MODELING OF PASSENGER WAITING TIME IN INTERMODAL STATION WITH CONSTRAINED CAPACITY ON INTERCITY TRANSIT

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1 9 MODELING OF PASSENGER WAITING TIME IN INTERMODAL STATION WIT CONSTRAINED CAPACITY ON INTERCITY TRANSIT S.K. Jason CANG Professor Deartment of Cvl Engneerng Natonal Tawan Unversty Tae, 067 Tawan Fax: E-mal: Srng C. SU Assstant Professor Deartment of Busness Admnstraton Chna Insttute of Technology 45, Sec. 3, Academa Rd., Nankang, Tae, 5 Tawan E-mal: d885007@ntu.edu.tw Abstract: Ths aer ams to develo a mathematcal model to analyze the assenger watng tme n an ntermodal staton n whch ntercty transt system served wth feeder buses. Passengers are categorzed nto two grous, named scheduled and random, resectvely. The analytc model s develoed for quantfyng the relatonshs of assenger watng tme to relablty of feeder bus servces and caacty of ntercty transt. The numercal results dentfy that mean watng tme of ntercty transt assengers s greater than half of the ntercty transt headway, wth the condton that feeder servce s rovded wth oor relablty; and on the contrary, mean watng tme wll be less than half of the ntercty transt headway. Moreover, to avod long watng tme, headway devaton of feeder buses should be less than 3.5 mnutes. In terms of ntercty transt caacty, t s suggested that at least 80 % of total arrval assengers should be satsfed. Keywords: ntermodal, transfer, watng tme, assenger behavors, headway.. INTRODUCTION When analyzng mean watng tme of an ntensve urban transt servce, assenger arrvals are assumed to follow unform dstrbuton; and thus consequently obtaned the outcomes of mean watng tme s half of the headway. owever, for the ntercty transt assengers, they tend to arrve at the staton close to the dearture tme, because of the regular schedule and relatvely long headway. Thus, the watng behavors are dfferent from those of urban bus transt assengers. Moreover, ntercty transt assengers wll go to ntermodal transt staton by varous feeder systems, e.g. mass rad transt, lght ral, bus, tax, auto, bcycle and even walk. Relablty of these feeder servces should have great nfluences on the assengers behavor of arrangng ther arrval tme to the ntermodal transt staton. If the urban bus servce s consdered as a major feeder system for the ntercty transt, assenger watng tme and relevant behavors for Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

2 0 the ntercty transt system should be nfluenced by the servce relablty of feeder buses. And of course, the caacty constrants of ntercty transt wll also make some macts on the assenger watng tme. For the urban bus systems, f consderng the nfluence of servce relablty, the mean random watng tme w r can be characterzed as Equaton (), where h reresents the mean headway and σ reresents the devaton of bus headways (Weldng, 957; Osuna & Newell, 97). w r h σ + ( ) h () Equaton () shows mean watng tme wll decrease wth the ncrease of bus servce relablty (.e., decrease of σ ). Turnqust (98) then tred to exand the study of Weldng. e dvded the assengers nto two categores: random and non-random, accordng to ther behavors of takng buses. Watng tme of the random assengers s calculated based on Equaton () and a log-normal dstrbuton of arrvals s assumed to calculate the watng tme of non-random assengers. owever, Turnqust dd not make further efforts on assenger behavors. Another study fnds that due to traffc congeston, relablty of servce usually decreases when feeder buses go nto the CBD area or ntercty transt staton area (Orloff & Ma, 975). They assume that the feeder bus headway follows normal dstrbuton, and let the devaton of headway reresents the relablty ndex of servce, then servce relablty of feeder system wll not only be affected by the route envronment but also by the congeston of ntercty transt staton area. In 995, Knoers & Muller show the watng tme of ntercty transt assengers wll be changed accordng to the arrval tme of feeder buses and dearture tme of ntercty transt. Exected mean transfer watng tme E (w) can be obtaned by the ntegral of assengers actual watng tme w ( ) and ther corresondng robablty, as shown n Equaton (). It mroves the accurate of watng tme calculaton. owever, caacty constrants and assenger behavors are not consdered n ther research. E ( w) w( ) Pr( ) d () Dstngushed and mroved from the former studes, ths aer ams to develo a mathematcal model for analyzng the ntegrated nfluence of feeder servce relablty, Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

