Limited number of berths can result in a subsequent bus stopping at the upstream of a bus stop when all berths are occupied. When this traffic phenomenon occurs, passengers waiting on the platform usually prefer walking to the stopped bus, which leads to additional walking time before boarding the bus. Therefore, passengers’ travel time consumed at a bus stop is divided into waiting time, additional walking time, and boarding time. This paper proposed a mathematical model for analyzing passengers’ travel time at conventional bus stop based on theory of stochastic service system. Fieldmeasured and simulated data were designated to demonstrate the effectiveness of the proposed model. By analyzing the results, conclusion was conducted that short headway can reduce passengers’ waiting time at bus stop. Meanwhile, the theoretical analysis explained the inefficiency of bus stops with more than three berths from the perspective of passengers’ additional walking time. Additional walking time will increase in a large scale when the number of berths at a bus stop exceedsthe threshold of three.
Bus stop is the connection between transit passenger and bus service system. Both the transit system and passenger satisfaction are under the influence of bus stop [
Besides the research on travel time on route level, a number of researchers have concentrated on the stoplevel travel time. Guenthner and Hamat studied the effect of complicated fare structure on dwell time [
Using theory of classical queueing, the waiting time can be deduced based on the distribution of arrival headway and service time, as well as the number of service counter [
We make the assumption that passengers of bus route
Passengers’ waiting process at bus stop.
When approaching a bus stop, the bus usually decelerates to dwell in a berth at the bus stop. It takes approximately 9 s for the bus to complete the whole deceleration process before entering the scheduled bus stops [
In this paper, passengers’ travel time at bus stops is divided into three parts: waiting time on the platform, additional walking time, and boarding time. Let
Theory of stochastic service system can be applied to conduct quantitative analysis for acquiring deep comprehension on the traffic mechanism introduced above. There exist two service processes, namely, buses entering berths and passengers boarding vehicles. In the former service process, a bus stop is the service counter. Similarly, buses are service counters when passengers are boarding vehicles. Based on queueing theory, the service time dwelling in a berth is assumed to obey negative exponential distribution. Buses and passengers arrive according to a Poisson process. It is worth noting that the service processes of vehicles entering berths and passengers boarding vehicles are different. When buses enter berths, the bus stop is always available. That is to say, the bus stop usually continues to be open to buses without prohibiting dwelling, which can be assumed as a classical queueing model. However, the practical process of serving passengers always becomes unavailable until the next vehicle comes. There exists absent service period due to headways between two adjacent buses. Classical queueing models cannot be applied to solve this problem. Instead, this case can be modeled with vacation queueing models. In the vacation queueing models, it allows servers to take vacations due to reasons such as taking a break and being checked for maintenance.
According to theory of vacation queueing, the waiting time in equilibriumstate can be divided into several independent random variables [
In general, the concrete arithmetic expression cannot be deduced easily from the complicated Markov process. Probability generating function and Laplace transform are introduced to simplify these processes. Thus,
We make the assumption that passengers of bus route
Limited dwell time and design capacity of vehicles result in the fact that the number of passengers loaded during the dwell time cannot exceed the maximum threshold
The average waiting time of passengers on the platform,
The average number of passengers loaded at the bus stop is
Limited number of berths can result in a subsequent bus vehicle stopping at the upstream of a bus stop if all berths are occupied. When this traffic phenomenon occurs, passengers waiting on the platform usually prefer walking to the stopped bus, which leads to additional walking time before boarding bus vehicle. Additional walking time consumed by passengers can be derived by dividing additional walking distance by passengers’ walking speed. Thus, the average additional walking time can be calculated as
This paper deduces the average additional walking distance,
The probability that a bus stops at the upstream of the scheduled bus stop is
The boarding time per passenger is under the influence of a number of factors [
The proposed travel time model at bus stop was validated against both fieldmeasured and simulated data. The fieldmeasured data was collected from 12 bus stops in Nanjing of China. Crosssectional analysis was conducted to compare the field data collected at bus stops with different number of berths. To ensure that the traffic and geometric characteristics at different types of bus stops were similar, the following criteria were applied during bus stop selection: (i) the selected bus stops are separated from bicycle lane to avoid the influence of bicycle stream; (ii) no access is located adjacent to the selected bus stops; (iii) distance from selected bus stops to the adjacent intersections is larger than 150 m; (iv) the selected bus stops have a larger boarding demand than alighting demand in the p.m. peak period; (v) exclude major transfer stops with large interchanging passenger demand.
Geometric characteristics of selected bus stops are shown in Table
Selected bus stops to collect field data.
