# Characteristics Analysis of Bus Stop Failure Using Automatic Vehicle Location Data

Authors: Rui Li; Xin Xue; Hua Wang
Journal: Journal of Advanced Transportation
 (2020)

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## Abstract

Queue forming behind a bus stop on an urban street is common and a traffic bottleneck usually occurs around the bus stop area. The bus stop failure means arriving buses cannot move into the bus stop due to limited capacity but have to wait for available loading areas. It is related with the transit operation level. Traditionally, the failure rate (FR), defined as the percentage of buses that arrives at the bus stop to find all loading areas occupied, is adopted in bus capacity analysis. However, the concept of FR is unable to quantitatively analyze failure characteristics in terms of its dispersion and uncertainty over time. Therefore, in this paper, we propose a new index called failure duration rate (FDR) to evaluate the bus stop failure, which can characterize waiting time for traffic delay calculation and capacity drop estimation. The automatic vehicle location data at eight bus stops in Wujiang District Suzhou, China, over 56 working days, are used to analyze the temporal characteristics of FR and FDR. We next examined the failed service duration characteristics during peak hours at the eight bus stops. Based on these characteristics analyses, we then proposed a Distribution Fitting and Cumulative Distribution Correlation (DF-CDC) approach to explore the correlation between FDR and FR at the same cumulative distribution function levels and validated the bus stop failure performance using the cross-validation method. The analysis results revealed that (i) FR fluctuates more significant than FDR, (ii) FDR is a more robust index than FR in describing the traffic characteristics incurred by bus stop failures, and (iii) FDR performs better in failure characteristics analysis than FR.


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## Body

## 1. Introduction

A bus stop serving a large number of bus lines can experience a condition known as bus stop failure due to limited capacity and high passenger demand, which will negatively affect the punctuality and reliability of bus service and also bring about delays to other traffic. The more frequently the bus stop failure takes place, the lower the transit system level of service (LOS) is [1]. However, irregular traffic flow characteristics concerned with bus stop failure are difficult to be captured and quantified because of its dispersion and uncertainty over time [2]. Without a doubt, bus stop failure will significantly affect traffic characteristics at bus berths and adjacent lanes. Typically, the failure rate is proposed for analyzing and evaluating the influence of bus stop failure at bus berths.Bus stop constitutes one potential bottleneck to interrupt traffic flow, which will deteriorate the level of transit operational service [3]. Transit operation parameters of buses served at loading areas, including dwell time [4, 5], headway [6, 7], capacity [8, 9], queue length [10, 11], and bunching characteristics [12, 13], are analyzed for evaluating the impact of bus stop failures. The failed service will increase bus waiting time for passenger boarding and alighting, and its impact can be measured by an index called failure rate FR, which is defined as the percentage of buses that arrive at stops to find all berths are full [1]. Wang et al. analyzed the correlation between failure rate and four kinds of transit dwell and arrival characteristics and proposed a diffusion approximation method [14]. As one of the desired level indexes of transit operation, the FR could assess the change of capacity and LOS of bus berths. Failure probability and dwell time variability can develop the function of bus queue length, which can better reflect the effects on bus stop capacity [15]. A parameter “Z” associating with the desired failure rate (under the assumption of standard normal distribution fitting) is developed to account for the fluctuation in bus dwell time in bus loading areas, and the design failure rates for urban and rural areas are recommended for estimating bus berth capacity [1]. The failure rate of curbside bus stops can be influenced by a serial of factors (such as bus arrival distribution type, bus arrival rate, bus berth maximal service rate, and bus service time variation), and the normalized capacity and incremental change (for multiberth stops) in capacity at different failure rate levels are proposed [8]. Moreover, the analysis process of failure rate is a poor proxy and suggests choosing the average waiting delay for evaluating the bus berth LOS is expressed comprehensively [16].The data associated with vehicle location are utilized for analyzing and predicting traffic flow characteristics, which can be effectively applied to transit operational characteristics analysis [17–20], bus schedule optimization design [21, 22], bus lane planning and control strategy [23–25], and transport network flow estimation [26–28]. Several findings of characteristics analysis for public transit LOS at bus stop appear in the relevant literature. A data platform for monitoring patterns of bus operation is developed, which is primarily composed of data acquired from the ITS system in Beijing, China. A multilevel framework for transit performance analysis is proposed considering several transit operational factors [29]. Based on automated vehicle location (AVL) data, lots of statistical parameters about travel times are analyzed for evaluating the performance of bus routes with transit priority facilities, and these tests indicate spatial and temporal characteristics are the most potent feature [30]. A regression method (using LS-SVM) is developed for exploring bunching patterns of buses halting at the stop area, and the headway irregularity pattern is analyzed using transit smart card data [31]. A probabilistic method considering the interference between buses, using the loading areas, is established for predicting travel time of buses using trajectory and ID card data, which can reflect buses’ dwell time distribution pattern well [32].Although several findings of bus stop failure analysis using the failure rate appear in the relevant literature, the duration time of bus stop failure is rarely mentioned. Additionally, little research has been observed using AVL data to analyze the characteristics of bus stop failure. In this paper, failure duration time is utilized for evaluating bus stop failure, and a measure called failure duration rate is proposed for failure analysis utilizing collected transit automated vehicle location (AVL) data. The characteristics analysis of bus stop failure using AVL data can provide valuable information for transit operation optimization to the public transit authority.The remainder of this research is organized as follows. InSection 2, a characteristic index called Failure duration rate FDR is developed for bus stop failure characteristic evaluation. Section 3 explores the failed duration characteristics at different failure levels based on the AVL data collected from eight bus stops in Wujiang District of Suzhou in China. In Section 4, a correlation analysis between FDR and FR is carried out by using a “Distribution Fitting and Cumulative Distribution Correlation (DF-CDC)” analysis approach. Section 5 concludes the paper.


## 2. Characteristic Analysis Indexes

### 2.1. Failure Rate

A bus stop failure occurs when a bus arrives at the loading areas but with no available berth to use. The failure rateFR, defined as the percentage of bus queuing for moving into bus berths occupied by other dwelling buses [14], could be formulated by(1)FR=Prn>b,where n is the number of buses halted at a bus stop and b is the number of bus berths.In general, transit vehicles’ dock at a bus stop (including the served and waiting buses) will obey the first-in-first-out rule and usually disperse in an independent manner to each other. The probability for the case without adequate berths at a bus stop could be calculated by(2)Prn>b=1−Prn=1−Prn=2−⋯−Prn=b.When the berths at a bus stop are all occupied by buses for passenger boarding and alighting, the number of active buses served at the bus stop is equal to the number of berthsn=b. And the probability of this kind of bus arrival can be approximated by the value of Qsn=b divided by Qs:(3)Prn=b≈Qsn=bQs,where Qsn=b is the occurrence times for n=b during a given observation period and Qs is the total number of arriving buses during the same time duration.Then, we have(4)Prn>b≈1−Qsn=1Qs−Qsn=2Qs−⋯−Qsn=bQs=Qs−Qsn=1+Qsn=2+⋯+Qsn=bQs.To facilitate our presentation, we denote the right hand of equation (4) by(5)Rf=Qs−Qsn=1+Qsn=2+⋯+Qsn=bQs.In other words,Rf stands for the FR of a bus stop, which reflects the level of failure (LOF) for bus loading areas.


