bunching behaviour in london buses - final draft single sided

Bunching behaviour in

London buses

David Bellinger

MSc in Transport Planning and Management, University of Westminster

2011

ii

Abstract The iBus system installed on London’s buses in 2009 provides a new source of data for the study of bus bunching, the phenomenon whereby two or more buses arrive at a stop simultaneously. A new metric to describe bunching is produced, and using this, the severity of bunching on 30 London bus routes is measured. Bunching by time of day is derived, showing clear peaks in the morning and the evening, with a somewhat different pattern in the school holidays. Bunching along the course of individual routes is also analysed: the curves produced generally fall into one of three specific forms. Comparison with simulations reveals that the rate at which bunching becomes more severe as buses progress down a route is likely to imply that one or more feedback mechanisms is at play in the system. Regression analysis relating bunching to various route factors is performed, revealing significant relationships between bunching and 12 route factors. Factor analysis groups these into three orthogonal components. The most successful predictor of bunching on a route is shown to be its modal frequency – an exponential relationship between the two variables is given. This model is tested on three hold-out samples, and is found to be successful, with an average error of 9%. Word count: 18,499 Acknowledgements Writing this dissertation has been an extremely rewarding experience, and I would like to thank the many people who have supported me in the process. Firstly, my dissertation supervisor Dr Andrew Cook has been incredibly supportive and encouraging, and I am indebted to him for the many helpful comments that have served to enhance the work significantly. Thanks should also go to Prof Peter White who offered useful advice during the initial formulation of the dissertation, and on possible areas to investigate during the course of it. The research would not have been possible without the co-operation of TfL, the analysis of whose data forms the meat of this dissertation. Thanks go to Simon Reed, and particularly Chris Fowler, who has been extremely helpful in introducing me to the people I have needed to liaise with, and asking them to help me as much as they could. These have included: Janet Brown, Greg Webster, Adam Mullineux, and Annelies de Koning. Thanks also go to Terry Butler of the Arriva North Control Room, who generously gave of his time to show me around the control room and explain to me the finer points of service control. Finally, special thanks must go to my family and particularly to my wife Ruth for all their fantastic support during the course of my MSc.

iii

Contents List of Figures v List of Tables vi Glossary of abbreviations and technical terms v ii Chapter 1 – Introduction 1 1.1 Overview 1 1.2 Aims 2 1.3 Structure 2 Chapter 2 – Literature review 3 2.1 Theoretical analysis 3 2.2 Experimental research 5 Chapter 3 – Background information 7 3.1 London’s buses 7 3.2 iBus 9 3.3 Service control 11 Chapter 4 – Methodology 14 4.1 Bunching metric 14 4.2 Route selection 16 4.3 Date selection 22 4.4 Data selection and cleaning 22 4.5 Production of the metrics 24 4.6 Simulations 25 4.7 Regression analysis 26 4.8 Factor analysis 28

iv

Chapter 5 – Analysis 29 5.1 Bunching by route 29 5.2 Bunching by time of day 32 5.3 Bunching along the route 37 5.4 Regression analysis 49 5.5 Factor analysis 56 5.6 Prediction of µ� for the hold out samples 57 Chapter 6 – Conclusion 58 6.1 Summary of results 58 6.2 Future research 65 References 67 Appendix 1 – Independent variables 69 Appendix 2 – Correlations between independent varia bles 70 Appendix 3 – SPSS factor analysis output 73

v

List of Figures

2.1 Number of stops before one would observe before buses would deviate from the schedule by a certain number of minutes (from Hill, 2003)

2.2 Headway variation against distance down route (from Abkowitz et al., 1986

3.1 Annual passenger-km on London buses (from TfL 2010a) 3.2 Changes in bus performance measures with time (from TfL 2010a) 3.3 Schematic of the iBus system (from TfL 2011b) 3.4 Countdown dot matrix display (from TfL 2011b) 3.5 Screenshot of the service control software used by route controllers

(from a personal communication with TfL) 4.1 Overview of selected routes 4.2 Selected bus routes in West London 4.3 Selected bus routes in South London 4.4 Selected bus routes in East London 4.5 Selected bus routes in North London 5.1 Evolution of the metrics over the 5 selected days 5.2 µ by time of day 5.3 µ by time of day with bus usage superimposed (partly from TfL 2007) 5.4 µ by time of day with road traffic volumes superimposed (partly from

Daft 2011) 5.5 µ by time of day during term time 5.6 µ by time of day during half term 5.7 µ by time of day in Inner London 5.8 µ by time of day in Outer London 5.9 µ-curve of the 54 towards Elmers End 5.10 µ-curve of the 341 towards Tottenham 5.11 µ-curve of the 63 towards Kings Cross 5.12 µ-curve of the 109 towards Brixton 5.13 µ-curve of the 18 towards Euston 5.14 µ-curve of the 29 towards Trafalgar Square 5.15 µ-curve of the 171 towards Catford, showing a sharp fall at the end of

the route 5.16 µ-curve of 171 towards Catford with the number of buses reaching each

stop superimposed 5.17 µ-curves of the 419 in both directions 5.18 µ-curve of the P13 towards Streatham 5.19 µ-curves of the 232 towards Neasden and the 487 towards Willesden 5.20 µ-curves of the 337 towards Richmond and the P13 towards New Cross 5.21 Reproduction of Figure 2.2 5.22 Graph of the coefficient of variation of the headway against distance

downstream (from a numerical example given in Adebisi, 1986) 5.23 µ-curve for a simulated bus route with a 5-minute headway 5.24 µ-curves for simulated routes with different headways 5.25 Comparison of µ-curves of a simulated 5 minute headway service with

an equivalent real service – the 24 towards Hampstead 5.26 Exponential model relating modal headway to µ� 5.27 Logarithmic model relating passenger numbers to µ� 5.28 Comparison of the actual values of the hold-out samples with the curve

of the regression model 6.1 Reproduction of Figure 5.3.

4 5 7 9 10 11 12 17 18 19 20 21 31 32 33 33 34 34 36 36 37 37 38 38 39 39 40 41 42 43 44 45 46 47 48 48 49 55 55 57 60

vi

List of Tables

6.2 Reproduction of Figure 5.9 6.3 Reproduction of Figure 5.11 6.4 Reproduction of Figure 5.13 6.5 Reproduction of Figure 5.25 6.6 Reproduction of Figure 5.28

61 61 61 62 64

4.1 Selected routes and their headways 4.2 Example of records of overtaking buses 5.1 Summary of the bunching metrics by route 5.2 Summary of the linear regression analysis for the three dependent

variables 5.3 Summary of the successful linear regression models 5.4 Rotated component matrix from the factor analysis 5.5 Summary of the performance of the regression model against the hold-

out samples 6.1 Reproduction of Table 5.1 6.2 Reproduction of Table 5.3

16 23 29 51 53 56 57 58 63

vii

Glossary of abbreviations and technical terms

AVL. Automatic vehicle location. A system that enables operators to track their buses. Modern systems generally use GPS and on-board devices such as odometers. AWT. Average waiting time. Defined as:

�� = ∑

�

� ∑

,

where Hi is the headway on run i. BREMS database. TfL’s passenger database formed from electronic ticket machine data. Coefficient of variation. The standard deviation of a variable divided by the mean of that variable. Curtailment. A service control strategy whereby buses are turned around early in order to fill a gap in the service in the opposite direction. EWT. Excess waiting time. The standard measure of on-time performance used by TfL on high frequency routes. Defined as:

�� = �� − ��. GPRS. General packet radio service. The data network used by mobile phones. GPS. Global positioning system. A satellite-based positioning system allowing precise knowledge of position. Headway. The time between successive arrivals of two buses at a particular point along their route. iBus. The AVL system used by TfL to track each bus in service in London. K. A parameter thought to be critical in bunching behaviour. Defined by Newell and Potts (1964) as: K =

��

�� .

