AN APPLICATION-ORIENTED MODEL OF PASSENGER WAITING TIME BASED ON BUS DEPARTURE TIME INTERVALS

0 0 0 0 AN APPLICATION-ORIENTED MODEL OF PASSENGER WAITING TIME BASED ON BUS DEPARTURE TIME INTERVALS y Huio Gong, Graduate Research Assistant MOE Key Laoratory for Transportation Complex Systems Theory and Technology, School of Traffic and Transportation, Beijing Jiaotong University Beijing 000 China Phone: -0-0; Fax: -0-0; Email: 0@jtu.edu.cn Xumei Chen, Ph.D. Professor (Corresponding Author) MOE Key Laoratory for Transportation Complex Systems Theory and Technology, School of Traffic and Transportation, Beijing Jiaotong University Beijing 000, P.R. China Phone: -0-0; Fax: -0-0; Email: tcxm@.net Lei Yu, Ph.D., P.E. Yangtze River Scholar of Beijing Jiaotong University and Professor of Texas Southern University College of Science and Technology, Texas Southern University 00 Cleurne Avenue, Houston, Texas 00 Phone: --00; Fax: --; Email: yu_lx@tsu.edu and Lijuan Wu, Graduate Research Assistant School of Traffic and Transportation, Beijing Jiaotong University Beijing 000 China Phone: -0-0; Fax: -0-0; Email: 00@jtu.edu.cn Sumitted for Presentation at the rd Transportation Research Board Annual Meeting and Pulication in the Transportation Research Record Washington, D.C. January 0 Word Count: (Text) + 0*(Tales) + 0* (Figures) = Sumission Date: Novemer, 0 TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 ABSTRACT Developing a reliale and practical method to estimate passenger waiting time ecomes a key issue in the evaluation of service quality for pulic transit systems. However, existing methods lack of applicaility in practices due to the need of the costly data collection. This paper develops an application-oriented model to estimate the waiting time as a function of us departure time intervals. First, distriutions of passenger arrival rates for two types of us stops are analyzed ased on field data collected in Beijing. Bus stops are classified into Type A and B, depending on whether they are connected with uran rail transit systems. The results show that the lognormal distriution has the est fit for Type A us stop, and gamma distriution provides the est fit for Type B us stop. Second, considering the convenience to extract the data of us departure times from existing intelligent transit systems, the relationship etween passenger arrival rates and us departure time intervals is analyzed. It is demonstrated that parameters of the passenger arrival rate distriution for oth two types of stops can e expressed y the average and CV (Coefficient of Variation) of us departure time intervals in functional relationships. Then, an application-oriented waiting time model is proposed. Finally, a model validation is conducted, resulting in the NMSE (Normalized Mean Square Error) of 0.0 and 0.0 for Type A and B stops, respectively. Thus, the proposed model is shown to provide a reliale estimation of the average passenger waiting time ased on only readily availale us departure time intervals. Key words: Average Passenger Waiting Time; Bus Departure Time Interval; Passenger Arrival Rate Distriution; NMSE TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 0 0 INTRODUCTION An efficient pulic transit system with a high quality of service is a crucial determinant to attracting more passengers. For transit trips, passenger waiting time at us stops is one of the most important performance measures to reflect the quality of the service level for pulic transit systems (). Moreover, modal choice decisions are more sensitive to the amount of waiting times than the time on oard (). The waiting time can e a factor that hinders the usage of the us transit. Thus, developing a reliale and practical estimation method of passenger waiting time is necessary, which is helpful for the transit agency to identify underlying elements for the long waiting time delay, and further implement improvement strategies and control measures. In this paper, the pulic transit system in Beijing is chosen for the case study. The pulic transit system in Beijing has two distinguishing characteristics in comparison with that of other cities. One is that there are a great numer of us stops due to an extensive pulic transit network of uses and a rapidly expanding suway system. Currently, there are a total of suway lines and us lines with more than,000 us stops (). Furthermore, waiting ehaviors at various us stops is different, depending on whether they are connected with uran rail transit systems. For the us stops connected with the uran rail transit system, most of the passengers disemarks from the rail system and the waiting ehaviors of them depend on the transfer time from rail to us system. Thus, the us stops connected with the uran rail transit system may e more likely to