Determining the Factors That Influence the Odds and Time to Streetcar Bunching Incidents

Determining the Factors that Influence the Odds and Time to Streetcar Bunching Incidents

Paula Diem Quynh Nguyen

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Civil Engineering University of Toronto

Determining the Factors that Influence the Odds and Time to Streetcar Bunching Incidents

Paula Diem Quynh Nguyen

Master of Applied Science

Department of Civil Engineering University of Toronto

2017 Abstract

Bunching is a common operational problem in surface transit systems with negative impacts on service quality and users’ perception. While many studies have focused on understanding the causes of bus bunching and developing strategies to mitigate its negative effects, there has been little research on streetcar bunching. This research aims at understanding the factors that impact the likelihood of streetcar bunching and to investigate the factors that impact the time to the initial bunching incident from the terminal. Focusing on Toronto’s streetcar lines, this study developed a binary logistic regression model and an accelerated failure time (AFT) model to address the first and second goals, respectively. Data from multiple sources, including the automatic vehicle location (AVL) system, were used to estimate the models. Headway deviations at the terminal and the usage of different vehicles types were two interesting factors found to increase odds of bunching and accelerate time to bunching.

Acknowledgments

Firstly, I’d like to thank my thesis supervisor, Dr. Amer Shalaby, for his expertise, advice, and vision throughout this foreign experience of academia for me. He has provided so much guidance and helped me stay on track while I got lost in the process and the mounds of data. As cliché as it sounds, I would not be here without him.

I want to express my deep appreciation and gratitude to Ehab Diab, who has shared, taught, and spared so much of his time during this process. I would have been hopeless without you on this journey. Despite all of my dim questions, he has remained smiling, patient, understanding, and encouraging.

From the Toronto Transit Commission, I’d like to thank Kenny Ling and Francis Li for providing the data that has enabled me to conduct this study.

I can’t forget to thank my family. My cousin, Katherine, was my biggest supporter in the pursuit of graduate studies. I wouldn’t have even submitted my application to the program without her encouragement. Of course, I wouldn’t have been successful without my parents, aunt, and sister and their unconditional love. Thank you, Christopher for the endless support you have given me. These words cannot express how grateful I am for all of you.

Thanks to the Auto Parts group for all of your help and support over these past two years. Whenever I had trouble with software or with concepts, you were always there for me. You guys provided comedic relief and support during those tough times. Without you, I wouldn’t be sitting here writing this.

Finally, I want to recognize and thank two of my undergraduate supervisors from the University of Windsor. Dr. Faouzi Ghrib always believed that I had the potential and was destined for higher education. Thank you for your faith in me. It was because of my undergraduate research experience with Dr. Chris Lee that inspired me to pursue graduate studies in transportation engineering. Thank you for your continuous support of my growth and development.

iii

Contents

Acknowledgments...... iii

List of Tables ...... vi

List of Figures ...... vii

1 Introduction ...... 1

1.1 Background ...... 1

1.2 Motivation ...... 2

1.3 Research Objective ...... 4

1.4 Methodology ...... 5

1.5 Thesis Structure ...... 6

2 Literature Review ...... 7

2.1 Bunching in Public Transit ...... 8

2.2 Streetcar Performance ...... 9

2.3 Transportation Applications on Time to an Event ...... 10

3 Study Context and Data ...... 12

3.1 Study Context...... 12

3.2 Data Description ...... 14

3.3 Data Processing ...... 16

3.4 Variable Definition ...... 19

4 Modelling Framework ...... 25

4.1 Binary Logistic Regression Model ...... 27

4.2 Survival Analysis ...... 29

5 Model Results and Discussion ...... 32

5.1 Descriptive Statistics/Data Trends ...... 32

5.2 Binary Logistic Regression Model Results...... 38

5.3 Accelerated Failure Time Model Results ...... 42

6 Summary and Conclusions ...... 46

6.1 Summary of Thesis ...... 46

6.2 Key Results ...... 46

6.3 Policy Implications ...... 47

6.4 Future Work ...... 48

7 References ...... 50

8 Appendix ...... 54

8.1 TTC Service Summary (January 3 – February 13, 2016) ...... 54

8.2 Stata Code ...... 55

List of Tables

Table 1 Average Daily Streetcar Ridership ...... 12

Table 2 TTC Fleet Details...... 13

Table 3 Streetcar Route Characteristics ...... 14

Table 4 Service Hours Defined by the TTC ...... 14

Table 5 Sample of Raw Data Provided by the TTC ...... 15

Table 6 Independent Variables Considered in Study ...... 20

Table 7 Variable Definitions ...... 23

Table 8 Descriptive Statistics of All Variables Used in Models ...... 32

Table 9 Summary Statistics of All Headways ...... 33

Table 10 Statistics on Time to First Bunch ...... 36

Table 11 Summary Statistics of Bunched Incidents ...... 38

Table 12 Logistic Regression Model Fit ...... 38

Table 13 Binary Logistic Regression Model Results ...... 41

Table 14 Comparison of Fits for the AFT Model ...... 42

Table 15 AFT Model Results ...... 45

List of Figures

Figure 1 Sample of TTC's Daily Performance Report ...... 3

Figure 2 Map of Streetcar Routes Included in Study ...... 16

Figure 3 Bunching Incident Definition ...... 25

Figure 4 Vehicle Naming System ...... 27

Figure 5 Sample Scheduled Time Distance Diagram ...... 34

Figure 6 Time Distance Diagram: Route 511 Southbound, AM Peak ...... 35

Figure 7 Time Distance Diagram: Route 511 Southbound, PM Peak ...... 36

Figure 8 Histogram of Time to First Bunching Incident ...... 37

Figure 9 Route Specific Examples of Time to First Bunching Incident Histogram ...... 37

Figure 10 Cox Snell Residual Graph ...... 42

vii

1 Introduction 1.1 Background

Public transit systems face many different operational problems and disruptions that can degrade the quality and reliability of service. One of the most common disruptions is vehicle bunching. Delays and inconsistencies in transit service can precipitate bunching. Bunching occurs when two or more consecutive vehicles on the same route are unable to maintain their scheduled headways and end up following each other too closely. In this scenario, the late bus usually carries more passengers, delaying it further and falling behind the schedule, while the other bus (the following bus) would serve fewer passengers, making it faster and ahead of schedule. Bunching, therefore, leaves some passengers stranded. Bunching causes serious challenges to both the passengers and operators. For the passengers, bunching causes longer or variable wait times and vehicle overcrowding due to uneven passenger distribution, which both contribute to reducing users’ satisfaction. For the transit agencies, bunching leads to increased costs due to the inefficient use of resources since overcrowded vehicles at the front of a bunch would be followed closely by near-empty ones. Bunching also impacts the overall image of transit agencies, making it harder to attract new transit riders and retain the existing ones (Diab, Bertini, & El-Geneidy, 2016; Hu & Shalaby, 2017). For all these reasons, bunching has been a popular topic in literature over the past two decades.

Bus bunching is a problem that is experienced globally, especially for transit agencies that run frequent service. The Metropolitan Transportation Authority, the transit operator in New York City, USA, was recently given an F grade for its poor bus service. Results from a survey show that scheduling problems are mostly due to the high volume of bunching incidences (Rivoli, 2017). Many other cities in various countries around the world are looking for solutions to their bus bunching problems including the UK and Singapore. The bus provider in Bristol, England is planning to implement exclusive lanes, smart ticketing, and traffic signal priority (Ashcroft, 2017) while Singapore plans to add more bus lanes and divide long routes into shorter ones (Erath, 2013) to reduce bunching and improve bus service. The Toronto Transit Commission (TTC), Toronto’s public transit provider, is one of the many operators that experience bunching and unreliability in its service (Kalinowski, 2014; Platt, 2014). 1

In the City of Toronto, both bus and streetcar systems often experience bunching. Although buses and streetcars share many similarities, one major difference between the two is that streetcars cannot overtake each other since they are limited to the path of their track infrastructure. This subtle difference makes bunching incidents more critical to the streetcar system quality than buses, which can overtake each other. In addition, the number of cities that are planning or constructing a new streetcar or light rail system is slowly growing. The increase in streetcar systems and the differences it has with bus systems require transit operators to have a better understanding of the factors that affect streetcar bunching.

1.2 Motivation

There are multiple motivations for this study:

1. Bunching is a well-known problem that is frequently experienced by the riders of City of Toronto’s streetcar system and needs to be addressed. 2. While the majority of studies focused on understanding the general factors that impact the likelihood of bunching, it is rare to find studies that explored the time to bunching for either bus or streetcar service. 3. There is a lack of studies and literature on streetcar performance and more specifically, streetcar bunching. 4. Streetcar and light rail systems are slowly becoming more popular and more widely implemented around the world.

Bunching is a well-known problem that is frequently experienced in the City of Toronto, not only along bus routes but even worse along streetcar routes (Kalinowski, 2014). The problem is exasperated in Toronto by the very high frequency of streetcar services relative to bus services. The TTC publishes a daily customer service scorecard (Figure 1) that shows how well it meets its target goals (TTC, n.d.-a). Its goal for the streetcar service is to provide on-time departures from end terminals at least 90% of the time. However, the streetcar service performance continues to hover around the 50-60% range, falling well short of its 90% goal.

Figure 1 Sample of TTC's Daily Performance Report

Therefore, in an effort to improve service, the TTC plans to replace its old fleet with over 200 Flexity vehicles on a gradual basis. The TTC believes that the higher capacity vehicle will reduce bunching on its network (TTC, n.d.-a). This is because the new vehicles will be operated with longer headways compared to the ones currently in effect while maintaining the same route capacity. The City of Toronto is also currently in the process of constructing the Eglinton Crosstown, a new high-end light rail line, and it is planning the addition of more light rail routes (Finch West LRT and Sheppard East LRT). Therefore, with the expected growth of the network and the city population, it is more critical now than ever to have a clear understanding of the factors that influence streetcar bunching to ensure a better service operation overall.

In an effort to find a solution to the streetcar bunching problem in Toronto, a systematic review of the academic literature was conducted. However, very little literature was found on streetcar bunching. There is also almost no literature on the time to a bunching incident. This is likely due to the fact that streetcars are an uncommon transit vehicle mode and are utilized in very few cities around the world. Even in Melbourne, Australia where the largest streetcar network exists, there are very few studies on streetcar performance compared to bus studies. However, many cities are now planning or in the construction stage of building new light rail or streetcar systems including Minneapolis, Kansas City, and Montreal. For example, Kansas City introduced a new

streetcar line in 2016 with further plans for expansion in the future (KCRTA, n.d.). New York City has also proposed a $2.5 billion plan to build a 16-mile streetcar line (Garfield, 2016). Streetcar technology may be mature, but it is evident that streetcar bunching is a topic that needs a more comprehensive study as streetcar and light rail systems become more popular around the world.

Because of the four main motivations listed above, this study was conducted. The results of this study will help transit agencies, including the TTC, minimize the occurrence of bus bunching and delay the onset of bunching if it occurs, which will likely lead to better service efficiency and higher rider satisfaction.

1.3 Research Objective

This study has two main objectives regarding streetcar bunching in the City of Toronto:

1. To understand the factors that impact the likelihood of streetcar bunching 2. To determine the factors that impact the time to the initial bunching incident from the terminal using historical automatic vehicle location (AVL) data provided by the TTC.

This work will touch on an important gap in the current literature; there are few studies conducted on streetcars in literature and few that investigate specifically streetcar bunching. Furthermore, time to bunching studies could not be found in the literature. This will be further discussed in Chapter 2. The availability of AVL data offer an excellent opportunity to conduct this research using the Toronto streetcar system as a case study. This work, besides it being beneficial for other transit agencies, is particularly important for the TTC due to their extensive efforts spent in improving the service (i.e., purchasing new streetcars and upgrading its fleet) as well as due to the increase in the public transit ridership and growth of the streetcar and light rail network.

If the factors that impact the likelihood of streetcar bunching and the factors that impact time to the initial bunching incident are well understood, these factors can then be used to predict if and when a bunching incident would occur. This would allow transit operators to act in a proactive manner instead of a reactive manner to bunching incidents, which is what normally happens. By being able to predict and potentially prevent bunching incidences, transit operators will be able 4

to improve efficiency, decrease operational costs, increase passenger satisfaction, and even improve the reputation of public transit and attract new riders.

1.4 Methodology

A systematic literature review was first conducted regarding bunching in both bus and streetcar service. This helped to identify gaps in the literature and determine the scope of the study. After the scope of the study was established, AVL data for the streetcar service in the City of Toronto was acquired from the TTC to address the research objectives. To analyze the TTC data, descriptive statistics (and visualizations), as well as two statistical models, were used. The descriptive statistics helped to show the magnitude of the problem as well some patterns in the data, while visualizations (time-space diagrams) assisted in visually identifying bunching incidents.