### 2.2. Failure Duration Rate

As the berths of a bus stop are occupied, the next arriving bus needs to queue at street lanes and exerts negative impacts on blocking movements of other vehicles along the same street lane. As a result, traffic delay goes up, and travel time reliability would be reduced [1]. It is an important and challenging task to analyze these adverse effects quantitatively. In general, the longer the bus stop failure service lasts, the worse the traffic efficiency evolves and deteriorates. It is worth pointing out that the severity of traffic deterioration in terms of traffic delay and road capacity reduction dramatically depends on the traffic blocking duration time.Thus, the failure duration rate that incorporates failure duration time is proposed for analyzing the LOF of bus stops. The failure duration time can be measured by the timespan (the waiting time of buses outside the stop) for all failed bus stopping services during a given time period. Specifically, it can be calculated by examining the arrival and departure time of buses using the loading areas of a bus stop. Then, the failure duration rateFDR can be formulated by(6)Rfd=tfdts=tfdt−tv,where tfd is the failure duration time (sec), ts represents the total occupancy time of using the bus stop (sec), and tv is the duration time for a vacant bus stop (sec).TheFDR can be interpreted as the ratio of waiting and blocking for arriving buses during a given time period. For a specific bus stop failure, the average duration time per failure tfdr can be formulated as(7)tfdr=tfdQs−Qsn=1+Qsn=2+⋯+Qsn=b.Then, the average duration rate per failureRfdr is obtained as(8)Rfdr=tfdrt−tv.


## 2.1. Failure Rate

A bus stop failure occurs when a bus arrives at the loading areas but with no available berth to use. The failure rateFR, defined as the percentage of bus queuing for moving into bus berths occupied by other dwelling buses [14], could be formulated by(1)FR=Prn>b,where n is the number of buses halted at a bus stop and b is the number of bus berths.In general, transit vehicles’ dock at a bus stop (including the served and waiting buses) will obey the first-in-first-out rule and usually disperse in an independent manner to each other. The probability for the case without adequate berths at a bus stop could be calculated by(2)Prn>b=1−Prn=1−Prn=2−⋯−Prn=b.When the berths at a bus stop are all occupied by buses for passenger boarding and alighting, the number of active buses served at the bus stop is equal to the number of berthsn=b. And the probability of this kind of bus arrival can be approximated by the value of Qsn=b divided by Qs:(3)Prn=b≈Qsn=bQs,where Qsn=b is the occurrence times for n=b during a given observation period and Qs is the total number of arriving buses during the same time duration.Then, we have(4)Prn>b≈1−Qsn=1Qs−Qsn=2Qs−⋯−Qsn=bQs=Qs−Qsn=1+Qsn=2+⋯+Qsn=bQs.To facilitate our presentation, we denote the right hand of equation (4) by(5)Rf=Qs−Qsn=1+Qsn=2+⋯+Qsn=bQs.In other words,Rf stands for the FR of a bus stop, which reflects the level of failure (LOF) for bus loading areas.


## 2.2. Failure Duration Rate

As the berths of a bus stop are occupied, the next arriving bus needs to queue at street lanes and exerts negative impacts on blocking movements of other vehicles along the same street lane. As a result, traffic delay goes up, and travel time reliability would be reduced [1]. It is an important and challenging task to analyze these adverse effects quantitatively. In general, the longer the bus stop failure service lasts, the worse the traffic efficiency evolves and deteriorates. It is worth pointing out that the severity of traffic deterioration in terms of traffic delay and road capacity reduction dramatically depends on the traffic blocking duration time.Thus, the failure duration rate that incorporates failure duration time is proposed for analyzing the LOF of bus stops. The failure duration time can be measured by the timespan (the waiting time of buses outside the stop) for all failed bus stopping services during a given time period. Specifically, it can be calculated by examining the arrival and departure time of buses using the loading areas of a bus stop. Then, the failure duration rateFDR can be formulated by(6)Rfd=tfdts=tfdt−tv,where tfd is the failure duration time (sec), ts represents the total occupancy time of using the bus stop (sec), and tv is the duration time for a vacant bus stop (sec).TheFDR can be interpreted as the ratio of waiting and blocking for arriving buses during a given time period. For a specific bus stop failure, the average duration time per failure tfdr can be formulated as(7)tfdr=tfdQs−Qsn=1+Qsn=2+⋯+Qsn=b.Then, the average duration rate per failureRfdr is obtained as(8)Rfdr=tfdrt−tv.


## 3. Characteristic Analysis

### 3.1. Data Collection

In this study, bus dwelling data are based on the AVL data provided by Wujiang Transit Agency in Suzhou, China. The AVL data span 56 consecutive working days from October 22, 2018, to January 9, 2019. The dataset of each day has a half-day bus dispatching time window, from 7:00 to 19:00, and there are nine bus routes and eight bay-type bus stops (see their geographic locations in Figure1). These eight test bus stops keep considerable distances to intersections (the average distance of 200 m), and thus the interaction between the bus bay and nearby intersections would be negligible.Figure 1
Locations of eight bus stops for AVL data collection.The details of the nine bus routes associated with each bus stop are given in Table1, and the headways of these transit routes range from 8 to 15 minutes.Table 1
The details of test bus stops.
No.Bus stop (number of bus berths)DirectionBus routes1Hengda Market West (1)S ⟶ NRoutes 708, 719, 732, 7332Hengda Market West (1)N ⟶ SRoutes 708, 719, 732, 7333Changan Rd. and Lianyang Rd. North (1)S ⟶ NRoutes 701, 710, 711, 720, 7414Changan Rd. and Lianyang Rd. North (1)N ⟶ SRoutes 701, 710, 711, 720, 7415Changban Rd. and Changan Rd. East (1)E ⟶ WRoutes 701, 710, 7116Changban Rd. and Changan Rd. East (1)W ⟶ ERoutes 701, 710, 7117Changan Rd. and Changban Rd. North (2)S ⟶ NRoutes 708, 719, 720, 732, 733, 7418Changan Rd. and Changban Rd. North (2)N ⟶ SRoutes 708, 719, 720, 732, 733, 741The dwelling time of buses serving these 9 routes at the 8 bus stops are extracted from the collected AVL data. Some records are provided in Table2. Based on the arrival and departure time of buses boarding and alighting passengers at the third bus stop, we can determine the bus failure characteristics. For example, the bus with the ID of SU-EU9353 serving Route 710 departed from the bus stop at 16:14:28, while the bus with the ID of SU-EU6029 serving Route 741 arrived at the same stop at 16:15:07, and the bus arriving later needed to wait outside the stop for 13 seconds (failure duration time) until bus no.710 left the stop.Table 2
Some buses arrival and departure time records extracted from AVL data.
Route no.Bus IDBus stopDate/timeArrival timeDeparture time701SU-EU9526No.3 bus stop2018-12-18 16:10:312018-12-18 16:11:11710SU-EU9353No.3 bus stop2018-12-18 16:13:342018-12-18 16:14:28741SU-EU6029No.3 bus stop2018-12-18 16:14:152018-12-18 16:15:07720SU-EU9582No.3 bus stop2018-12-18 16:19:152018-12-18 16:20:08701SU-EU9379No.3 bus stop2018-12-18 16:20:302018-12-18 16:21:02711SU-EU9359No.3 bus stop2018-12-18 16:22:162018-12-18 16:23:19


### 3.2. Temporal Characteristics Analysis

The number and duration of buses stop failure (per hour) at bus stops are determined based on AVL data. We here use equations (5) and (6) to calculate the hourly FR and FDR of the eight bus stops. Figure 2 plots the hourly time-varying characteristics of Rf and Rfd over 56 working days of the No. 1 bus stop (672 hourly Rf data and Rfd data in total). It can be observed that Rf and Rfd in the morning (7:00–9:00) and evening peak hours (17:00–19:00) are higher than those in nonpeak hours. Overall, the mean value of Rf is more significant than that of Rfd in most times.Figure 2
The time-varying characteristics of no.1 bus stop failure.
(a)(b)We then look at the median value of the hourlyRf and Rfd. As we can see in Figure 3, there are highly similar patterns of Rf and Rfd during peak hours for all the eight bus stops. The reason might be that there is high travel demand in both passengers and vehicles are in at peak hours, and thus bus stop failures occur more frequently, especially because of a long time for passengers to board and alight.Figure 3
Bus stop failure time-varying characteristics of test bus stops. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No.8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)