have passengers arrive in groups, which is different from the randomly arrival of the us stops not connected with the uran rail transit system. Here in this research, us stops are classified into two types. One is connected with the uran rail transit system (Type A), and the other one is not connected with the uran rail transit system (Type B). The other characteristic is that new technologies, such as the automatic vehicle locating y GPS (Gloal Position System) and the automatic fare collection y IC card (i.e., Smart Card), have een widely used in the pulic transit system of Beijing. Both GPS and IC card systems can provide transit agencies with enormous amounts of data, such as the us arrival and departure times at each us stop. Thus, the GPS and IC card systems in Beijing can provide on-line data, which support the transit agency to estimate passenger waiting time and further evaluate the pulic transit service in a timely manner. In addition, the uran us transit system in Beijing has short headways to satisfy the high transit demand, usually less than 0 minutes intervals in peak hours. Further, the us arrival and departure times at each stop are irregular everyday especially during peak hours, due to the traffic congestion. The passenger arrival distriution is affected more or less y the short and irregular us departure time intervals. Accordingly, the proposed model should incorporate the impact from such short and irregular us departure time intervals. Hence, the ojective of this paper is to develop a reliale passenger waiting model, which can e applied to typical us stops in Beijing with us departure time intervals extracted from the GPS and IC card data. In the following sections, existing studies on passenger waiting time models are first reviewed. Then, the paper analyzes the distriution of passenger arrival rates for two types of us stops ased on field data collected in Beijing. Susequently, the relationship etween the TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 0 0 distriution of passenger arrival rates and us departure time intervals is analyzed. Further, an application-oriented waiting time model for two types of us stops is proposed y the method of integral calculus. Finally, this study conducts a validating analysis of the proposed model y comparing the results from the model with the field data. Some conclusions are summarized at the end of the paper. LITERATURE REVIEW Passenger waiting time models have een studied since 0s. In traditional passenger waiting time models from early studies, the average passenger waiting time was derived as the sum of one half of the average headway of uses and the ratio of the headway variance to twice the average headway, y assuming a random passenger arrival (). Such a model indicated that the mean waiting time of passengers decreases as the reliaility of the us service increases (). Similarly, the relationship etween the average passenger waiting time and us headway was widely examined later in passenger waiting time studies. The work of Seddon and Day() was among the early studies on passenger waiting time models (). They otained a function for the us passenger waiting time y analyzing uses in Leeds, UK (), which was formulated in Equation (): w =. + 0.µ () where w is the average waiting time (minute); and µ is the headway of uses studied (minute). A similar result was found y O Flaherty and Mangan(0) in Manchester, UK (), as shown in Equation (): w =. + 0.µ () Some studies also indicated that waiting ehaviors are different depending on different us headways (, ). For the us transit service with small headways, passengers rarely need to consult schedules since vehicles arrive frequently. Therefore, these passengers arrive at the stop at a random rate. In contrast, for longer headways, passengers generally learn schedules in advance in order to reduce their waiting times. These passengers arrive at the stop near the departure time. Compared with previous literatures, Knoppers and Muller() were the first who proposed a precise calculation method for passenger waiting time with integral calculus (0), as shown in Equation (). + Ew ( ) = wp ( )Pr( pd ) p () Equation () shows that the expected mean passenger waiting time E(w) can e otained y integrating passengers actual waiting time w(p) with the corresponding proaility function Pr(p), where the variale p represents the actual punctuality deviation of feeder vehicle arrivals. Since then, there were also some studies that developed passenger waiting time models ased on distriution curves of passenger arrival rates and the integral calculus method. Luethi et al.(00) proposed a model for passenger arrival rates that comine a uniform distriution with a Johnson S B distriution (). Normal, lognormal and triangular distriutions TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 0 had all een used to calculate passenger waiting time (, ). Guo et al.