Type  Site  Stop  Direction 



Online  1  Yuquan Station  WE  2  4 
2  Fuzhuo Station  NS  2  5  
3  Hunan Station  WE  3  6  
4  Daxinggong Station  NS  3  7  
5  Jimingsi Station  EW  4  7  
6  Taipingmen Station  WE  4  8  


Offline  7  Fuzimiao Station  WE  2  7 
8  Zongtongfu Station  EW  2  7  
9  Beijing Dong Station  WE  3  7  
10  Xuanwuhu station  NS  3  8  
11  Gulou Station  SN  4  9  
12  Xinjiwkou station  NS  4  10 
Note: NS and WE mean from north to south and from west to east, respectively.
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The simulated data was generated from the simulation models developed for all the selected bus stops. A widely used microscopic simulation package VISSIM 5.4 was used in this study to develop simulation models for the selected bus stops. To more accurately reflect actual traffic characteristics at the selected bus stops, several issues should be considered seriously: (i) parameters of traffic facilities such as public transit stops, platform, and vehicles in the simulation models were defined according to the real condition; (ii) multiple waiting area of platform was set to simulate the actual queueing phenomenon, and routing decisions for passengers were located in the area of the platform edge; (iii) pedestrian inputs were added as an origin for boarding passengers according to the statistical data; (iv) define start and destination areas for passengers of each bus route to measure the travel time at bus stops.
The travel time at bus stop estimated with the proposed model in this paper was compared with the data measured in the field and simulation models. The mean absolute percent error (MAPE) was applied to compare the differences among the estimated, fieldmeasured, and simulated travel time at selected bus stops. The MAPE value can be obtained as
The MAPE values between the estimated and fieldmeasured or simulated average travel time of
The MAPE value between the estimated and fieldmeasured or simulated average travel time of
With the proposed model estimating passengers’ travel time at bus stops, sensitivity analyses were conducted to identify the effects of crucial parameters on the passengers’ travel time at bus stops.
The effect of arrival rate on the average waiting time is illustrated in Figure
Average waiting time with different arrival rate of buses.
In Figure
Additional walking time at different dwell times for various berths,
Additional walking time at different arrival rate of buses for various berths,
Passengers’ travel time at bus stops is modeled based on the theory of queueing and probability. The proposed model divides the travel time at bus stops into three parts: waiting time on the platform, additional walking time, and boarding time. Fieldmeasured and simulated data were obtained to demonstrate the effectiveness of the proposed model. Analysis was conducted to identify the effects of crucial parameters such as arrival rate of buses, dwell time, and number of berths at the selected bus stops on passengers’ travel time at bus stops. Based on the results of data analysis, the following conclusion can be conducted.
The waiting time on the platform results mainly from the unavailable service period during headways between two adjacent buses. The service period does not significantly affect the waiting time. Therefore, the emphasis of strategy to decrease the waiting time should be placed on the headways, not on the boarding time.
The additional walking activity occurs when no berth at the scheduled bus stop is available for the approaching bus. However, it does not mean that implementing more berths can eliminate the additional walking time. On the contrary, as analysis suggests, the average additional walking time increases in a large scale when the number of berths at a bus stop exceeds the threshold of three. Analysis also shows that the additional walking time increases with an increase in headways between two adjacent buses. Increase in dwell time can, however, lead to a decline in the additional walking time. The theoretical analysis explains the inefficiency of bus stops with more than three berths from the perspective of passengers’ additional walking time, which is significant in the design of the bus stop.
Bus stops are usually placed near signalized intersections. Further research is needed to identify the effect of signalized intersections on passengers’ travel time at the nearside bus stops. Furth and SanClemente [
General distribution of boarding time
Average additional walking distance
The joint probability that the embedded point is the
Space headway between stopped vehicles
Length of a bus
Number of buses dwelling at the stop at time
The probability generating function of additional queue length due to vacation effect
The maximum threshold number of passengers loaded during the dwell time
The joint probability that the embedded point is the starting of serving passengers and that
Average travel time consumed by passengers at a bus stop
Additional average walking time
Average boarding time
Average waiting time
The average service time per passenger
Walking speed of passengers
The headway duration
The Laplace transform of headway duration
Waiting time of a classical queueing model in equilibriumstate
Waiting time in equilibriumstate of the corresponding vacation queueing model
The derivative waiting time due to vacation policies
The derivative waiting time due to vacation policies
The Laplace transform of the corresponding variables in equilibriumstate
The rate of Poisson process of passengers of bus route
Arrival rate of bus.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was sponsored by the Major State Basic Research Development Program of China (no. 2012CB725400) and the Key Project of National Natural Science Foundation of China (no. 51338003). The authors would like to thank the survey team from the School of Transportation in Southeast University for their assistance in collecting and processing field data.