Travel Time Reliability Analysis Considering Bus Bunching: A Case Study in Xi’an, China

Abstract

Bus bunching occurring at stops has an unstable impact on bus travel time. In order to evaluate urban bus travel time effectively, the travel time reliability (TTR) addressing bus bunching is analyzed. This paper focuses on the delayed time caused by bus bunching in the dwelling process at bus stops and uses the coefficient of variation of time headway to evaluate the degree of bus bunching. Moreover, the travel time deviation (TTD) indicator and travel time on-time accuracy (OTA) model are proposed to evaluate the bus TTR. The proposed model is used to analyze 113 runs of a bus route in Xi’an city, China. Real-time GPS data are used to analyze the operation of each run from the origin to the destination stops. The results show that 74.34% of the runs are delayed. When the value of TTD is higher than |0.1|, 64.2% of runs are delayed with bus bunching. Based on the measuring of OTA in two situations, the value of TTR considering bus bunching is reduced by 20%. In addition, the number of stopping routes at peak periods has a significant impact on the occurrence of bus bunching. The research results would have practical implications for the operation and management of buses.

1. Introduction

Travel time has been generally recognized as an important consideration for travelers when making travel mode and route choice decisions [1]. Since urban roads are completely exposed to the environment, any kind of unstable factors will have an impact on the system. In addition, more and more people pay important attention to the reliability of travel time rather than just travel time. Therefore, the importance of travel time reliability (TTR) is higher than travel time savings in some cases [2]. In reliability assessment, travel time is used to reflect the service levels of transportation networks from the perspective of travelers [3]. Travel time reliability is defined as the probability of successful travel from the origin to the destination within a given service level [4]. It is a significant component of transportation reliability and is identified as a key roadway mobility measure [5]. Therefore, it is important to evaluate and guarantee the reliability of travel time [6].

There are many types of research analyzing the TTR from the modeling of and designing of methods and quantitative indices to estimate or improve TTR, such as bus passenger flow [7], road capacity [8], and transit signal [9]. The above studies on TTR are all explored from the perspective of optimization models or parameter modification. However, most of the above works do not consider the influence of bus stop designations and the microscopic phenomenon at stops. Actually, buses operating on urban roads are known to be an unstable system. Once disturbed, it will cause unstable travel time for passengers. Although there are fixed timetables for bus departures, the scheduling of buses may be disturbed in practice because of the influences of the actual travel time of each run, and the consequences of travel time instability are usually caused. Hence, it is necessary to consider the influence of bus stop operation for the reliability of travel time.

Bus bunching refers to adjacent buses on the same route stopping at the same stop simultaneously [10]. It frequently appears at bus stops and leads to unreliable transit service for passengers in densely populated cities with unstable consequences on travel time. Many studies focus mainly on the prediction and optimization of bunching. For example, some control strategies [11] and models [12] for adjusting bus running speeds are studied. The previous studies have reached valuable results in alleviating bus bunching. The time instability of bus bunching at stops has a significant influence on TTR. Due to the occurrence of the bus bunching, the dwell time at stops in the entire travel of the bus will be greatly deviated from the timetable, which will reduce the reliability of the bus travel time. However, it is found that few studies analyze the effect of bus bunching on TRR. As a kind of undesirable phenomenon that occurs frequently in the actual operation of urban buses, the effect of the bunching should be taken into account. Understanding how to accurately estimate the variability of travel time, especially for stops with bus bunching, would be important for both bus operators and passengers.

To sum up, the evaluation of the reliability of bus travel times is one of an important and effective way to measure the level of bus services. This work aims to develop an evaluation model to analyze the TTR of urban buses, considering bus bunching so as to improve the evaluation and estimation of bus travel time. First, to analyze the TTR, the actual travel time is divided into running time and dwell time depending on the operating status of the bus; this paper focuses on the delayed time of dwell time caused by bus bunching in the dwelling process of buses at stops. Then, the travel time deviation (TTD) is designed to describe the difference between actual travel time and timetable and the level of stable operation for one bus route is expressed. Furthermore, the degree of bunching is measured as the coefficients of variation of time headway and the on-time accuracy (OTA) model is proposed to measure the reliability of travel time considering bus bunching, which can demonstrate the deviation degree and time headway of buses. Finally, according to the results, the main factors significantly affecting bus bunching are analyzed.

The remainder of this paper is organized as follows. In Section 2, the travel time deviation and OTA model measuring the TTR considering bus bunching are proposed. The case study description and data process are detailed in Section 3. Then, the results and discussion of the case study are presented in Section 4. Finally, conclusions are reached in Section 5.