(0) compared normal, exponential, lognormal and gamma distriutions while fitting arrival rates of passengers transferring from the rail to uses. The results showed that lognormal and gamma distriutions have the est fit for direct transfer and non-direct transfer passengers, respectively (). From the aove literature review, it can e found that almost all of existing studies showed that headways of uses significantly affect the average passenger waiting time (). Most of existing waiting time models were ased on us headways and the distriution of passenger arrival rates. However, in order to determine the distriution of passenger arrival rates, it is difficult and also costly to collect field data of passenger arrival times for different types of us stops. Therefore, this research is aimed at developing an application-oriented passenger waiting time model. This model is intended to overcome shortcomings of the traditional waiting time model y avoiding manually collecting a large numer of field data on passenger arrival rates. This model can also estimate passenger waiting time for different types of stops. DATA COLLECTION AND PREPARATION In order to collect valid data, a preliminary investigation for selecting survey locations is conducted and survey locations are selected on the asis of three criteria. The first criterion is that the us stop should have an independent waiting platform serving only one us line, which can help the surveyor identify the passenger who plans to go aoard the us of the surveyed us line. The second criterion is that there should e a relatively high demand at the us stop for the surveyed us line to ensure an adequate sample size. Finally, for Type A us stop, the surveyed us stop should not e far away from the uran suway station. For Type B us stop, there should e no uran suway station connecting with the surveyed us stop within an acceptale walking distance. Based on the aove criteria, a total of four us stops, including two Type A us stops and two Type B us stops on four us lines were selected (see Tale ). The survey was conducted on weekdays during evening peak hours from :00 to :00 p.m.. Passenger arrival times at us stops and the us arrival/departure time were collected. All the time data were recorded in the format of hh:mm:ss. TABLE Date Collection Frequency Stop type Bus stop (on) Bus line Survey date (trips/peak hour) Xizhimen Line #0 th, th, th April 0,, Type A Dongwuyuan Line #0 th, th, th June 0,, Baishiqiao Line # th, th, th April 0,, Type B South Liuliqiao Line # th, th, th June 0,, Based on the collected data, passenger arrival time, passenger waiting time, and us departure time intervals can e calculated. The relationship among these parameters is shown TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. as Figure. Passenger arrival time t_arrival(i-,k) T_arrival(i,k) Passenger arrival time t_arrival(i,k) T_interval(i,k) T_waiting(i,k) Passenger arrival time t_arrival(i+,k) (i-) th us (i) th us (i+) th us 0 0 0 Bus arrival time t_arrival(i-) FIGURE Illustration of passenger waiting time. In Figure, the passenger arrival time is calculated according to the previous us departure time. The waiting time can e calculated y sutracting the passenger arrival time from the us departure time. The us departure time interval is equal to the departure time difference etween two consecutive uses, as shown in Equations () to (): T _ arrival(, i k) = t _ arrival(, i k) t _ departure( i ) () T _ waiting(, i k) = t _ departure() i t _ arrival(, i k) () T _ interval( i,) i = t _ departure() i t _ arrival() i () th th where T _ arrival(, i k ) is the arrival time of k passenger waiting for the i us; T _ waiting(, i k ) is the waiting time of k th passenger waiting for the i th us; and T _ interval( i-, i ) is the us departure time interval etween the ( i -) th th and i uses. In this paper, the data including, and,0 valid records of passengers for Xizhimen us stop on us line #0 and Baishiqiao us stop on us line #, which represent the Type A us stop and Type B us stop in Beijing respectively and meet the sample size, is used for the model development. And the rest of data is used for the validation of the proposed model. In the following analysis, the data on passenger arrival times are mainly used to determine the passenger arrival rate distriutions, and the data on us departure times are used to identify the relationship etween the passenger arrival distriution and the us departure time interval. METHODOLOGY Bus departure time t_departure(i-) Bus arrival time t_arrival(i) Bus departure time t_departure(i) Bus arrival time t_arrival(i+) Bus departure time