µ. The preferred measure of bunching for this dissertation. Defined as:

µ = 1 −! − "

!� − " ,

where H is the headway, Hs is the scheduled headway, and α is the cut-off point at which buses are considered 100% bunched (set at 30 seconds in this dissertation). µ-curve. µ plotted against stop number for a particular route. SWT. Scheduled waiting time (i.e. half the scheduled headway). TfL. Transport for London. The public body responsible for planning and managing the public transport network in London.

1

Chapter 1 - Introduction 1.1 Overview Bus bunching is the phenomenon whereby two or more buses arrive at the same point in quick succession. It is not thought to be simply a result of random variation in initial headways, but of the following positive feedback effect. Assuming a steady rate of passenger accumulation at stops, a bus which is delayed between two stops will have more passengers to pick up at the second stop than would otherwise be the case. The extra delay in picking up these extra passengers causes the bus to run a little later, so that more passengers will be waiting at the next stop, and so on down the route, with the bus becoming later and later. The next bus after the late one (assuming this bus is not subject to an identical initial delay) will in turn have progressively fewer passengers to pick up at each stop, and so will gradually become more and more ahead of schedule, catching up on the delayed bus. Two additional factors increase the magnitude of the positive feedback loop. Firstly, since the second bus of a bunched pair has fewer passengers on board, it is therefore less likely to stop at bus stops with no waiting passengers in order to allow passengers on the bus to alight. Secondly, the extra passengers travelling on the leading bus of a bunched pair add to its weight significantly – a half load of passengers on a double decker bus increases its weight by around 3 tonnes or 28%1. This additional weight causes slower average speeds in leading buses compared with following buses, further exacerbating bunching. Bunching is an undesirable phenomenon, both from the point of view of passengers using the service, and from that of service operators. From passengers’ point of view, the more regular a bus service is, the less time is wasted in the allowances made for irregularity. These time savings can be monetised in the usual way, allowing for the fact that waiting time is generally acknowledged to be valued at around 1.6 times in vehicle time.2 Thus when passengers assess the relative merits of different modes of transport, service regularity is a key concern. Since bus operators seek to attract passengers, it is therefore important to them to minimise bunching, and various techniques have been developed to do so. These include the provision of infrastructure (such as bus lanes), careful planning of timetables, and the use of real time service control to split up paired buses. However, such measures are expensive to implement, and are by no means perfectly successful. There is, therefore, a clear economic motivation for planners to seek to understand bunching behaviour, in order that it can be minimised with a minimum of capital and operational cost, and that passengers can use bus services with a minimum waste of their time. The iBus system in London offers a new opportunity for studying real world bus bunching. iBus, fully implemented in 2009, incorporates an automatic vehicle location (AVL) system which tracks the position of the 8,500 buses in use in London. Each vehicle keeps detailed records of its position as it progresses along its route, which is stored centrally for analysis. With this data it is possible to know to the nearest second the time at which a given bus reaches a given stop. This makes iBus an ideal data source for the study of bunching, at an extremely fine-grained level of detail.

1 DfT, 2009, using figures for a Scania NUD / Alexander Double Deck with air conditioning. Assumes a maximum load of 101 passengers with an average weight of 65 kg (around 10.24 stone). 2 Balcombe, et al., 2004.

Chapter 1 - Introduction

2

1.2 Aims

1. To produce a satisfactory metric with which to measure bunching. 2. To use iBus data to establish the severity of bunching on the routes being

studied, according to the metric derived for Aim 1. 3. To use the metric to investigate how bunching varies with time of day, and

how it develops down a route. 4. To ascertain whether bunching occurs because of the positive feedback effect

discussed above, or whether it is simply a result of natural variation in bus speeds.

5. To use statistical tests to assess which factors of a route are most closely associated with bunching.

6. To construct an empirical model using the iBus data that uses the factors from Aim 5 to predict how severe bunching will be on a given route.

7. To test this model against one or more hold-out samples to ascertain how successful it is at predicting bunching.

8. To compare the results obtained in this dissertation with those of earlier studies, summarised in the literature review.

1.3 Structure The remainder of this dissertation will be structured as follows. Chapter 2 will give an overview of the relevant literature in the study of bus bunching, including both theoretical analyses and empirical studies making use of AVL data. Chapter 3 will present some background information about the nature of London’s bus system, including iBus and service control. Chapter 4 will give a detailed account of the methodology used to select and analyse the data. Chapter 5 will present the results of the analysis. Finally, Chapter 6 will summarise the findings of the research, and suggest ways in which it could be extended.

3

Chapter 2 - Literature review 2.1 Theoretical analysis The extensive range of theoretical analyses of bunching in transit vehicles, often in the mathematical or scientific journals, is broadly acknowledged to have been instigated by Newell and Potts (1964). In this seminal paper, the authors derive a model which depends crucially on K, defined as: K = ��

�� .

Behaviour of buses within the model is such that headways between buses are unstable, from the frame of reference of a particular bus run. If any disturbance causes a bus’s headway to become shorter or longer than scheduled at a particular stop, this deviation is always magnified as the bus progresses down the route. The behaviour of the whole system following a disturbance (as opposed to that of individual bus runs) depends on the value of K within the system. If K < �

�, the

system is stable, and headways at the affected stop will return to normal, more quickly as K decreases in magnitude. If

�

�< K < 1, the system is unstable, and the

headway disturbance will be amplified as successive buses reach the stop. The authors suggest that the tendency of buses to bunch can be reduced by drivers reacting to seeing preceding buses ahead, by buses skipping stops due to overcrowding, and by empty buses overtaking full ones. They note that the importance of keeping K small in order to minimise the impact of disturbances implies that the best way to minimise bunching within the system is to maximise the passenger loading rate. On this evidence therefore, the fact that passenger loading rates on articulated buses are somewhat higher than on conventional buses, would lead one to predict that bunching might be less prevalent on routes served by articulated buses than those served by conventional buses. Daganzo (2009) constructs a different model with similar implications. The characteristic equation of this model is:

��, �� ≈ ��, � + �� + ��, � − ��, � − ! + "�, ��,

where ��, � is the arrival time of bus n at stop s, �� is the controlled time between stop s and s+1,

�� is the expected number of passenger arrivals in segment (s, s+1) during the average marginal delay induced by one boarding move (which is exactly equivalent to K as defined in Newell and Potts, 1964),

H is the scheduled headway, and "�, �� is the random variation in running time in segment (s, s+1). It can be seen that this equation implies a positive feedback loop given a disturbance in the running time. If (��, � − ��, �) > H (i.e. the headway is greater than scheduled between buses n and n-1 at stop s), then the arrival time of bus n at stop s+1 will become later, since by definition �� > 0. If (��, � − ��, �) < H, then bus n will arrive at stop s+1 more quickly. In short, late buses become later and early buses become earlier. The larger the value of ��, the stronger the ‘attraction’ or ‘repulsion’ between consecutive buses. The author observes that the probability density function of the noise term, "�, ��, has ‘high tails’, i.e. there is a relatively

Chapter 2 – Literature review

4

high chance of extreme variation. On this basis slack time of four standard deviations of the noise term is recommended to be scheduled at control points. Hill (2003) produces a simple model which shows clearly that the tendency of buses to bunch can only be compensated for by adjusting the velocities of buses. Again, the critical parameter (which he calls µ) in the severity of bunching is equivalent to K in Newell and Potts, 1964. Hill observes that for bunching to be prevented, bus drivers need to react immediately to deviations from the scheduled headway, speeding up when behind, and slowing down when ahead. He observes that in practice drivers tend only to react when they can see a preceding bus ahead of them, and this is why bunching occurs. However, this is not the case in London, where extensive control mechanisms are employed (see Section 3.3). Hill runs some simulations to test how quickly one might expect to see disturbances from headways form as a bus progresses down its route. Figure 2.1 shows how many stops one would expect the bus to have to travel before deviating from the schedule by a certain number of minutes. The simulation predicts that one might expect lower frequency buses to serve hundreds of stops before a bunching impact is observed due to passenger boardings (though of course deviations from the schedule can occur for other reasons).