2. Literature Review

Research on the reliability of transportation systems began in the 1980s and, since then, much attention has been paid to the connectivity reliability, travel time reliability, and capacity reliability [13]. Among them, the reliability of travel time is used to reflect the service level of the transportation network from the perspective of travelers since the travel time is one of the most important travel characteristics considered as the main factor affecting personal activities and travel choice behavior [3].

In previous studies, some evaluating indexes and methods are proposed to estimate TTR. For instance, Long developed a reliability index to analyze how the index is included in the traffic signal design and fluctuations in travel time estimation and the results show that appropriate headway can improve the reliability of a public transport service based on the proposed index [14]. The TTR indicators were developed to optimize and assess the traffic management and operation under different traffic conditions [15]. Chepuri used generalized extreme value to estimate TTR and employed the reliability indexes to establish the LOS and the results show that the 95th percentile travel time and buffer time are the most effective performance indicators to measure the travel time variability [16]. Indices related to the standard deviations of travel time and the coefficients of variation are often used to measure travel time variability [17]. Moreover, in the study of the travel time reliability model, Wang proposed a new method to calculate the travel time reliability of road networks and found two coefficients can reduce the degree of reliability [18]. Ni et al. evaluated and improved the bus travel time using the mixed logit model and developed a reasonable allocation of the inter district bus departure plan to improve the reliability of public transport systems [19]. Jamous and Balijepalli adopted a probability measure to calculate the TTR for private cars and bus routes on traffic network and proposed a joint modelling method to evaluate the roadwork diversion schemes [20]. Kathuria et al. used the variability of travel time to measure travel time reliability and set a three-level travel time variability analysis model from multi-angles to improve the conventional level of service (LOS) criteria [21]. Ghader et al. used cumulative prospect theory (CPT) to analyze the influence of TTR [22]. Kidando et al. used a non-parametric random effect regression to analyze how serious crashes impact the TTR [23]. Tufuor et al. improved the TTR model to estimate the travel time distribution [24]. The above studies on TTR focus on optimizing models or modifying parameters. Few studies evaluate the TTR addressing the bus stop designation and/or the microscopic phenomena at stops such as bus bunching.

Current research on bus bunching focuses mainly on the alleviation of bunching and the optimization of bus scheduling methods considering the bunching problem. For example, to avoid the bus bunching, Daganzo and Pilachowski developed an adaptive control scheme to adjust the bus cruising speed in real-time such that the bus travels with a regular headway [25]. Deng et al. proposed a real-time speed control model considering signalized intersection delays and roadway conditions to optimize the bus headway deviations [12]. To reduce the effects of bunching, Delgado et al. proposed a mathematical programming model to control the operating vehicle and optimize the total delay of travel time [26]. Wang and Sun developed dynamic and flexible holding control strategies for a bus route and proposed a multi-agent deep reinforcement learning framework to avoid bus bunching [11]. A holding strategy based on cooperative control is proposed to analyze the effect of corridor service on bus bunching [27]. Phillips et al. designed a control strategy to prevent bus bunching and reduce travel time and its variability such that higher reliability was provided to the passengers [28]. The above research mainly focus on avoiding bus bunching. However, it is found that there are few examples in the research addressing the TTR influenced by bus bunching.

The analysis of bus travel time reliability is an obvious and effective method to measure the level of bus services. This paper aims to develop a TTR evaluation model for urban public transport considering bus bunching to improve the evaluation and estimation of bus travel time.

3. Methodology

Bus Travel Time Deviation and On-Time Accuracy Model

In this paper, the bus travel time is divided into two parts: running time and dwelling time, which is shown in Equation (1). A certain route that the bus operates between two adjacent stops and the operation states are shown in Figure 1.

The running time is the time spent between two stops while the bus travels at its cruising speed and the dwelling time refers to the time period from state A to state B. The bus travel time spent at the stop is described as Equation (2).

(1)$T_A=T_R+T_W$

(2)$T_W=t_c+t_{oc}+t_p+T_d$

where $T_A$ describes the actual travel time; $T_R$ means the bus running time that takes on the specified road of the route; $T_W$ is the time spent in the speed changing and dwell process at the stop; $t_c$ is the time spent during acceleration and deceleration when the bus is entering and leaving the stop; $t_{oc}$ is the delay time caused by the bus opening and closing the door at the stop and is estimated by $max\left\{t_{open},t_{close}\right\}$; $t_p$ is the time caused by the passengers getting on and off at stop and its estimated value is $max\left\{t_{on},t_{off}\right\}$; and $T_d$ is the delay time.