t_departure(i+) Determination of Passenger Arrival Rate Distriutions for Different Types of Bus Stops In order to determine the passenger arrival rate distriution, extreme value, exponential, lognormal, gamma and normal distriutions are chosen to fit the passenger arrival rate. R-Square and the Sum of Squared Errors (SSE) are used to compare the goodness-of-fit of each distriution. The distriution fitting analyses for oth Type A us stop and Type B stop are completed y using the DFITTOOL (Distriution Fitting Tool) of MATLAB. Passenger Arrival Rate Distriution for Type A Bus Stop For Xizhimen us stop,, records of passengers of line #0 are valid. The records without a us arrival time efore the passenger arrival time or without a us departure time TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. after the passenger arrival time during the period of survey are not valid and eliminated. For Line #0, the maximum us departure time interval at Xizhimen us stop is, seconds, while the average is seconds. More than.% of passengers arrived at 00 seconds or less, as shown in Figure. FIGURE Proaility density distriution of passenger arrival time during evening peak hours for Type A us stop. 0 Figure illustrates passenger arrival rates during evening peak hours (:00 p.m.-:00 p.m.) using extreme value, exponential, lognormal, gamma and normal distriution fits. The results for oth R-Square and SSE indicate that the lognormal distriution provides the est quality of fit for Type A us stop with a R-Square of 0.00 and a SSE of 0.00, which is similar to the results of arrival rate distriutions of direct transfer passengers conducted y Guo et al.(). The proaility density distriution of passenger arrival times for Type A us stop can e expressed y Equation () as ln t µ f( t) = exp( ( ) ) σt π σ () where f () t is the proaility density distriution of passenger arrival times for Type A us stop; t is the passenger arrival time variale; and μ and σ are the mean and standard deviation 0 of the lognormal proaility distriution of the passenger arrival time, in Figure the value of μ and σ are.0 and 0. respectively. Passenger Arrival Rate Distriution for Type B Bus Stop For Baishiqiao us stop, a total of,0 records of passengers are valid. The maximum us departure time interval at Baishiqiao us stop is seconds and the average is 0 seconds. More than.% of passengers arrived at 00 seconds or less, as shown in Figure. TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. FIGURE Proaility density distriution of passenger arrival time during evening peak hours for Type B us stop. Figure shows passenger arrival rates during evening peak hours (:00-:00) y using extreme value, exponential, lognormal, gamma and normal distriution fits. As shown in Figure, the Gamma distriution has the est fit for Type B us stop with a R-Square of 0. and a SSE of 0.00, which is similar to the results of arrival rate distriutions of non-direct transfer passengers conducted y Guo et al.(). The proaility density distriution 0 of passenger arrival times for Type B us stop can e written y Equation () as: t f( t) t α = exp( ) α β Γ( α) β () where f () t is the proaility density distriution of the passenger arrival time for Type B us stop; t is the passenger arrival time variale; and α and β are parameters of the proaility distriution. The mean and standard deviation of the passenger arrival time for Type B us stop can e calculated y αβ and αβ. In Figure the value of α and β are.0 and.0 respectively. Relationship etween Passenger Arrival Rate Distriutions and Bus Departure Time Intervals 0 In existing waiting time models, a large numer of field data on passenger arrival times need to e collected manually in order to determine the distriution of passenger arrival rates. Such data collection is difficult and also costly, especially for different us lines at different types of us stops. Consequently, the relationship etween passenger arrival rate distriutions and us departure time intervals is analyzed ecause us departure time intervals can e extracted from existing intelligent transit systems and have an impact on passenger arrival rates according to Equation (). The relationship etween passenger arrival rate distriutions and us departure time intervals might e expressed y certain analytical functions. The us departure time interval represents the primary characteristic for the pulic TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 0 transit scheduling and is also used to estimate the median passenger waiting time in transportation models (). Results of previous studies identified that the us departure time interval is the most important influencing factor of passenger arrival rates in microscopic distriutions analysis (). Thus, a sensitivity analysis of the us departure time interval on