Figure 2.1. Source: Hill, 2003.

Turnquist (1981) examines the importance of various factors in reducing the standard deviation of journey times (which is likely to imply a reduction in bunching). His simulations indicate that vehicle frequency is a dominant factor in headway variability. He also observes that reducing variation along individual links of the route will reduce variation over the whole route, and that priority for buses at signalised junctions should therefore be expected to reduce journey time variability. His simulations verify this, along with the positive impact that exclusive bus lanes can have on journey time variation. Adebisi (1986) presents a model of headway variance dependent on two factors. The first he calls the route factor – the conditions of the highway, including traffic. The second is the bus loading factor, which is dependent on demand, and also on the route factor. The author observes that where the route factor predominates in determining the extent of headway variance, it is possible to design in infrastructural features to reduce variance, for example bus lanes. Where the bus loading factor dominates, headway-based control schemes, such as holding at stops, are likely to be more appropriate. Adebisi also predicts that abrupt changes in demand, such as those that occur in the transition between peak and off-peak, are likely to cause increases in headway variation. He presents a brief numerical example giving the


5

headway’s coefficient of variation along the route of a putative bus service. This will be returned to in Chapter 5. Abkowitz et al. (1986) provide a treatment that is in a sense half way between theoretical and experimental, by using Monte Carlo techniques to produce an empirical model of headway variance as a function of mean headway and journey time variance. An interesting prediction of this model is that headway variation increases non-linearly with distance down the route – see Figure 2.2.

Figure 2.2. Source: Abkowitz et al., 1986. The curve of interest is the one to the extreme left (the ot her two

indicate headway variation following implementation of a specific holding strategy at stops 4 and 10). 2.2 Experimental research Since the advent of advanced GPS-based automatic vehicle location (AVL) systems in the last two decades or so, many studies have sought to analyse bus performance using the generated data. Many of these projects have investigated bunching insofar as it affects service reliability, though few have focused particularly on bunching itself. Peng et al. (2008) analysed AVL data from a one-week period on Bus Route 20 in Chicago, a service with a peak headway of around 7 minutes. This study aimed to examine patterns in long gaps, defined as headways over 50% longer than scheduled headways, and in bunching, defined as two buses arriving within one minute of each other. The methodological approach was to carry out regression analyses on large gaps and bunching, to ascertain whether a bunch or a large gap at one time point gave rise to a bunch or large gap at the following time point. A total of 9 time points were used. The study found that there was no significant correlation between bunching at one time point and bunching at the following time point, and concluded that bunching does not propagate down the route of this service. Similar analysis of long gaps found that there was a significant relationship linking one time-point to the next, and that therefore long gaps do propagate down the route of the service. However, it was found that the long gaps tended to diminish as individual services progressed down the route. These findings are in contrast to the prediction of the theoretical models detailed in the previous section, which is that bunches and large gaps in individual services will become more severe as the bus moves down the route. A related finding is that the overall proportion of long gaps increases with distance along the route, in both directions. Unfortunately, an equivalent result for bunching is not presented.


6

Strathman et al., (2002) studied data from 15 bus routes in Portland, Oregon to find relationships between several independent variables and journey time. An interesting implication of this study is that the identity of bus drivers has a significant effect on journey times, accounting for 17% of journey time variation. Several attributes of drivers were tested for significance, but only driver experience was found to be significant: on average each month of experience was found to decrease running time by 0.57 seconds. The study also notes that there is some seasonal variation in journey times, raising the question of whether this might be the case in bunching. Another study by the same group (Strathman et al., 1999) of 8 routes in Portland, Oregon also produced some interesting results. Of particular relevance to this dissertation is the emergence of extreme bunching in the evening peak of bus services, a phenomenon noted by several other studies. It is discovered that a 15% decrease in bunching occurred following the introduction of a bus despatching system using real-time AVL, a more basic forerunner of TfL’s iBus system. Abkowitz and Engelstein (1983) as cited in Abkowitz and Tozzi (1987) created a regression model from data from buses in Cincinnati, Ohio. Their model, later successfully tested on data from South California and Boston, took several independent variables and predicted segment journey time. Independent variables included: segment length, number of signalised junctions, number of boardings/alightings, percentage of route with on-street parking, and dummy variables for particular time periods. Since journey time and bus bunching are fairly closely related, the success of this model would appear to suggest that a similar model might be derived for bus bunching. Kulash (1971) as cited in Abkowitz et al. (1987) presents a study of a bus route in Boston. This shows that the variance of headways at the beginning of the route (0.8 min2) is far exceeded by that midway down the route (26.1 min2). This is evidence in favour of propagation of disturbances down a bus route. Kulash also observes that the headway distribution downstream on the route approaches an exponential distribution. Abkowitz et al. (1987) also make some important observations concerning the causes of service unreliability. They argue that while the positive feedback loop arising from passenger boardings is a factor in variability of headways, a more important factor is simply the variation in journey times. They cite a simulation study by Bly and Jackson (1974) in which journey time variation had a far greater impact than bus stop dwell time variation on headway variation. Feng and Figliozzi (2011) undertook a study of the Route 15 in Portland, Oregon. Their definition of bunching is that two buses are bunched if the headway between them is less than three minutes. Their work suggests that bunching may propagate down the route, and that holding strategies at control points may serve to mitigate bunching. They also observe that bunching becomes rather more severe during times of higher frequency service – though this should not be too surprising given their definition of bunching.

7

Chapter 3 – Background information 3.1 London’s buses London has a fleet of around 8,500 buses1. On a typical week day, around 7,000 of these are in scheduled service, operating on around 700 routes. Each week day, around 6 million passenger trips are completed on the network. Bus services are managed by London Bus Services Limited (known as London Buses), part of Surface Transport, one of the three main directorates of Transport for London (TfL). TfL is the integrated body responsible for the planning and management of London’s transport system. It is a publicly-owned body, responsible to the Mayor of London and the Greater London Assembly. London’s bus network is extensive. There are around 19,500 bus stops, and it is estimated that more than 90% of Londoners live within 400 metres of a bus stop. The network has seen considerable investment in the last decade, and this has been rewarded with high levels of passenger growth. In the decade from 1999/2000 to 2009/2010, annual passenger-km grew by 81%2 - see Figure 3.1.

Figure 3.1. Annual bus passenger-km on TfL buses. So urce: TfL 2010a, Figure 2.4, p. 47. Note that a new

estimation method was introduced in 2007/2008. Several different types of vehicle are used, ranging from small ‘midi-buses’ to double-deckers and articulated or ‘bendy’ buses. These have been subject to some controversy in recent years, with the current Mayor of London, Boris Johnson, having issued the promise to remove articulated buses from London’s streets in the run up to his 2008 election. Articulated buses are perceived as unsafe and costly3, though there is limited evidence to support this. At the time of the stop-data obtained for this study, 5 articulated routes had been converted to double decker services. One has been converted since (the 25), and the remaining 6 will be by the end of 2011.

1 Information in this section is largely derived from TfL sources (TfL 2010c, 2010d and 2009). 2 TfL (2010a), Table 2.5, p. 46. 3 See for example Cohen, 2008 and Kisiel, 2009.