The bus bunching is judged according to the actual time headway of two adjacent runs. The time headway at a stop is described as the difference in arrival time between the adjacent former and later buses of the same route and stop, as formulated in Equation (3).

(3)$T{H}_{i,r}=t_{i,r+1}-t_{i,r}$

where $T{H}_{i,r}$ is the current time headway at stop $i$ of run $r$; $t_{i,r+1}$ denotes the arrival time of bus $r+1$ at stop $i$; the $t_{i,r}$ is the arrival time of bus $r$ at stop $i$. In addition, this model uses a coefficient of variation of time headway to evaluate the degree of bus bunching, which can measure the bus stop operation conditions, and its impact on the surrounding roads. The detail is shown as Equation (4):

(4)$CV\left(T{H}_i\right)=\frac{\sqrt{\frac{1}{f_m}{{\displaystyle \sum }}_{i=1}^{f_m}{(T{H}_{i,r}-\overline{T{H}_{i,r}})}^2}}{\overline{T{H}_{i,r}}}$

where $f_m$ denotes the total number of bus runs in one day. According to the above-mentioned coefficient of variation of the headway, the degree of the bus bunching can be described as the different levels and is shown in Table 1 [29].

For degree A, the route can provide very punctual services for passengers. For degree B, most buses deviate a little from the initial headway. For degree C, most buses deviate from the initial headway more seriously. For degree D, the headways have irregular bunching occur for some of the routes. For degree E, the time headway is very irregular and the total number of bunching buses roughly accounts for less than $\frac{1}{2}$ of the total number of buses. For degree F, the bus bunching happens frequently.

Thus, the travel time model is modified by considering bus bunching. Specifically, the dwell time is divided into two categories to describe the different situations of stop $i$. The situation is normal when the number of buses waiting at a stop is within the capacity of this stop. When the capacity of a stop is exceeded, the bus bunching is assumed to be occurred. The dwell processing time of normal and bus bunching situation are provided in Equation (5).

(5)$\{\begin{array}{l}{T}_{Wn}=t_c+t_{oc}+t_{pn}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ T_{Wnb}=t_c+t_{oc}+t_{pb}+T_d\end{array}$

where $T_{Wn}$ is the dwell processing time in normal situation; $T_{Wnb}$ is the dwell processing time in a bus bunching situation; $t_{pn}$ is the passengers boarding time when passengers get on and off the bus under normal conditions; and $t_{pb}$ is the passengers boarding time when this stop has bus bunching. This paper assumes that the passenger flow is evenly distributed among the same route runs arriving at the same stop at the same time. T_d is the delay time caused by the bus bunching at the stop.

Actually, the travel time for travelers is unstable whether higher or lower than the scheduled time. More and more people might pay more attention to time control rather than the travel time itself when arranging their trips. In this article, the degree of the variation of the actual arrival time is used to evaluate the reliability of bus travel time instead of the length of the total travel time.

In the previous study [13], TTR is described as a probability indicator that is constrained by the difference between actual time and expected time under the minimum acceptable service level $L_0$. In order to describe the TTR with bus bunching and evaluate the TTR at each stop accurately, this paper uses time headway to express the bunching factor. It can show the probability of deviation between the current time headway at each stop and the initial time headway. This definition can evaluate the reliability of the arrival time at any stop and is shown in Equation (6).

(6)$P_b=P\{H_i>\frac{1}{2}{H}_0|L\ge L_0\}$

where $P_b$ is the probability of a bunching situation that could measure the TTR, $H_i$ is the current time headway at stop $i$, and the $H_0$ is the initial time headway.

Comparing the actual travel time of each run with the timetable of the routes, the delay rate is used to measure the degree of a route arriving on time. The bus delay rate $P_d$ is described in Equation (7).

(7)$P_d=\frac{f_d}{f_m}$

where $f_d$ describes the number of the bus runs which arrived delayed.

Based on the calculation of the delay rate of the above bus travel time, the influence of bunching is considered. The $Q_b$ is defined to describe the proportion of delayed runs caused by bunching and illustrate the impact of bunching at stops on bus travel time, which is shown in Equation (8).

(8)$Q_b=\frac{f_b}{f_d},f_b\in f_d$

where $f_b$ is the number of late runs caused by bus bunching.

This paper uses the degree of travel time deviation to measure the difference between the actual travel time and timetable. The travel time deviation (TTD) is shown in Equation (9).

