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1 Modellg pssegers rrvls t publc trsport stops Ľudml Jáošíková, Mrt lvík Dept of Trsportto Networks, Fcult of Mgemet cece d Iformtcs, Uverst of Žl, Uverztá, 6 Žl, lovk Republc; Phoe (4-4) 53 4, F (4-4) E-mls: Mrt.lvk@st.fr.uz.sk Abstrct The pper presets sttstcl eplorto of pssegers rrvls t bus stops urb publc trsport. Frst, t descrbes the methodolog whch s ppled o the urb publc trsportt sstem, where the followg codtos re met: () pssegers re fmlr wth the tmetbles, () the vehcles ru o tme, d () the cpct of the vehcles s suffcet. The methodolog s demostrted o the urb publc trsport Žl, lovk Republc. The correlto lss of the surve performed t trsportto stops Žl revels tht there s correlto betwee the wtg tme d hedw. The reltoshp betwee ths vrbles c be modelled b ler fucto or better b logrthmc fucto. The Kolmogorov-mrov d the ch-squre tests ccept the hpothess tht the Gumbel mmum dstrbuto s the sutble mthemtcl model of pssegers rrvls rtes. The proposed models re ecessr the opertos reserch d smulto methods used for publc trsport plg. Kewords: publc trsport, pssegers wtg tme, correlto lss, regresso lss.. Itroducto The pper descrbes the results of our reserch med t the behvour of pssegers urb publc trsport, prtculrl t ther rrvls t bus stops. The reserch gol ws:. to determe whether there ests reltoshp betwee pssegers wtg tme d the le frequec (hedw betwee the successve buses);. f the wtg tme depeds o the hedw, to determe the mthemtcl model of ths depedece; 3. to propose sutble mthemtcl model of pssegers rrvls rte t stop. The tme pssegers sped t the stops wtg for bus hs bee regrded s mportt crtero of the publc trsport sstem qult (see for emple Osu d Newell, 97; Dgzo, 997; Aver, 4; F d Mchemehl, 4; Álvrez et l., ). Therefore, t s mportt to kow the spects ffectg pssegers rrvls t the stopss order to desg effectve publc trsport sstem or to mprove ts qult. Moreover, descrpto of rrvl ptters b mthemtcl models s ecessr f oe wts to use sophstcted methods for publc trsport plg, such s opertos

2 reserch methods (F d Mchemehl, 4, Jáošíková et l.,, Jáček d Kohá, ) or computer smulto (Erth et l., 3). There s wdespred belef tht the wtg tme depeds o the le frequec. Tht s wh severl studes regrdg publc trsport qult ol frequec s cosdered (e.g. Hesher et l., 3; Ebol d Mzzull, 8, 8b). Publshed mthemtcl models of wtg tme cosder both the hedw d the rrvl tme (or wtg tme) s rdom vrbles. The most smple model s bsed o the ssumpto tht pssegers do ot kow the tmetbles, the rrve to the orgl stop rdoml, d the me wtg tme s proportol to the hedw (versel proportol to the le frequec, respectvel). Uder the ssumpto tht pssegers rrve t costt rte, the wtg tme s fucto of the me hedw d ts vrce: EH D H E W, () E H where E(W) deotes the me wtg tme, E(H) the me hedw d D(H) the hedw vrce. Whe trsportto servce opertes wth log hedws, pssegers do ot rrve t stops rdoml but the ted to rrve few mutes before the pled vehcle deprture. Prevous studes performed Europe the 97s (for the surve, see Lueth et l., 6) d the U..A. (F d Mchemehl, 4) were med t the determto of the mmum hedw wth o-rdom rrvl ptter d the model for the reltoshp betwee the wtg tme d the hedw. The hedw threshold vred from 5 to mutes d the models were ler or qudrtc. The recet Europe stud ws crred out Zürch b Lueth et l. (6). Pssegers were supposed to belog to oe of two groups: those who were fmlr wth the schedule d those who dd ot kow the schedule. As cosequece, the uthors suggest rrvl rte model tht combes the uform dstrbuto for formed pssegers wth the shfted Johso B dstrbuto for uformed pssegers. The Johso dstrbuto s shfted wth smll vlue due to the observto tht some pssegers rrve short tme fter the vehcle deprture. The reported shre of these pssegers s qute hgh (from 5 to 6 %). The uthors epl pssegers erl rrvls b the fct tht pssegers do ot trust the servce relblt d rel o regulr del but the the do ot ctch the bus. Pssegers wtg the whole perod would be observed lso f the fled to bord due to the suffcet vehcle cpct. Regrdg the depedece of the me wtg tme o the hedw, Lueth et l. propose the logrthmc model.. Methodolog The frst gol of our stud ws to fd out whether the wtg tme depeds o the hedw eve the cse of relble servce wth the suffcet cpct of vehcles d wth pssegers beg fmlr wth the schedules. We proceeded from the stuto the lovk Republc, whch s smlr to most Europe coutres, where the publc trsport users re well-formed. The tmetbles re vlble t the stops s well s o the Iteret, so pssegers re ble to obt schedule formto lmost everwhere b usg ew formto techologes.