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

Pas

seng

er k

ilom

etre

s (m

illio

ns)

Chapter 3 – Background information

8

London buses operate on a wide range of headways from 2-3 minutes in the peak on busy routes, to 24 hours on very low demand routes. Services are classified as high frequency when modal headways are 12 minutes or less, and low frequency when they are 15 minutes or more (there are no headways between 12 and 15 minutes because of clockface running). Reliability is measured differently for the two types of service. For high frequency services, the measure used is excess waiting time (EWT). This is a measure of how many passenger-minutes are wasted in waiting for late services, defined by

�� = �� − ��, where AWT is the average actual waiting time, and SWT is the scheduled waiting time (i.e. half the scheduled headway). The average wait time is defined as:

�� = ∑ ��

�

2 ∑ ��

where Hi is the headway on run i. This formula takes into account the fact that more passengers will be waiting at the end of a long gap than at the end of a short gap. For low frequency services, ‘on time performance’ is used. A bus is considered on time if it departs up to 2½ minutes early or up to 5 minutes late, with respect to the published schedule. The percentage of buses departing on time can then be used to gauge a route’s performance. 3.1.1 Regulatory framework While the rest of Great Britain’s bus operations were deregulated in 1986, and subsequently largely privatised, London was exempted from this process. Instead, bus services in London were tendered, such that contracts for particular routes were awarded to operators able to run the most successful service at the lowest cost (known as ‘Gross Cost’ contracts). Initially, publicly owned subsidiaries of London Buses competed with private companies for these services, but these were all privatised by 1994. From the mid-1990s, ‘Net Cost’ contracts were also awarded, in which the revenue risk was transferred to the operator. In 2001, both forms of contract were replaced by ‘Quality Incentive Contracts’ (QICs). QICs are similar to Gross Cost contracts in that TfL retain the revenue. However, there are also financial rewards and penalties for operators linked to quality of service. London Buses are responsible for planning routes and specifying service frequencies and vehicle types. Routes are tendered separately every 5-7 years, and a review of each route is undertaken by London Buses before tender. Contracts are awarded based on costs, but also taking into account other factors, including safety and competition. Contracts are generally 5 years in length, but with a possible 2 year extension if performance levels are met (see below). Incentives are in the following forms:

1. Reliability performance payments. Reliability targets are set by London Buses for each individual route during the tender specification process. The relevant measure is defined differently for low frequency and high frequency routes.

o For high frequency routes the unit of measure is 0.1 minute change in EWT.

o For low frequency routes, the unit of measure is a 2.0 percentage point change in on time performance percentage.

Bonus payments of 1.5% of the overall contract price are made for every whole unit of measure above the target level for that route, to a maximum of 15%. Penalty payments of 1% of the overall contact price are imposed for every unit of measure below the specified route target, to a maximum of 10%.


9

2. Contract extensions. An ‘extension threshold’ related to reliability of service is set at the outset of the tender. If the operator’s performance exceeds this threshold, they are automatically eligible for a two-year extension to the contract, which they are able to accept or decline.

3. Quality performance payments. These are further incentive payments based on the standard of driving, and the interior and exterior presentation of vehicles.

4. Lost km adjustments. In general, the number of bus-km operated in a given period is less than the number of scheduled bus-km. Some of these ‘lost km’ will be due to circumstances under the operator’s control – e.g. staff shortages, mechanical failure, etc. The operator is fined the same proportion of the contract price as the proportion of lost km to scheduled km. Note that the operator is not fined for lost km beyond their control, e.g. due to unusually bad traffic conditions.

Clearly, the above measures act as a significant incentive to operators to keep services as regular as possible. Bunching is therefore a serious concern of controllers, as its corollary, long gaps, can result in a significant loss of revenue for operators. Figure 3.2 shows how the introduction of QICs in 2001 had a major impact on reliability in London Buses.

Figure 3.2. Change in reliability measures with tim e. Source: TfL 2010a, Table 4.7, p. 111.

3.2 iBus The iBus system, fully operational in Spring 2009, is an Automatic Vehicle Location (AVL) system. The iBus project cost around £117m, and was developed by Siemens AG. It uses several complementary technologies to determine the location of each bus in the London network, to an accuracy of around 10 metres. It does this using:

• Global Positioning System (GPS) • Odometer (measuring distance travelled) and gyroscope (measuring

orientation), coupled with mapping software. These mechanisms are contained within an on board unit on each bus, shown as ‘G’ in Figure 3.3. The on-board unit fulfils several functions, including the following:

1. It sends a GPRS message to a central server (K in Figure 3.3) every 30 seconds, containing information on the bus’s GPS location. This information is

60%

65%

70%

75%

80%

85%

0

0.5

1

1.5

2

2.5

On

tim

e p

erf

orm

an

ce

EW

T i

n m

inu

tes

EWT On time performance


10

fed in real time to route controllers, who use it to regulate the service (see Section 3.3).

2. It logs the GPS location every second on its internal memory. This log is then uploaded wirelessly every evening to a server within the garage when the bus returns. This data is in turn uploaded to the London Reporting Datamart database, where it is available for subsequent analysis of route performance, etc.

These two functions are key to this dissertation. 1. concerns how buses are regulated by the controllers, and 2. is the source of the bus stop data which is analysed in Chapter 5.

Figure 3.3. Schematic of the iBus system. Source: TfL 2011b.

Several other functions that iBus carries out should be noted. Firstly, the on-board unit matches GPS co-ordinates with bus stop mapping data, and provides on-board audio announcements and dot matrix displays of the next bus stop. Secondly, the central positional data is fed to the Countdown system, which consists of dot matrix displays at selected busy bus stops, informing waiting passengers of expected


11

arrival times of the next buses (see Figure 3.4 below). Before the introduction of iBus, Countdown relied on radio beacon technology – it was intended that the introduction of iBus would make the information considerably more accurate. Thirdly, iBus allows buses to ‘talk to’ traffic signals, either through the central system, or directly. This allows a late running bus to request priority through the signalised junction, which can be granted under certain conditions. One might expect this to have a lessening effect on bunching.

Figure 3. 4. Countdown dot matrix display. Source: T fL, 2011b.

TfL is currently undertaking the technically challenging task of integrating the electronic ticket machines on each bus with the iBus system. This would enable TfL to know exactly how many people board each bus at each stop. Currently this is estimated using on-bus surveys, which are costly, and provide a snapshot of boarding profiles on a route renewed only every five years. It would have been fascinating to have passenger data for the purposes of this dissertation, in order to gain a better understanding of the interaction between passenger loadings and bunching. 3.3 Service control 4 Bunching is an undesirable phenomenon from the point of view of both the operator and the passenger. The more severe the bunching, the more inefficient the use of the operator’s expensive resources. Compare a route served by severely bunched pairs of buses with one served by regularly spaced single buses. The former could offer the same frequency as the latter, but with twice as many buses, and therefore double the costs (though with double the capacity). Since an extra bus on a London route costs around £120,000 per year in capital and operational costs, it can be seen that bunching is potentially a very expensive problem for London bus operators. Highly variable headways are also very inconvenient for passengers, and this could cause some of the passenger base to seek alternative modes of transport. In addition, the fact that in an uneven service most passengers will arrive at a stop in a long gap, means that more passengers will tend to travel on overcrowded leading buses – another off-putting factor. Given the above facts, operators around the world go to considerable effort to implement steps to minimise bunching. Strategies can be broadly grouped into infrastructural, planning, and operational. Infrastructural measures include bus lanes, ‘bus gates’ (traffic signals giving buses priority on pulling out of stops) and traffic signal priority. All of these exist in the London network.

4 The content in this section is derived largely from the author’s discussions with Terry Butler, manager of the Arriva North control room in Dalston, North London (Butler, 2011).