### 3.3. Failure Duration Analysis

Figure4 shows the relationship between Rf and Rfd. For each bus stop, 224 data during the morning and evening peak hours (7:00 to 9:00 and 17:00 to 19:00) are considered.Figure 4
The relationship analysis betweenFDR vs. FR. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No.8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)From Figure4, the hourly FDR shows a weak positive correlation to the FR for all the eight bus stops. Overall, the fitting parameters between Rf and Rfd vary remarkably among bus stops sites. For example, there are the lowest R-square (0.31) for no.1 bus Stop and the highest R-square (0.661) for no. 4 bus stop. In addition, the statistical relationship between Rf and Rfd of 1-berth stops (no.1 through no.6 bus stops) is weaker than that of the two 2-berth stops (no.7 and no.8 bus stops).As mentioned above, it is difficult to determine a well-fitted failure duration rate function (for the 8 bus stops) using failure rate directly due to the dispersion. Taking no.1 bus stop as an example,Rfd is increased from 13.3% to 33.0% as Rf changes from 24% to 25%. But Rf increases from 16.7% to 43.3% as Rfd increases from 24% to 25%. It indicates the significant dispersion. To decrease the dispersion of failure duration rate in analysis, the failure rate of bus stops is divided into sections with an interval length of 5%. Rf5% is defined as the 5% section level of failure rate at a bus stop, which range from 5% to 10% Rf5%∈5%,10%. Then, the mean, standard deviation (S.D.), and coefficient of variation (C.V.) of failure duration rate for these 8 bus stops are clustered and calculated at a different level of failure rate. The mean, S.D., and C.V. of failure duration rates at different failure rate level of these eight test bus stops are compared in Table 3. When the frequency of bus stop failure is less than 5 at a level, the failure duration rate is not calculated and identified as “not available (N/A)” in our analysis. Because the span of these 8 bus stops is different, each bus stop has some “not available (N/A)” at a corresponding failure rate level.Table 3
Failure duration rate of test bus stops at different failure rate levels.
Failure rate levelNo.1 bus stopNo.2 bus stopNo.3 bus stopNo.4 bus stopNo.5 bus stopNo.6 bus stopNo.7 bus stopNo.8 bus topRf5%MeanN/AN/AN/AN/A0.07100.05040.03820.0464S.D.N/AN/AN/AN/A0.05080.02890.02250.0236C.V.N/AN/AN/AN/A0.71550.57320.58780.5081Rf10%MeanN/AN/A0.08960.08850.09600.08070.08350.0797S.D.N/AN/A0.03750.03600.04310.03830.03970.0353C.V.N/AN/A0.41880.40700.44880.47500.47500.4430Rf15%Mean0.1443N/A0.15410.11040.11550.11330.11180.1070S.D.0.0502N/A0.05440.03550.04800.05200.03100.0461C.V.0.3477N/A0.35270.32180.41590.45870.27690.4306Rf20%Mean0.17350.13800.16300.14050.13380.13040.14220.1450S.D.0.04980.05540.05100.03780.04940.04990.04290.0510C.V.0.28710.40150.31290.26880.36880.38280.30150.3516Rf25%Mean0.21040.20750.19040.18600.17760.16640.17180.1754S.D.0.05400.06610.05320.04500.05330.05760.04300.0497C.V.0.25660.31860.27940.24170.30020.34590.25050.2833Rf30%Mean0.22310.21590.22360.19750.16980.18320.19390.1780S.D.0.04950.05580.05540.03700.04020.06100.04600.0470C.V.0.22170.25830.24790.18720.23680.33280.23730.2640Rf35%Mean0.22150.25060.25690.24210.2641N/A0.22820.2328S.D.0.04780.05390.05460.04580.0608N/A0.04050.0470C.V.0.21600.21510.21250.18900.2304N/A0.17730.2019Rf40%Mean0.27190.28280.28850.2587N/AN/AN/AN/AS.D.0.05320.06090.05120.0543N/AN/AN/AN/AC.V.0.19570.21540.17740.2098N/AN/AN/AN/ABased on the mean, S.D., and C.V. of failure duration rates at different failure rate level, we then analyze the failure duration characteristics. Figure5 presents the average value (for all these eight bus stops) of mean, S.D., and C.V. for failure duration rate at different failure rate levels. The mean and S.D. of Rfd increase with the increasing failure rate level, and the mean has a higher and faster growth rate than that of S.D. The mean of Rfd is increased from 5.2% to 27.5% as failure rate level increases from 5% to 40%. And the S.D. of Rfd is increased from 3.1% to 5.5% as the same span of failure rate level. However, the favorable trend does not hold for the C.V. of Rfd, as illustrated in Figure 5. The C.V. of Rfd is decreased from 59.6% to 19.9% as failure rate level changes from 5% to 40%.Figure 5
Average mean, S.D., and C.V. ofFDR for eight bus stops.Figure6 displays the average value of the mean, S.D., and C.V. for failure duration rate of six 1-berth stops and two 2-berth stops. The average value of the mean, S.D., and C.V. for failure duration rate of the single-berth stops is more significant than that of the 2-berth stops.Figure 6
Average mean, S.D., and C.V. ofFDR for two types of bus stops.Figure7 displays the correlation relationships between FDR per failure Rfdr and the FR for the eight bus stops. It is shown that the FR is insensitive to Rfdr. For the no.1 bus stop, Figure 7(a) reveals that Rfdr is increased from 1.5% to 5.4% and from 1.9% to 3.4% when the FR climbs up from 21% to 22% and from 34% to 35%, respectively. Thus, the span of Rfdr is significant at the low level of failure rate, and the volatility of Rfdr becomes more and more slight with the increasing failure rate. Besides, the diversity correlation relationship between Rfdr and Rf is influenced by the number of berths of loading areas. Figure 7 shows that Rfdr is increased from 1.2% to 3.8% and from 1.5% to 5.4%, respectively, as the failure rate increase from 21% to 22% for no.7 bus stop (2-berth type) and no.1 bus stop (1-berth type). Therefore, compared with single-berth bus stops, the two 2-berth bus stops have less Rfdr at the same level of failure rate.Figure 7
FDR per failure vs. FR for test bus stops. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No.8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)For the different levels of failure rate (range from 5% to 40%), the value of the mean, S.D., and C.V. forRfdr at the bus stops are calculated and presented in Figure 8. The average values of the mean, S.D., and C.V. for Rfdr reveal a significant negative correlation with the FR level. When the FR ranges from 30% to 40%, the three statistics for Rfdr are not very sensitive to the FR level, and there are no obvious fluctuations. For the mean value of Rfdr, the maximum variation is merely 0.1% when the FR level falls into the range of 30% to 40%. Therefore, the dispersion of Rfdr decreases sharply with the FR level (especially when the FR is greater than 30%), which implies that the average FDR per failed is comparatively stable. In Figure 9, there is a similar trend of C.V.Figure 8
Average mean, S.D., and C.V. ofFDR per failure for eight bus stops.Figure 9
Average mean, S.D., and C.V. ofFDR per failure for two types of bus stops.For the different levels of failure rate (range from 5% to 40%), the value of the mean, S.D., and C.V. forRfdr for six 1-berth stops and two 2-berth stops are calculated and depicted in Figure 9. The three statistics for Rfdr at different failure rate levels for the single-berth stops is larger than that of the 2-berth stops, which shows higher stability at the 2-berth bus stops.