3 As t ws sd before, both the wtg tme (or pssegers rrvl tme, respectvel) d the hedw re rdom vrbles. The depedece of the rdom vrbles s the mtter of the correlto lss. The mesure of the depedece betwee two rdom vrbles d Y s so clled correlto coeffcet ρ(,y). It s obted b dvdg the covrce of the two vrbles b the product of ther stdrd devtos. The correlto coeffcet rges from to. The vlue of mples tht there s o ler correlto betwee the vrbles. If ρ(,y), the d Y re depedet. The vlue of mples tht the ler equto descrbes the reltoshp betwee d Y perfectl, wth ll dt pots lg o the le for whch Y creses s creses. The vlue of mples tht ll dt pots le o the le for whch Y decreses s creses. To fd ρ(,y) we eed to kow the probblt dstrbutos of the vrbles. However, the probblt dstrbutos re ukow prctce. We usull hve observed or mesured ol relzto of the smple,.e. vlues (, ). These vlues c be used to estmte ρ(,y) b the Perso correlto coeffcet: R, Y, () where d re rthmetc mes of the vlues d, respectvel. Becuse some propertes of ths estmte deped o the probblt dstrbuto of the pr (, Y), other estmte of ρ(,y) ws desged, whch does ot deped o the jot dstrbuto of (, Y). It s clled the perm correlto coeffcet, d t s bsed o the rks of vlues d. The vlues d re sorted o-decresg order: d, respectvel. Let us deote the rks of the vlues d b r d q, respectvel. The smlrt betwee rks r d q shows the reltoshp betwee the vlues d. The perm correlto coeffcet s gve b the followg term: 6 R r q (3) The estmted correlto coeffcets R,Y d R re lmost lws dfferet from zero. Therefore sttstcl test should be preformed to verf whether ther vlue s sttstcll sgfct. The ull hpothess H : = (correlto s sgfct) s tested gst the ltertve hpothess H:. everl tests wth dfferet test crter re vlble. The test crter re fuctos of the estmted correlto coeffcet. The frst test s bsed o the ssumpto tht the smple of the prs (, ) for =,..., comes from the two-dmesol orml dstrbuto wth the correlto coeffcet. The test crtero T R, Y (4) R, Y hs the tudet s t dstrbuto wth degrees of freedom uder the ull hpothess. The secod test s bsed o the sme ssumpto s the orml dstrbuto. The test uses the Fsher z trsformto tht coverts the Perso correlto coeffcet to the vrble Z. The formul for the trsformto s: 3

4 Z R, Y l (5) R, Y Z s ppromtel ormll dstrbuted wth the me l d the stdrd devto. 3 The test crtero * 3 R, Y Z l (6) R, Y hs the stdrd orml dstrbuto N(,) uder the ull hpothess. If the ssumpto of the bove metoed tests s ot met, the oprmetrc test for the perm correlto coeffcet c be used. The vlue R s s compred wth the tbulted crtcl vlue r. If R r α, the the ull hpothess s ccepted. Further, the cofdece tervl of the correlto coeffcet c be clculted usg Fsher Z vrble. The (-) % cofdece tervl of the theoretcl correlto coeffcet s defed s follows: z z tgh Z, tgh Z, (7) 3 3 where z s the vlue of the verse dstrbuto fucto of the stdrd orml dstrbuto N(,) t the pot, d the fucto tgh() s the hperbolc tget: e e tgh. e e I the cse tht rdom vrbles d Y re depedet, oe c descrbe ther reltoshp b regresso fucto. The most smple form of the regresso fucto s the ler fucto = + b, where d b re ukow coeffcets tht eed to be estmted. The most commo method of estmto s the lest squres method. Usg ths method we get the followg estmtes: b The pot estmtes (8) d (9) should be cotrsted wth cofdece tervls. These c be costructed uder the followg ssumptos:. vlues re depedet;. for ever the vlues hve the orml dstrbuto N(, ) wth me = + b d vrce detcl for ll. The estmtes d b hve lso the orml dstrbuto wth the followg chrcterstcs: (8) (9) 4