12

Planning strategies to mitigate bunching include building in sufficient recovery time at each end of the bus route such that the vast majority of services are able to adhere to their scheduled departure time. However, there is a balance to be struck here, as too much recovery time results in expensive resources being underused. Operational service control is a complex undertaking. With the advent of AVL in the last decade or so, and the iBus system in London in 2009, the task has been simplified somewhat. Before these tools became available, controllers were deployed along the route, and sought to regulate the headways of buses leaving a particular stop. iBus now offers centrally-based controllers real time information on the position of the buses under their management. In London, one controller is typically responsible for anything from one very busy route to three less busy routes. Information is conveyed to them in the form of a colour display showing buses moving around the route (see Figure 3.5). As stated in the previous section, the position of each bus is updated roughly every 30 seconds. Controllers are normally former drivers themselves, and so have a good practical understanding of service control issues.

Figure 3.5. Screenshot of the software used by the r oute controllers. Buses are colour-coded according

to adherence to schedule. Source: personal communic ation with TfL. Various strategies to minimise bunching along a route are employed. They include:

1. Holding an early-running bus at a particular point to even out headways. 2. Instructing a late-running bus to skip several stops in order to catch up with

the preceding bus. 3. Turning a bus around before it reaches the end of the route in order to fill

in a ‘hole’ in the service in the opposite direction (known as ‘curtailment’).


13

4. Reducing the speed between stops of early-running buses. Note that instructing an early-running driver to increase her speed is avoided for safety reasons.

5. Sending out a bus to begin part-way along the route. Strategies 1 and 3 are clearly unpopular with passengers on the bus, while 2 is unpopular for those waiting at skipped stops, and used in London only under exceptional circumstances. Regularly implementing any of strategies 1-3 is therefore likely to have a detrimental impact on ridership. Keeping buses in reserve to fill in gaps in the service, as in strategy 5, involves increased operating costs for bus operators. In London, the most common of these strategies is strategy 4. This is popular with controllers and drivers, because it is not easily perceived by the passengers of the bus in question. Indeed, the strategy is, to an extent, automatic, in that the iBus system also includes a display in every driver’s cab indicating how many minutes ahead or behind schedule the driver is. Strategy 3 is also widely used by controllers, though care has to be taken to avoid reducing capacity where the stops at the end of the route have high demand. For example, one would not want to curtail many buses terminating at Victoria station during the morning peak, as this would lead to an unmanageable build-up of passengers at Victoria. In addition, each lost bus-mile costs operators around £7-8 as a result of quality incentive contracts. This is one simple illustration of the many conflicting considerations that controllers have to take into account while regulating their route(s). This complexity explains why automatic service control software is not yet in use – judgment plays a key role. One final observation to make is that Butler (2011) emphasises strongly the role that driver behaviour can have in influencing service control. Despite their job title, controllers are only able to exert control if drivers cooperate with their instructions. This can be difficult to achieve where instructions conflict with the driver’s own wishes, or where passengers become restive, for example if a bus is held at a particular stop. This somewhat gives the lie to the highly sophisticated service control strategies advocated in the mathematical literature (for example in Hill, 2003). Given that it is impossible to describe mathematically a driver’s state of mind, seeking to devise a strategy based on controlling vehicle speed very precisely seems rather futile, at least until the advent of driverless buses. These observations gain evidential support from the Strathman et al. study described in Section 2.2, in which bus driver identity predicted 17% of journey time variation (Strathman et al., 2002).

14

Chapter 4 – Methodology 4.1 Bunching metric In a practical examination of bunching, where aims include describing observed levels of bunching, and predicting its severity, it is extremely important to decide on the right definition of bunching itself. A key consideration is that bunching is a time-based, as opposed to a distance-based, phenomenon. From a passenger’s point of view, the important thing is how closely spaced in time buses are rather than in distance. Thus the metric should be predicated on bus timings rather than observed distances. This has important ramifications for the type of data required to compute the metric (see Section 4.3). A second key consideration to highlight is that a study of bunching is concerned with successive headways of buses, rather than adherence to a schedule. For example, it is possible for every bus on a particular route to arrive 10 minutes late, and yet for no bunching to be taking place. Thus, a metric should take into account headways rather than lateness with respect to a schedule. A number of different approaches are possible. 1. The standard metric for measuring bus punctuality, and the one favoured by

London Buses, is excess waiting time (EWT – see Section 3.1). This is unsuitable for measuring bunching, as it measures excess waiting time over and above what is expected. EWT could be high even with no bunching (for example if one bus is removed from a service), and can be low even with severe bunching (for example if extra buses are introduced).

2. Standard deviation of observed headway variations from the scheduled headway. This suffers from a similar problem to EWT. If scheduled headways are all identically longer (or shorter) than scheduled, this metric would be non-zero even if headways are all regular.

3. The coefficient of variation of headways, C, defined as:

� = �� ℎ��

�� ℎ�� ℎ��.

This is recommended in The Transit Capacity and Quality of Service Manual (Transportation Research Board, 2003). While one would expect this to increase with increased bus bunching, it suffers from the same weaknesses as Metrics 1 and 2, and therefore will not be used.

4. The proportion of bunched buses given a certain critical headway below which all buses are considered bunched. This can be:

a. Absolute. For example, the Chicago Transit Authority describes buses arriving within 60 seconds of the previous bus as bunched (Cham, 2006). This will be expressed as Hc, where c is the cut-off point below which buses are considered bunched.

b. Relative to the scheduled headway. For example, the proportion of buses running at headways less than or equal to 25% of the intended headway (as suggested in Bellei and Gkoumas, 2010). This will be expressed as hc, where c is the proportion of the headway below which buses are considered bunched.

5. Metric 4 would be suitable for describing bunching, but it is a little crude in that buses are described as bunched or not given a certain arbitrary cut off point. For example, for H60, a bus that arrives 59 seconds after its predecessor would be considered bunched, while one arriving 61 seconds after its predecessor would not be. One can derive a new more continuous metric

Chapter 4 - Methodology

15

which might be more suitable for the purposes of this dissertation in the following way.

a. Intuitively, one might suggest that buses running at exactly the scheduled headway should be considered 0% bunched. Conversely, buses arriving at exactly the same time, i.e. with a headway of 0 seconds should be considered 100% bunched. This suggests the following for Metric, M:

� = 1 −�

��

,

where H is the actual headway (the time between the arrival of the previous bus and that being considered) and Hs is the scheduled headway. The fraction in this metric is known as the headway ratio, HR (see, for example, Strathman, et al., 1999). It can be seen that as H→Hs, M→0 (or 0%); and as H→0, M→1 (or 100%). This metric is what was intuitively wished for. M for a stop could then be found by averaging values of M for buses arriving at the stop. M for a route could in turn be found by averaging over all of the stops on the route. In doing so it would be necessary to impose the condition that M ≥ 0, so that negative values of M for headways larger than Hs are set to 0, and do not cancel out positive values of M from bunched buses.

b. A criticism of Metric 5a is that the condition of H = 0 for 100% bunching is too strong, that it rules out some buses that one would intuitively consider bunched. For example, if two buses arrive at a stop only 10 seconds apart, it might be considered that they are 100% bunched. Metric 5a can be modified slightly to take this into account as follows:

µ = 1 −� − �

�� − � ,

where α is the time beyond which buses are considered 100% bunched1 (e.g. 10 seconds). µ behaves in a similar way to M: as H→Hs, µ→0 (or 0%); and as H→α, µ→1 (or 100%). Again, this is what one would intuitively wish. We must now impose the condition 0 ≤ µ ≤ 1 so that when H < α, µ is set to 1: buses cannot be more than 100% bunched. Note that µ→M as α→0, so the choice of a small value of α compared with Hs and H will mean that there is little difference between Metrics 5a and 5b. While the choice of α is arbitrary as with the cut-off in Metrics 4a and 4b, the use of a sliding scale type metric means that the arbitrary choice is far less crucial. In this dissertation, α will be set to 30 seconds, a value at which buses might sensibly be considered 100% bunched.