## 3.1. Data Collection

In this study, bus dwelling data are based on the AVL data provided by Wujiang Transit Agency in Suzhou, China. The AVL data span 56 consecutive working days from October 22, 2018, to January 9, 2019. The dataset of each day has a half-day bus dispatching time window, from 7:00 to 19:00, and there are nine bus routes and eight bay-type bus stops (see their geographic locations in Figure1). These eight test bus stops keep considerable distances to intersections (the average distance of 200 m), and thus the interaction between the bus bay and nearby intersections would be negligible.Figure 1
Locations of eight bus stops for AVL data collection.The details of the nine bus routes associated with each bus stop are given in Table1, and the headways of these transit routes range from 8 to 15 minutes.Table 1
The details of test bus stops.
No.Bus stop (number of bus berths)DirectionBus routes1Hengda Market West (1)S ⟶ NRoutes 708, 719, 732, 7332Hengda Market West (1)N ⟶ SRoutes 708, 719, 732, 7333Changan Rd. and Lianyang Rd. North (1)S ⟶ NRoutes 701, 710, 711, 720, 7414Changan Rd. and Lianyang Rd. North (1)N ⟶ SRoutes 701, 710, 711, 720, 7415Changban Rd. and Changan Rd. East (1)E ⟶ WRoutes 701, 710, 7116Changban Rd. and Changan Rd. East (1)W ⟶ ERoutes 701, 710, 7117Changan Rd. and Changban Rd. North (2)S ⟶ NRoutes 708, 719, 720, 732, 733, 7418Changan Rd. and Changban Rd. North (2)N ⟶ SRoutes 708, 719, 720, 732, 733, 741The dwelling time of buses serving these 9 routes at the 8 bus stops are extracted from the collected AVL data. Some records are provided in Table2. Based on the arrival and departure time of buses boarding and alighting passengers at the third bus stop, we can determine the bus failure characteristics. For example, the bus with the ID of SU-EU9353 serving Route 710 departed from the bus stop at 16:14:28, while the bus with the ID of SU-EU6029 serving Route 741 arrived at the same stop at 16:15:07, and the bus arriving later needed to wait outside the stop for 13 seconds (failure duration time) until bus no.710 left the stop.Table 2
Some buses arrival and departure time records extracted from AVL data.
Route no.Bus IDBus stopDate/timeArrival timeDeparture time701SU-EU9526No.3 bus stop2018-12-18 16:10:312018-12-18 16:11:11710SU-EU9353No.3 bus stop2018-12-18 16:13:342018-12-18 16:14:28741SU-EU6029No.3 bus stop2018-12-18 16:14:152018-12-18 16:15:07720SU-EU9582No.3 bus stop2018-12-18 16:19:152018-12-18 16:20:08701SU-EU9379No.3 bus stop2018-12-18 16:20:302018-12-18 16:21:02711SU-EU9359No.3 bus stop2018-12-18 16:22:162018-12-18 16:23:19


## 3.2. Temporal Characteristics Analysis

The number and duration of buses stop failure (per hour) at bus stops are determined based on AVL data. We here use equations (5) and (6) to calculate the hourly FR and FDR of the eight bus stops. Figure 2 plots the hourly time-varying characteristics of Rf and Rfd over 56 working days of the No. 1 bus stop (672 hourly Rf data and Rfd data in total). It can be observed that Rf and Rfd in the morning (7:00–9:00) and evening peak hours (17:00–19:00) are higher than those in nonpeak hours. Overall, the mean value of Rf is more significant than that of Rfd in most times.Figure 2
The time-varying characteristics of no.1 bus stop failure.
(a)(b)We then look at the median value of the hourlyRf and Rfd. As we can see in Figure 3, there are highly similar patterns of Rf and Rfd during peak hours for all the eight bus stops. The reason might be that there is high travel demand in both passengers and vehicles are in at peak hours, and thus bus stop failures occur more frequently, especially because of a long time for passengers to board and alight.Figure 3
Bus stop failure time-varying characteristics of test bus stops. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No.8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)


## 3.3. Failure Duration Analysis

Figure4 shows the relationship between Rf and Rfd. For each bus stop, 224 data during the morning and evening peak hours (7:00 to 9:00 and 17:00 to 19:00) are considered.Figure 4
The relationship analysis betweenFDR vs. FR. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No.8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)From Figure4, the hourly FDR shows a weak positive correlation to the FR for all the eight bus stops. Overall, the fitting parameters between Rf and Rfd vary remarkably among bus stops sites. For example, there are the lowest R-square (0.31) for no.1 bus Stop and the highest R-square (0.661) for no. 4 bus stop. In addition, the statistical relationship between Rf and Rfd of 1-berth stops (no.1 through no.6 bus stops) is weaker than that of the two 2-berth stops (no.7 and no.8 bus stops).As mentioned above, it is difficult to determine a well-fitted failure duration rate function (for the 8 bus stops) using failure rate directly due to the dispersion. Taking no.1 bus stop as an example,Rfd is increased from 13.3% to 33.0% as Rf changes from 24% to 25%. But Rf increases from 16.7% to 43.3% as Rfd increases from 24% to 25%. It indicates the significant dispersion. To decrease the dispersion of failure duration rate in analysis, the failure rate of bus stops is divided into sections with an interval length of 5%. Rf5% is defined as the 5% section level of failure rate at a bus stop, which range from 5% to 10% Rf5%∈5%,10%. Then, the mean, standard deviation (S.D.), and coefficient of variation (C.V.) of failure duration rate for these 8 bus stops are clustered and calculated at a different level of failure rate. The mean, S.D., and C.V. of failure duration rates at different failure rate level of these eight test bus stops are compared in Table 3. When the frequency of bus stop failure is less than 5 at a level, the failure duration rate is not calculated and identified as “not available (N/A)” in our analysis. Because the span of these 8 bus stops is different, each bus stop has some “not available (N/A)” at a corresponding failure rate level.Table 3
Failure duration rate of test bus stops at different failure rate levels.
Failure rate levelNo.1 bus stopNo.2 bus stopNo.3 bus stopNo.4 bus stopNo.5 bus stopNo.6 bus stopNo.7 bus stopNo.8 bus topRf5%MeanN/AN/AN/AN/A0.07100.05040.03820.0464S.D.N/AN/AN/AN/A0.05080.02890.02250.0236C.V.N/AN/AN/AN/A0.71550.57320.58780.5081Rf10%MeanN/AN/A0.08960.08850.09600.08070.08350.0797S.D.N/AN/A0.03750.03600.04310.03830.03970.0353C.V.N/AN/A0.41880.40700.44880.47500.47500.4430Rf15%Mean0.1443N/A0.15410.11040.11550.11330.11180.1070S.D.0.0502N/A0.05440.03550.04800.05200.03100.0461C.V.0.3477N/A0.35270.32180.41590.45870.27690.4306Rf20%Mean0.17350.13800.16300.14050.13380.13040.14220.1450S.D.0.04980.05540.05100.03780.04940.04990.04290.0510C.V.0.28710.40150.31290.26880.36880.38280.30150.3516Rf25%Mean0.21040.20750.19040.18600.17760.16640.17180.1754S.D.0.05400.06610.05320.04500.05330.05760.04300.0497C.V.0.25660.31860.27940.24170.30020.34590.25050.2833Rf30%Mean0.22310.21590.22360.19750.16980.18320.19390.1780S.D.0.04950.05580.05540.03700.04020.06100.04600.0470C.V.0.22170.25830.24790.18720.23680.33280.23730.2640Rf35%Mean0.22150.25060.25690.24210.2641N/A0.22820.2328S.D.0.04780.05390.05460.04580.0608N/A0.04050.0470C.V.0.21600.21510.21250.18900.2304N/A0.17730.2019Rf40%Mean0.27190.28280.28850.2587N/AN/AN/AN/AS.D.0.05320.06090.05120.0543N/AN/AN/AN/AC.V.0.19570.21540.17740.2098N/AN/AN/AN/ABased on the mean, S.D., and C.V. of failure duration rates at different failure rate level, we then analyze the failure duration characteristics. Figure5 presents the average value (for all these eight bus stops) of mean, S.D., and C.V. for failure duration rate at different failure rate levels. The mean and S.D. of Rfd increase with the increasing failure rate level, and the mean has a higher and faster growth rate than that of S.D. The mean of Rfd is increased from 5.2% to 27.5% as failure rate level increases from 5% to 40%. And the S.D. of Rfd is increased from 3.1% to 5.5% as the same span of failure rate level. However, the favorable trend does not hold for the C.V. of Rfd, as illustrated in Figure 5. The C.V. of Rfd is decreased from 59.6% to 19.9% as failure rate level changes from 5% to 40%.Figure 5
Average mean, S.D., and C.V. ofFDR for eight bus stops.Figure6 displays the average value of the mean, S.D., and C.V. for failure duration rate of six 1-berth stops and two 2-berth stops. The average value of the mean, S.D., and C.V. for failure duration rate of the single-berth stops is more significant than that of the 2-berth stops.Figure 6
Average mean, S.D., and C.V. ofFDR for two types of bus stops.Figure7 displays the correlation relationships between FDR per failure Rfdr and the FR for the eight bus stops. It is shown that the FR is insensitive to Rfdr. For the no.1 bus stop, Figure 7(a) reveals that Rfdr is increased from 1.5% to 5.4% and from 1.9% to 3.4% when the FR climbs up from 21% to 22% and from 34% to 35%, respectively. Thus, the span of Rfdr is significant at the low level of failure rate, and the volatility of Rfdr becomes more and more slight with the increasing failure rate. Besides, the diversity correlation relationship between Rfdr and Rf is influenced by the number of berths of loading areas. Figure 7 shows that Rfdr is increased from 1.2% to 3.8% and from 1.5% to 5.4%, respectively, as the failure rate increase from 21% to 22% for no.7 bus stop (2-berth type) and no.1 bus stop (1-berth type). Therefore, compared with single-berth bus stops, the two 2-berth bus stops have less Rfdr at the same level of failure rate.Figure 7
FDR per failure vs. FR for test bus stops. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No.8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)For the different levels of failure rate (range from 5% to 40%), the value of the mean, S.D., and C.V. forRfdr at the bus stops are calculated and presented in Figure 8. The average values of the mean, S.D., and C.V. for Rfdr reveal a significant negative correlation with the FR level. When the FR ranges from 30% to 40%, the three statistics for Rfdr are not very sensitive to the FR level, and there are no obvious fluctuations. For the mean value of Rfdr, the maximum variation is merely 0.1% when the FR level falls into the range of 30% to 40%. Therefore, the dispersion of Rfdr decreases sharply with the FR level (especially when the FR is greater than 30%), which implies that the average FDR per failed is comparatively stable. In Figure 9, there is a similar trend of C.V.Figure 8
Average mean, S.D., and C.V. ofFDR per failure for eight bus stops.Figure 9
Average mean, S.D., and C.V. ofFDR per failure for two types of bus stops.For the different levels of failure rate (range from 5% to 40%), the value of the mean, S.D., and C.V. forRfdr for six 1-berth stops and two 2-berth stops are calculated and depicted in Figure 9. The three statistics for Rfdr at different failure rate levels for the single-berth stops is larger than that of the 2-berth stops, which shows higher stability at the 2-berth bus stops.