In order to achieve the thesis research objectives, two models were used. The first model explores the odds of a headway becoming a bunching incident, irrespective of the location of the incident. This model is also used to identify similarities with previous research done in the literature regarding bus bunching. The knowledge gained from this model could help transit operators formulate policies and strategies to reduce the occurrence of bunching incidents. The second model estimates the impact of external and internal factors on time to the initial bunching incident, given that one occurs. The later a bunching incident occurs, the better it is for the operator; in other words, it is extremely useful to formulate policies that delay the onset of bunching and its detrimental effects as far down the line as possible knowing that bunching cannot be eliminated completely. The second model can help inform and guide such strategies. The combined results of the two models can help inform policies that minimize the occurrence of bunching and delay their onset if they do occur.

Over six million vehicle location data points were observed over the study period. The data were processed to remove duplicates and erroneous points. Once the data were processed, the data were analyzed to see if there were any identifiable trends prior to building the models. The first objective of the study, which is to understand the factors that impact the likelihood of streetcar bunching incidences, is addressed using a binary logistic regression model. An accelerated failure time (AFT) model is used to address the second objective of the study, which is to 5

determine the internal and external factors that influence the time from terminal to the first bunching incident. AFT models are used within survival analysis, which is a method typically used in medicine to determine the time until failure or death. This type of model was chosen for this study because it was deemed most appropriate for the objective of understanding time to a bunching incident. A detailed description of the methodology will be discussed in Section 4 of this study.

1.5 Thesis Structure

This study is organized into 6 chapters. Following this introduction, which presents the background, motivation, and objectives of the study, chapter 2 will present a literature review on relevant bunching and streetcar studies as well as studies using the time to an event models. Chapter 3 discusses the study context and area while chapter 4 describes the modeling framework. Chapter 5 reports the findings of models and discusses the interpretation of these results. Lastly, chapter 6 concludes the study and discusses the implications of the results found and the next steps that can be applied going forward.

2 Literature Review

The objective of this section is to have a thorough understanding and awareness of what studies have been conducted regarding the factors that influence bunching and the prediction of bunching incidences in both bus and streetcar networks. In addition, this section will identify the gaps in the literature and highlight why this study is necessary and valuable. Since streetcars are not widely used across the world, it is expected that there is a limited number of studies on streetcars compared to buses. Furthermore, any models that have been developed for bus systems are likely to able to be applied to streetcar systems. Therefore, a literature review of bus bunching was completed for this study despite the fact that this study focuses on streetcars only.

A systematic methodology was used to review the academic literature regarding bus bunching. The search only includes full articles published after January 1995 up to May 2016. This date restriction was chosen to capture articles that involved the use of AVL or automatic passenger count (APC) technology to assist in operation disruption detection and prediction. Transit agencies began adopting automatic vehicle monitoring (AVM) technology more in the mid- 1990’s despite the fact that AVL systems were available in the 1970’s (TRB, 1997). This means AVL and APC technology only began to gain popularity within the last 20 years. Furthermore, only articles written in English are considered. Conference proceedings were included in this review since there were very few articles found regarding this topic.

The methodology involves searching some of the most comprehensive engineering and transportation academic databases. The databases that were used include Web of Knowledge, Compendex, and TRID. TRID is the world’s largest and most comprehensive bibliographic resource on transportation research information, containing more than one million records of references in the field of transportation research (TRID, 2017). The search is restricted to articles in engineering and transportation disciplines only. Once the search was complete, a second search was conducted by going through the references of the relevant articles found from the first search. The articles from the list of references are subjected to the same search criteria and restrictions as the first search.

Multiple searches were conducted within the ‘title’ search field. The main keywords that were used in every search were (Bus OR Streetcar OR Tram OR Transit). In the search regarding bus 7

bunching, the following words were used in separate searches in combination with the main keywords: bunching, bunch(ing) detection, disruption detection, incident detection, operational disruption, bunch prediction, headway delay, headway deviation, headway disruption, and headway variation. The same keywords are used in searches at all three databases.

2.1 Bunching in Public Transit

A sizable body of the transit literature has focused on bus bunching in terms of generating and proposing several holding strategies to reduce bunching once it has occurred. For example, Daganzo has developed several studies that provide theoretical holding techniques and other corrective actions to deal with bus bunching (Daganzo, 2009; Xuan, Argote, & Daganzo, 2011). Daganzo and Pilachowski proposed a control strategy whereby bus speeds are adjusted to maintain headways and consequently, reduce bus bunching (Daganzo & Pilachowski, 2011). Similarly, Moreira-Matias et al. and Liang et al. have developed different theoretical methods to handle bunching once it has occurred (Liang, Zhao, Lu, & Ma, 2016; Luis Moreira-Matias, Ferreira, Gama, Mendes-Moreira, & De Sousa, 2012).

Moreira-Matia et al. was one of the few who developed a prediction framework for bunching in real time (L. Moreira-Matias, Gama, Mendes-Moreira, & Freire de Sousa, 2014). Moreira- Matias suggested using machine learning to predict link travel times by using real-time travel link times. Using the predicted link travel times, headways can also be predicted. The headway residuals, defined to be the difference between the actual and estimated headway, are then used to develop a probability density function that describes the headway at the stops on a route. When a new headway value arrives for each stop, the probability for bus bunching to occur is calculated and a bus bunching score is given the stop. This framework was applied to bus routes in Portugal and resulted in predictions with high accuracy but mediocre precision. (Nair et al., 2015) developed a real-time bus bunching prediction system that was applied in a pilot program in Miami, USA. This system uses the abundance of archived AVL data and machine learning to predict bunching incidences based on the main concept that history repeats itself. The result of the bunching prediction ranges between 68-80% accuracy.

Despite these previous efforts, there is little that has been done on understanding the causes and factors that impact bus bunching. In fact, only a few studies can be found in the literature that 8

investigated bus bunching using statistical analyses. One of them is done by (Mandelzys & Hellinga, 2010), where they attributed late arrivals and early departures to fluctuating travel times between stops and dwell times (usually a measure of passenger volume) at stops. Long travel times and dwell times were found to cause late arrivals. Likewise, shorter travel times and short dwell time were found to cause early departure. These late arrivals and early departures occurring at previous stops and current stops were the events that provoked bus bunching to occur. These characteristics were also attributed to bus bunching in (Feng & Figliozzi, 2011; Shi An, Xinming Zhang, & Jian Wang, 2015). (Diab et al., 2016) developed a bus bunching model that was used to investigate several factors such as passenger volume, delay at the start, and their impact on the probability of bunching. In contrast to work that has been done regarding bus bunching, it is rare to find studies that explored both the impact of internal and external factors on streetcar bunching. The following subsections will explore the general efforts that have been done regarding streetcar performance and transportation applications on time to an event.

2.2 Streetcar Performance

Other researchers have focused on understanding the impacts of several factors on streetcar service performance, but not specifically on streetcar bunching. For example, Currie has generated multiple articles regarding the impacts of different factors on streetcar service performance. He explored streetcar safety (Naznin, Currie, & Logan, 2017), weather impacts (Mesbah, Lin, & Currie, 2015), and dwell times (Currie, Delbosc, Harrison, & Sarvi, 2013; Currie, Delbosc, & Reynolds, 2012) and compared streetcar performance in different countries (Currie, Burke, & Delbosc, 2014; Currie & Shalaby, 2007). He also discussed how transit signal priority (TSP) handled bunched streetcar vehicles (Currie & Shalaby, 2008). (Ling & Shalaby, 2005) developed a reinforcement learning approach to control streetcar bunching. With this very little research on streetcar service operations, and even lesser on streetcar bunching, a better understanding of the streetcar service bunching is needed. With the availability and the accuracy of AVL data, it is now possible to investigate streetcar bunching, while isolating the effects of different influential variables on the service.

2.3 Transportation Applications on Time to an Event

Time to an event studies are most commonly utilized in the field of medicine to analyze time to death. In engineering applications, time to an event studies are used to analyze time to failure of a system or product. However, time to event studies has recently grown in transportation applications. In the transportation realm, this type of analysis has been commonly applied to the infrastructure aspect of transportation such as road pavement, vehicle tires, and rail tracks. The studies of pavement failure are popular because of the growing area of paved surfaces and the high costs of maintenance and resurfacing. (Hong & Mikhail, 2016) studied the impact of pavement composition, vehicular traffic, and the environment had on pavement lifetime. Others conducted studies on the impacts of maintenance and rehabilitation methods on pavement service life (Chen, Williams, Marasinghe, Omundson, & Schram, 2014; Dong & Huang, 2015). Dong et al. found that preventative treatments on the pavement would not be effective in poor environments and conditions. (Wu, 2011) studied tire failures and revealed that tires are likely to age faster under higher loads, mileage, and temperatures.

Bus travel times have also been predicted using this type of analysis. In this case, the time to an event would be the time to reach a downstream stop. (Yu, Wood, & Gayah, 2017) used operational and weather data in real time to predict travel time to a future stop and found that a survival analysis model provided travel time predictions with smaller variances than a linear regression model. Yu et al. was able to predict travel times as well as the uncertainty associated with these travel times with the idea that this would help control or alleviate user expectations.

Time to an event modeling has been used in the aviation industry to study unique topics such as life expectancy of airline pilots after retirement and commitment of airlines to alliances. Besco et al. studied the life expectancy of pilots after the mandatory retirement age of 60 in the U.S. to find if there was evidence for the assumption that there was a lowered life expectancy for airline cockpit crews (Besco, Sangal, Nesthus, & Veronneau, 1995). From the sample set, Besco et al. found that retired airline pilots had a life expectancy of 5 or more years compared to the U.S. general population of 60-year-old white males. (Gudmundsson & Rhoades, 2001) used this modeling to understand which factors contributed to the commitment or separation of airlines to these airline alliances.

One of the more popular topics where survival analysis has been applied is in traffic incident duration. Weng et al. and Louie et al. used survival analysis in incident duration in public transit. Weng et al. found factors such as power and signal cable failure as well as incidents that involve a casualty contribute to longer subway incident durations (Weng, Zheng, Yan, & Meng, 2014). Survival analysis was applied to the disruption duration in the TTC’s subway system (Louie, Shalaby, & Habib, 2017) and found that train type and time of day were some of the statistically significant factors on incident durations. Giuliano, Chimba et al, and Zhang et al. all conducted studies on incident durations on freeways. They all wanted to understand the impact of different factors on incident durations; some wanted to understand the impacts of lane closures and time of day, while others had specific factors of interest such as abandoned and disabled vehicles left within the roadway (Chimba, Kutela, Ogletree, Horne, & Tugwell, 2014; Giuliano, 1989; Zhang et al., 2014). All of these studies concluded with factors that were statistically significant to the incident durations that they were studying using the time to an event modeling.

The application of survival analysis in streetcars or bunching incidents has not been found in the literature. Thus, this type of analysis was not found in the even narrower field of streetcar bunching incidents. The absence of research on estimating the time until a bunching incident occurs and the factors that impact this provided the opportunity for this study to be conducted. Survival analysis seemed to provide promising results in many studies mentioned above and has much potential for its application in this study to investigate the impact on the time it would take for two vehicles to bunch.

3 Study Context and Data 3.1 Study Context

The City of Toronto reported a population of 2.8 million in 2015 and has projected the population to reach 3.7 million by 2041. To serve this large population, the TTC operates a bus, subway, and streetcar networks. The TTC streetcar system is one of the largest in North America. In 2016, the TTC operated 11 streetcar routes, covering 338 km (TTC, n.d.-b). Approximately 300,000 passengers ride the streetcars on a typical weekday. Table 1 lists the different streetcar routes and their daily ridership. The routes are ranked among all surface routes, which includes both bus and streetcar routes.

Table 1 Average Daily Streetcar Ridership

All-Day Typical Business Day Ridership for Surface Routes (as of December 31, 2016) Rank Route # Route Name All-Day Ridership 1 504 King 64,579 5 510 Spadina 43,804 6 501 Queen 43,464 8 506 Carlton 39,601 9 512 St. Clair 38,113 12 505 Dundas 32,410 28 511 Bathurst 21,433 54 509 Harbourfront 9,903

The TTC operates streetcars every ten minutes or better. Headways range from 2 to 10 minutes with the average being approximately 4 minutes on weekdays during the morning and afternoon peak periods. Overnight streetcar service is provided for 4 of the routes at much longer headways. Headways vary on different routes as well as throughout the different time periods of the day. The TTC aims to adhere to its scheduled trips rather than its scheduled headways.