the distriution parameters is conducted to examine the relationship etween the passenger arrival rate distriution and the us departure time interval. In the sensitivity analysis, two indicators, the average us departure time interval and the coefficient of variation (CV) of us departure time intervals at us stops, are chosen, which reflect the service level and the service reliaility (, ) of different us lines. Based on the results of determining the passenger arrival distriution, parameters μ, σ and parameters α, β are chosen to represent characteristics of the passenger arrival rate distriution of different types of us stops. In analyzing the impact of different us departure time intervals on distriution parameters, all of the surveyed data including hours and hours for two types of us stops are selected. Line #0 s survey on th June 0 is less than two hours, thus only an hour s data is selected. Then an hour s data was used as a group for the analysis, thus groups of data sets for Type A us stop and groups of data sets for Type B us stop are used. The average and CV of us departure time intervals in each data group are calculated, and parameters of the passenger arrival rate distriution in each data group are calculated y the DFITTOOL of MATLAB. Relationship Analysis for Type A Bus Stop Because parameters μ and σ of the lognormal distriution are the mean and standard deviation of the lognormal proaility distriution of passenger arrival rates, which are determined y the mean and CV of the passenger arrival time, we assume that parameters μ and σ are also influenced y the average and CV of us departure time intervals. The regression analysis etween the distriution parameters μ, σ and the average and CV of us departure time intervals is conducted as shown in Figures and, respectively. Figure shows that the relationship etween the distriution parameters μ and the average of us departure time intervals can e represented y a quadratic regression function with a R-Square of 0. as shown in Equation () µ µ + µ + () =-0.0000 0.00. where μ is the distriution parameter of passenger arrival rates for Type A us stop; and the average of us departure time intervals. µ is TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 Parameter μ.... μ = -0.0000μ + 0.00μ +. R² = 0. Data. Quadratic Regression. 00 0 00 0 00 Average of Bus Departure Time Intervals(s) FIGURE Relationship etween average of us departure time intervals and parameter μ. Figure shows that the underlying relationship etween the distriution parameters σ and the CV of us departure time intervals can e characterized y a linear function with a R-Square of 0. as shown in Equation (0) σ = 0.0cv + 0. (0) where σ is the distriution parameter of passenger arrival rates for Type A us stop; and the CV of us departure time intervals. Parameter σ.. 0. σ = 0.0cv + 0. R² = 0. 0. Data 0. Linear Regression 0. 0. 0. 0. 0. 0. CV of Bus Departure Time Intervals FIGURE Relationship etween CV of us departure time intervals and parameter σ. cv is Relationship Analysis for Type B Bus Stop Before analyzing the impact of different us departure time intervals on distriution parameters α and β, we derive mathematical relationships etween the mean, standard deviation of passenger arrival times and the parameters α and β according to the existing relationship in the Gamma distriution. This existing relationship is as follows. µ p = αβ () TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 σp = αβ () where µ is the mean of the passenger arrival time; and p σ is the standard deviation of the p passenger arrival time. Then, parameters α and β can e represented y the mean and standard deviation of the passenger arrival time as α = ( ) () cv p β = µ cv () p p 0 0 where cv = σ / µ is the coefficient of variation of the passenger arrival time. p p p Based on the relationship etween parameters α and β and the mean and CV of the passenger arrival time in Equations () and (), we assume that the parameters α and β have a similar underlying relationship with the average and CV of us departure time intervals. The regression analysis etween the distriution parameters α and β and the average and CV of us departure time intervals is conducted as shown in Figures and, respectively. Factors of / cv and µ cv are considered as two new indicators in the following analysis. Figure illustrates that the relationship etween the distriution parameter α and the indicator / cv can e represented y a quadratic regression function with a R-Square of 0. as shown in Equation () Parameter α... 0. 0 α = 0.00( ) 0. 0. cv + + () cv α = -0.00(/cv ) + 0.(/cv ) + 0. R² = 0. 