## 4. Correlation Analysis

As discussed inSection 3, it is found that it is not so easy to establish a satisfactory relationship between the FDR and FR, fRfd,Rf, via linear regression models. In this section, we propose a “Distribution Fitting and Cumulative Distribution Correlation (DF-CDC)” method for an in-depth and more reasonable analysis of the correlation between Rfd and Rf.


### 4.1. Distribution Fitting

The distribution fitting analysis is regarded as a useful approach to mine characteristics of transit operational parameters from the probabilistic perspective [33]. A unified probability distribution is explored that can well fit the essence of Rf and Rfd, utilizing the probability and statistical analysis methods. That is, we aim to understand how well a candidate distribution is fitted with predicted parameters for Rf and Rfd. Typically, chi-squared (χ2), Kolmogorov–Smirnov (K-S), and Anderson–Darling (A-D) tests could be used for assessing the goodness-of-fit of our analysis results. In this paper, the K-S test statistic at a significance level of 0.05 is adopted for the goodness-of-fit test based on the data of 8 bus stops on peak periods per workday (224 data per stop). 36 probability distributions listed in Table 4 are chosen for hypothesis analysis. Table 4 shows the number of rejections for the 36 possible candidates. The results reveal that the distributions of Error, Gen. Extreme Value, Gen. Logistic, Logistic, and Normal could be selected as the candidate distributions for the correlation analysis between Rfd and Rf.Table 4
Candidate distributions fitting results for bus failure characteristics.
No.DistributionNumber of rejectionsFRFDR1Beta442Burr213Cauchy764Dagum205Erlang536Error007Fatigue Life348Gamma209Gen. Extreme Value0010Gen. Gamma1011Gen. Logistic0012Gen. Pareto2113Gumbel max6214Gumbel min7615Hypersecant2216Inv. Gaussian4317Johnson SB2018Johnson SU6819Kumaraswamy4420Laplace7621Logistic0022Log-Logistic3123Lognormal3424Log-Pearson 35625Nakagami1026Normal0027Pearson 55628Pearson 62129Pert4230Phased Biexponential8631Phased Bi-Weibull3132Rayleigh4533Rice2134Triangular4435Uniform4236Weibull21Note. 5 distributions highlighted in grey shading are selected as the well-fitted candidate distribution of Rf and Rfd.The five well-fitted candidate probability distributions (Error distribution, Gen. Extreme Value distribution, Gen. Logistic distribution, Logistic distribution, and Normal distribution) are redeemed for analyzing the goodness-of-fit of fittedFR and FDR at the bus stops. After estimating the parameters of these distributions (using the Probability density function for distributions shown in Table 5), K-S test results (P value) for the five candidate probability distributions are plotted in Figure 10. It can be seen that the goodness-of-fit measured by P values for FDR fitted distribution is much better than that for the FR.Table 5
AverageP value of FR and FDR at test bus stops for well-fitted candidate distributions.
DistributionProbability density function (PDF) and sample spaceAverageP valueFRFDRErrorfxσ,k,ξ=c1σ−1exp−c0zkc0=Γ3/k/Γ1/k1/2c1=kc0/2Γ1/kz≡x−ξ/σσ>0, ξ≤x<+∞0.513760.74114Gen. Extreme Valuefxσ,k,ξ=exp−1+kz−1/k1+kzk+1/−k/σ,k≠0,exp−z−exp−z/σ,k=0,σ>0, ξ≤x<+∞0.682240.87865Gen. Logisticfxσ,k,ξ=1+kz−1−1/k/σ1+1+kz−1/k2,k≠0,exp−z/σ1+exp−z2,k=0,σ>0, ξ≤x<+∞0.477260.65518Logisticfxσ,ξ=exp−z/σ1+exp−z2σ>0, ξ≤x<+∞0.259960.40434Normalfxσ,ξ=exp−1/2x−ξ/σ2/σ2πσ>0, ξ≤x<+∞0.470320.68553Figure 10
Well-fitted candidate distributionsP value comparison for bus failed characteristics. (a) Failure rate (%). (b) Failure duration rate (%).
(a)(b)In Table5, the means of P value for Rf and Rfd at the eight bus stops for the five candidate distributions are also given. Gen. Extreme Value distribution is the best one in terms of P value (with 0.68224 of FR and 0.87865 of FDR) in distribution fitting for the Rf and Rfd.