For the purposes of this dissertation, the following metrics will be used:

• µ • H60 • H30 • h0.25

µ is considered the most useful of these metrics, because of its continuous nature. It will therefore be favoured over the others in the analysis.

1 Alternatively, α could be set to the minimum physically possible headway between two successive buses, or to the average dwell time of buses at a stop.


16

4.2 Route selection As the aim of the dissertation was to examine in detail the bunching behaviour of London’s buses, and it was thought that bunching would be most prevalent among high-frequency buses, it was decided that the first routes to be selected should be those of the highest frequency. Data on route frequencies were obtained from London Bus Routes (2011), and sorted by frequency in Excel. From this, the nine most frequent routes were chosen, having weekday headways of 4, 5 or 5-6 minutes (see top nine rows of Table 4.1). As there were a further 23 routes with headways of 6 minutes, routes from this group were selected in order to be representative of London geographically (see Figures 4.1 - 4.5). This was done by plotting the routes already selected on a map, and choosing new routes to fill in gaps. In this way 6 routes with headways of 6 minutes were added to the previous 9. These high frequency routes would, it was hoped, provide some evidence of bunching behaviour. However, it was also of interest to investigate the relationship between frequency and bunching behaviour, so a further 16 routes were selected with progressively lower frequencies, again such that they would be geographically representative of London.

Route number Headway (mins) 18 4 12 5 24 5 41 5 94 5 25 5-6 29 5-6 253 5-6 254 5-6 63 6 87 6 109 6 279 6 472 6 H37 6 205 7-8 147 8 171 8 184 10 333 10 341 10 388 10 54 12 153 12 337 12 272 15 419 15 487 15 166 20 232 20 P13 20

Table 4.1.


17

Figure 4.1. Overview of selected routes. Note that Route 166, depicted in light blue at the extreme so uth

of the map, does not appear in Figure 4.3, as it wa s later dropped (see Section 4.4.3).


18

Fig

ure

4. 2

. Sel

ecte

d bu

s ro

utes

in W

est L

ondo

n.


19

Fig

ure

4. 3

. Sel

ecte

d bu

s ro

utes

in S

outh

Lon

don

.


20

Fig

ure

4. 4

. Sel

ecte

d bu

s ro

utes

in E

ast L

ondo

n.


21

Fig

ure

4.5.

Sel

ecte

d bu

s ro

utes

in N

orth

Lon

don.

Not

e th

at a

lth

ough

onl

y 4

rout

es a

ppea

r he

re, s

ever

al n

orth

ern

rout

es

appe

ar in

Fig

ures

4.2

and

4.4

, so

Nor

th L

ondo

n is

not

und

er-r

epre

sent

ed.


22

4.3 Date selection To maximise the chances of bunching being observed, weekdays were preferred to weekends, in the same way that high frequency routes were preferred to low frequency. In order to minimise the effects of any anomalous traffic conditions, days in successive weeks were chosen. For the same reason, different days in the successive weeks were chosen. The dates selected were:

• 7th February 2011 (Monday) • 15th February 2011 (Tuesday) • 23rd February 2011 (Wednesday) • 3rd March 2011 (Thursday) • 11th March 2011 (Friday)

It should be noted that the 23rd of February was in the state school half term, when traffic levels and bus demand are typically somewhat lower than in term time. This shall be returned to in Section 5.2. 4.4 Data selection and cleaning As discussed in Section 4.1, the data required are time point data, i.e. the times at which buses reach certain way-points. As London Buses’ focus is on bus stops, their time-point data use bus stops as way-points. Fortunately, this is perfectly adequate for the purposes of this dissertation. Data were obtained from the London Reporting Datamart database via a terminal in the TfL Palestra building, in comma separated variable form, as a separate file for each day. These data were uploaded onto a personal computer, and imported into Microsoft Access as five separate tables, one for each day. Each table contained a row for each stop made by each bus on each of the 31 routes selected: around 370,000 rows per table. Relevant fields were:

• Shortdesc. This is the route number of the bus. • Direction. Either 1 or 2 to indicate the direction of the bus. • Tripnr. This is the running order of the bus, the first bus being numbered 1,

the second 2, etc. Buses in a particular direction are either all odd or all even. • Stoppointid. This is a unique number identifying a particular bus stop. • Stopsequence. This indicates where on the route a bus stop comes, e.g. 10

for the 10th stop. Note that this is different for each direction of a route. • Scheduleddistance. This indicates the distance (in metres) between the stop

represented by this row and the previous stop. • Sched Dist In Trip. This indicates cumulative distance (in metres) along the

route. • Observed Arrival Time. The time to the nearest second at which the bus

arrived at the stop. • Observed Departure Time. The time to the nearest second at which the bus

departed from the stop. • Observed Headway. The time (in seconds) between the departure of this bus

and the departure of the previous bus from this stop. • Scheduled Headway. The intended time (in seconds) between the departure

of this bus and the departure of the previous bus from this stop. • Operatorid. This is a unique number identifying which operator runs the bus

route.


23

• Patternid. This indicates which stopping pattern a bus is following. This can vary slightly according to time of day.

• Nightbus. Bus stops scheduled between midnight and 05:00 are considered to be night-time stops. These are marked with a 1 within this field; others are marked with a 0.

• Observed. This binary value indicates whether the bus was logged by the system at this particular stop – 1 if yes, 0 if no.

• Unscheduled. This binary value indicates whether the bus was on an unscheduled run or not, 1 for unscheduled, 0 for scheduled.

The data were ‘cleaned’ by querying the tables for all relevant fields to exclude lines (i.e. stop events) according to the following conditions:

• Tripnr > 2. This was to prevent the first runs in either direction being included in the analysis. Scheduled Headways for these runs were set at zero, which would have distorted the headway-based metrics. This excluded 0.66% of the data.

• Scheduled Headway ≠ 0. This was to prevent any further Scheduled Headways with values of zero distorting the metrics. These may have occurred due to glitches in the system. This excluded 1.56% of the data.

• Observed = 1. This caused only observed stops to be included in the analysis. Stops with Observed = 0 had mostly empty fields. This excluded 6.29% of the data.

• Nightbus = 0. As some routes ran a night bus service and some did not, this condition was imposed so that only day-time runs would be considered, to allow fair comparison between routes. This excluded 3.03% of the data.

• Unscheduled = 0. This prevented unusual stopping patterns from distorting the results. This excluded 0.86% of the data.

It should be noted that it was not necessary to adjust for overtaking (as recommended in, for example, Trompet, et al., 2011), as this is done automatically by iBus. For example, consider Table 4.2, showing records taken from Route 109 in Direction 1 (towards Brixton) at stop 20 on the 11th March.

Tripnr Observed Departure Time Headway

33 07:18:18 6:14

35 07:21:06 2:48

37 07:22:59 1:53

39 07:26:35 3:36

41 07:49:07 3:03

43 07:40:01 13:26

45 07:46:04 6:03

47 07:55:08 6:01

49 07:57:39 2:31 Table 4.2.

This shows the departures (headways are calculated using departures) of successive buses on this service from this stop. Observe that Trip Number 43 has overtaken Trip Number 41 and arrives at the stop first. Next comes Trip Number 45, and finally Trip Number 41. The crucial fact is that the headway of Trip Number 41 is calculated using the Departure Time of the previous bus to depart from that stop, not from the previous bus in the running order. Fortunately, this is exactly what is required for this dissertation, as the headway-based metrics are concerned with headways between successive buses, rather than between scheduled buses.