5 5, D E () D b b E b, () where () The vrce s ot kow d t must be estmted b b (3) The correspodg (-) % cofdece tervls re the s follows:, t t (4), t b t b b (5) where t s the crtcl vlue of the tudet t dstrbuto wth degrees of freedom. Usg these cofdece tervls we c test the hpothess bout ozero vlues of the coeffcet. It mes tht regrdg the coeffcet, the ull hpothess H : = (the coeffcet s sgfct) s tested gst the ltertve hpothess H:. The ull hpothess s ccepted f the cofdece tervl cots,.e., t t. A smlr test c be performed for the coeffcet b of the regresso fucto. The vrblt of the observed vlues c be mesured through dfferet sums of squres: (6) reg (7) err (8) where b s the vlue o the regresso le. s the totl sum of squres of devtos of the mesured vlues from ther me, reg s the regresso sum of squres, lso clled the epled frcto of vrce, d err s the resdul sum of squres, lso clled the uepled frcto of vrce. It holds err reg. The rto

6 reg R (9) s so clled coeffcet of determto. R s sttstcs tht wll gve some formto bout the goodess of ft of model. It tkes vlues betwee. The better the ler regresso fts the dt comprso to the smple verge, the closer the vlue of R s to oe. A R of dctes tht the regresso le fts the dt perfectl. I ddto to the R sttstcs, the model vldto c be doe through the F-test o the sttstcl sgfcce of the regresso model. reg err Let us deote reg d err, where p s the umber of regresso p p prmeters. The regresso model s cosdered to be sttstcll sgfct f reg s sgfctl greter th err. The ull hpothess H : reg = err (the regresso model s sgfct) s tested gst the ltertve hpothess H: reg > err. The test crtero reg F () err hs the Fsher-edecor dstrbuto wth the prmeters v = p - d w = - p uder the ull hpothess. The pproch derved for the ler regresso model c be ppled for some oler models s well, for emple for the logrthmc fucto = l() + b. The method for the prmeter estmto s the sme; the ol ecepto s tht the logrthm l( ) s used sted of. 3. Cse stud As cse stud for pssegers rrvls d wtg tme modellg we chose the urb publc trsport the ct of Žl. Žl s mddle-szed ct stuted the orth-wester prt of the lovk Republc. It hs 8,494 hbtts (b ), d covers the re of 8 km. The trsportto servce the ct s provded b the trsportto opertor Doprvý podk mest Žl, s.r.o. (DPMŽ). Durg the d, 8 trollebus les d bus les operte. At ght, the ct re s served b bus le. The dt for the lss were collected t 6 stops Žl o weekds durg the morg pek d off-pek perods (from 6: to :). The stops were selected ccordg to the followg crter: Pssegers re ot supposed to chge les t the stop. The stop must be bus eough to eble collectg suffcet dt. The dt were collected b hd,.e. b observg pssegers rrvls t stops d recordg the psseger s rrvl tme, the umber of the le tke b the psseger, d the vehcle deprture tme. 6

7 3. The results of the correlto lss Usg the mesured dt we wt to determe whether there ests reltoshp betwee pssegers wtg tme (rdom vrble Y) d the le hedw (rdom vrble ). The sze of the smple used the followg clcultos s = 467. The reltoshp s mesured b the Perso d perm correlto coeffcets tht were computed usg the terms () d (3), d tke the vlues R,Y =.34 d R =.759. Both coeffcets re dfferet from zero, whch mes tht the wtg tme d the hedw re relted b the mootoc fucto. Ths fdg c be verfed b the test of sgfcce of the theoretcl correlto coeffcet. The ull hpothess H : = (correlto s sgfct) s tested gst the ltertve hpothess H:. t the level of sgfcce =.5. I the frst two tests wth the Perso correlto coeffcet t s ssumed tht the smple of prs (,Y) comes from the orml dstrbuto. As regrds the vrble (the le hedw), t tkes ol severl vlues, most ofte, 5,, d 3 mutes tht re commo publc trsport operto d therefore t s mpossble to mke test o ts probblt dstrbuto. The smple of Y ws tested o the probblt dstrbuto for prtculr hedws (see ecto 3.3). Although the ch-squre test fled to reject the hpothess bout the orml dstrbuto for some, for ll dt together the hpothess ws rejected t the sgfcce level =.5. Although the ssumpto of the frst two tests ws ot proved, ll three tests metoed ecto were performed. The vlue of the test crtero ccordg to (4) s T =.99. It s greter th the crtcl vlue of the tudet dstrbuto for the sgfcce level.5 d - degrees of freedom (.965), therefore we reject the ull hpothess H d ccept the ltertve hpothess H tht d Y re depedet rdom vrbles. The vlue of the test crtero ccordg to (6) s Z * =.97. It s greter th the crtcl vlue of the stdrd orml dstrbuto for the sgfcce level.5 (.96), therefore we reject the ull hpothess H d ccept the ltertve hpothess H tht d Y re depedet rdom vrbles. The sme outcome s obted usg the thrd oprmetrc test wth the perm correlto coeffcet. The vlue R s =.759 s greter th the tbulted crtcl vlue r =.9, so the ull hpothess s rejected. Further, the cofdece tervl of the Perso correlto coeffcet c be clculted. Accordg to (7), the 95% cofdece tervl s.44,.. Ths tervl cludes the Perso correlto coeffcet R,Y =.34. Ths fct cofrms the correlto betwee the wtg tme d the hedw. 3. The results of the regresso lss To specf the depedece mthemtcll, regresso fucto c be derved, whch descrbes the depedece of the pr of rdom vrbles (,Y). Usg the lest squre method, ler d logrthmc fuctos were proposed, further the sgfcce of coeffcets ws vestgted d the qult of both models ws emed usg the F-test. The ler fucto ws estmted s = The 95% cofdece tervls of the coeffcets re:.9,.47, b.77, Noe of the 7