(9)$TTD=\frac{T_A-T_T}{T_T}$

where $T_T$ is the planned travel time of the run and $T_A$ is the observed travel time of the bus run.

Finally, setting the travel time On-Time Accuracy (OTA) model to measure the bus runs whether they arrive at each stop on time, the detail is shown in Equation (10).

(10)$\{\begin{array}{l}\mathit{OTA}=\frac{\sum _{r=1}^{f_m}\left\{\frac{T_{Ar}-T_T}{T_T}\right\}}{f_m}·\frac{\sum _{i=1}^n\sqrt{\frac{1}{f_m}\sum _{r=1}^{f_m}(T{H}_{i,r}-{\overline{TH}}_{i,r}^2)}}{n-\overline{T{H}_0}},r\in m\\ {\mathit{OTA}}_b=P_d\frac{\sum _{r_b}\frac{T_{A{r}_b}-T_T}{T_T}\}}{f_b}·\frac{\sum _{i=1}^n\sqrt{\frac{1}{f_b}\sum _{r_b}(T{H}_{i,r_b}-{\overline{TH}}_{i,r_b}^2)}}{n·\overline{T{H}_{0,r_b}}},r\in r_b\end{array}$

where $T{H}_0$ is the initial time headway (departure interval) of the route; $T_{Ar}$ is the actual travel time of all run $r$; $T_{A{r}_b}$ is the actual travel time of runs $r_b$ where bunching happened; $T{H}_{i,r_b}$ is the time headway of the stop corresponding to the bunched runs; and $T{H}_{0,r_b}$ is the initial time headway of the bunching runs.

In order to highlight the impact of bunching on the evaluation of TTR, this model takes into account the ratio of delay caused by bunching. The degree of bus bunching is evaluated through the standard deviation and mean value of the time headway. Moreover, the degree of dispersion of travel time is also considered in OTA to assess the TTR.

As a public transportation mode, buses are subject to many restrictions, but this study assumes that all bus drivers and buses have the same characteristics. In addition, the buses of each run are usually the same type and it is assumed that the performance of each bus does not change during the operation. The bus travel time model is established based on the above assumptions.

4. Case Study

4.1. Subsection

Xi’an is located in the central part of the Guanzhong Plain in China. As one of the world-famous ancient civilization capitals, it is the political, economic, and cultural center of the Shaanxi Province. The Xi’an city center is about 12 square kilometers, which is the core area of Xi’an culture, tourism, entertainment, and commerce. Almost every main road has bus routes and stops. Three layers of the center area, two urban road levels, and a bus route are established in the ArcMap and shown in Figure 2. Moreover, many routes share the same stops. Therefore, there are some problems at stops, for example: (1) the number of stopping spaces at stops are not enough for buses arriving at the same time; (2) more than one vehicle of the same route arrives at the same stop at same time; (3) due to the unstable bus arrival time, the passengers are gathering at the stops; and (4) bus bunching occurs frequently in this area. This paper aims to analyze the impact of bus bunching at stops on the bus travel time reliability.

Through data collection and analysis, 91 routes and 134 stops pass through the study area. The number of routes stopping at stop ranges from 1 to 21 and the distribution is shown in Figure 3. It can be seen that more than half of the stops have more than five stopping routes. This paper mainly analyzes the dwell time of these hot stops that serve more than 15 bus routes.

In order to choose a representative case route, the route 604 passes through not only the above-mentioned hot stops but also the center area and suburbs (outside the Second Ring Road, the red dotted line in Figure 4) of the city. This route can therefore represent the operation of buses in different areas of the city. The distribution of each stop (every stop is represented in numerical order) are shown in Figure 4.

4.2. Original Data Processing

The GPS data are extracted from the public transportation professional database and the original data are in dmp format. The data are imported into an Oracle11g database to export in csv format for analysis. The selected GPS data are shown in Table 2.

There are 113 runs in a day for route 604 and the operation of each vehicle from the original stop to the destination stop is analyzed based on real-time GPS data. Table 3 shows the arrival, departure, and total travel times of different runs of this route.

Many stops have a propensity for bunching to occur since they serve a large number of routes. The bunching is determined by comparing the GPS locations of buses to the coordinates of the stops, which are shown in Table 4.

4.3. Field Data Processing

The article divides the bus travel time into running time and dwelling time. When calculating the running time, the time included in the bus speed changing and dwell process should be removed. In addition, the dwelling time is composed of bus speed changing time when entering and leaving the bus stop, passenger boarding time, bus door opening and closing time, and delay time. Therefore, in addition to the real-time GPS data in the database, it is also necessary to go to the bus stops and dispatch to investigate and obtain the requirements for the running speed in the different urban areas.