24

In total, 10.20% of the data were excluded in the cleaning process, varying from 9.13% to 10.63% over the five days. These totals are smaller than the sum of those for the cleaning conditions, as there is some overlap between the conditions, e.g. a bus that is both unscheduled and a night bus. 4.5 Production of the metrics 4.5.1 By route µ was produced for each line of the five tables, and then mean values of µ were calculated for each bus route on each day. The mean across the five days, µ�, was then produced by combining the means for the five days. Strictly speaking, these are not the true means, as there may have been a slightly different number of trips on each route on each day. However, this is very unlikely to have had a significant impact on the results, and was considerably easier to achieve computationally than the true mean, given the data were in separate tables. H60, H30, and h0.25 were all produced similarly to each other. A dummy variable was created such that the dummy variable was set at 1 if the relevant bunching criterion was met, and set at 0 otherwise. The average values of the metrics for each route on each day were then calculated by dividing the sum of this dummy variable by the sum of the Observed field (i.e. the total number of stops for that route on that day). This gave an average for each route on each day, and these were combined to find the means for the route (H!"#, H!$#, and h�#.&'). The same note of caution applies to these means as to that for µ�, with respect to a possible very slight deviation from the true mean. 4.5.2 By time of day Mean values of µ were produced in 15 minute intervals by averaging values across routes. This was done by querying the Observed Arrival Time field for the minutes part of the time, dividing this by 15, and taking the integer part of the resulting number. Results were analysed by day, and as an overall average. 4.5.3 By stop number Mean values of the metrics at each bus stop along each route in each direction were produced. The data first had to be filtered so that only the main stopping pattern was used in the analysis. Otherwise, a particular stop number on one run might not correspond to that on another. To ascertain which stopping pattern dominated each route, a query was run showing how many stops each bus made on each stopping pattern. In general, one stopping pattern far outweighed all the others, and the data were duly filtered. However, on Route 166, the runs were split roughly in half between two stopping patterns, one a shortened version of the other (corresponding to the bus starting/terminating in Banstead rather than Epsom). This caused several insurmountable problems, and Route 166 was therefore discarded. µ was chosen as the variable to be plotted against stop number, because of the fact that it is more continuous than the other three metrics, and therefore yielded smoother curves – even when floating averages were used.


25

4.6 Simulation In order to understand fully the results of the analysis with respect to µ, and in particular to form a judgment of whether the positive feedback mechanism is at play (see Aim 4), it was considered necessary to have some understanding of the behaviour of µ in a system without positive feedback. Basic simulations were therefore carried out for this purpose. The simulations were run on Excel spreadsheets, using values drawn from a normal distribution, generated using the ‘rv.normal’ function in SPSS. 100 bus runs were included in each simulation, with buses moving down a route with 50 stops, each spaced 300m apart. A different set of simulations was run for four different headways: 5 minute, 8 minute, 12 minute and 20 minute. Initial headways were considered to be normally distributed, with the standard deviation of this distribution being chosen such that the initial value of µ was similar to that observed in reality. Speeds on each leg were also considered to be normally distributed, with means and standard deviations chosen with reference to the real routes being studied, by means of a simple query of the Access database. Only average leg speeds were used, rather than a combination of travel time and dwell time. As both of these are likely to be normally distributed, using normally distributed average speeds should produce identical results, and was simpler to achieve. As overtaking was allowed in the simulation, headways were reordered before the calculation of µ so that the headway in question was between two consecutive buses, rather than two consecutive runs (see Table 4.2 and associated comments). It should be emphasised that these simulations are basic in nature. In particular they are subject to the following limitations:

• Only one mean headway is used, whereas in reality this varies somewhat with time of day.

• Speeds on each leg of the route were all based on the same mean and standard deviation. In reality, some legs would be particularly slow, and some particularly variable.

• The assumption that initial headways are normally distributed may be a poor one. Initially, the value of the standard deviation of initial headways was estimated with reference to a query of the Access database (as with the speeds), but this led to an initial value of µ that was far too high. It was therefore decided to calibrate the standard deviation used to produce the initial headways such that the initial value of µ in the simulations should roughly match that in reality. The fact that a value of the standard deviation close to reality in combination with an assumed normal distribution led to an initial value of µ that was far too high, leads one to conclude that the distribution of initial headways is not normal. It may be that the distribution is skewed such that there are more headways that are longer than the average than shorter.

• There is no variation in mean speed from run to run. Evidence from other studies suggests that there may be some run time variability due to driver behaviour (e.g. Strathman et al., 2002). However, given that speeds of London buses are generally extremely constrained by traffic conditions, this may not be a major deficiency in the simulations.

• Overtaking is instantaneous in the model used here, with no time penalty for overtaking events. In reality, one bus might follow another for some time before having an opportunity of overtaking.


26

4.7 Regression analysis 4.7.1 Independent variables Factors thought to be likely to have an impact on bunching behaviour were ascertained from the literature review, from discussions with staff at TfL, and from a priori consideration of how bunching arises. Relevant data were provided by various sources within TfL, or derived from the Access database. Taking them one by one:

• Route length. Provided by TfL. • Proportion of route on bus lanes. Provided by TfL. • Total traffic signals along the route. Provided by TfL. • Mean distance between stops. Derived from route length and total bus stops,

a simple query of the Access database. • Passenger numbers. These were obtained by querying the TfL BREMS

database, which is formed from electronic ticket machine data. The figures are in passenger trips per day, averaged over the 5 week period in question. It should be noted that the BREMS passenger volume data for articulated routes are considered by TfL to be significant understatements of the real case. They advised that when the Route 149 was recently converted from articulated to double-decker, a four-fold increase in BREMS passenger volume data was observed. On this basis, the passenger numbers for the three articulated routes in this dissertation (12, 25 and 29) were increased by a factor of 4.

• Passengers per route-km. Derived from passenger volumes and route length. • Scheduled bus-km. Obtained from the TfL BREMS database. • Modal headway. This was deemed the most appropriate average to use to

represent headways. It was obtained by a simple query of the Access database. It was pleasing to note that the headways thus obtained corresponded exactly to those obtained from the London Bus Routes website (London Bus Routes, 2011) for each route (see Table 4.1).

• Mean journey time. This was a relatively complex task, ultimately accomplished by querying the database for the difference between the Observed Arrival Time at the final stop and the Observed Departure Time from the first stop. These were then averaged over each day by route, and over the five days to provide a final figure.

• Standard deviation of journey time. Once the journey times had been derived in order to find the mean speed, this was a relatively straightforward query. Again, note that the final figure is an average over the five days.

• Coefficient of variation of journey time. Derived by dividing the standard deviation by the mean.

• Mean speed. This was obtained by dividing the journey distance by the journey time for each trip, and taking the mean of the result.

• Average stop dwell time. This was found by subtracting the arrival time from the departure time for each line, and taking the mean of the result.

• Average dwell time per passenger. This was obtained by summing the total dwell time, and dividing by total passenger numbers.

• K. As defined in Newell and Potts, 1964 (see Section 2.1). The arrival rate was found by dividing the total passenger volumes by the hours of operation multiplied by the number of stops along the route. This gave arrival rates per unit time per stop. The boarding rate was found by taking the inverse of average dwell time per passenger (see above).


27

• Articulated. A dummy variable of 1 if the route is served by articulated buses, and 0 otherwise. This information was obtained from London Bus Routes, 2011.