8 tervls cots, therefore both coeffcets d b re sgfct. The F-test sttes the ull hpothess H : reg = err (the regresso model s sgfct) gst the ltertve hpothess H: reg > err. The vlue of the test crtero ccordg to () s F = 8.5. It s greter th the crtcl vlue of the Fsher-edecor dstrbuto wth the prmeters v = d w = 465 for the sgfcce level.5 (F.95 (,465) = 3.86), therefore we reject the ull hpothess H d ccept the ltertve hpothess H tht the epled frcto of vrce s sgfctl greter th the uepled frcto of vrce. The coeffcet of determto for ths ler model s R =.8. The logrthmc fucto ws estmted s =.93l() The 95% cofdece tervls of the coeffcets re:.93,.937, b -.857,.95. The cofdece tervl for coeffcet b cots, but ths ol mes tht the bsolute prt of the fucto m be zero, however the tpe of the fucto s stll logrthmc. The vlue of the F-test crtero ccordg to () s F = It s greter th the crtcl vlue of the Fsher-edecor dstrbuto (F.95 (,465) = 3.86), therefore we reject the ull hpothess H d ccept the ltertve hpothess H tht the logrthmc model s sttstcll sgfct. The coeffcet of determto s R =.3. The coeffcets of determto for both regresso models re qute smll. It mes tht the vlues o the regresso curve re fr w from the observed vlues. The reso s tht for ech hedw there were lot of dfferet wtg tmes observed, whch c lso be see Fgures d. I ccordce wth Lueth et l. (6), the logrthmc depedece seems to be better ppromto of the reltoshp betwee the emed rdom vrbles. 5 Wtg tme [m] 5 5 Wtg tme L. reg Le hedw [m] Fgure. Ler regresso 8

9 5 Wtg tme [m] 5 5 Wtg tme Log. reg Le hedw [m] Fgure. Logrthmc regresso 3.3 The dstrbuto of the rrvl rtes The et step our reserch ws to specf the dstrbuto of pssegers rrvls for the most commo hedws (, 5,, d 3 mutes). The rdom vrble s ow the tme elpsed betwee the deprture of the prevous vehcle d the rrvl of the psseger. The frequec dgrms of the pssegers rrvl tmes were costructed for ech hedw. The dgrms suggested tht the Gumbel mmum dstrbuto would be sutble model. The formul for the probblt dest fucto (PDF) of the Gumbel mmum dstrbuto s f ep ep () b b b for (-,); (-,) s the locto prmeter d b > s the scle prmeter. Usg the Kolmogorov-mrov d the ch-squre tests we c ccept the hpothess tht the rrvls follow the Gumbel mmum dstrbuto. Moreover, for the 5-mute hedw lso the orml dstrbuto ws ccepted b the tests. For llustrto, Fgures 3 to 6 dspl the reltve frequeces of pssegers rrvls d the PDF of the Gumbel mmum dstrbuto for four most commo hedws. As t c be see, the locto prmeter of the PDF strogl depeds o the hedw, sce pssegers ted to rrve t the bordg stop few mutes before the pled bus deprture. Ol couple of pssegers rrve t the begg of the perod. o we c coclude tht most pssegers re fmlr wth the tmetbles d djust ther rrvls to the schedule. 9