### 4.2. Cumulative Distribution Correlation

The probability density function of Gen. Extreme Value distribution is utilized for fitting the hourlyFR and FDR distributions for the eight bus stops. The results are provided in Table 6.Table 6
Parameters of fittedFR and FDR at test bus stops.
Bus stopFittedFR parametersFittedFDR parametersσkξσkξNo.1 bus stop−0.30340.081160.28627−0.304320.062810.19447No.2 bus stop−0.427990.085920.35594−0.311850.077680.22823No.3 bus stop−0.148160.082910.20818−0.214020.070470.15087No.4 bus stop−0.289090.100290.26163−0.254430.071850.16056No.5 bus stop−0.245150.096760.15746−0.184590.069120.10188No.6 bus stop−0.221780.075110.11501−0.112380.054390.06722No.7 bus stop−0.202090.085640.17765−0.17150.062540.1101No.8 bus stop−0.219550.081230.15239−0.133280.060370.09181Based on the calculated parameters of the fitted Gen. Extreme Value distribution in Table6, the cumulative distribution function (CDF) curve of Rf and Rfd for these test bus stops can be determined. From the fitted CDF curve, the fitted value at a different level of CDF can be recorded. The fitted value at a different level of CDF can be recorded using the fitted CDF curve, and the actual value can be determined by analyzing the ranking level based on the sorted 224 data collected at each test bus stop. For analyzing the accuracy of the fitted CDF value, the relative error between the actual and fitted value of Rf and Rfd for test bus stops at a different level of CDF is examined. Furthermore, 17 critical levels of CDF, ranging from 10% to 90% (with 5% of interval length), are selected for verifying. The relative error between the actual and fitted value of Rf and Rfd for the critical CDF level at the 8 test bus stops are presented in Tables 7 and 8.Table 7
Relative error between actual and fittedFR for test bus stops at different CDF levels.
Level of CDFNo.1 bus stopNo.2 bus stopNo.3 bus stopNo.4 bus stopNo.5 bus stopNo.6 bus stopNo.7 bus stopNo.8 bus stopPc10%0.720.224.053.628.6710.430.1532.67Pc15%3.012.431.913.1826.521.337.191.37Pc20%1.970.083.131.022.2415.977.900.48Pc25%0.951.370.762.275.8710.430.840.09Pc30%1.851.800.333.935.634.263.134.10Pc35%0.551.262.020.460.285.361.703.53Pc40%0.550.090.070.390.502.791.300.97Pc45%2.320.170.330.831.231.381.762.13Pc50%0.090.431.221.662.070.991.250.45Pc55%1.181.100.300.252.111.011.234.05Pc60%0.681.520.753.142.182.870.291.56Pc65%0.120.420.742.391.793.623.203.32Pc70%0.521.062.661.711.763.772.942.30Pc75%0.030.250.830.032.001.612.091.19Pc80%1.850.570.470.350.401.211.922.48Pc85%0.090.633.552.681.772.300.062.79Pc90%0.470.380.553.633.944.190.251.69Note:Pc10% represents the 10% CDF level (the 10th percentile of the fitted FR), ranking in ascending order ranging from 0 to 1.Table 8
Relative error between actual and fittedFDR for test bus stops at different CDF levels.
Level of CDFNo.1 bus stop (%)No.2 bus stop (%)No.3 bus stop (%)No.4 bus stop (%)No.5 bus stop (%)No.6 bus stop (%)No.7 bus stop (%)No.8 bus stop (%)Pc10%1.600.465.161.9412.4834.544.820.89Pc15%0.680.880.322.494.1411.603.995.04Pc20%0.832.612.381.827.780.140.648.65Pc25%1.842.323.422.797.600.850.643.29Pc30%2.311.040.361.299.653.320.081.98Pc35%0.900.091.121.113.081.172.031.43Pc40%0.341.090.170.290.401.061.843.81Pc45%0.770.230.760.272.883.611.025.75Pc50%0.480.620.422.564.192.811.472.73Pc55%0.481.360.761.974.282.280.361.36Pc60%1.320.291.061.082.901.331.050.21Pc65%0.960.690.481.900.522.101.591.61Pc70%0.441.320.172.141.110.471.760.04Pc75%0.320.851.212.650.160.111.780.45Pc80%0.020.272.470.513.550.040.153.59Pc85%0.050.753.982.377.963.191.560.56Pc90%1.510.420.313.624.070.651.931.04Expect for a tiny minority of critical CDF level, relative errors between the actual and fitted value ofRf and Rfd are always less than 10% at these test bus stops, as shown in Tables 7 and 8. Therefore, Gen. Extreme Value distribution performs well in fitting CDF value for Rf and Rfd at these test bus stops, and the accuracy and reliability for the fitted value are convincing.For the distribution pattern ofRf and Rfd at test bus stops, Gen. Extreme Value distribution can be well fitted. Moreover, they also have a positive correlation relationship (as shown in Figure 4). Therefore, it can be considered that the fitted value ofRf and Rfd at the same CDF level perform equivalently in failure characteristics analysis. Figure 11 presents the fitted curves and critical level values of Rf and Rfd, which are fitted and utilized Gen. Extreme Value distribution for no.1 bus stop. In Figure 11, the “star (with pink color)” and “triangle (with red color)” display the fitted values of Rf and Rfd at these 17 critical CDF levels, and the “star” and “triangle” connected by a dotted line is defined as an equal correlation pair of Rf and Rfd for the corresponding critical CDF level.Figure 11
Correlation between fittedFR and FDR at the same CDF level using Gen. Extreme Value distribution.Figure12 reveals the relationship for the 136 equal correlation pair of Rf and Rfd, including the pairs for 17 critical CDF levels at 8 test bus stops. A quite strong linear regression expression (with 0.98 of R-square) for the fitted couples of Rf and Rfd can be observed, which can reflect the significant correlation relationship between fitted Rf and Rfd.Figure 12
Gen. Extreme Value distribution fittedFDR vs. FR correlation analysis.


### 4.3. Correlation Performance Evaluation

The crossvalidation method [34] for Rfd prediction based on AVL data is adopted for analyzing the correlation between Rfd and Rf. A four-step procedure for predicting Rfd at a certain cumulative distribution ranking level is illustrated as below. Firstly, the observed Rfd and Rf in peak hours of 56 workdays at 75% bus stops of the total 8 test bus stops are selected randomly as the modeling datasets, and the data of the rest two bus stops are defined as predicted datasets. Secondly, the probability density distribution of Rfd and Rf (for the selected six bus stops in modeling datasets) are fitted using Gen. Extreme value distribution, and the fitted Rfd and Rf are recorded based on their probability density functions at critical cumulative distribution levels, respectively. Thirdly, linear regression expression is developed based on these fitted Rfd and Rf for the six bus stops (in modeling dataset) at critical cumulative distribution levels. Finally, the observed Rf (for two bus stops in prediction dataset) at corresponding critical cumulative distribution levels are determined, and the predicted Rfd can be calculated using the linear regression model (as formulated in Step 3). The prediction accuracy and reliability of Rfd can be determined by comparing the actual and predicted value.It is obvious that there are 28 different combinations of modeling and prediction datasets, as the rules described in Step 2. There are seven estimated values ofRfd for different cumulative distribution levels at each bus stop, calculated from these 28 different division plans. Based on crossvalidation, the predicted value and relative error of Rfd for different cumulative distribution levels at 8 test bus stops are calculated, and the results are shown in Figure 13.Figure 13
Results ofFDR estimation for test bus stops at CDF levels. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No. 8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)In Figure13, the mean value of predicted relative errors for seven times Rfd prediction at different cumulative distribution levels is represented by a solid blue line. According to the results, these test bus stops have more accurate predicted values (the relative error is less than 15%) at most cumulative distribution levels, expect for low cumulative distribution levels (less than 15%). The prediction results of Rfd for different cumulative distribution levels at no.1 bus stop perform well (the relative error is not more than 5%) in general. Also, for most of the bus stops (no.2, no.4, no.5, no.6, no.7, and no.8 bus stop), the relative error of predicted Rfd is diminished gradually, as the cumulative distribution sort decreases. No. 3 and no. 4 bus stops serve five bus routes, more than other bus stops. In general, the more bus routes are served at a bus stop, the more complicated buses arrival patterns are. Therefore, the average predicted relative error values of no. 3 and no. 4 bus stops will be larger than other bus stops.To make an easy comparison, the average relative error ofRfd for 1-berth bus stops and 2-berth bus stops are plotted in Figure 14. It can be observed that, for both types of stops, the average relative error shows a decreasing trend. In addition, the volatility is insensitive when the cumulative distribution level is higher than 30%, and the average relative error is less than 8% in general. According to the results, the proposed “Distribution Fitting and Cumulative Distribution Correlation (DF-CDC)” can develop a significant correlation relationship between the failure rate and failure duration rate.Figure 14
Comparison ofFDR estimation relative error at CDF levels.