0 0 0 /CV of Bus Departure Time Intervals FIGURE Relationship etween /CV of us departure time intervals and parameter α. Data Quadratic Regression TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 Figure shows that the relationship etween the distriution parameter β and the indicator index µ cv can e characterized y a linear function with a R-Square of 0. as shown in Equation () β = + () 0.ucv.0 where α and β are the parameters of the passenger arrival rate distriution for Type B us stop; cv = σ / µ is the coefficient of variation of us departure time intervals; µ is the average deviation of us departure time intervals; and σ is the standard deviation of us departure time intervals. Parameter β 00 00 00 00 00 00 0 β = 0.u cv +.0 R² = 0. FIGURE Relationship etween u*cv of us departure time intervals and parameter β. Passenger Waiting Time Model for Different Types of Bus Stops Passenger Waiting Time Model for Type A Bus Stop For a given us departure time interval, the standard proaility density function of the passenger arrival time for Type A us stop can e expressed y Equation () as f() t gh () t = () i hi f ( τ ) 0 where gh i () t is the standard proaility density function of the passenger arrival time for Type th A us stop within a us departure time interval h i ; h i is the departure time interval of the i us; and t is the arrival time variale for passengers varied from 0 to h i. Thus, the average waiting time for a given us departure time interval can e derived using Equation () as Data Linear Regression 0 00 00 00 00 000 u*cv of Bus Departure Time Intervals d τ hi i = 0 i hi t W ( ) ( ) () h h tg td () where W( h i ) is the average waiting time within a us departure time interval h i. TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 Finally, the average waiting time of all passengers can e otained as shown in Equation (). s.t. W f () t n hi n ( h ) ( ) 0 i t d hi t W h i i= f 0 ( τ ) d i= τ = = n ln t µ f t σt π σ µ µ µ σ = 0.0cv + 0. cv = σ / µ ( ) = exp( ( ) ) =-0.0000 + 0.00 +. where W is the average waiting time of all passengers; and n is the numer of us departure time intervals for Type A us stop. Passenger Waiting Time Model for Type B Bus Stop By using the same method, we can otain the average waiting time model of all passengers for Type B us stop as shown in Equation (0). W = n i= hi 0 hi t d hi t ( ) n 0 f () t f ( τ ) d α t f( t) = t exp( ) α β Γ( α) β s.t. α = 0.00( ) 0. 0. + + (0) cv cv β = 0.ucv +.0 cv = σ / µ VALIDATION In order to validate the applicaility of the proposed models, this section analyzes and compares results from the case study using field data and results from the proposed waiting time model. The proposed model is applied to Dongwuyuan us stop on line #0 (Type A), and South Liuliqiao us stop on line # (Type B), respectively. First, we otain us departure time intervals and passenger waiting times during each us departure time interval in each surveyed period (evening peak hours on th, th, th June 0). Then, the average and CV of us departure time intervals at each period can e calculated. Thus, we can get the average passenger waiting times at Dongwuyuan stop and South Liuliqiao stop in each us departure time interval y the proposed model, respectively, as shown in Figure. A total of and data sets for Type A and Type B us stops are used for validation, respectively. τ n () TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. Average passenger waiting time (s) 00 00 00 00 00 00 00 00 00 0 Field vs. estimated values of average passenger waiting time at Dongwuyuan stop (a) Field value Estimated value 0 00 00 00 00 000 00 00 Bus departure time interval (s) 0 Average passenger waiting time(s) 00 000 00 00 00 00 0 Field vs. estimated values of average passenger waiting time at South Liuliqiao stop 0 00 000 00 000 Bus departure time interval (s) () FIGURE Field vs. estimated values of average passenger waiting time in each us departure time interval for two us stops. To evaluate the proposed model, NMSE (Normalized Mean Square Error) is used in this paper, which can evaluate the average relative discrete degree etween the field and estimated values (), as shown in Equation (). ( w w ) i = n real model NMSE = n () w w i= real model Field value Estimated value where w real is passenger waiting time in the field, and w model is passenger waiting time TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 estimated y the model, and w real and w model are the mean of the field and estimated passenger waiting time, respectively. In an accurate model, the NMSE should e close to 0. Based on existing studies, NMSE <0. is an acceptale limit (). For the calculation results in this paper, the NMSEs in three days evening peak hours at Dongwuyuan stop and South Liuliqiao stop are 0.0 and 0.0, respectively. The average passenger waiting time in the field and that estimated y the proposed model at Dongwuyuan stop and South Liuliqiao stop are very close with relative errors of -.