The TTC has a fleet of approximately 250 streetcar vehicles using a combination of three different types which are listed in order of increasing capacity: a standard vehicle (Canadian Light Rail Vehicles - CLRV), an articulated vehicle (Articulated Light Rail Vehicles - ALRV), and a new low-floor articulated vehicle (Flexity Outlook) that was just introduced in 2014. The

ALRV looks similar to the CLRV but uses two light rail vehicles connected to each other using articulation. The vehicles used in an ALRV are shorter than the CLRV and thus, only provide 1.5 times more capacity than the ALRV. The Flexity, however, looks completely different and has a larger capacity than the other two vehicles. One key feature of the Flexity that may have an impact on bunching is that it has a low floor. This feature may assist passengers in alighting and boarding and is likely to reduce dwell times especially during peak times. This may also influence the likelihood of bunching incidents. Table 2 presents the details of the TTC’s fleet and the capacities of each type of vehicle.

Table 2 TTC Fleet Details

Quantity Seated Maximum Vehicle Type in Fleet Capacity Capacity

CLRV 165 46 74

ALRV 43 61 108

Flexity 33 70 130

The majority of streetcar routes operate on a shared right of way. However, there are a few that operate on a dedicated right of way and some on a hybrid right of way with portions that are dedicated and others shared. Stops on streetcar routes are placed all on one side (either near or far side) or use a combination of stop placements. Three routes (Route 505, 506, and 511) have all of its stops placed on the near side in both directions, while Route 512 has all of its stops placed on the far side in one direction. The other routes have stops placed on both the near side and far side. Table 3 below summarizes the key characteristics of the streetcar routes included in the study. AM peak headways, based on the TTC January 2016 Service Summary (TTC, 2016), have the shortest headways out of all the time periods. A full matrix of the headways of all streetcar routes in all time periods can be found in the appendix.

Table 3 Streetcar Route Characteristics

Route Route Weekday AM Peak Right of Average Stop Stop Vehicles Used # Length (km) Headway (mins) Way Spacing (m) Placement 501 48.86 ALRV/CLRV 5 Shared 256.12 Combination 504 25.62 ALRV/CLRV 4 Shared 234.70 Combination 505 21.47 CLRV 6 Shared 233.91 All near side 506 29.63 CLRV 4 Shared 245.88 All near side 509 5.95 CLRV/Flexity 5 Exclusive 332.48 Combination 510 12.33 Flexity 3.5 Exclusive 331.06 Combination 511 7.03 CLRV 4 Shared 298.22 All near side 512 13.99 CLRV 3 Exclusive 271.31 Combination

The TTC defines service hours differently for weekdays compared to weekends. This is important to note while analyzing and comparing the results between weekdays and weekends. The difference between the service hours for weekdays and weekends is that the early morning and midday periods are shorter and the afternoon period is longer during the weekend. Table 4 presents the specific service hours. The duration between 1:00 AM and 1:30 AM as well as 5:30 AM and 6:00 AM that has been undefined. These undefined periods are considered to be overnight periods in this study.

Table 4 Service Hours Defined by the TTC

MONDAY TO FRIDAY SATURDAY AND SUNDAY Morning peak period 06:00 – 09:00 Early morning 06:00 – 08:00 Midday 09:00 – 15:00 Morning 08:00 – 12:00 Afternoon peak period 15:00 – 19:00 Afternoon 12:00 – 19:00 Early evening 19:00 – 22:00 Early evening 19:00 – 22:00 Late evening 22:00 – 01:00 Late evening 22:00 – 01:00 Overnight 01:30 – 05:30 Overnight 01:30 – 05:30

3.2 Data Description

Although the TTC currently operates 11 routes, when the data were collected back in 2016, the TTC was only operating 10 routes. The additional route was added shortly after the data collection had occurred. More than 6 million global positioning system (GPS) observations were collected from the TTC’s AVL system for the 10 streetcar routes between January 24 and 30 in

2016. The data were collected for one whole week starting on Sunday and ending on Saturday to capture the differences between weekdays and weekends. The selected week had mild and clear weather with minimal streetcar track construction, closures or service diversions. Route 509 and 511 share tracks for a portion of the trip that experienced track construction during data collection. This simply resulted in the terminal station being relocated to the closest loop for these two routes. TTC’s AVL system records vehicle location at 20-second intervals. A sample of the raw data is shown in Table 5. Each row in the data is considered a data point or an observation. Each observation in the raw data contains information about the date, time of the day given in seconds, route number, vehicle number, run number, along with the latitude and longitude. Every route has at least one loop where the streetcar is able to turn back into the opposite direction prior to the terminal station. It is possible that some of the routes have multiple branches. However, the only route where the branch was identified in the data was for Route 501, which is the longest streetcar route in Toronto.

Table 5 Sample of Raw Data Provided by the TTC

MESSAGE_DATETIME ROUTE # VEHICLE # RUN # LATITUDE LONGITUDE 2016-01-24 7:38:40 511 1037 99 43.666634 -79.411118 2016-01-24 7:39:00 511 1037 99 43.666634 -79.411118 2016-01-24 7:39:20 511 1037 99 43.666634 -79.411118 2016-01-24 7:39:40 511 1037 99 43.666634 -79.411118 2016-01-24 7:39:40 511 8145 2 43.666382 -79.411064 2016-01-24 7:40:00 511 1037 99 43.666634 -79.411118 2016-01-24 7:40:00 511 8145 2 43.666382 -79.411064 2016-01-24 7:40:20 511 1037 99 43.666634 -79.411118

The TTC provided data for the following routes: 501, 502, 503, 504, 505, 506, 509, 510, 511 and 512. However, after assessing the data, it was determined that route 502 and 503 would not be included in the analysis since it was being operated solely by buses due to a shortage of streetcar vehicles at the time. Since this study focuses on streetcar bunching, the data from Route 502 and 503 would not be useful or relevant if it was included. Route 514 did not exist at the time of data collection. AVL data were also collected for the overnight routes. Overnight streetcar service is only available for the 501, 504, 506, and 510 routes. Figure 2 shows a map of the routes that were included in the study.

Figure 2 Map of Streetcar Routes Included in Study

Data for external factors that were considered in the study such as vehicular traffic volumes, number of transit signal priority (TSP), and number of through intersections were taken from the City of Toronto’s Open Data catalogue. Other factors that were extracted from the data catalogue include number of right turns, number of left turns, pedestrian volume, and number of signalized approaches in each segment. Vehicular traffic and pedestrian volumes are 8 hour counts conducted at signalized intersections typically between 7 AM and 6:30 PM (City of Toronto, n.d.). The remaining factors are simply counts of infrastructure assets, which are then merged with the segments of each route, which is discussed below.

3.3 Data Processing

Each route was divided into multiple segments according to the stop location, with an average distance of 265 meters each, using ArcGIS. Each segment is assigned a unique identifier number and contains related information such as branch number and route number. The dataset contained many erroneous points and therefore, only GPS points that were within 200 meters of the centerline of the road were kept. This was completed by creating a buffer around each route

in ArcGIS. The latitude and longitude were converted into x and y coordinates so that distance could be calculated later on. The remaining GPS points were then assigned to a segment of the route based on the shortest distance to the road centerline. Thus, each observation now contained data regarding the segment it was located in and the corresponding segment characteristics. This data set was exported to Excel for further processing. Each route’s dataset was contained in separate files. This was done to keep minimize the file size and assist in processing as it would require tremendous amounts of computer processing power to handle millions of data observations at once.

Excel was used to extract useful information from the GPS points and assign GPS points into trips. This process was a very time-consuming procedure. Since this procedure had to be applied to multiple datasets, it was coded into visual basic within Excel to expedite the process and ensure that the same processing procedure was being applied to all of the data. The following steps were coded into a macro:

1. Extract the time and date into individual cells for easier and quicker manipulation 2. Convert time into number of seconds in a day. Thus, time ranges from 0 to 86400 seconds. 3. Sort data in chronological order by date, vehicle number, and time. 4. Remove any duplicate data points. These points would have to have the same date, time, vehicle number, run number, and coordinates. 5. Assign a unique identifier number to each data observation. 6. Calculate the start and end time in each segment. 7. Calculate distance (in meters) travelled between each observation using the distance

formula, . This distance formula is based on Pythagorean’s Theorem and is typically applied on flat surfaces. This formula was used in this study despite the fact that the Earth is curved due to the fact that each observation is 20 seconds apart and the distance that could be travelled within 20 seconds by a streetcar would be small. It is likely that if the curve of the Earth was considered in the calculation of the distance travelled, the difference would be negligible. Thus, the curvature of the Earth was not taken into account in the calculation in this study. 8. Calculate time (in seconds) between each observation. 17

9. Calculate speed travelled (in km/h) between each observation. 10. Remove any observations where the time difference is more than 10 minutes or less than 0 minutes and any observations where the speed is greater than 70 km/h or less than 0 km/h. It is unlikely that a streetcar would be able to travel over 70 km/h within the city, especially when the majority of the routes operate on a shared right of way. Moreover, the typical speed limit is 50 km/h on local roads in Toronto. If there is more than a 10 minute difference between two consecutive observations, it is likely due to the fault of AVL system. 11. Each segment travelled by a different vehicle and run number is assigned a unique segment ID number. The segment ID number will assist in sorting the observations into complete trips. 12. Using a pivot table in Excel, trips are identified. A trip is considered to have consecutive segment numbers and do not necessarily have to start and end at the segments that contain the terminal station. Trips that contained only 1 or 2 segments were removed as these would be too short to be considered a trip. 13. The segment where the layover occurs is identified as well as the number of segments travelled and direction. For routes that travel in the east-west direction, trips travelling in the eastbound direction are assigned a value of 0 and westbound direction a value of 1. Similarly, trips travelling in the southbound direction are assigned a value of 0 and in the northbound direction a value of 1. 14. Total trip duration, trip start time and trip end time is calculated for each trip.

Both models required the identification of bunching incidences in the data. A bunching incident is defined to be when the actual headway between two vehicles is less than half of the scheduled headway. Bunching incidents were isolated at the segment level by comparing the time that vehicles left the segment. Streetcar vehicle lengths range from 15 to 30m. When there are two or more vehicles bunched, the length of the total vehicles would range from at least 30-60m or more. This makes the segment level a reasonable level of precision for bunching incident identification. The distance between stops is very similar to the average length of each segment and thus, this could be considered bunching at the stop level. Bunching is identified using the following steps:

1. The data are sorted by day, segment and then time. Within a segment, the time between consecutive vehicles (headway) is calculated. 2. The actual headway is compared to half of the scheduled headway for the time period of travel. If it is less than half of the scheduled headway, it is then extracted and added to a new database of bunching incidents. When a bunching incident involves multiple vehicle or headways, only the first pair of vehicles or first headway that is less than half of the schedule headway is extracted and added into the new database.

The data on external factors taken from the City of Toronto’s Open Data catalogue were consolidated for each segment of the route by fellow researcher, Bo Wen. For example, there can be many signalized intersections in one segment. Each segment will thus have information from multiple intersections.

The scheduled headways were extracted from the TTC January 2016 service summary. The scheduled departures of the streetcar network was downloaded from the General Transit Feed Specification (GTFS) for January 2016 and used to compare actual departures with scheduled ones. This was later used to calculate time deviation between actual and scheduled departure times.

3.4 Variable Definition

Table 6 lists all of the variables that were considered in this study. All of the variables were tested in the models but may not have been kept in the model due to insignificance or collinearity. For example, route length and average stop distance were not used in either model since the route number captures these characteristics already. Studying bunching incidences can be complex due to the fact that multiple vehicles can be involved in one incident. Therefore, it was decided in this study to consider characteristics of both vehicles involved in a bunching incident. This is further discussed in chapter 4, but it is mentioned here to be able to understand the variables that were considered for the study. The first vehicle in a bunching incident is considered as the lead vehicle (L) and the second vehicle is considered the following vehicle (F). Information about the vehicle ahead of the lead vehicle is also used to better understand the bunching phenomenon; this vehicle is designated as the ‘lead+1 or L+1’ vehicle.

Table 6 Independent Variables Considered in Study

Control Internal External

Time Period Right of Way Number of Left Turns Route Length Number of TSP Number of Right Turns Average Stop Distance Stop Placement Number of Through Intersections Route # Following & Lead Headway Number of Signalized Ratio Intersections Trip Direction Lead & Lead +1 Headway Ratio Number of Pedestrian Crossings Weekday/Weekend Vehicle Combination Type Average Vehicle Volume Cumulative Transfer Points Scheduled Headway Average Pedestrian Volume Scheduled Headway2 Distance from Union Station Short-turn trip Binary variables representing deviation of Following and Leading vehicle from scheduled headway Interaction variable between route and binary deviation variable

Table 7 below summarized the variables used in both models and the values that are assigned to the variables.