## 4.1. Distribution Fitting

The distribution fitting analysis is regarded as a useful approach to mine characteristics of transit operational parameters from the probabilistic perspective [33]. A unified probability distribution is explored that can well fit the essence of Rf and Rfd, utilizing the probability and statistical analysis methods. That is, we aim to understand how well a candidate distribution is fitted with predicted parameters for Rf and Rfd. Typically, chi-squared (χ2), Kolmogorov–Smirnov (K-S), and Anderson–Darling (A-D) tests could be used for assessing the goodness-of-fit of our analysis results. In this paper, the K-S test statistic at a significance level of 0.05 is adopted for the goodness-of-fit test based on the data of 8 bus stops on peak periods per workday (224 data per stop). 36 probability distributions listed in Table 4 are chosen for hypothesis analysis. Table 4 shows the number of rejections for the 36 possible candidates. The results reveal that the distributions of Error, Gen. Extreme Value, Gen. Logistic, Logistic, and Normal could be selected as the candidate distributions for the correlation analysis between Rfd and Rf.Table 4
Candidate distributions fitting results for bus failure characteristics.
No.DistributionNumber of rejectionsFRFDR1Beta442Burr213Cauchy764Dagum205Erlang536Error007Fatigue Life348Gamma209Gen. Extreme Value0010Gen. Gamma1011Gen. Logistic0012Gen. Pareto2113Gumbel max6214Gumbel min7615Hypersecant2216Inv. Gaussian4317Johnson SB2018Johnson SU6819Kumaraswamy4420Laplace7621Logistic0022Log-Logistic3123Lognormal3424Log-Pearson 35625Nakagami1026Normal0027Pearson 55628Pearson 62129Pert4230Phased Biexponential8631Phased Bi-Weibull3132Rayleigh4533Rice2134Triangular4435Uniform4236Weibull21Note. 5 distributions highlighted in grey shading are selected as the well-fitted candidate distribution of Rf and Rfd.The five well-fitted candidate probability distributions (Error distribution, Gen. Extreme Value distribution, Gen. Logistic distribution, Logistic distribution, and Normal distribution) are redeemed for analyzing the goodness-of-fit of fittedFR and FDR at the bus stops. After estimating the parameters of these distributions (using the Probability density function for distributions shown in Table 5), K-S test results (P value) for the five candidate probability distributions are plotted in Figure 10. It can be seen that the goodness-of-fit measured by P values for FDR fitted distribution is much better than that for the FR.Table 5
AverageP value of FR and FDR at test bus stops for well-fitted candidate distributions.
DistributionProbability density function (PDF) and sample spaceAverageP valueFRFDRErrorfxσ,k,ξ=c1σ−1exp−c0zkc0=Γ3/k/Γ1/k1/2c1=kc0/2Γ1/kz≡x−ξ/σσ>0, ξ≤x<+∞0.513760.74114Gen. Extreme Valuefxσ,k,ξ=exp−1+kz−1/k1+kzk+1/−k/σ,k≠0,exp−z−exp−z/σ,k=0,σ>0, ξ≤x<+∞0.682240.87865Gen. Logisticfxσ,k,ξ=1+kz−1−1/k/σ1+1+kz−1/k2,k≠0,exp−z/σ1+exp−z2,k=0,σ>0, ξ≤x<+∞0.477260.65518Logisticfxσ,ξ=exp−z/σ1+exp−z2σ>0, ξ≤x<+∞0.259960.40434Normalfxσ,ξ=exp−1/2x−ξ/σ2/σ2πσ>0, ξ≤x<+∞0.470320.68553Figure 10
Well-fitted candidate distributionsP value comparison for bus failed characteristics. (a) Failure rate (%). (b) Failure duration rate (%).
(a)(b)In Table5, the means of P value for Rf and Rfd at the eight bus stops for the five candidate distributions are also given. Gen. Extreme Value distribution is the best one in terms of P value (with 0.68224 of FR and 0.87865 of FDR) in distribution fitting for the Rf and Rfd.


