streams urthermore the capacity uder this coditio is real type 37 or ad with geeral values the iteratio procedure for estimatig yields a b c d 0; 0 38 with [ ] a b [ ] c [ ] d ype 3 lae combiatios see igure b ad 6 A B Stop lie igure 6 Parameters for ype 3 of ort traffic laes

15 Nig Wu 5 ype 3 ort laes is symmetrically to ype ort laes Echagig ide left-tur ad righttur i the euatios for ype ort laes yields the correspodig euatios for ype 3 ort laes or eample aalogously to Euatios 30 3 ad 3 oe obtais here s type3 P 39! [ ] Ps ma type3 ma 40 4 type3 lared lae at mior approaches ight flared lae at mior approaches see igure 7 G right igure 7 ight flared approach A special applicatio of Euatios 3 ad 40 is the so-called flared laes igure 7 or the right flared approach right-tur stream passes by the left-tur/through stream the followig relatioips are valid: 0 right Accordigly oe obtais the postulate P s right ma! [ ] right right right right right

16 apacity of Shared-Short aes at Usigalised Itersectios 6 Solvig this euatio for right oe obtais 4 right right right right ad right right i right right right 43 or right it becomes rechts rechts 44 eft flared lae at mior approaches igure 8 G left igure 8 eft flared approach Aalogously for the left flared approach left-tur stream passes by the right-tur/through stream igure 8 oe obtais 45 left left left left ad for left left left 46

17 Nig Wu 7 or 0 the Euatios 43 ad 45 yield right right 0 left left Agai oe obtais the ared lae formula from Harders 4 Mied flared lae at mior approaches see igure 9 igure 9 Mied flared approach igures 7 ad 8 ow the two possibilities a flared approach ca be used by vehicles However it is ot easy to forecast how real-world vehicles would use the flared approach Here oly the behavior of the drivers i the through vehicles is decisive for the calculatio because the right- ad left-tur vehicles always pass by each other at a flared approach he decisio of a through driver whether he passes by a waitig left-tur vehicle or by a waitig right-tur vehicle determies the cofiguratio of the flared approach If a through driver passes o the left of a waitig right-tur vehicle the approach is a right flared approach because the left-tur vehicles must pass by the waitig right-tur vehicle also If a through driver passes o the right of a waitig left-tur vehicle the approach is a left flared approach because the right-tur vehicles must pass by the waitig left-tur vehicle also If a through driver arrives while aother through vehicle is waitig o the stop lie the approach could also be cosidered a right flared approach because i this case oly the right-tur vehicles may drive by o the right As a approimatio oe ca assume that the probabilities whether the approach is used as a left or right flared approach are proportioal to the correspodig degree of saturatios of the traffic streams Accordigly a euatio for estimatig the capacity of the flared approach with mied cofiguratio

18 apacity of Shared-Short aes at Usigalised Itersectios 8 which treats the approach both as a left flared approach ad a right flared approach ca be represeted by see igure 9 right left mi 47 accordig to the degree of saturatios respectively Isertig Euatios 4 ad 45 ito Euatio 47 ad settig left right oe obtais 0 mi 48 where 0 is the capacity of the ared lae for the case 0 correspodig to Harders formula 4 or the Euatio 48 yields 0 mi 49 igure 0 presets a compariso of capacity icreases caused by the flarig of the approach or this compariso the orth boud approach of a stadard itersectio is calculated he capacities of the

19 Nig Wu 9 separate traffic streams are obtaied accordig to the Germa Highway apacity Maual he traffic flow of this itersectio is ow i igure he calculatio yields the parameters ad 005 for the subect approach hese parameters characterized ualitatively most of the real traffic coditios at approaches to usigalized itersectios actor of capacity icrease [-] G left flarig mied flarig right flarig legth of flared area [veh] igure 0 Icreases of capacity caused by flarig igure raffic flow of the test eample i [vph]

20 apacity of Shared-Short aes at Usigalised Itersectios 0 learly oe ca recogize that the left flarig causes a much greater capacity icrease compared with the right flarig or the left flarig the eample has a capacity icrease of 38% at left compared to left 0 or the right flarig it is barely 6% he mied use of the flared area which is more realistic tha the pure left ad/or right flarig delivers approimately 8% icrease i capacity he value of the mied flarig correspods very well to the measuremets i the techical report from Kyte et al 8 see there igure 87 ESING HE HEOY BY SIMUAION o test the derived theory differet combiatios of ared/ort laes are simulated i the style of KNOSIMO 7 Altogether 95 traffic-flow ad lae variatios were simulated he simulatio results are preseted i igure together with the theoretical values he ey statistical values of this compariso are assembled i able which ows that the relatioips betwee both parameters are arrowly correlated Accordigly the correctess of the derived theory is verified alculated capacity [vph] Simulated capacity [vph] igure compariso of calculated ad simulated capacities