The weekday variable is a simple binary variable that indicates whether the trip occurred on a weekday or weekend. A weekend trip is assigned a value of 0 and a weekday trip is assigned a value of 1. Since the data were collected over 1 full week starting with Sunday and ending on Saturday, it is obvious that the majority of the trips in this study are weekday trips.

As can be seen in the map of the routes (Figure 2), not all routes travel solely in one direction. There are a few routes that travel in an east-west direction as well as a north-south direction. The trip direction variable is a binary variable, which can be assigned a value of 0 for east or south bound travel or a value of 1 for west or north bound travel.

The vehicle combination variable captures the difference in capacity between consecutive vehicles was also used. Many of the routes operate only one type of vehicle. However, there are some routes that utilize multiple vehicle types, especially now that the TTC is in the process of updating its fleet. The major difference between the different vehicle types is the number of

passengers each type can hold. This categorical variable is assigned a value of 0 when the vehicle type of both vehicles in a headway is the same. When the following vehicle has a larger capacity than the leading vehicle, the variable is assigned a value of 1. When the leading vehicle has a larger capacity than the following vehicle, the variable is assigned a value of 2.

The time period variable is a categorical variable. The TTC has defined six different time periods for both weekdays and weekends. Because the majority of the routes do not have overnight service and service during the late evening becomes less frequent, trips during these periods have been combined with the early evening period for a combined period, labeled as simply ‘evening’ period. This reduces the total number of periods from 6 to 4. The time periods are assigned values in chronological order from 1 to 4.

The TTC operates 4 overnight routes and they are all included in the study. These routes numbers use the 300 series instead of the 500 series. In this study, the routes are not differentiated between overnight and the rest of the day. The 500 series route numbers are used in this study. The route number variable captures the route length, right of way, and average stop distances, which are unique to each route.

The following & lead headway ratio is a ratio of the actual headway between the following and lead vehicles and the scheduled headway. A similar ratio is calculated for the actual headway between the lead and lead+1 vehicles and the scheduled headway. These ratios capture the deviation from scheduled headway. Scheduled headway and the squared term of the scheduled headway are also variables used in this study. The squared term of the scheduled headway is included to account for possible non-linear relationships if such a relationship existed.

Stop combination is a binary variable that captures the location of stops in each route. It is generally unlikely that stops are all placed on one side, either near or far. However, it was found that there are routes that do have stops placed all on one side. The majority of the routes typically use a combination of far and near sides though. If a combination of stop placements is used, this variable is assigned a value of 1. If all of the stops are placed on one side only, the variable is assigned a value of 0.

Many of the external variables are discrete variables. These variables were only used in the second model, which focused on understanding in detail the impact of the factors on the time to the initial bunching incident. The cumulative TSP, signalized intersections or approaches, and pedestrian crossings represent the number of each between the terminal station and the bunching location. The cumulative value of these factors was used instead of individual values at each segment because the cumulative effects are likely to show the real impact of such a variable on the time to bunching. (Vehicular) Traffic volume is captured as an ordinal variable and is measured during an 8 hour count period. The level of traffic is determined as a percentage of the highest traffic volume. The lowest level of traffic falls between 0 and 33% of the highest volume and is given a value of 1. A medium level of traffic falls between 34 and 66% and given a value of 2. The highest level of traffic is assigned a value of 3 and falls between 67-100% of the highest traffic volume. The traffic volume ranges from 0 to 9300 vehicles. The range of the lowest level is 0 to 3100, medium level is 3101 to 6200, and highest level is 6201 to 9300.

Short-turning a vehicle is a method that the TTC utilizes to address bunching once it has occurred. Short-turning is known to occur frequently on some of the streetcar routes. It may be unintuitive to include a variable capturing whether a vehicle was short-turned or not since short- turning is applied to help treat bunching incidences and thus, is assumed to be beneficial in reducing the likelihood of bunching. The short-turning binary variable is included to confirm this assumption. A value of 1 is assigned if the lead vehicle is short-turned or a value of 0 is assigned otherwise.

A set of headway deviation dummy variables were used in the first model to reveal the impact of different combinations of headway deviations on bunching. Headway deviation is measured at the terminal and is categorized into three classes: shorter than scheduled headway, same as scheduled headway, or longer than scheduled headway. Headways that fall between 80-120% of the scheduled headway are defined to be the same as scheduled headway or on time. Headways that are less than 80% are defined to be shorter than scheduled headway and those greater than 120% are longer than scheduled headway. These values were arbitrarily chosen but seemed logical as some tolerance is required in defining on-time performance. Shorter headways can also be interpreted as vehicles leaving earlier than expected. Likewise, longer headways can be interpreted as vehicles leaving later than expected. With 3 different classes, there are 9 different 22

possible pairs: both are shorter than schedule headway; both are on time; both are longer than scheduled; one is shorter and one is longer; one is shorter and one is on time; and lastly one is longer and one is on time. The last three pairs occur twice each since the leading headway could be shorter and the following headway is longer or vice versa.

An interaction variable between Route 501 and the short/short headway deviation combination is included in the model due to the fact that route 501 experiences a lot of short/short headway deviations and skews the results of the short/short variable. The variable will capture the variances in the 501 route.

Table 7 Variable Definitions Variable Description Name

Weekday/Weekend Weekend (0) or weekday (1) Trip Direction Eastbound/Southbound (0) or Westbound/Northbound (1) Vehicle Following & leading are same vehicle type = 0 Combination Following vehicle capacity is larger than leading vehicle capacity = 1 Following vehicle capacity is smaller than leading vehicle capacity = 2 Time Period AM Peak=1, Midday=2, PM Peak=3, Evening = 4 Route # Streetcar route number; it captures route characteristics such as route length, right of way and average stop distance Following & Lead Ratio of actual F, L vehicle headway to the scheduled headway Headway Ratio Lead & Lead+1 Ratio of actual L, L+1 vehicle headway to the scheduled headway Headway Ratio Scheduled Headway Scheduled headway between vehicles in minutes Scheduled Headway2 Squared value of scheduled headway

Cumulative TSP Number of intersections equipped with transit signal priority between the terminal and bunching location Stop Combination Stop placement at route level: if same stop (all near or all far side) placement (0), Combination of near and far side stops (1) Cumulative Number of pedestrian crossings between the terminal and the bunching location Pedestrian Crossing Cumulative Number of signalized intersections between the terminal and the bunching Signalized location Approaches Variable Description Name Traffic Volume Traffic volume is defined to be a proportion of the highest volume. Low volume 23

(0-33% of highest volume) (0), medium volume (34-66%) (1), high volume (67- 100%) (2) Lshort Leading vehicle is not short turned from the opposite direction (0), leading vehicle is short turned (1) Short/Short Actual headway between F, L is shorter than scheduled headway and actual headway between L, L+1 is shorter than scheduled headway at terminal Short/On Time Actual headway between F, L is shorter than scheduled headway and actual headway between L, L+1 is the same as scheduled headway at terminal Short/Long Actual headway between F, L is shorter than scheduled headway and actual headway between L, L+1 is longer than scheduled headway On Time/Short Actual headway between F, L is the same as scheduled headway and actual headway between L, L+1 is shorter than scheduled headway at terminal On Time/On Time Actual headway between F, L is the same as scheduled headway and actual headway between L, L+1 is the same as scheduled headway at terminal On Time/Long Actual headway between F, L is the same as scheduled headway and actual headway between L, L+1 is longer than scheduled headway at terminal Long/Short Actual headway between F, L is longer than scheduled headway and actual headway between L, L+1 is shorter than scheduled headway at terminal Long/On Time Actual headway between F, L is longer than scheduled headway and actual headway between L, L+1 is the same as scheduled headway at terminal

Long/Long Actual headway between F, L is longer than scheduled headway and actual headway between L, L+1 is longer than scheduled headway at terminal Route 501 x An interaction variable between trips that belong to Route 501 and also Short/Short experience shorter than scheduled headways

4 Modeling Framework

The objectives of this study is to understand the general factors that impact the likelihood of streetcar bunching as well as to investigate in more detail the external and internal factors that impact the time to bunching. The Transit Capacity and Quality Service Manual (TCQSM) has defined bunching to be where two or more vehicles on the same route arrive together or in close succession, followed by a long gap between vehicles. More specifically, the TCQSM defines bunching using the coefficient of variation of headways; this coefficient is calculated by dividing the standard deviation of headways by the mean of headways. When the coefficient of variation of headways is larger than .40 and scheduled headway is 10 minutes or less, bunching is likely to happen (National Academies of Sciences, 2013). Since there is no defined standard for the headway threshold, bunching has been defined differently by researchers and transit operators around the world. Some use absolute values of time while others use a percentage of scheduled headway. For example, the San Francisco Municipal Transit Agency has defined bunching to be when the following vehicle arrives less than 2 minutes after the leading vehicle or less than 1 minute if the route has a scheduled headway of 5 minutes or less. (Feng & Figliozzi, 2011) used an arbitrary value of three minutes. In this study, a bunching incident is defined to be when the actual headway between two vehicles is less than half of the scheduled headway. This was chosen since all of the streetcar routes used in this study are run quite frequently (equal or less than 10 minute headways) and vary throughout the day. Figure 3 is a simplified illustration of how a bunching incident is detected at the segment level. Actual headways are calculated at the end of a segment and then compared to scheduled headways.

Figure 3 Bunching Incident Definition 25

It is important to note that the unit of analysis in this study is the headway between consecutive vehicles. In other words, this study aims to determine the factors that cause a headway to become less than half of the scheduled headway. It also aims to understand which factors contribute to the time it takes for the actual headway to become half of the scheduled headway. This is noted because the data are organized in this manner. After the data have been processed, the identified bunching incidences are stores in a new file. Each row in the spreadsheet gives details on the characteristics related to that headway such as the vehicles involved with that headway, time of day, location, and more.

To assist in understanding the dynamic factors that influence the streetcar bunching phenomenon, information about the previous headway of a bunching occurrence is also used. The headways (or the vehicles involved with these headways) are labeled as shown in Figure 4 to better understand the methodology. The vehicle in question is labeled as Following (F) vehicle, the vehicle in front of it is labeled as a Leading (L) vehicle, and the vehicle prior to the Leading is labeled as Leading+1 (L+1) vehicle. If a bunching incident was observed between F and L, the headway between L and L+1 was considered as a predictor in the models. The underlying rationale is that if the Leading+1 vehicle is leaving the terminal early and the Leading vehicle is slightly late, the latter vehicle will likely pick up more passengers in addition to its normal load, leading to more delays for itself. Meanwhile, the Following vehicle (vehicle in question) will find fewer passengers to serve along the route even if it is leaving the terminal on time, which will likely increase the odds of bunching with the Leading vehicle at a point down the line. Furthermore, the time to the initial bunching incident is defined to be the time it takes from the terminal station for the first bunching incident (as opposed to subsequent bunching incidents) in a trip to occur. Since streetcars cannot overtake one another, the study focuses on the location when any pair of consecutive streetcars first form a bunch on the route. The time is measured from the instant the Following streetcar leaves its route terminal to the instant it first catches up with the Leading streetcar.

Figure 4 Vehicle Naming System

The following sections discuss the models used in this study: a binary logistic regression model and an accelerated failure time model. Each model addresses different aspects of bunching and allows for the research objectives to be addressed. The binary logistic regression model estimates the odds of bunching while the AFT model estimates the time to bunching. Together, the results from both models can contribute to building a mathematical model that can predict the likelihood of bunching and how quickly bunching will occur.

4.1 Binary Logistic Regression Model

The first model is a binary logistic regression model that investigates the effects of different factors on the likelihood of streetcar bunching. This model was chosen because the dependent variable of interest, whether the headway will bunch or not, is dichotomous. Linear regression would not be an appropriate choice since the dependent variable is not a continuous variable, but a binary one. The logistic regression model does not assume that there is a linear relationship between the dependent and independent variables. Furthermore, it assumes that each observation is independent and that the error is not normally distributed. Logistic regression uses a different method to estimate the parameters compared to linear regression and thus, is able to give more accurate results. The binary logistic regression model helps to answer the following question: will it or will it not bunch?