10 ,3 Reltve frequec/pdf,5,,5,,5 Frequec GumbelM (7.6,.9) Tme [m] Fgure 3. Pssegers rrvls -mute hedw Reltve frequec/pdf,8,6,4,,,8,6,4, Tme [m] Frequec GumbelM (.,.76) N(9.6,3.54) Fgure 4. Pssegers rrvls 5-mute hedw

11 ,4 Reltve frequec/pdf,,,8,6,4, Frequec GumbelM (5.5,4.9) Tme [m] Fgure 5. Pssegers rrvls -mute hedw,5 Reltve frequec/pdf,,5,,5 Frequec GumbelM (6.5,3.9) Tme [m] Fgure 6. Pssegers rrvls 3-mute hedw 4. Coclusos We proposed methodolog where sttstcl methods re used for the eplorto of pssegers rrvls t bus stops urb publc trsport. The m results of our reserch re s follow:. Pssegers wtg tme d le hedw re correlted rdom vrbles.. The reltoshp betwee ths vrbles c be modelled b ler fucto or better b logrthmc fucto.

12 3. The Gumbel mmum dstrbuto s the sutble mthemtcl model of pssegers rrvls; the locto prmeter of the PDF strogl depeds o the le hedw. The results c be geerlzed for ever publc trsport sstem whch the users re well-formed. The proposed models re ecessr the opertos reserch d smulto methods used for mprovg publc trsport qult. Ackowledgemets Ths reserch ws supported b the cetfc Grt Agec of the Mstr of Educto of the lovk Republc d the lovk Acdem of ceces uder project VEGA /339/3 Advced mcroscopc modellg d comple dt sources for desgg sptll lrge publc servce sstems. The uthors wsh to thk studets who helped wth dt collecto: Jozef Peck, Iv Urbčová, d Alžbet Jáošíková. Refereces Álvrez, A., Csdo,., Gozález Velrde, J.L. d Pcheco, J. () A computtol tool for optmzg the urb publc trsport: rel pplcto. Jourl of Computer d stems ceces Itertol, 49(), pp Aver, E. (4) A cumultve prospect theor pproch to pssegers modelg: wtg tme prdo revsted. Jourl of Itellget Trsportto stems, 8 (4), pp Dgzo, C.F. (997) Fudmetls of Trsportto d Trffc Opertos. New York: Elsever cece Ic. Ebol, L. d Mzzull, G. (8) A stted preferece epermet for mesurg servce qult publc trsport. Trsportto Plg d Techolog, 3 (5), pp Ebol, L. d Mzzull, G. (8b) Wllgess-to-p of publc trsport users for mprovemet servce qult. Europe Trsport\Trsport Europe, 38, pp Erth, A., v Eggermod, M.A.B., Foure, P.J. d Chkrov, A. Decso support tools trsport plg: from reserch to prctce. Pper preseted t the 3th wss Trsport Reserch Coferece, Asco, 4 6 Aprl 3. F, W. d Mchemehl, R.B. (4) Optml Trst Route Network Desg Problem: Algorthms, Implemettos, d Numercl Results. Report No. WUTC/4/6744-, Aust: Ceter for Trsportto Reserch, Uverst of Tes t Aust. Hesher, D.A., topher, P. d Bullock, P. (3) ervce qult developg servce qult de the provso of commercl bus cotrcts. Trsportto Reserch Prt A 37, pp Jáček, J. d Kohá, M. () Wtg tme optmsto wth IP-solver. Commuctos : scetfc letters of the Uverst of Žl, (3A), pp Jáošíková, Ľ., Kohá, M., Bltoň, M. d Techm, D. () Optmzto of the urb le etwork usg mthemtcl progrmmg pproch. Itertol Jourl of ustble Developmet d Plg, 7(3), pp

13 Lueth, M., Wedm, U. d Nsh, A. (6) Psseger rrvl rtes t publc trsport sttos. Workg pper do:.399/ethz , Zürch: Isttute for Trsport Plg d stems, ETH Zürch. Osu, E.E. d Newell, G.F. (97) Cotrol strteges for delzed publc trsportto sstem. Trsportto cece, 6 (), pp Author Bogrphes Ľudml Jáošíková s Assocte Professor of Trsportto d Commucto Techolog the Deprtmet of Trsportto Networks t the Uverst of Žl, lovk Republc. Her reserch terests clude trsportto plg d trvel behvour. he focuses o the developmet of mthemtcl models d pplcto of opertos reserch methods. Mrt lvík s studet of the Msters degree progrmme Iformto sstems t the Fcult of Mgemet cece Iformtcs, Uverst of Žl. He s terested probblt theor d sttstcs. 3