## 4.2. Cumulative Distribution Correlation

The probability density function of Gen. Extreme Value distribution is utilized for fitting the hourlyFR and FDR distributions for the eight bus stops. The results are provided in Table 6.Table 6
Parameters of fittedFR and FDR at test bus stops.
Bus stopFittedFR parametersFittedFDR parametersσkξσkξNo.1 bus stop−0.30340.081160.28627−0.304320.062810.19447No.2 bus stop−0.427990.085920.35594−0.311850.077680.22823No.3 bus stop−0.148160.082910.20818−0.214020.070470.15087No.4 bus stop−0.289090.100290.26163−0.254430.071850.16056No.5 bus stop−0.245150.096760.15746−0.184590.069120.10188No.6 bus stop−0.221780.075110.11501−0.112380.054390.06722No.7 bus stop−0.202090.085640.17765−0.17150.062540.1101No.8 bus stop−0.219550.081230.15239−0.133280.060370.09181Based on the calculated parameters of the fitted Gen. Extreme Value distribution in Table6, the cumulative distribution function (CDF) curve of Rf and Rfd for these test bus stops can be determined. From the fitted CDF curve, the fitted value at a different level of CDF can be recorded. The fitted value at a different level of CDF can be recorded using the fitted CDF curve, and the actual value can be determined by analyzing the ranking level based on the sorted 224 data collected at each test bus stop. For analyzing the accuracy of the fitted CDF value, the relative error between the actual and fitted value of Rf and Rfd for test bus stops at a different level of CDF is examined. Furthermore, 17 critical levels of CDF, ranging from 10% to 90% (with 5% of interval length), are selected for verifying. The relative error between the actual and fitted value of Rf and Rfd for the critical CDF level at the 8 test bus stops are presented in Tables 7 and 8.Table 7
Relative error between actual and fittedFR for test bus stops at different CDF levels.
Level of CDFNo.1 bus stopNo.2 bus stopNo.3 bus stopNo.4 bus stopNo.5 bus stopNo.6 bus stopNo.7 bus stopNo.8 bus stopPc10%0.720.224.053.628.6710.430.1532.67Pc15%3.012.431.913.1826.521.337.191.37Pc20%1.970.083.131.022.2415.977.900.48Pc25%0.951.370.762.275.8710.430.840.09Pc30%1.851.800.333.935.634.263.134.10Pc35%0.551.262.020.460.285.361.703.53Pc40%0.550.090.070.390.502.791.300.97Pc45%2.320.170.330.831.231.381.762.13Pc50%0.090.431.221.662.070.991.250.45Pc55%1.181.100.300.252.111.011.234.05Pc60%0.681.520.753.142.182.870.291.56Pc65%0.120.420.742.391.793.623.203.32Pc70%0.521.062.661.711.763.772.942.30Pc75%0.030.250.830.032.001.612.091.19Pc80%1.850.570.470.350.401.211.922.48Pc85%0.090.633.552.681.772.300.062.79Pc90%0.470.380.553.633.944.190.251.69Note:Pc10% represents the 10% CDF level (the 10th percentile of the fitted FR), ranking in ascending order ranging from 0 to 1.Table 8
Relative error between actual and fittedFDR for test bus stops at different CDF levels.
Level of CDFNo.1 bus stop (%)No.2 bus stop (%)No.3 bus stop (%)No.4 bus stop (%)No.5 bus stop (%)No.6 bus stop (%)No.7 bus stop (%)No.8 bus stop (%)Pc10%1.600.465.161.9412.4834.544.820.89Pc15%0.680.880.322.494.1411.603.995.04Pc20%0.832.612.381.827.780.140.648.65Pc25%1.842.323.422.797.600.850.643.29Pc30%2.311.040.361.299.653.320.081.98Pc35%0.900.091.121.113.081.172.031.43Pc40%0.341.090.170.290.401.061.843.81Pc45%0.770.230.760.272.883.611.025.75Pc50%0.480.620.422.564.192.811.472.73Pc55%0.481.360.761.974.282.280.361.36Pc60%1.320.291.061.082.901.331.050.21Pc65%0.960.690.481.900.522.101.591.61Pc70%0.441.320.172.141.110.471.760.04Pc75%0.320.851.212.650.160.111.780.45Pc80%0.020.272.470.513.550.040.153.59Pc85%0.050.753.982.377.963.191.560.56Pc90%1.510.420.313.624.070.651.931.04Expect for a tiny minority of critical CDF level, relative errors between the actual and fitted value ofRf and Rfd are always less than 10% at these test bus stops, as shown in Tables 7 and 8. Therefore, Gen. Extreme Value distribution performs well in fitting CDF value for Rf and Rfd at these test bus stops, and the accuracy and reliability for the fitted value are convincing.For the distribution pattern ofRf and Rfd at test bus stops, Gen. Extreme Value distribution can be well fitted. Moreover, they also have a positive correlation relationship (as shown in Figure 4). Therefore, it can be considered that the fitted value ofRf and Rfd at the same CDF level perform equivalently in failure characteristics analysis. Figure 11 presents the fitted curves and critical level values of Rf and Rfd, which are fitted and utilized Gen. Extreme Value distribution for no.1 bus stop. In Figure 11, the “star (with pink color)” and “triangle (with red color)” display the fitted values of Rf and Rfd at these 17 critical CDF levels, and the “star” and “triangle” connected by a dotted line is defined as an equal correlation pair of Rf and Rfd for the corresponding critical CDF level.Figure 11
Correlation between fittedFR and FDR at the same CDF level using Gen. Extreme Value distribution.Figure12 reveals the relationship for the 136 equal correlation pair of Rf and Rfd, including the pairs for 17 critical CDF levels at 8 test bus stops. A quite strong linear regression expression (with 0.98 of R-square) for the fitted couples of Rf and Rfd can be observed, which can reflect the significant correlation relationship between fitted Rf and Rfd.Figure 12
Gen. Extreme Value distribution fittedFDR vs. FR correlation analysis.


## 4.3. Correlation Performance Evaluation

The crossvalidation method [34] for Rfd prediction based on AVL data is adopted for analyzing the correlation between Rfd and Rf. A four-step procedure for predicting Rfd at a certain cumulative distribution ranking level is illustrated as below. Firstly, the observed Rfd and Rf in peak hours of 56 workdays at 75% bus stops of the total 8 test bus stops are selected randomly as the modeling datasets, and the data of the rest two bus stops are defined as predicted datasets. Secondly, the probability density distribution of Rfd and Rf (for the selected six bus stops in modeling datasets) are fitted using Gen. Extreme value distribution, and the fitted Rfd and Rf are recorded based on their probability density functions at critical cumulative distribution levels, respectively. Thirdly, linear regression expression is developed based on these fitted Rfd and Rf for the six bus stops (in modeling dataset) at critical cumulative distribution levels. Finally, the observed Rf (for two bus stops in prediction dataset) at corresponding critical cumulative distribution levels are determined, and the predicted Rfd can be calculated using the linear regression model (as formulated in Step 3). The prediction accuracy and reliability of Rfd can be determined by comparing the actual and predicted value.It is obvious that there are 28 different combinations of modeling and prediction datasets, as the rules described in Step 2. There are seven estimated values ofRfd for different cumulative distribution levels at each bus stop, calculated from these 28 different division plans. Based on crossvalidation, the predicted value and relative error of Rfd for different cumulative distribution levels at 8 test bus stops are calculated, and the results are shown in Figure 13.Figure 13
Results ofFDR estimation for test bus stops at CDF levels. (a) No.1 bus stop. (b) No.2 bus stop. (c) No.3 bus stop. (d) No.4 bus stop. (e) No.5 bus stop. (f) No.6 bus stop. (g) No.7 bus stop. (h) No. 8 bus stop.
(a)(b)(c)(d)(e)(f)(g)(h)In Figure13, the mean value of predicted relative errors for seven times Rfd prediction at different cumulative distribution levels is represented by a solid blue line. According to the results, these test bus stops have more accurate predicted values (the relative error is less than 15%) at most cumulative distribution levels, expect for low cumulative distribution levels (less than 15%). The prediction results of Rfd for different cumulative distribution levels at no.1 bus stop perform well (the relative error is not more than 5%) in general. Also, for most of the bus stops (no.2, no.4, no.5, no.6, no.7, and no.8 bus stop), the relative error of predicted Rfd is diminished gradually, as the cumulative distribution sort decreases. No. 3 and no. 4 bus stops serve five bus routes, more than other bus stops. In general, the more bus routes are served at a bus stop, the more complicated buses arrival patterns are. Therefore, the average predicted relative error values of no. 3 and no. 4 bus stops will be larger than other bus stops.To make an easy comparison, the average relative error ofRfd for 1-berth bus stops and 2-berth bus stops are plotted in Figure 14. It can be observed that, for both types of stops, the average relative error shows a decreasing trend. In addition, the volatility is insensitive when the cumulative distribution level is higher than 30%, and the average relative error is less than 8% in general. According to the results, the proposed “Distribution Fitting and Cumulative Distribution Correlation (DF-CDC)” can develop a significant correlation relationship between the failure rate and failure duration rate.Figure 14
Comparison ofFDR estimation relative error at CDF levels.


## 5. Conclusion

In order to analyze the characteristics of bus stop failure, we propose a new measurement calledFDR and make a comparison to the traditional index of the FR. Compared with the FR, the proposed FDR is capable to quantitatively assess the impact of bus stop failure on traffic efficiency. Based on the collected AVL data associated with the eight bus stops in Wujiang District of Suzhou, we make an in-depth analysis of the characteristics and correlation of the FR and FDR. Some insightful findings are summarized as follows:(i)
It can be observed that the values ofFR and FDR in morning and evening peak hours are greater than those during off-peak hours. The value of the FR is usually larger than that of the FDR across all the eight bus stops.(ii)
It is found that there is a positive correlation between theFR and the FDR. However, the R-square values of the linear regressions fluctuate dramatically among different bus stops. The results also indicate that the FDR is more robust than the FR in describing transit system status and traffic characteristics.(iii)
We find that Gen. Extreme Value distribution could be well used for the fittings of both theFR and FDR and the proposed “Distribution Fitting and Cumulative Distribution Correlation (DF-CDC)” method works well in determining the fitted values of the FR and FDR at the critical CDF level that reflects a significant correlation between the FR and FDR.Future works could be extended in two aspects. First, more AVL and other source data in other cities could be collected and used for analyzing the failure characteristics of bus stops. Second, based on the data-driven analysis of bus stop failure characteristics, we could make some scenario analysis to find out the most important factors, such as the number of bus stop berths and the number of lanes or the passenger demand.



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*Source: 8863262-2020-07-01.xml*