Binary logistic regression estimates the odds of a successful outcome, typically coded as 1, based on one or more explanatory variables (The Pennsylvania State University, 2017). The failed outcome is therefore coded as 0. In this case, if bunching occurred, it was coded as 1. Otherwise, it was coded a value of 0. It is important to note that this method estimates the odds not the probability. The probability of an event can be simply described as the number of times 27

this event can happen divided by the total number of outcomes. In contrast, the odds is the number of times the event can occur divided by the number of times the event does not occur (Wasserstein, 2014). It is commonly reported as a ratio. To give an example to illustrate the difference, the probability of getting heads on a coin toss is ½ or 0.5, while the odds of getting heads is 1:1. The general model of binary logistic regression is described by the following equation:

Where  = odds ratio

 is the binary dependent variable, Y=1 if it is a success and Y=0 if it is a failure

 is the set of independent variables, which can be discrete or continuous or a combination of the two types

The goodness of fit of the model is measured by the Nagelkerke R square value. The model was run using SPSS software for its ease of use. If the odds ratio is greater than 1, this indicates that the odds of bunching will increase as the independent variable increases. If the odds ratio is less than 1, the odds of bunching are decreased as the independent variable increases.

Many variables were tested but were eliminated from the model due to insignificance such as schedule deviation for both the following and leading vehicle as well as headway ratios. Squared terms of some independent variables were used to account for a possible non-linear relationship between each variable and the dependent variable if such a relationship existed. In addition to the route number, direction, day of the week, and time of day variables, other variables that were utilized in the model included the short turn variable, the variable representing the vehicle type combinations, the scheduled headway, and the headway deviation dummy variables.

Prior to actually performing the analysis, there are expectations of how the various factors will impact the odds of bunching. It is expected that a weekday should increase the odds of bunching since service on weekdays are much more frequent and number of passengers should be higher due to commuters. Short turning is expecting to reduce the odds of bunching. Otherwise, the strategy of short turning would be rendered ineffective. Despite the fact that all routes are

operated on a 10 minute or less headway, it is also expected that specific routes will increase the odds of bunching compared to other routes. The routes that are expected to increase the odds of bunching include the 504 and 510 due to their shorter headways compared to the other routes as well as the 501 for its long route. When the leading vehicle has a higher passenger capacity than the following vehicle, it is expected that this situation should increase odds of bunching. This is because the leading, larger vehicle will be able to pick up many more passengers, which will leave few passengers for the following vehicle. Therefore, the following vehicle will have shorter dwell times and will be able to catch up with the leading vehicle more easily. Both AM and PM peak periods are expected to increase the odds of bunching due to the frequent service and high passenger demand during these time periods. It is obvious that the longer the scheduled headway, the lower the odds of bunching should be. Lastly, the expectation for the headway deviation dummy variable is that when the following headway is shorter than the scheduled headway, this should increase the odds of bunching.

4.2 Survival Analysis

The reason survival analysis was considered to address the study of time to bunching was due to the fact that other attempted models proved to be inadequate. A linear regression model was first used to model time to bunching. However, the fit was poor and resulted in a very low R2 value. An ordinal logit model was also attempted, but similarly, this resulted in a very low ρ2 value. Survival analysis was then considered.

Survival analysis is a statistical method where the outcome variable of interest is the time until an event occurs (Kleinbaum & Klein, 2005). This time until an event occurs is usually referred to as survival time. The event in question can vary from negative events such as death or failure to positive events such as returning back to work after a surgical procedure. In this study, the event is the time to bunching. In survival analysis, it is also important to understand what a hazard function is. The hazard function can be described as the probability that an event occurs at time T given that an individual has survived up to time T. It can be considered as the instantaneous event rate, analogous to the concept of instantaneous velocity.

There are two different but popular models used in survival analysis: the Cox proportional hazards (PH) model and the accelerated failure time model. The Cox PH model assumes that the 29

effect of the independent variables acts multiplicatively on the hazards while the AFT model assumes that the effect of the independent variables acts multiplicatively on the survival time. The model type used in this second part was decided based on this key assumption. As this study is interested in understanding the effects of independent variables directly on the survival time, the AFT model was chosen to be applied. Another way to interpret the assumption of the AFT model is that it assumes that the effect of the independent variables accelerate or decelerate the survival time by some constant referred to as the acceleration factor. The AFT model helps to answer the following question: If it bunches, which factors increase or decrease the time to bunching?

The AFT model is estimated using the following equation (Jenkins, 2008):

Where  T is survival time  β is a vector of parameters  z is the error term.

This model assumes a linear relationship between the log of the survival time and the independent variables, X. This equation can be rewritten as:

= u

Where  μ =  u = z/σ. u determines which sort of AFT model describes the distribution of T.  σ is a scale factor related to the shape parameter of the hazard function

Some of the most commonly used AFT models are the exponential, weibull, loglogistic, lognormal, and generalized gamma. Each of these models has different scale factors as well as different assumptions. A simple way of interpreting the parameters of the model is to look at whether it is a negative or positive value. A negative coefficient indicates an accelerated time to

bunch (failure) or a reduction in the survival time. A positive coefficient would indicate the opposite: a decelerated time to bunch or an increase in survival time. The acceleration factor is determined by exponentiating the coefficient, eβ (Kleinbaum & Klein, 2005).

This model was estimated using Stata since survival models cannot be specified in SPSS. The main difference between the two softwares is that Stata uses a coding interface instead of a graphic interface, which is used by SPSS.

This AFT model was used to explore the impact of both internal and external factors on the time to the first bunching incident for pairs of successive streetcars. The time to bunching is calculated from the time the following vehicle leaves the terminal to the time the following vehicle first bunched with the leading vehicle. In this model, only bunched trips were used. External factors such as traffic volume and the number of signalized intersections were included in this model. The internal factors discussed above were also included in the model. The variables utilized in the final model include the following: day of week, trip direction, route number, vehicle type combination, scheduled headway and its squared value, headway ratio between the following and leading vehicles, headway ratio between the leading and lead+1 vehicle, number of cumulative TSP, stop placement combination, number of cumulative pedestrian crossing, number of cumulative signalized approaches, and lastly, the vehicular traffic volume.

It is expected that a weekday would accelerate time to bunching due to the more frequent service than on weekdays. Similar to the bunching odds model, it is expected that during both AM and PM peak periods, bunching will take a shorter time to occur because of the decreased headways during these periods. When the leading vehicle has a higher passenger capacity than the following vehicle, this should decrease the survival time. Furthermore, shorter headways and low headway ratios should accelerate time to bunching. In regards to the external variables, it is expected that the number of cumulative TSP will accelerate the time to bunching since it will give streetcar vehicles priority and thus, allow them to catch up to each other more easily especially on exclusive right of ways. The number of pedestrian crossings, signalized intersections, consistent stop placement, and high vehicular traffic volume is expected to delay time to bunching since these factors should present interruptions between vehicles.

5 Model Results and Discussion 5.1 Descriptive Statistics/Data Trends

Table 8 shows the mean and standard deviation of all of the variables used in both models in this study. The average scheduled headway was 5.24 minutes with a standard deviation of 1.98 minutes. The average stop distance is 269 meters with a deviation of 35 meters. The majority of these variables is not continuous and is actually categorical. This should be taken into consideration when trying to interpret the statistics of these variables.

Table 8 Descriptive Statistics of All Variables Used in Models

Variable Mean Standard Deviation Weekday 0.77 0.42 Trip Direction 0.51 0.50 L Short Turned 0.11 0.32 Vehicle Combination 0.28 0.45 Time period 2.68 1.10 Scheduled Headway 5.24 1.98 Short/Short .36 .480 Short/On Time .07 .252 Short/Long .16 .365 On Time/Short .07 .258 On Time/On Time .03 .178 On Time/Long .05 .222 Long/Short .12 .322 Long/On Time .05 .212 Long/Long .09 .289 Route 501 x Short/Short .19 .393 FL HeadRatio 91.77 175.40 LL1 HeadRatio 124.86 436.21 Cumulative TSP 15.46 11.50 Stop Combination 0.77 0.42 Cumulative Pedestrian Crossing 6.27 4.98 Cumulative Signalized Approaches 69.81 58.40 Vehicle Volume Category 1.43 0.56 Average Stop Distance 269.12 35.15 Route Length 10625.85 4435.23 Right of Way 0.69 0.46 Number of Trips 30492

Table 9 shows the summary statistics of the trips used in the study and the percentage of bunched headways per route. If a headway experiences bunching in any segment (i.e. less than half the scheduled headway), it is considered a bunched headway. In total, about 30,500 headways were included in the analysis. The majority of the analyzed headways occurred on weekdays. Out of the total number of headways, approximately a quarter of them were involved in a bunching incident. Route 504, with the highest ridership in Toronto (65,000 riders per day), experiences the highest number of bunched headways (38.9%). The evening time period has the most number of headways due to the fact that it is a combination of two time periods (Late Evening and Overnight) as mentioned above.

Table 9 Summary Statistics of All Headways Direction Day Time Period

EB/ WB/ Week Week AM Mid PM Even- Grand Bunching Route % bunch SB NB end day Peak day Peak ing Total Events

501 3894 3880 1006 6768 1282 2242 1602 2648 7774 2141 27.5% 504 2918 2662 543 5037 1156 1367 1284 1773 5580 2171 38.9% 505 1313 1279 399 2193 423 791 505 873 2592 508 19.6% 506 1154 1080 260 1974 482 750 470 532 2234 839 37.6% 509 1212 1210 409 2013 331 732 610 749 2422 877 36.2% 510 1711 1715 554 2872 430 1213 779 1004 3426 741 21.6% 511 1242 1197 354 2085 432 724 483 800 2439 415 17.0% 512 2034 2004 468 3570 742 1183 864 1249 4038 65 1.6% Total 15478 15027 3993 26512 5278 9002 6597 9628 30505 7757 25.4% % 50.7% 49.3% 13.1% 86.9% 17.3% 29.5% 21.6% 31.6% na na na

A sample of trips taken from the morning peak period on a Monday of Route 511 going southbound was arbitrarily chosen to be graphed on a time distance diagram (Error! Reference source not found.) to provide an illustration of the data. On the Y-axis is the distance from the terminal, where the terminal is located at 0 meters and the last stop of Route 511 is almost 4000m (4km) away from the terminal. Time in seconds is shown on the X-axis. The AM peak period ranges from 21600 to 32400 seconds. Figure 7 shows trips from the same route and direction, but in the afternoon peak period instead. Each of the points in the trip is a stop location. As can be seen in Error! Reference source not found., there are trips that do not start

at 0m but start later in the route path and end at the terminal station. Likewise, there are trips that start at 0m but do not end at terminal station. These could potentially be short turned trips. The last 3 trips do not seem to arrive at terminal station because they reached it after the morning peak period and thus, that portion of the trip was not graphed. Ideally, the trips in the diagram should all be parallel to each other if they were able to maintain a constant headway between

each other. Figure 5 shows this ideal situation.

Direction of Travel of Direction

Figure 5 Sample Scheduled Time Distance Diagram

However, because of all the different factors that are involved in each trip, the spacing between each line varies quite a lot. The departure and arrivals at the terminal stations are not consistent. Many of the lines also end up converging and then diverge afterward. Bunching incidents occur at locations where the lines converge. The red circles highlight some of the bunching incidents. It is possible that the lines are able to diverge afterward due to traffic lights, intersection delays, or a variable number of passengers. Otherwise, the lines would continue to travel with a smaller space between them compared to the space when they started at the terminal.

Route 511: Monday Southbound, AM Peak Trips

Distance from terminal (m) terminal from Distance Direction of Travel of Direction

Time (sec)

Figure 6 Time Distance Diagram: Route 511 Southbound, AM Peak

Figure 7 shows the trips on the same route and direction but in the PM peak period. The PM peak period ranges from 54000 to 68400 seconds. Almost all of the trips start and begin at the terminal stations. Only the first and last two trips in the diagram have been cut off and not displayed due to the time period range. There is one trip that extends past the 3700m mark. This is also likely due to an error in the data collection and is removed from the data. For the most part, the trips generally begin with consistent spacing between each other, but slowly begin to converge towards each other shortly after the 2000m mark. The second model will specifically look into which factors make these lines (headways) converge sooner or later.

Route 511: Monday Southbound, PM Peak Trips

Distance from terminal (m) terminal from Distance Direction of Travel of Direction

Time (sec)

Figure 7 Time Distance Diagram: Route 511 Southbound, PM Peak

Figure 8 displays a histogram of the distribution of times to the first bunching incident. This graph indicates that the time to bunching is positively skewed. Histograms for some individual routes are provided in Figure 9. Similarly, they also show a positive skew. Table 10 summarizes the general statistics of the time to the first bunch for all of the routes and the sample routes.