3 assenger behavors, and caacty constrant of ntercty transt on assenger watng tme for ntercty transt systems (e.g., ntercty bus, conventonal ral, and hgh seed ral), and dentfyng the otental benefts of ntercty transt users by ncrease of feeder servce relablty. erenafter, the aer frst dscusses assumtons of model, then analyzes the nfluences of dfferent feeder and ntercty transt servce scenaros, and fnally makes some useful suggestons to system oeraton accordng to the numercal case.. MODELING PASSENGER WAITING TIME In ths study, relablty of feeder bus servces, assenger behavors and servce characterstcs of ntercty transt are taken nto consderaton to develo the ntercty assenger watng tme model. Concet of system oeraton s shown as Fgure. Some basc assumtons of ths model are lsted as follows: Oeraton of feeder and man servces (.e. ntercty trans s consdered as a contnuous flow. Feeder buses have a good scale of fleet. There s ntensve assenger demand n servce area. Intercty transt system s always oerated romtly. The n-staton walkng and rocessng tme are neglected, e.g., the watng tme reresents total transfer tme. Passengers are categorzed nto two grous: - Scheduled assengers: assengers who have decded to rde secfc transt or even have the seats reserved already. - Random assengers: the others not belong to scheduled assengers. Orgn Feeder Buses Intermodal Staton Intercty Transt Destnaton Access Tme (Urban Travel Tme) Watng Tme Intercty Travel Tme Total Travel Tme Fgure. System Toology Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

4 Accordng to the characterstcs of these two tyes of assengers, the robablty dstrbuton functons of assenger arrvals and mean watng tme model of ntercty transt assengers wll be derved ste by ste. Detaled defnton and reasonable values of varables and arameters deloyed n the model are lsted as n Table. Notaton Defnton Table. Varables and Parameters a Access behavor arameter of random assengers A Total arrval assengers by feeder buses durng t 0 ~ c Crush caacty of feeder buses (assengers/veh), ncludng 55 standee and seated C Seat caacty of ntercty transt (seats/ ) 70 Reasonable Value h eadway of feeder buses 0 mn eadway of ntercty transt 30 mn Planned arrval tme n ntermodal transt staton I Relablty factor of feeder bus nformaton system 0.8 k Percentage of scheduled assengers 80% m r Mssed ercentage of addtonal watng cycle r, due to the caacty constrant of ntercty transt n Arrval number of feeder buses durng the ntercty transt headway N ( Aggregated arrval robablty densty functon of ntercty transt assengers N g ( Arrval robablty densty functon of scheduled assengers N h ( Arrval robablty densty functon of random assengers Passengers lanned unctualty devaton to the ntercty transt 5 mn staton s Addtonal watng cycles for ntercty transt W Mean watng tme of ntercty transt assengers α Access behavor arameter of scheduled assengers α Access behavor arameter of random assengers β ( Delayed-access arameter of random assengers γ Scheduled assengers reduced-robablty factor for rdng earler feeder buses µ q Mean arrval tme of feeder buses σ q eadway devaton of feeder buses - ndex of servce relablty 5 mn ρ Average load factor of feeder buses 0.9 Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

5 3. Dstrbuton of Passenger Arrvals Scheduled Passengers Suose that reresents headway of ntercty transt and h reresents headway of feeder buses, and t s assumed the revous transt dearted n t 0. Then the analytc erod of tme wll le n t 0 ~. Addtonally, suose that assengers wll arrve at the staton n mnutes before the dearture tme of transt to make sure that they wll not mss the transt. The oeraton characterstcs of ntermodal staton, relablty of transt and feeder servces wll all affect. For whch, t comes the corresondng analytc tme, as shown n Equaton (3). 0 < (3) In Equaton (4), n reresents the arrval number of feeder buses wthn t 0 ~. n reresents the arrval number of feeder buses wthn, as shown n Equaton (5); and n reresents the arrval number of feeder buses wthn tme erod ~, whch can be obtaned by Equaton (6). n h + (4) n + h n n n (5) (6) Suose that the arrvals of feeder buses follow normal dstrbuton, and Equaton (7) shows the arrval robablty densty functon of the th q bus. Where µ q means the exected arrval tme of the th q feeder bus, as shown more detaled n Equaton (8) and (9), and σ q s the headway devaton of th q feeder bus. ( t µ q ) σ q f q e (7) π σ q µ ( ) h ~ n (8) P Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