21 Nig Wu Multiple correlatio coefficiet [-] 0993 ertaity [-] 0985 Adusted certaity [-] 0985 Stadard errors [vph] 44 Observatios [-] 95 able Key statistical values of the compariso SUMMAY he theory derived here delivers a geeral approach for estimatig the capacity of ared/ort laes at usigalized itersectios his theory cosiders the legth of ort laes ad fills a gap i the curret calculatio procedures for usigalized itersectios he derivatio of this theory presumes that the ueuig systems at usigalized itersectios ca be approimately cosidered as M/M/-ueuig systems his could lead to a mior deviatio of the results from reality he simulatio results however ow that this deviatio is egligible ad ot statistically sigificat or practical applicatios Euatios ad 48 are most importat With these three euatios the capacity of mior approaches at usigalized itersectios with left right ad mied flarig ca be determied i the simplest ad most eact way or ared/ort laes with arbitrary lae combiatios a geeral implicit euatio for estimatig the capacity is give Euatios 0 ad or the solutio of the implicit Euatio 0 the Newto iteratio method ca be used Euatio With this procedure all possible ared/ort-lae combiatios at usigalized itersectios ca be determied without large epeses If a woreet program is available oe ca also use the so-called SOVE by Ecel to solve the Euatio 0 As a summary all possible cofiguratios of ared/ort laes ad their solutios are assembled i able More eamples for the theory ca be see i the wor from Wu 9

22 apacity of Shared-Short aes at Usigalised Itersectios or estimatig the capacity of ared laes it is presupposed that the traffic flows of all sub-streams icrease proportioally to their origial traffic flows All sub-streams were multiplied by the same factor It is also possible however to determie the capacity of a particular sub-stream by usig fied traffic flows for all other sub-streams he theory ould be epaded to itersectios with traffic sigals

23 Nig Wu 3 able apacity of ared - ort laes ase igure Euatio for Notatio Euatio 3 iteratio procedure A i m i i i m m m or all i Euatio eplicit Geeralized case with oe level of sub-streams A i i i i B i B B i i ii mi i i i i i i ii ii ii mi mi mi Euatio iteratio procedure Geeralized case with two levels of sub-streams m m m m B m 3 A Stop lie Euatio 9 iteratio procedure or i Euatio 6 eplicit 4 A B Euatio 38 iteratio procedure Stop lie 5 G right Euatio 4 eplicit ight flared lae 6 left G Euatio 45 eplicit eft flared lae 7 Euatio 48 eplicit Mied flared lae i is directly available

24 apacity of Shared-Short aes at Usigalised Itersectios 4 EEENES HM Highway apacity Maual rasportatio esearch Board Special eport 09 ew editio Waigto 994 D-HM Germa Highway apacity Maual Verfahre für die Berechug der eistugsfähigeit ud Qualität des Verehrsablaufes auf Straße Deutsches HM Schriftereihe "orschug Straßebau ud Straßeverehrstechi" Heft Woreet for calculatig capacities at usigalised itersectios Merblatt zur Berechug der eistugsfähigeit vo Kotepute ohe ichtsigalalage Herausgeber: orschugsgesellschaft für Straße- ud Verehrswese 99 4 J Harders Die eistugsfähigeit icht sigalgeregelter städtischer Verehrsote Schriftereihe "Straßebau ud Straßeverehrstechi" Heft W Siegloch Die eistugsermittlug a Kotepute ohe ichtsigalsteuerug Schriftereihe "Straßebau ud Straßeverehrstechi" Heft N Wu A Approimatio for the Distributio of Queue egths at Usigalised Itersectios Proceedig of the secod Iteratioal Symposium o Highway apacity Sydey 994 I Aceli ed volume pp M Grossma Methode zur Berechug ud Beurteilug vo eistugsfähigeit ud Verehrsualität a Kotepute ohe ichtsigalalage uhr-uiversität Bochum Schriftereihe ehrstuhl für Verehrswese Heft Kyte M et al apacity ad level of service at usigalised itersectios ial report volume - wo-way stop-cotrolled itersectios Natioal cooperative highway research program Proect 3-46 Dec 995

25 Nig Wu 5 9 N Wu apacity of Shared-Short aes at Usigalised Itersectios I Michael Kyte ed: Proceedigs of the hird Iteratioal Symposium o Itersectios Without raffic Sigals Portlad 997

26 apacity of Shared-Short aes at Usigalised Itersectios 6 ab - Key statistical values of the compariso ab - apacity of ared/ort laes iga - Possible ueues at the approaches of usigalised itersectios igb - ombiatio forms of ort traffic laes ig - elatioip betwee a ared lae ad its sub - streams ig3 - elatioip betwee ared laes ad their sub - ad sub - sub streams ig4 - Parameters for the type of ort traffic laes ig5 - Parameters for the type of ort traffic laes ig6 - Parameters for the type 3 of ort traffic laes ig7 - ight flared approach ig8 - eft flared approach ig9 - Mied flared approach ig0 - raffic flow of the test eample ig - Icreases of capacity caused by flarig ig - compariso of calculated ad simulated capacities