Table 10 Statistics on Time to First Bunch Statistics on Time to First Bunch All Routes (min) Route 510 (min) Route 504 (min) Mean 21.20 12.99 26.56 Mode 16.00 5.33 13.00 Median 6.67 11.33 21.67 Standard Deviation 16.58 7.69 16.18

Figure 8 Histogram of Time to First Bunching Incident

Figure 9 Route Specific Examples of Time to First Bunching Incident Histogram 37

Out of the over 30500 headways, approximately 7700 bunching incidents occurred. The summary statistics of these 7700 bunching incidents are shown in Table 11. Interestingly, the bunching incidents occur heavily in one direction over the other. The distribution of bunching incidents over the different time periods is almost equal, but the midday time period surprisingly has the highest number of bunching incidents.

Table 11 Summary Statistics of Bunched Incidents Direction Time Period Route EB/ SB WB/NB AM Mid PM Even Grand Peak day Peak ing Total 501 1316 825 402 669 488 582 2141 504 1161 1010 457 631 595 488 2171 505 260 248 69 216 131 92 508 506 0 839 226 321 197 95 839 509 436 441 76 305 269 227 877 510 496 245 95 342 157 147 741 511 236 179 122 136 85 72 415 512 34 31 13 26 24 2 65 Grand 3939 3818 1460 2646 1946 1705 7757 Total 50.78% 49.22% 18.82% 34.11% 25.09% 21.98%

5.2 Binary Logistic Regression Model Results

In this model, headways that have experienced bunching are coded as “1” and those that have not experienced bunching are coded as “0”. The results of this model are reported in Table 13. Variables that are found to be statistically significant of at least 90% are bolded in the table. The model fit reported in Table 12 shows that it has a Nagelkerke R Square value of 0.59, which indicates that 59% of the variance has been explained by the model. This R square value is comparable to other binary logistic models that investigate on-time performance (Diab et al., 2016; Surprenant-Legault & El-Geneidy, 2011).

Table 12 Logistic Regression Model Fit

-2 Log Cox & Snell R Nagelkerke R likelihood Square Square 18934.586a .401 .592

The route number variable, which has been included in the model as a control variable, shows a significant coefficient. This is expected since each route has different right of way characteristics, length, as well as the average stop distance. As shown in the descriptive statistics, Route 512 experienced the least amount of bunching, and therefore this model shows that all other routes have higher odds of bunching compared to this route. If the reference route was a different route, the results would have differed. Route 512 was arbitrarily chosen as the reference. This was also done in the AFT model for consistency.

The model also indicates that the odds of bunching are higher on weekdays compared to the weekend. This is expected as there is an increase in ridership, frequency and traffic congestion during the week than the weekend. In addition, the midday, PM peak, and evening time periods were found to increase the odds of bunching compared to the AM peak. The increased chances of bunching frequency in the midday and evening peaks are likely due to the combined effect of the relatively high streetcar frequencies with lower volumes of the general traffic.

Interestingly, the model shows that when the following vehicle has a greater capacity than the leading vehicle, this reduced the odds of bunching by 24%. This can be explained by the fact that since the following vehicle has a higher capacity, it will be able to hold more passengers and thus have a longer dwell time as well as total travel time. These longer times will prevent it from catching up with the leading vehicle. However, when the following vehicle has a lower capacity than the leading vehicle, the odds of bunching are increased by 124%. This is due to the fact that the leading vehicle will likely have longer dwell times, making it easier for the following vehicle to catch up and bunch with it. To summarize, both of the previous cases indicate that vehicles with higher capacity are slower, and therefore they bunch with the following ones while increasing the headway gap with the leading ones.

The model indicates that for every minute that scheduled headway is increased, the odds of bunching is reduced by 44%, which is expected and was found in the bus literature (Diab et al., 2016). Therefore, schedule design plays a big role in bunching for streetcar service, and transit agencies should address this problem. This can be done by providing higher volume vehicles with longer headways, which is currently the TTC’s plan (TTC, n.d.-a). The dummy variable Lshort was added to the model to understand the effects of short-turning on bunching incidents.

On Routes 504 and 510, up to 20% of the vehicles are short-turned. The strategy of short- turning is used in streetcar operations to address a serious effect of bunching occurrence, namely the long gaps in service downstream of bunched vehicles. It is assumed that the TTC short- turning procedure is only implemented when there is a long gap ahead of a streetcar bunch extending into the opposite direction. The model indicates here that when the leading vehicle is short-turned, it decreases the odds of bunching by 64%. This is logical since the following vehicle will still have to go to the terminal and run back in the opposite direction, which will create a gap between originally bunched trips.

With respect to headway deviation at terminals, only five of the nine combinatory dummy variables were found to be significant. A pattern can be noted with the significant combinatory headway deviation variables: when the following vehicle has an actual headway that is shorter than the scheduled headway, the odds of bunching is increased and when the following vehicle has an actual headway that is longer than the scheduled headway, the odds of bunching is reduced. The headway deviation combination that increases the odds of bunching the most is when the following vehicle has a shorter headway and the leading vehicle has a longer headway at the terminal, increasing the odds of bunching by 146%. This scenario essentially represents when the leading vehicle is delayed at the start and the following vehicle leaves early at the start. When the leading vehicle is delayed, it is likely to pick up more passengers, thus experiencing longer dwell times. When the following vehicle leaves the terminal early, it has fewer passengers to pick up and therefore can easily catch up to the leading vehicle. In contrast, when the following vehicle has a longer headway and the leading vehicle has a shorter headway, this situation provides the greatest reduction in the odds of bunching out of all the cases where the following vehicle has a longer headway.

Table 13 Binary Logistic Regression Model Results 95% Confidence Coefficie Odds Wald Significance Interval nt Ratio Lower Upper Wkday 2.15 2450.65 0.00 8.62 7.92 9.39 Trip direction 0.32 72.73 0.00 1.37 1.28 1.47 Lshort -1.02 253.45 0.00 0.36 0.32 0.41 Vehicle Combination (Reference to same vehicle type for both following and leading vehicles) FVehCap > -0.27 18.60 0.00 0.76 0.67 0.86 LVehCap FVehCap < 0.33 32.36 0.00 1.39 1.24 1.56 LVehCap Time Period (Reference to AM Peak) Mid Day 0.78 183.44 0.00 2.19 1.95 2.45 PM Peak 0.18 10.17 0.00 1.20 1.07 1.34 Evening 0.94 145.62 0.00 2.56 2.19 2.98 Route Number (Reference to Route 512) Route 501 8.16 2121.09 0.00 3494.14 2469.15 4944.62 Route 504 3.12 547.37 0.00 22.62 17.42 29.37 Route 505 3.88 696.07 0.00 48.58 36.40 64.82 Route 506 4.94 1190.14 0.00 139.23 105.19 184.31 Route 509 3.88 747.04 0.00 48.53 36.73 64.10 Route 510 2.03 212.45 0.00 7.61 5.79 9.99 Route 511 2.49 305.49 0.00 12.05 9.11 15.92 Scheduled Headway -0.59 938.62 0.00 0.56 0.53 0.58 Headway Deviation Combination (Reference to On Time/On Time) Short/Short 0.00 0.00 0.96 1.00 0.83 1.22 Short/On Time 0.18 2.68 0.10 1.20 0.97 1.48 Short/Long 0.38 14.26 0.00 1.46 1.20 1.77 On Time/Short -0.04 0.11 0.74 0.96 0.78 1.20 On Time/Long 0.05 0.18 0.67 1.05 0.84 1.32 Long/Short -0.68 40.58 0.00 0.51 0.41 0.63 Long/On Time -0.51 16.88 0.00 0.60 0.47 0.77 Long/Long -0.27 6.26 0.01 0.76 0.62 0.94 Route 501 x -24.59 0.00 0.96 0.00 0.00 na Short/Short Constant -0.45 6.29 0.01 0.64 na na

5.3 Accelerated Failure Time Model Results

A linear regression model as well as an ordinal logit model were developed to try to investigate the impact of the internal and external factors on the time to bunching. However, both of these models resulted in very low R squared and ρ squared values. Therefore the results of those models are not reported in this thesis. Instead, an AFT model was used to study the time to the first bunch along the route. Since many different distributions can be used for the AFT model, each distribution had to be tested. The distribution with the Cox Snell plot that is closely aligns with a line of slope 1 is considered to be the best model specification. Although not perfectly linear, Figure 10 shows the Cox Snell Residuals graph for the log-logistic distribution which had the plot closest to a slope of 1.

Figure 10 Cox Snell Residual Graph

Furthermore, when comparing the Akaike Information Criterion (AIC) values for each distribution, the log-logistic distribution was found to have the best fit and was thus chosen for this model. The log-logistic distribution had the lowest AIC value (as can be shown by Table 14) at 14907 compared to the log-normal, weibull, and exponential distributions.

Table 14 Comparison of Fits for the AFT Model Distribution Log Likelihood AIC Log-logistic - 7407.915 14907.24 Log-normal -7718.669 15487.22 Weibull -7462.586 14975.17 Exponential -9206.41 18460.8 42

The output of this model is reported in Table 15. Only bunching incidents were used in this model. Bolded variables indicate statistical significance of at least 90%. The reference variables are kept the same as in the first model to allow for comparison. A negative coefficient of a variable in this model indicates that as the magnitude of the associated variable increases, the departing (following) streetcar from the terminal will catch up with the leading streetcar (i.e. creating a bunching incident) sooner than later compared to the baseline scenario. In other words, a negative coefficient indicates an accelerated time to bunch (failure) or a reduction in the survival time. A positive coefficient would indicate the opposite, meaning that it would prolong the time to a bunching incident or extend its survival time. Again, the acceleration factor can be determined by exponentiating the coefficient, eβ (Kleinbaum & Klein, 2005).

Since the weekday variable has a negative coefficient, this means that on weekdays initial bunching incidents, when they happen, take place sooner (relative to the departure time of the following vehicle from the terminal) compared to weekends. The acceleration factor of e-0.038 = 0.96 indicates that the survival time or time to bunch on a weekday is 0.96 as large as on a weekend. However, this was not found to be statistically significant. This indicates while the odds of bunching are higher during weekdays (according to the previous model), these weekday bunches do not necessarily take a shorter time to occur (according to this model).

Compared to the time periods to the AM peak, the results show that during the midday, PM peak, and evening periods, initial bunching incidents take longer to happen compared to the AM peak. The PM peak indicates that the time to bunching is increased by a factor of 1.67 compared to the AM peak. Again, the route numbers are included in the model as control variables. Regardless of the vehicle type combinations, when the vehicle capacities are different, the model indicates that the time to initial bunch will be accelerated compared to when they are the same vehicle type for both following and leading vehicles. This is expected because the differences in capacities will impact the dwell times and thus, time to bunching. Therefore, while the previous model indicates a difference in the impact of the size of the vehicle on the probability of bunching, this model shows that when bunching occurs, it occurs quicker when combinations of different vehicle types are involved compared to the case of only one type of vehicle along the route.

A unit increase in other internal variables such as scheduled headway, headway ratio between actual and scheduled headway for the following and leading vehicle, as well as the cumulative number of TSP-equipped intersections all indicated they would cause a longer time for the initial bunching incident to occur. For every additional minute of scheduled headway, the survival time is 1.11 times longer. Increasing the number of TSP-equipped intersections and headway ratio do not increase the time to initial bunching as much as increasing the scheduled headway, but they still do prolong the time to initial bunching. This is logical since increasing the headway ratio would imply an increase in the actual headway which is likely to prolong the time for two consecutive streetcars to meet in a bunching incident. The increased number of TSP delaying time to bunch may be explained by the fact that TSP can be conditional and will only grant vehicles priority when they meet the specific conditions. This means that not all vehicles will be given priority and therefore, do not have the opportunity to catch up to each other as easily, which will delay time to bunching. However, the model shows that a combination of different stop placements will accelerate the time to initial bunching compared to when stops are placed all on the same side, whether it be far or near side. This could be because when stop placements are alternated, they can still be between two consecutive intersections (i.e. far side stop followed by a near side stop) thus allowing the following vehicle to catch up more easily with the leading one.

In terms of the external factors, the cumulative number of pedestrian crossing and signalized approaches also accelerate the time to initial bunching. This is likely due to the effect of signalized approaches on interrupting streetcar movements and thus, allowing following vehicles to catch up easily to vehicles that have been stopped by traffic signals or crossings. On routes that do not have dedicated right of way, streetcars must interact with vehicular traffic. High vehicular traffic actually increases the survival time by a factor of 1.30. This may sound counterintuitive but makes sense because the more traffic there is, the more vehicles there will likely be between successive streetcars. A microsimulation model of streetcar operation in Toronto found similar results (Shalaby, Abdulhai, & Lee, 2003). This increased number of traffic vehicles between streetcars will increase the time it takes for a bunching incident to occur.