6 4 µ j + j h j ~ n (9) Wth the dstrbuton of feeder bus arrvals, the assenger arrvals can be obtaned va t. For the scheduled assengers, t s assumed that they can obtan some nformaton from the feeder bus nformaton system. owever, relablty of bus nformaton system s stll not good enough, thus a arameter for relablty of feeder bus nformaton system s set to be between 0.0 ~.0. Meanwhle, because of the unrelablty of feeder servces, some of the scheduled assengers wll take the earler feeder buses to arrve at the ntermodal staton on ther exected tme and to mnmze the robablty of mssng the antcated transt. Snce t s a lanned behavor, the arrval number of these assengers wll ncrease when t close to the dearture tme of ntercty transt, and vse versa. In order to characterze ths assenger behavor, let the arrval robablty densty functon of feeder buses multled by a reduced-robablty factor γ, as shown n Equaton (0) and (). γ { I[ + ( ) h]} α ~ n (0) γ 0 j ~ n j () In Equaton (0), α reresents the assenger behavor factor corresondng to feeder bus servces. When the relablty of feeder servces s oor, the assengers wll tend to take the earler buses and then α declnes. When α aroaches zero, the arrval dstrbuton of scheduled assenger follows normal dstrbuton. Based on the former study, α s set to be (Chang & su, 00). Let the arrval dstrbuton of ndvdual feeder bus multled by the corresondng γ and take the summaton, t can be obtaned as Equaton (), whch reresents the aggregate arrval functon of scheduled assengers. g( n n γ f ( t; ( ) h, σ ) + γ f ( t; + j h, σ ) () j j j To be notced, few assengers may mss the lanned ntercty transt and have to wat for next transt because of the severely delay of feeder buses. owever, snce the servce of entre system s contnuous, ths secal case wll also haen n revous ntercty transt servce nterval, thus these cases for outlers can be eventually neglected. Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

7 5 Furthermore, let g ( dvded by the area below ts functon curve, t then turns out to be a robablty densty functon of scheduled assengers, as shown n Equaton (3). N g ( g( g( dt 0 g( (3) g( dt Random Passengers For the random assengers, snce arrve by feeder buses, too; thus Equaton (4) reresents the dstrbuton of random arrvals. f ( n n f ( t; ( ) h, σ ) + f ( t; + jh, σ ) (4) j j owever, due to the relatvely long headway of ntercty transt, f the assengers arrve at the transt staton and fnd t s stll early for dearture of transt servces, assengers may leave the staton for other actvtes and then come back n a secfc tme nterval before the next dearture. In order to characterze ths behavor, a arameter named as delayed-access arameter β ( s deloyed, as shown n Equaton (5). For the value of α, as also based on the former study, s set to be (Chang & su, 00). t α β( [ + a( ] t [0, t [ ], ] (5) Smlarly, accordng to Equaton (4) and (5), arrval functon and robablty densty functon of random assengers can be obtaned, as shown n Equaton (6) and (7), resectvely. h( β ( f ( (6) N h ( h( h( dt 0 h( h( dt (7) Wth both of the robablty densty functons of scheduled and random assengers, let k reresents the roorton of scheduled assengers and k reresents the roorton of random ones. Combnng Equaton (3) and (7), the robablty densty functon of total Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

8 6 ntercty transt assenger can be obtaned as Equaton (8). N( k N ( + ( k) N ( (8) g h. Caacty Constrant on Intercty Transt In Equaton (9), A reresents the total assenger arrvals durng tme erod 0 ~. And n Equaton (0), due to the caacty constrants of ntercty transt, m r reresents the mssed ercentage for addtonal watng cycle r. n q ) q 0 A n c ρ f ( t dt (9) m r A r C A r C r ~ s A (0) 0 A< r C.3 Mean Watng Tme Snce the robablty and behavor equatons have develoed, mean watng tme of ntercty transt assengers can be obtaned by Equaton (). W s N ( m t dt + N t mr r + t 0 )( ) ( ) [( ) ] 0 ( dt () 3. NUMERICAL STUDY After develong the watng tme model, reasonable values (see Table ) may be aled to t and to obtan the arrval dstrbuton of scheduled and random assengers, as shown n Fgure. For scheduled assengers, t exhbts that when t < 0, arrvals nearly zero, when t 0 ~ 5, arrvals gently ncrease, and when t 5 ~ 0, arrvals boom radly. For the random ones, arrvals ncrease sgnfcantly n the erod of t 0 ~ 5 and declne after t > 0. Because t s assumed that f random assengers reach the ntermodal transt staton too early before the transt deartures, they wll not wat and temorarly leave the staton untl some secal tme before the transt deartures, thus the arrval rates of random assengers are nearly zero before t 0. When t > 8 ( mnutes before dearture of ntercty trans, random arrvals emerges, and manly ntensve n 0 mnutes before the Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