In the process of acceleration and deceleration, this part of the time will be different for different types of buses. Therefore, in order to be realistic and reduce errors in the calculation, this paper uses a follow-up survey on the case route bus vehicle to calculate the acceleration and deceleration time. The average acceleration and deceleration at each stop and the time for getting on and off passengers are shown in Table 5.

Based on the field observations (of one complete run of bus route 604), the average deceleration time of the bus on this route is 10.45 s during the deceleration process and the mean value of acceleration time during the acceleration process is 11.69 s. Under the regulations, the bus running speed is related to the geographical location. Specifically, the mean value of the bus running speed outside the second ring road is 35 km/h and inside the second ring road is 25 km/h. Therefore, the acceleration and deceleration distance of all bus stops at the different urban areas in this case study could be calculated using Equation (11).

(11)$S_c=\frac{1}{2}\left(V_{in}+V_{out}\right)\left(t_a+t_d\right)$

where $S_c$ is the distance due to speed change; $V_{in}$ and $V_{out}$ are the speed of the bus running on the inbound and outbound Second Ring Road; $t_a$ is the mean value of the bus acceleration time; and $t_d$ is the mean value of the deceleration time.

Then, after removing the distance calculated in dwell time, the average running time of the bus can be calculated more accurately, substituting Equation (12) to obtain the running time.

(12)$T_R=\frac{\overline{L}-S_c}{\overline{V_R}}$

According to field observation and previous research [30], the time of opening and closing the door ($t_{oc}$) is 2 s and the time of passengers getting on and off at the stop is around 16.8 s. Under normal conditions, the dwell time should be 19 s.

The time headways at each stop are used to determine when bunching occurs, Table 6 shows an extract of the calculated time headways. Bunching is considered to have happened when the time headway difference is less than half of the initial time headway value.

5. Results and Discussion

As a result, a total of 732 effective headway values for each bus at all stops are obtained and 297 runs occurred with bus bunching. Then, Equation (4) is used to calculate the time headway coefficient of variation $CV\left(T{H}_i\right)$ for each stop. Then, after judging the degree of bus bunching for each stop, the results are shown in Table 7. According to the bunching degree results, the beginning of this route is located in the suburbs and between the Second and Third Ring Roads. There are few surrounding buildings and passenger vehicles on the roads. Therefore, the stops in this area have a higher level of bus bunching. After stop 5, the route passes through large residential communities. Thus, there are more vehicles and pedestrians on the road, and the number of passengers at the bus stops is also increased. So, the degree of bunching gets worse gradually. Starting from stop 10, the route will pass through many office workplaces, commercial buildings, shopping malls, and hotels in the city’s high-tech zone. This location is the transportation hub of the South Second Ring Road, where the traffic is more congested than the other parts of the route. After stop 15, the route starts to pass through the middle schools, universities, research institutions, design institutes, and the surrounding residential areas. Although the population is quite large, the number of trips is relatively small, the bus bunching at the stop is more relaxed than the former area. The end of the route enters the center area of the city and the road infrastructure is old, especially after stop 23. In addition, the route begins to pass the city’s famous tourist spots and the population and vehicles increase sharply. Therefore, it will cause more serious bunching. It can be seen from Table 7 that the degree of bunching is high.

On average, there are 113 bus runs in one day in the survey. According to the standard value in the timetable, there were 84 runs with delays in arrivals at stops among them, 66 runs with bus bunching, and 53 delayed runs also with bus bunching. The bus delay rate obtained by Equation (7) is 74.34%. Moreover, for the delayed runs caused by bunching compared with the total delayed runs the value is 63.1%. Finally, a new indicator to measure the reliability of travel time considering bus bunching is expressed by TTD and evaluation model OTA, which can be calculated by Equation (10). Therefore, through the above calculation and analysis, it can be concluded that the bus travel time is seriously affected by bus bunching at stops during the dwelling process.

The coefficient of the variation of the time headways at each stop can be seen in Figure 5, in it, the abscissa is all the stops of the whole route and the ordinate is the headway variation coefficient. The red straight line is the coefficient of variation of the initial time headways at the start of the runs and the blue broken line is the time headway variation coefficient of each stop. This figure shows the changes in bus bunching at different stops in the case route as measured by the coefficient of variation of time headways. By judging the bunching degree according to the $CV\left(T{H}_i\right)$ in the previous chapter, it can be concluded that the coefficient of variation in almost all stops is E, including the initial time headway. So, the timetable of this route is set unreasonably, which could cause undesirable phenomena.