Table 15 AFT Model Results 95% C.I.for Coefficient Standard Variable z P>z Coefficient (β) Error Lower Upper Wkday -0.038 0.02 -1.55 0.12 -0.09 0.01 Trip direction 0.044 0.02 2.99 0.00 0.02 0.07 TimePeriod (Reference to AM Peak) Midday 0.129 0.02 5.89 0.00 0.09 0.17 PM Peak 0.154 0.02 7.28 0.00 0.11 0.20 Evening 0.066 0.03 2.54 0.01 0.02 0.12 Route (Reference to Route 512) 501 -0.196 0.10 -1.97 0.05 -0.39 0.00 504 0.639 0.09 6.87 0.00 0.46 0.82 505 0.286 0.11 2.68 0.01 0.08 0.50 506 0.109 0.11 1.04 0.30 -0.10 0.32 509 -0.180 0.10 -1.84 0.07 -0.37 0.01 510 0.162 0.10 1.71 0.09 -0.02 0.35 511 -0.078 0.10 -0.77 0.44 -0.28 0.12 VehCombination (Reference to same vehicle type for both) FVehCap > LVehCap -0.079 0.02 -3.67 0.00 -0.12 -0.04 FVehCap < LVehCap -0.084 0.02 -4.30 0.00 -0.12 -0.05 SchedHead 0.101 0.05 2.22 0.03 0.01 0.19 SchedHead2 -0.011 0.00 -3.16 0.00 -0.02 0.00 FLHeadRatio 0.002 0.00 18.04 0.00 0.002 0.002 LL1HeadRatio 0.000 0.00 -0.44 0.66 0.00 0.00 CumTSP 0.077 0.00 23.79 0.00 0.07 0.08 StopComb -0.373 0.13 -2.84 0.01 -0.63 -0.12 CumPedCross -0.030 0.00 -7.09 0.00 -0.04 -0.02 CumSigApp -0.006 0.00 -10.97 0.00 -0.01 -0.01 Traffic Volume Cat (Reference to low traffic volume category) Medium Volume -0.012 0.02 -0.74 0.46 -0.04 0.02 High Volume 0.267 0.04 6.84 0.00 0.19 0.34 Constant 1.909 0.16 11.97 0.00 1.60 2.22

6 Summary and Conclusions 6.1 Summary of Thesis

The research objectives of this thesis were developed from the gaps in the literature on bunching, specifically, streetcar bunching and the need for a better understanding of this operational problem as light rail and streetcar systems become more popular and widely installed around the world. This study was carried out using a week’s worth of AVL data collected in January 2016. These millions of GPS points were processed into trips to be able to analyze the headways. The key unit of measure in this study is the headway. Each headway has a potential of bunching as it fluctuates from the scheduled headway. The objectives of this study were to understand the factors that increased or decreased the odds of a headway bunching and to investigate in detail the factors that impact the time for a headway to bunch. The binary logistic regression model and AFT model were employed to analyze the two objectives. The models used many variables such as day of the week, trip direction, vehicle type, scheduled headway, number of cumulative TSP, and more. The results of the model give the transit operator key factors to focus on when attempting to reduce the odds of bunching and time to bunching. The key results of the model are discussed in detail below.

6.2 Key Results

The overall results indicate that transit operators of streetcar systems should pay more attention to headway deviations at terminals particularly on weekdays. Not only are the odds of bunching higher on weekdays than on weekends, the time to bunching is shorter on weekdays than weekends. No matter what the previous headway was, if the following headway is longer than the scheduled headway, the odds of bunching are reduced. However, this will present a nuisance to passengers as well as to the headway that occurs after the following headway. The model found that longer scheduled headways will reduce the odds of bunching and together with higher headway ratios, will lengthen the time to a bunching incident, if one occurs.

The models found that odds of bunching are increased in the midday, PM peak, and evening time periods. However, when bunching does occur, the time to bunching is shorter in the AM peak. This means that the TTC needs to focus on reducing the odds of bunching in the midday, PM

peak, and evening time periods, but in the AM peak period, it should focus on prolonging the time to bunching. This finding is specific to the TTC and is likely to be different for other cities as geographic and local factors will influence the impact of this variable.

Despite the fact that the TTC does not have a protocol for short turning its vehicle, the procedure has found to be effective in reducing bunching. This means that the employees that the TTC has employed to make decisions on which and when vehicles should be short turned has proven to be effective albeit decisions are likely made based on field experience only.

When the following vehicle has a greater capacity than the leading vehicle, this reduces the odds of bunching, but in general, when there are combinations of different vehicles types, this will accelerate time to bunching. This finding emphasizes the need for a uniform fleet for transit operators if they want to reduce the odds of bunching. This may be a difficult requirement in reality as technology advances and older vehicle models become obsolete. This may also be difficult for transit operators to achieve due to high costs of new vehicles if they want to expand or update their fleet.

During the planning process, stop locations should also be considered carefully since different stop placements cause initial bunching incidents to occur sooner than later. The cumulative presence of TSP, when programmed effectively, can delay the onset of bunching. Furthermore, the cumulative number of pedestrian crossings and signalized approaches have been found to accelerate the time to bunching. Heavy traffic volume delays the onset of initial bunching, but this may also cause longer than anticipated travel times, which will also be a nuisance to passengers.

6.3 Policy Implications

The results found from the models have many implications for the TTC and other transit operators. It would be best if the TTC focused on the factors that could provide the most improvement (decreased odds of bunching and longer time to bunch) for both parties (operator and passenger) such as scheduled headway adherence and changes in the fleet for consistency. To reduce the likelihood of bunching occurrence, transit operators should try to ensure that headways at the terminal are not shorter than scheduled headway. Ensuring this or increasing the

scheduled headway will reduce the odds of bunching and even lengthen the time to a bunching incident. The TTC should try to eliminate the headway deviation combination that causes the highest increase in bunching odds. The combination that causes this is when the following vehicle has a short headway and the leading has a long headway. There are other combinations that increase the odds of bunching as well, but if the TTC had to choose which combination to focus on, it should be this one.

The TTC is already in the process of upgrading its fleet; this should provide consistency in vehicle combinations and reduce the odds and time to bunching. Although short turning has been effective for the TTC in reducing the odds of bunching, the TTC should develop a short turning protocol that will assist in providing continued effective resolution of bunching incidences. A documented protocol will be very beneficial for the TTC especially when experienced employees resign or retire from the employer and can no longer provide the judgment or expertise that is required to make a decision on when and which vehicles should be short turned.

If not already done so, the TTC should be able to give input to the municipality’s planning department on changes or new installations of pedestrian crossing and signalized approaches. It should also give deeper consideration to the number of TSP and stop placements when changing or designing new routes. Vehicular traffic volume is difficult for the TTC to control unless its routes have dedicated right of ways, which can be very expensive to add to its current infrastructure. However, this is a factor that the TTC should consider when designing new routes and must choose which type of right of way to give the route.

6.4 Future Work

One limitation of this study is that only one week of data were used in the analysis. This did not allow for capturing of the impact of construction, special events, or weather conditions on bunching. A future study could utilize a data set which spans an extended time period to investigate and understand the impacts of these aspects.

Since it is rare to find streetcar bunching models in the literature, this paper provides valuable insights into streetcar bunching. Nevertheless, with additional data such as passenger volume or

method of payment, which were not available for this paper, the models can be improved to provide more information to streetcar operators. The results from this study can be combined to build a real-time predictive model for bunching, which can allow transit operators to act proactively with expected bunching incidents. Such a model would be able to warn operators of potential and upcoming bunching incidents and the time it would take for the bunching incident to occur with a given accuracy.

The methodology used in this study could also be applied to bus systems to investigate the time to bunching for buses as this has not been explored before. Similarly, combining the significant results from this model with bunching likelihood studies will allow for a predictive model to be built. This model can have different factors from the streetcar model such as number of lanes on the road, a variable to capture if street parking exists on the route or a variable to capture bus overtaking.

The results and future work from this study provide great potential for streetcar operators. Armed with the knowledge gained from this study, operators can make informed decisions when trying to improve streetcar services or when planning and building new streetcar routes. This will allow operators to make evidence-based decisions instead of ad-hoc ones and therefore, they would be able to develop actual procedures or decision-making processes to prevent and reduce bunching. Analogous to a screening procedure developed to give patients early treatment in an attempt to extend their life, a real-time predictive bunching model could “detect and cure” vehicles from bunching and extend its time away from the terminal to provide an efficient transit service to the public.

7 References Ashcroft, E. (2017, August 4). MetroBus promises to ease ‘banana bus bunching’ in the city. Retrieved September 12, 2017, from http://www.bristolpost.co.uk/news/bristol- news/metrobus-promises-ease-banana-bus-274626 Besco, R. O., Sangal, S. P., Nesthus, T. E., & Veronneau, S. (1995). Longevity and Survival Analysis For a Cohort of Retired Airline Pilots (p. 10 p.). Retrieved from http://www.faa.gov/data_research/research/med_humanfacs/oamtechreports/1990s/media /AM95-05.pdf Chen, C., Williams, R. C., Marasinghe, M. G., Omundson, J. S., & Schram, S. A. (2014). Survival Analysis for Composite Pavement Performance in Iowa (p. 14p). Transportation Research Board. Retrieved from https://trid.trb.org/view/1289533 Chimba, D., Kutela, B., Ogletree, G., Horne, F., & Tugwell, M. (2014). Impact of Abandoned and Disabled Vehicles on Freeway Incident Duration. Journal of Transportation Engineering, 140(3), Content ID 04013013. City of Toronto. (n.d.). Traffic Signal Vehicle and Pedestrian Volumes - Data catalogue - Open Data | City of Toronto. Retrieved August 29, 2017, from https://www1.toronto.ca/wps/portal/contentonly?vgnextoid=417aed3c99cc7310VgnVCM 1000003dd60f89RCRD Currie, G., Burke, M., & Delbosc, A. (2014). Performance of Australian Light Rail and Comparison with U.S. Trends. Transportation Research Record: Journal of the Transportation Research Board, (2419), pp 11–22. Currie, G., Delbosc, A., Harrison, S., & Sarvi, M. (2013). Impact of Crowding on Streetcar Dwell Time. Transportation Research Record: Journal of the Transportation Research Board, (2353), pp 100-106. Currie, G., Delbosc, A., & Reynolds, J. (2012). Modeling Dwell Time for Streetcars in Melbourne, Australia, and Toronto, Canada. Transportation Research Record: Journal of the Transportation Research Board, (2275), pp 22–29. Currie, G., & Shalaby, A. (2008). Active Transit Signal Priority for Streetcars: Experience in Melbourne, Australia, and Toronto, Canada. Transportation Research Record: Journal of the Transportation Research Board, 2042, 41–49. https://doi.org/10.3141/2042-05 Currie, G., & Shalaby, A. S. (2007). Success and Challenges in Modernizing Streetcar Systems: Experiences in Melbourne, Australia, and Toronto, Canada. Transportation Research Record: Journal of the Transportation Research Board, (2006), pp 31-39. Daganzo, C. F. (2009). A headway-based approach to eliminate bus bunching: Systematic analysis and comparisons. Transportation Research Part B: Methodological, 43(10), pp 913-921. Daganzo, C. F., & Pilachowski, J. (2011). Reducing bunching with bus-to-bus cooperation. Transportation Research Part B: Methodological, 45(1), pp 267-277. Diab, E., Bertini, R., & El-Geneidy, A. (2016). Bus Transit Service Reliability: Understanding the Impacts of Overlapping Bus Service on Headway Delays and Determinants of Bus 50

Bunching (p. 18p). Presented at the Transportation Research Board 95th Annual Meeting, Washington DC, United States. Retrieved from http://docs.trb.org/prp/16-2059.pdf Dong, Q., & Huang, B. (2015). Failure Probability of Resurfaced Preventive Maintenance Treatments: Investigation into Long-Term Pavement Performance Program, (2481), pp 65–74. Erath, A. (2013, March 24). How to solve the problem of bus bunching [Text]. Retrieved September 13, 2017, from http://www.straitstimes.com/singapore/how-to-solve-the- problem-of-bus-bunching Feng, W., & Figliozzi, M. (2011). Empirical findings of bus bunching distributions and attributes using archivedAVL/APC bus data. In ICCTP 2011: Towards Sustainable Transportation Systems - Proceedings of the 11th International Conference of Chinese Transportation Professionals (pp. 4330–4341). American Society of Civil Engineers (ASCE). https://doi.org/10.1061/41186(421)427 Garfield, L. (2016, November 2). New York City’s $2.5 billion streetcars for Brooklyn and Queens - Business Insider. Retrieved August 27, 2017, from http://www.businessinsider.com/new-york-city-streetcars-brooklyn-queens-2016-11 Giuliano, G. (1989). Incident Characteristics, Frequency, and Duration on a High Volume Urban Freeway. Transportation Research Part A: Policy and Practice, 23A(5), 387–396. Gudmundsson, S. V., & Rhoades, D. L. (2001). Airline Alliance Survival Analysis: Typology, Strategy and Duration. Transport Policy, 8(3), 209–218. Hong, F., & Mikhail, M. (2016). Survival Analysis of Rigid Pavement Expected and Remaining Service Life Based on Long-Term Pavement Performance Sections (p. 15p). Transportation Research Board. Retrieved from https://trid.trb.org/view/1392882 Hu, W. X., & Shalaby, A. (2017). Use of Automated Vehicle Location Data for Route- and Segment-Level Analyses of Bus Route Reliability and Speed. Transportation Research Record: Journal of the Transportation Research Board, (2649), pp 9–19. Jenkins, S. P. (2008). Survival Anaylsis. Institute for Social and Economic Research, University of Essex, Colchester. Kalinowski, T. (2014, August 8). TTC wants to put some POP in your streetcar ride, with alldoors boarding by Jan. 1. The Toronto Star. Retrieved from https://www.thestar.com/news/gta/2014/08/08/ttc_wants_to_put_some_pop_in_your_stre etcar_ride_with_alldoors_boarding_by_jan_1.html KCRTA. (n.d.). Midtown/UMKC Streetcar Extension Resources – KCRTA. Retrieved July 29, 2017, from http://kcrta.org/streetcar/ Kleinbaum, D. G., & Klein, M. (2005). Survival Analysis: A Self-Learning Text. Springer Science & Business Media. Liang, S., Zhao, S., Lu, C., & Ma, M. (2016). A self-adaptive method to equalize headways: Numerical analysis and comparison. Transportation Research Part B: Methodological, 87, pp 33-43.