9 7 dearture of ntercty transt. Fnally, for the dstrbuton of total assenger arrvals, t s shown as Fgure 3. For aggregated assengers, arrvals reach hgh when 5 ~ 5 mnutes before the dearture of ntercty transt, and then declne Random Passengers Arrvals (%) Scheduled Passengers t (mn) Fgure. Arrval of Passengers wth Dstnct Behavors Arrvals (%) t (mn) Fgure 3. Arrval of Total Passengers Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

10 8 4. SENSITIVITY ANALYSIS Based on the watng tme model, t s found that the roorton of scheduled assengers, relablty of feeder buses, and caacty of ntercty transt all lay sgnfcant roles on mean watng tme. Therefore, a senstve analyss for each of the mortant arameters s made n ths secton. 4. Caacty of Intercty Transt In Fgure 4, t s found that caacty of ntercty transt lays an mortant role on mean watng tme. owever, when the caacty s greater than the assenger arrvals A, the sared caacty becomes a waste, snce t cannot mrove mean watng tme any more. For ths case, at least C / A 0.63 ~ 0. 8 s suggested to be suled. W (mn) W C (seats/) Fgure 4. Passenger Watng Tme vs. Caacty of Intercty Transt 4. eadway Devaton of Feeder Buses Suose that the headway devaton σ stands for relablty ndex of the feeder buses, relaton of mean watng tme to σ s shown as Fgure 5. It exhbts that mean watng tme ncreases wth the headway devaton. Mean watng tme s very senstve to the change of relablty of feeder buses when σ s greater than 5 mnutes, and become less senstve when t s less than 5 mnutes. For ths case, at most σ 3.5 ~ 4. 6 mnutes are recommended, when σ > 4.6 mnutes, mean watng tme ncrease dramatcally. Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

11 9 30 W (mn) W σ (mn) Fgure 5. Passenger Watng Tme vs. eadway Devaton of Feeder Buses 4.3 Percentage of Scheduled Passengers Fgure 6 shows the relaton of mean watng tme to roorton of scheduled assengers. When k ncrease, the longer watng tme t wll be. Thus, t s found that for the scheduled assengers, they wll suffer more than that of the random ones, f the feeder servce s not relable. For the Fgure 7, t exhbts the contrast stuaton,.e., wthout caacty constrant on ntercty transt. It s found that, comarng to Fgure 6, k makes no sgnfcant nfluences on watng tme. Ths verfes agan, for any negatve factors n entre servce, the scheduled assengers always damaged more. W (mn) W k (%) Fgure 6. Passenger Watng Tme vs. Percentage of Scheduled Passengers Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

12 0 6 4 W0.5 W (mn) k (%) Fgure 7. Passenger Watng Tme vs. Percentage of Scheduled Passengers Note: wthout caacty constrant of ntercty transt 5. CONCLUSIONS AND RECOMMENDATIONS The conventonal assumton on watng tme,.e., watng tme s half of the headway or mean watng tme s some fxed multler of headway, s not comletely to recsely reresent the assenger watng tme n an ntermodal transt staton. Ths study exhbts that the watng tme can be modeled by consderng the relablty of feeder servce, assenger behavors, and system characters of ntercty transt. Accordng to the numercal results, t s found that assengers tend to arrve at 5 ~ 5 mnutes before the dearture of ntercty transt. For the senstvty analyss on feeder relablty, t dentfes that mean watng tme of ntercty transt assengers s greater than half of the transt headway f the feeder servce s rovded wth oor relablty; and for the contrast, t s less than half of the transt headway as long as the feeder servce s rovded wth good relablty. Moreover, t should be emhaszed that, for any negatve factors n feeder or ntercty servces, the scheduled assengers always affected more on average watng tme. For the caacty of ntercty transt, t s suggested to satsfy at least 80 % of total arrval assengers, f 00 % s not avalable. In general, the ntercty transt would really mrove ts servce qualty only f the feeder servce also rovdes good relablty. In ths study, watng tme of random assengers s only calculated based on ther re-arrval tme, so that ther watng tme s less than that of scheduled assengers. Moreover, eak and off-eak factors are not taken nto consderatons n ths aer, they should be dscussed n further study. Journal of the Eastern Asa Socety for Transortaton Studes, Vol.5, October, 003

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