It can be seen that the buses on this route are generally in a serious bunching degree and the coefficient of variation of time headway of most stops is different from the initial ones. Stop 4 can especially be seen to be far lower than the initial headway variation coefficients. It is located in the suburbs and the surrounding buildings are less than the center area. Few passengers choose this place as the destination and almost no one gets on and off the bus. Most runs leave from this stop without stopping. Stops 11 and 13 are located at the intersection of the main looping road in the city and there are a large number of surrounding buildings. Due to the strong attraction of this area, more passengers choose it as the destination, which aggravates the bunching. For route 604, from the suburbs to the city center, stops that are closer to the center serve more routes. Most stops in the center have more than 7 routes and the most served stops have 21 routes. The stops with a large number of stopping routes have a more serious degree of bunching, indicating that the excessive numbers of routes per stop have a negative impact on the operation of the stop.

It can be seen at the stops in suburban areas, that there are fewer cases of bus bunching and more relaxed traffic conditions. After gradually entering the center area, the phenomenon of bunching is increasing and the traffic situation is worsening, indicating that the different types of urban areas in the city have an impact on the bunching.

Figure 6 shows waiting times according to the time of day. The bus runs can be classified according to four conditions: (1) Runs with delays on arrival, (2) Runs with bus bunching, (3) Runs with both delays and bunching (both happened), and (4) Normal runs (both not happened). These are shown in Figure 6 with different markers. First, it should be pointed out that there are only 16 runs with no travel time delay and no bus bunching (the normal operation), which means that there are critical problems in bus operation. Second, 53 runs have both of arrival delay and bus bunching, accounting for almost half of the total bus runs. Among them, the $T_w$ of 42 runs is beyond a half of TTD; this result means that the bus bunching can greatly cause the travel time delay and reduce the service level of buses. This phenomenon means that the number of passengers has a significant influence on $T_w$, especially in the morning and evening peak periods. In addition, the abscissa also represents the departure time sequence of each run. The morning peak period is defined as 7:00–9:00 a.m. and the evening peak period is 17:00–19:00 p.m. Runs 10–27 are started at the morning peak and runs 88–102 start at the evening peak.

According to the trends of the runs, it can be found that stops between 8–29 and 80–104 are associated with travel time delay and bus bunching. Meanwhile, these runs are all started during the morning or evening peak hours. Thus, it is concluded that the occurrence of bunching and time delay runs is correlated with the peak periods.

Figure 7 shows the number of bunching in each run, it can be found that four runs have bunching at more than 15 stops, the highest up to 18 stops, during the four most bunched runs. According to the time sequence of the runs, the frequency of bus bunching in the morning and evening peak periods is significantly higher than that in the off-peak hours. These phenomena indicates that the increased traffic flow and travel demand have important impact on the occurrence of bus bunching.

The abscissa of Figure 8 is the run number in the time sequence and the ordinate is TTD measuring the difference between the actual travel time and timetable. The red line in the figure is the standard case. Through the analysis, the deviation degree of the actual travel time and the timetable for different runs at different times can be obtained. Due to the early morning hours of the first eight runs, there are few passenger vehicles on the road and the buses run smoothly. The dwell time is within a normal range, because there are almost no passengers boarding at these stops. As a result, the actual travel time will be lower than the standard time. This phenomenon also appeared at the later evening time. During the morning and evening peak periods, the actual travel time significantly deviates from the timetable. The evening peak may not be conducive to driving because of the darkness and the delay time range is larger than the morning peak. During the off-peak hours, the travel time is relatively close to the timetable and fluctuates in a small range. In addition, after analyzing the actual travel time and initial timetable of bunching runs, the time headway of 49 runs are more than 5 min and these runs account for 74.24% of the number of bunching runs. Finally, the TTD was calculated according to Equation (9). This simply measures the reliability of the travel time from the length of time and the result is 0.13. Moreover, this paper uses the 90 percentage to show that the results of travel time are unreliable. Because the travel time will arrive early and late, the value will appear negative. Hence, the value of TTD is higher than |0.1|, there are 64.2% runs delayed with bus bunching. The OTA is computed according to Equation (10) which is used to measure the travel time reliability considering bus bunching. Among them, the value of OTA for all the runs are 0.49 and the value of OTA for the runs with bunching is 0.41. Additionally, through measuring the OTA in two situations, the value of travel time reliability considering bus bunching is reduced by 20%. This result shows that the bus bunching has a significate influence on travel time reliability and it is necessary to include this factor in reliability measurement.