% and.0%,.% and.00%, -.% and.% in each evening peak hours, which are considered to e reasonale, as shown in Tale. Tale Field vs. Estimated Values of Average Passenger Waiting Time in Each Period for Two Bus Stops Bus stop Time Field value(s) Estimated value(s) Relative error 0 0 th June_p.m. peak 0.. -.% Dongwuyuan stop th June_ p.m. peak...% th June_ p.m. peak 0.. -.% th June_ p.m. peak...0% South Liuliqiao th June_ p.m. peak 0. 0..00% stop th June_ p.m. peak...% From the aove analysis, it can e shown that passenger waiting time model for oth Type A and Type B us stops can e used to calculate passenger waiting time with a high accuracy. Moreover, in practice, the us departure time at each us stop can e extracted from the GPS and IC card systems, which are installed in uses, so the us departure time intervals are availale inputs to the proposed passenger waiting time model. Therefore, the proposed passenger waiting time model in this paper is more targeted on applications, which has a good potential to e applied in estimating average passenger waiting times for different types of us stops in Beijing. CONCLUSIONS Based on field data in Beijing, an application-oriented model of the average passenger waiting time for different types of us stops is proposed, which is a function of us departure time intervals. As the only input of the proposed model, us departure time intervals can e easily extracted from the GPS and IC card data. Therefore, the proposed model has a good applicaility in estimating average passenger waiting times for different types of us stops in Beijing. Main findings in this study can e summarized as follows:. Distriutions of passenger arrival rates for oth Type A and Type B us stops were analyzed ased on field data collected in Beijing, and it was found that lognormal distriution had the est fit for Type A us stop, and the gamma distriution provided the est fit for Type B us stop. TRB 0 Annual Meeting Paper revised from original sumittal.

Gong, Chen, Yu, and Wu. 0 0 0 0. The relationship etween the passenger arrival rate distriution and the us departure time interval was analyzed. It was shown that distriution parameters of the passenger arrival rate for oth Type A and Type B us stops could e expressed y the average and CV of us departure time intervals in functional relationships.. An application-oriented model of average passenger waiting time for oth Type A and Type B us stops was proposed y the method of integral calculus. The model was validated y comparing the field value and estimated value of passenger waiting time. The validation results showed that the proposed model had a high accuracy with a relative error no more than %. In future studies, the influence of the next us arrival time information service on the passenger waiting ehavior at us stops need to e considered. The classification of us stops can e further expanded to refine the proposed model. A study on waiting time models for suuran us lines with longer us departure time intervals (often more than minutes) can e conducted. In addition, for the model application in other cities, a caliration and validation study of the proposed passenger waiting time model is needed. ACKNOWLEDGMENTS Authors acknowledge that this paper is prepared ased on National Basic Research Program of China. (No. 0CB0), Program for New Century Excellent Talents in University (NCET--0), the Fundamental Research Funds for the Central Universities (No. 0JBM0), and NSFC #0. This research is partially supported y the National Science Foundation (NSF) under grant #. REFERENCES. Wardman, M. A Review of British Evidence on Time and Service Quality Valuations. Transportation Research Part E, Vol., No. -, 00, pp. 0-.. Guo, S., L. Yu, X. Chen, and Y. Zhang. Modeling Waiting Time for Passengers Transferring from Rail to Buses. Transportation Planning and Technology, Vol., No., 0, pp. -0.. Beijing Transportation Research Center. Beijing Transport Annual Report. Beijing Transportation Research Center, 0.. Welding, P. I. The Instaility of a Close Interval Service. Operations Research Quarterly, Vol.,, pp.-.. Osuna, E. E. and G. F. Newell. Central Strategies for an Idealized Pulic Transport System. Transportation Science, Vol.,, pp. -.. Seddon, P. A. and M. P. Day. Bus Passenger Waiting Time in Great Manchester. Traffic Engineering and Control, Vol.,, pp. -.. O Flaherty, C. A. and D. O. Mangan. Bus Passenger Waiting Time in Central Areas. Traffic Engineering and Control, Vol., 0, pp. -.. Bowman, L. A. and M. A. Turnquist. Strategies for Improving Reliaility of Bus Transit Service. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies, Washington, TRB 0 Annual Meeting Paper revised from original sumittal.

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