Ling, K., & Shalaby, A. S. (2005). A Reinforcement Learning Approach to Streetcar Bunching Control. Journal of Intelligent Transportation Systems, 9(2), pp 59-68. Louie, J., Shalaby, A., & Habib, K. N. (2017). Modelling the impact of causal and non-causal factors on disruption duration for Toronto’s subway system: An exploratory investigation using hazard modelling. Accident Analysis and Prevention, 98, 232–240. https://doi.org/10.1016/j.aap.2016.10.008 Mandelzys, M., & Hellinga, B. (2010). Identifying Causes of Performance Issues in Bus Schedule Adherence with Automatic Vehicle Location and Passenger Count Data. Transportation Research Record: Journal of the Transportation Research Board, (2143), pp 9-15. Mesbah, M., Lin, J., & Currie, G. (2015). “Weather” Transit is Reliable? Using AVL Data to Explore Tram Performance in Melbourne, Australia. Journal of Traffic and Transportation Engineering (English Edition), 2(3), pp 125-135. Moreira-Matias, L., Ferreira, C., Gama, J., Mendes-Moreira, J., & De Sousa, J. F. (2012). Bus Bunching detection: A sequence mining approach (Vol. 960, pp. 13–17). Presented at the Workshop on Ubiquitous Data Mining, UDM 2012 - In Conjunction with the 20th European Conference on Artificial Intelligence, ECAI 2012, August 27, 2012 - August 31, 2012, CEUR-WS. Moreira-Matias, L., Gama, J., Mendes-Moreira, J., & Freire de Sousa, J. (2014). An Incremental Probabilistic Model to Predict Bus Bunching in Real-Time. In Advances in Intelligent Data Analysis XIII 13th International Symposium, IDA 2014, 30 Oct.-1 Nov. 2014 (pp. 227–38). Springer International Publishing. https://doi.org/10.1007/978-3-319-12571- 8_20 Nair, R., Bouillet, E., Gkoufas, Y., Verscheure, O., Mourad, M., Yashar, F., … Transportation Research Board. (2015). Data as a resource: real-time predictive analytics for bus bunching (p. 23p). Retrieved from http://docs.trb.org/prp/15-2129.pdf National Academies of Sciences, E. (2013). Transit Capacity and Quality of Service Manual, Third Edition. https://doi.org/10.17226/24766 Naznin, F., Currie, G., & Logan, D. (2017). Streetcar Safety from Tram Driver Perspective (p. 15p). Presented at the Transportation Research Board 96th Annual Meeting, Washington DC, United States. Retrieved from http://docs.trb.org/prp/17-05635.pdf Platt, B. (2014, July 29). The Pulse: Bus bunching the problem for North York riders. The Toronto Star. Retrieved from https://www.thestar.com/news/gta/2014/07/29/the_pulse_bus_bunching_the_problem_for _north_york_riders.html Rivoli, D. (2017, August 14). Rider advocacy groups hit MTA, DOT with poor grades on buses. Retrieved September 12, 2017, from http://www.nydailynews.com/new-york/rider- advocacy-groups-hit-mta-dot-poor-grades-buses-article-1.3408780 Shalaby, A., Abdulhai, B., & Lee, J. (2003). Assessment of streetcar transit priority options using microsimulation modelling. Canadian Journal of Civil Engineering, 30(6), 1000–1009. https://doi.org/10.1139/l03-010

Shi An, Xinming Zhang, & Jian Wang. (2015). Finding Causes of Irregular Headways Integrating Data Mining and AHP. ISPRS International Journal of Geo-Information, 4(4), 2604–18. https://doi.org/10.3390/ijgi4042604 Surprenant-Legault, J., & El-Geneidy, A. M. (2011). Introduction of a Reserved Bus Lane: Impact on Bus Running Time and On-Time Performance. Transportation Research Record: Journal of the Transportation Research Board, (2218), pp 10-18. The Pennsylvania State University. (2017). 6.2 - Binary Logistic Regression with a Single Categorical Predictor | STAT 504. Retrieved September 1, 2017, from https://onlinecourses.science.psu.edu/stat504/node/150 TRB. (1997). AVL Systems for Bus Transit: A Synthesis of Transit Practice. TRID. (2017). About TRID | Information Services. Retrieved August 30, 2017, from http://www.trb.org/InformationServices/AboutTRID.aspx TTC. (n.d.-a). TTC. Retrieved July 28, 2017, from http://www.ttc.ca/About_the_TTC/Projects/New_Vehicles/New_Streetcars/FAQ/FAQ_G eneralInformation.jsp TTC. (n.d.-b). TTC Section One. Retrieved May 21, 2017, from http://www.ttc.ca/About_the_TTC/Operating_Statistics/2016/section_one.jsp Wasserstein, R. (2014, December 14). Odds or Probability? Retrieved September 1, 2017, from http://senseaboutscienceusa.org/know-the-difference-between-odds-and-probability/ Weng, J., Zheng, Y., Yan, X., & Meng, Q. (2014). Development of a subway operation incident delay model using accelerated failure time approaches. Accident Analysis & Prevention, 73, pp 12-19. Wu, J. (2011). Survival Analysis of Real-World Tire Aging Data (p. 9p). National Highway Traffic Safety Administration. Retrieved from http://www- esv.nhtsa.dot.gov/Proceedings/22/isv7/main.htm Xuan, Y., Argote, J., & Daganzo, C. F. (2011). Dynamic bus holding strategies for schedule reliability: Optimal linear control and performance analysis. Transportation Research Part B: Methodological, 45(10), 1831–1845. Yu, Z., Wood, J. S., & Gayah, V. V. (2017). Using survival models to estimate bus travel times and associated uncertainties. Transportation Research Part C: Emerging Technologies, 74, pp 366-382. Zhang, L., Shi, Y., Yang, W., Liu, P., Rao, Q., & Transportation Research Board. (2014). Survival Analysis-Based Modeling of Urban Traffic Incident Duration: Shanghai Case Study, China (p. 18p). Retrieved from https://trid.trb.org/view/1289349

8 Appendix 8.1 TTC Service Summary (January 3 – February 13, 2016)

8.2 Stata Code

. import excel "C:\Users\user\Documents\All_bunch_incidents_ext_filter.xlsx", sheet("a > ll") firstrow

. stset Fmin

failure event: (assumed to fail at time=Fmin) obs. time interval: (0, Fmin] exit on or before: failure

7757 total obs. 0 exclusions

7757 obs. remaining, representing 7757 failures in single record/single failure data 164463 total analysis time at risk, at risk from t = 0 earliest observed entry t = 0 last observed exit t = 91

. streg wkday Ftripdir i.VehCombA i.TimePeriod ib(512).Route SchedHead SchedHead2 FLHe > adRatio LL1HeadRatio CumPedCross CumSigApp CumTSP StopComb i.Vehvolcat, dist(loglogi > stic)

failure _d: 1 (meaning all fail) analysis time _t: Fmin Fitting constant-only model:

Iteration 0: log likelihood = -13046.444 Iteration 1: log likelihood = -10264.2 Iteration 2: log likelihood = -10092.263 Iteration 3: log likelihood = -10082.585 Iteration 4: log likelihood = -10082.571 Iteration 5: log likelihood = -10082.571

Fitting full model:

Iteration 0: log likelihood = -10082.571 (not concave) Iteration 1: log likelihood = -8022.3783 Iteration 2: log likelihood = -7866.1312 Iteration 3: log likelihood = -7417.0018 Iteration 4: log likelihood = -7407.9251 Iteration 5: log likelihood = -7407.9149 Iteration 6: log likelihood = -7407.9149

Loglogistic regression -- accelerated failure-time form

No. of subjects = 7757 Number of obs = 7757 No. of failures = 7757 Time at risk = 164462.95 LR chi2(24) = 5349.31 Log likelihood = -7407.9149 Prob > chi2 = 0.0000

. streg wkday Ftripdir i.VehCombA i.TimePeriod ib(512).Route SchedHead SchedHead2 FLHe > adRatio LL1HeadRatio CumPedCross CumSigApp CumTSP StopComb i.Vehvolcat, dist(loglogi > stic)

failure _d: 1 (meaning all fail) analysis time _t: Fmin

Fitting constant-only model:

Fitting full model:

Loglogistic regression -- accelerated failure-time form

No. of subjects = 7757 Number of obs = 7757 No. of failures = 7757 Time at risk = 164462.95 LR chi2(24) = 5349.31 Log likelihood = -7407.9149 Prob > chi2 = 0.0000

_t Coef. Std. Err. z P>|z| [95% Conf. Interval]

wkday -.0376551 .0242968 -1.55 0.121 -.0852761 .0099658 Ftripdir .0444026 .0148645 2.99 0.003 .0152688 .0735364

VehCombA 1 -.0787867 .0214778 -3.67 0.000 -.1208824 -.036691 2 -.0838915 .0195086 -4.30 0.000 -.1221276 -.0456554

TimePeriod 2 .1288428 .0218631 5.89 0.000 .085992 .1716937 3 .1544698 .0212284 7.28 0.000 .1128629 .1960768 4 .0656324 .0258108 2.54 0.011 .0150441 .1162206

Route 501 -.1964486 .0997438 -1.97 0.049 -.3919428 -.0009543 504 .6386224 .0929342 6.87 0.000 .4564747 .8207701 505 .2862854 .1066301 2.68 0.007 .0772944 .4952765 506 .1091632 .1051792 1.04 0.299 -.0969841 .3153106 509 -.1796104 .0976135 -1.84 0.066 -.3709293 .0117086 510 .1619145 .0948524 1.71 0.088 -.0239929 .3478218 511 -.0784813 .1016433 -0.77 0.440 -.2776985 .1207359

SchedHead .1012382 .045575 2.22 0.026 .0119129 .1905636 SchedHead2 -.0107773 .0034152 -3.16 0.002 -.0174709 -.0040837 FLHeadRatio .0016915 .0000938 18.04 0.000 .0015077 .0018754 LL1HeadRatio -3.98e-06 9.13e-06 -0.44 0.663 -.0000219 .0000139 CumPedCross -.0301214 .0042486 -7.09 0.000 -.0384486 -.0217942 CumSigApp -.0061994 .0005653 -10.97 0.000 -.0073072 -.0050915 CumTSP .0772594 .0032472 23.79 0.000 .0708949 .0836239 StopComb -.3729285 .1314982 -2.84 0.005 -.6306603 -.1151967

Vehvolcat 2 -.0116743 .0158221 -0.74 0.461 -.042685 .0193364 3 .266856 .0389922 6.84 0.000 .1904327 .3432793

_cons 1.908968 .1594456 11.97 0.000 1.596461 2.221476

/ln_gam -1.071194 .0097778 -109.55 0.000 -1.090359 -1.05203

gamma .342599 .0033499 .336096 .3492279