6. Conclusions

This paper aims to evaluate the reliability of urban bus travel time by addressing the delay time caused by bus bunching in the dwell process at stops. The coefficients of variation of time headway among all runs and the degree of bunching at stops are measured. By analyzing the differences between initial and actual time headway of each run, it is observed that only several runs keep the same time headway as those of the initial. An evaluation model OTA that considers bus bunching for TTR analysis is proposed. The results show that the influencing factors of roads and stops affect the arrival time significantly. The conclusions were drawn as follows.

The bus bunching has serious effects on the buses running in densely populated cities and reduces the level of service of buses. Different from the establishment of the TTR evaluation model in previous studies, this paper takes the bus bunching into account in the analysis of the TTR. As a frequently occurring phenomenon at bus stops, bus bunching has a serious negative impact on bus operation, increasing the deviation of travel time and reducing the reliability of travel time. In the case study, once the deviation of standard time is higher than 10%, the travel time is considered unreliable, and the results show that more than half of the delayed runs with bus bunching. After measuring by the OTA model, it is found that the impact of bus bunching considered will have almost 20% deviation to the results. This shows that the bus bunching has a serious negative impact on bus operation and it increases the deviation of the travel time and reduces the travel time reliability.

There is a relationship between time periods and bus bunching, the number of bus bunching occurrences at peak hours is more than at off-peak hours. According to the GPS data, the actual travel time deviates from the standard time with a small fluctuation during off-peak hours and the buses are operated with little bunching. During the morning and evening peaks, the buses are operated with serious bus bunching and travel time delays. Additionally, the actual travel time is obviously deviated from the timetable. From the above results, it is concluded that the bus travel time greatly deviated from the standard during the peak periods that carry the most passengers of a day. This will result in low passenger satisfaction, high travel time delays, and poor bus transit operation.

The different geographical locations of the stops will also have an impact on bus bunching. The stops in different geographical locations will have a different number of stopping routes. The number of stopping routes is positively correlated with bus bunching occurrences. When buses pass through suburban areas or the underdeveloped places with fewer passengers, the bunching degree and travel time delay of each run is alleviated. However, when the bus enters the urban center area, many routes need to stop at the same stop, and the bus bunching degree becomes serious. Therefore, the location and number of stopping routes of stops have negative impacts on the dwell process of buses.

This article considers microscopic factors for bus travel time reliability evaluation; it can provide guidance for bus operating companies and city government departments to improve public transport service levels from two aspects: the suitable timetables and the reasonable number of stopping routes in urban center area. However, in this study, the time headway is used as a measure of whether the vehicle is in tandem or not and the travel time of buses in a running state is assumed to be fixed. These assumptions guarantee the desired results, but there is room for improvement in accuracy. To improve the limitation in this paper, the future works could focus on: (1) it is necessary to consider other factors correlated with bus bunching at stops to further investigate the unreliability of bus travel time; and (2) find and analyze the factors affecting the reliability of travel time in bus operation during the bus running process part.

Author Contributions

The authors confirm contribution to the paper as follows: Conceptualization: Y.Z. and H.X.; Methodology: Y.Z.; Data curation: H.X. and X.F.; Writing—original draft: Y.Z.; Validation: Q.-C.L.; Writing—review and editing: Q.-C.L.; Formal analysis: X.F. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

3rd Party Data. Restrictions apply to the availability of these data. Data was obtained from Xi’an Transportation Bureau and are available from the authors with the permission of Xi’an Transportation Bureau.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. The constitution of dwelling time.

Figure 2. Case study area. (a) Xi’an City; (b) Center area; (c) Bus routes distribution.

Figure 3. The distribution of stopping route at stops in the center area of Xi’an.

Figure 4. Stops distribution of route 604.

Figure 5. The CV(THi) distribution for stops.

Figure 6. The TW distribution of runs with bus bunching and delays.

Figure 7. The number of bunching stops in each run.

Figure 8. The distribution of travel time deviation.

Table 1

Degree of bus bunching.

Degree	CV(THi)	Statement
A	0.00–0.21	Very consistent with the initial headway
B	0.21–0.30	Slight deviation from the initial headway
C	0.30–0.39	Severe deviation from the initial headway
D	0.39–0.52	Irregular headway, some bus bunching
E	0.52–0.74	Bus bunching frequent
F	0.74	Most of bus bunching

Table 2

The GPS data.

Bus ID	Real-Time	Longitude	Latitude	Speed (km/h)
116155	6:01:04	108.9643	34.26102	30.48
116153	6:01:05	108.9387	34.26078	24.45
116160	6:01:05	108.9189	34.24378	30.65
116160	6:01:05	108.9189	34.24378	30.65
116152	6:01:06	108.8404	34.23993	0
116155	6:01:07	108.9641	34.26103	32.67
116155	6:01:08	108.964	34.26103	32.72
116357	6:01:08	108.8402	34.24301	0
116155	6:01:09	108.9639	34.26104	29.41
……	……	……	……	……

Table 3

Each run data.

Bus ID	Run	Departure Time	Arrival Time	Travel Time
116152	1	6:00:05	6:51:11	0:51:06
116354	2	6:04:43	6:57:35	0:52:52
116362	3	6:11:40	7:09:06	0:57:26
116357	4	6:15:51	7:13:10	0:57:19
116363	5	6:24:09	7:15:38	0:51:29
116356	6	6:33:10	7:37:46	1:04:36
116361	7	6:39:59	7:43:26	1:03:27
116157	8	6:53:22	8:03:44	1:10:22
116355	9	6:55:56	8:04:39	1:08:43
……	……	……	……	……
116364	113	20:57:58	22:11:15	1:13:17

Table 4

Location of each stop.

Stop ID	Stop Longitude	Stop Latitude	Number of Routes at Stop
1	108.8400463	108.8404	4
2	108.8420668	108.8432	9
3	108.8457358	108.847	10
4	108.8570179	108.858	11
5	108.8660371	108.868	8
6	108.8720935	108.8741	8
7	108.8764403	108.8784	9
8	108.8848875	108.8869	10
9	108.8903408	108.8923	17
……	……	……	……
27	108.9616233	108.9636	17

Table 5

The situation of speed change and boarding at stops.

Stop ID	Time of Deceleration (td)	Boarding Time	Number of Boarding Passengers	Time of Acceleration (ta)
1	9.25	24.16	3	9.46
2	8.89	5.4	0	10.35
3	8.36	16.18	3	12.31
4	6.98	25.18	5	13.2
5	6.01	29.84	11	13.78
6	8.53	8.48	2	10.23
7	13.16	36.28	1	10.91
8	6.81	15.95	3	10.7
9	7.93	12.9	2	15.26
……	……	……	……	……
27	15.91	31.15	9	8.04

Table 6

Each time headway in the case study.

Run (r)	TH 1,r	TH 2,r	TH 3,r	TH 4,r	TH 5,r	TH 6,r	TH 7,r	TH …,i
1	0	0	0	0	0	0	0	0
2	0:08:32	0:08:32	0:09:32	0:10:12	0:09:40	0:08:31	0:07:41	……
3	0:02:15	0:04:07	0:02:51	0:03:03	0:01:42	0:05:20	0:03:09	……
4	0:04:52	0:04:22	0:04:38	0:04:30	0:05:31	0:08:55	0:09:17	……
5	0:06:07	0:04:45	0:06:17	0:04:47	0:06:18	0:07:09	0:07:57	……
6	0:10:44	0:10:44	0:09:42	0:10:11	0:12:29	0:06:03	0:06:41	……
7	0:01:33	0:02:44	0:04:44	0:04:38	0:04:38	0:04:25	0:04:48	……
8	0:05:56	0:06:36	0:07:40	0:07:59	0:05:10	0:05:33	0:03:12	……
9	0:05:06	0:09:11	0:06:08	0:07:19	0:07:00	0:06:49	0:07:29	……
……	……	……	……	……	……	……	……	……
113	0:09:13	0:08:42	0:11:33	0:08:31	0:09:11	0:12:25	0:12:46	……

Table 7

The number of bunching runs and the degree of bus bunching at each stop.

Stop ID	The Number of Bunching Runs	The Degree of Bus Bunching	Stop ID	The Number of Bunching Runs	The Degree of Bus Bunching
1	2	D	15	3	C
2	3	C	16	14	D
3	3	B	17	13	D
4	2	B	18	19	E
5	4	D	19	12	D
6	5	D	20	15	D
7	7	D	21	15	D
8	9	D	22	19	D
9	6	C	23	18	D
10	9	E	24	23	D
11	30	F	25	14	E
12	12	D	26	13	D
13	7	E	27	16	E
14	4	C	-	-	-

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