• Operators. It was originally intended to use dummy variables to represent service control strategies used by the operators. However, these were very similar across the operators (see TfL, 2011a), so it was decided to represent the operators themselves, to ascertain whether a difference in performance could be observed. Four dummy variables were used to allow for representation of the operator of a service, set at 1 if a service was run by the relevant operator and 0 otherwise. The 30 services were run by a total of eight operators, but of these only four operators ran more than two services. Only these four operators were represented in the regression analysis:

o Arriva o Go Ahead Group o Stagecoach o RATP

Six routes were run by operators other than the four above. 4.7.2 Hold-out samples Before the analysis was carried out, a hold-out sample of 10% (i.e. three routes out of thirty) was produced. This was done using the random number generator in Microsoft Excel. The following conditions were imposed:

• No route served by articulated buses would be included in the hold-out sample. As there were only three routes served by articulated routes in the sample as a whole, it was felt that if one or more were taken out of the regression process, it would be difficult to discern what effect, if any, articulated buses have on bunching behaviour.

• A minimum of two routes with the same modal headway would remain in the sample to be analysed (see Table 4.1). This was to ensure that no headway would be under-represented in the analysed group.

• No two routes with the same modal headway would be included in the hold-out sample. This was to ensure a spread of routes to test the regression model on.

The three routes selected for the hold-out sample under these conditions were:

• 41 • 63 • 341

4.7.3 Analysis The analysis was carried out in PASW Statistics 18 (more commonly known as SPSS). Each bus route was represented by one line or case, with fields representing the independent variables detailed in the Section 4.5.1, as well as µ� itself. It was also decided to include two new measures derived from µ which might be more closely correlated with one or more of the independent variables. These two new measures were:

• ∆µ, defined as the difference between µ at the end of the route and µ at the beginning of the route. This is a measure of how much bunching arises over the course of the route.

• ∆µ/10N, where N is the number of stops along a route in one direction. In words, this is the average increase in µ per 10 stops. This is a measure of how bunching increases, normalised for the number of stops in the route.


28

Both of these were calculated for both directions of each route, and then an average taken of the two directions. It was initially intended for a multiple regression to be carried out. A correlation matrix for the independent variables was duly produced, to uncover any underlying colinearities. It had been expected that there would be several significant correlations between the independent variables. Indeed, several compound variables (for example passengers per route-km) had been constructed with a view to combining significantly correlated variables. However, the correlation matrix revealed that nearly all of the independent variables were significantly correlated to nearly all of the other independent variables! This rendered multiple regression virtually impossible, and instead regressions between individual independent variables and the three dependent variables were carried out. Regressions were carried out examining the relationship between each independent and each dependent variable for all 27 routes together (i.e. an aggregate analysis). Linear regressions were the default form. However, scatter plots of µ� against each independent variable were examined, and other forms of relationship (e.g. inverse, exponential, logarithmic) were explored where the form of the scatter plot suggested them. Linear relationships were retained unless a significant improvement was found using the non-linear model. 4.8 Factor analysis Principal component analysis with varimax rotation was carried out in PASW Statistics 18. The variables chosen for the analysis were those which had produced significant models predicting µ�, i.e. those in Table 5.3. Variable factor weightings with magnitudes < |0.4| were excluded from the component matrix and the rotated component matrix for clarity. Variable factor weightings were also sorted by factor weight for each component. The full output is available in Appendix 3.

29

Chapter 5 – Analysis 5.1 Bunching by route Table 5.1 presents a summary of the analysis of the 30 selected bus routes according to the four metrics selected in Chapter 4. It should be noted that the metrics are presented in percentage form. Recall that the bar over each metric in the table indicates that it is an average over all stops and all 5 days for that particular route.

Metric µ� �� .��

Route Headway (mins)

Value (%) Rank Value (%) Rank Value (%) Rank Value (%) Rank

18 4 32.16 1 15.59 1 8.87 1 14.56 1

94 5 29.43 2 11.65 2 6.19 2 13.90 2

253 5 27.73 3 10.19 3 5.90 3 13.15 3

25* 5 27.53 4 7.64 8 4.08 8 11.31 6

12* 5 27.21 5 8.20 5 4.35 7 11.14 7

254 5 27.07 6 8.79 4 5.51 4 12.36 4

63 6 26.89 7 8.02 6 4.69 5 11.89 5

279 6 24.67 8 7.77 7 4.61 6 10.33 8

41 5 23.38 9 6.99 9 3.68 9 8.92 9

29* 5 23.11 10 4.84 12 2.45 12 7.83 11

24 5 22.64 11 5.74 10 2.98 11 7.78 12

205 8 21.93 12 3.94 14 2.31 13 8.48 10

171 8 21.18 13 3.53 15 1.85 15 7.56 13

472 6 20.53 14 5.30 11 2.98 10 7.41 14

H37 6 19.93 15 4.05 13 2.11 14 6.31 15

87 6 19.32 16 2.35 18 1.21 20 4.52 19

109 6 19.09 17 2.58 16 1.26 18 4.44 20

147 8 17.52 18 2.42 17 1.38 17 4.78 17

388 10 17.47 19 1.98 20 1.26 19 4.75 18

153 12 16.98 20 1.98 19 1.38 16 5.78 16

341 10 16.71 21 1.87 21 1.08 21 4.05 21

54 12 14.44 22 1.40 22 0.81 22 3.34 22

184 10 14.18 23 1.35 23 0.79 23 3.00 23

333 10 13.55 24 0.56 26 0.29 27 2.12 26

337 12 12.15 25 0.85 24 0.50 24 2.45 24

232 20 11.76 26 0.67 25 0.39 25 2.44 25

487 15 9.66 27 0.24 29 0.14 29 0.73 29

272 15 9.15 28 0.24 30 0.13 30 0.89 28

P13 20 9.07 29 0.39 27 0.38 26 0.91 27

419 15 7.01 30 0.25 28 0.22 28 0.43 30 Table 5.1. Bus routes with their metrics in percent ages and ranks, sorted by µ� rank. * indicates that the

route is served by articulated buses.

Chapter 5 – Analysis

30

5.1.1 Comparison of rankings by metrics It can be seen that the metrics agree on the identity and ranking of the three most bunched routes, the 18, 94 and 253. The rankings of the four metrics for another four routes (the 41, 54, 184 and 341) are also in total agreement. The two routes with the highest standard deviation of metric rankings are the 472 and the 153 (both with standard deviations of 2.06). In the case of the 472, the rankings of H�� and H�� are both particularly high. This may simply be a statistical anomaly, whereby there happened to be a particularly high number of buses arriving at just over the cut off points for these metrics (e.g. several buses arriving 59 or 29 seconds after the previous bus). However, the very high number of data (around 65,000 records for this route) would lead one to suspect that an anomaly is unlikely. An alternative explanation is that instances of less severe bunching are relatively rarer in the 472 than in other routes, or put another way, bunching is more severe when it does occur. This might be a reflection of the relative skill of the route controllers, or perhaps a feature of the route which makes the bunching positive feedback loop more effective, such as a longer average boarding time on this service (see Section 2.1). In fact, the possibility of long boarding times being the cause is ruled out by the fact that boarding times for the 472 are below average – see Appendix 1. In the case of the 153, it is the rankings of H��, and h��.�� that are the highest. As the cut-off point for the second of these metrics at the modal headway is around 3 minutes, it is hard to understand why the rankings for these two should be similar, while that for H�� which has a cut-off in between the two is of a lower ranking. There were some anomalies in the last 2 stops in Direction 1 of this route, which might explain this unexpected result. The standard deviations of the rankings for the other routes are all fairly small (below 2), suggesting that the metrics are all measuring roughly the same phenomenon, which is encouraging.


31

5.1.2 Evolution of the metrics over the 5 sample days From Figure 5.1 it can be seen that the metrics all display similar behaviour over the five sample days: an initial rise, followed by a dip during half term (i.e. the 23rd February), and finally a further rise. Again, this indicates that the metrics are measuring similar phenomena. Since the analysis that follows focuses on µ, it is encouraging to note that it agrees with more conventional measures of bunching, while offering more fine-grained analysis due to its continuous nature.

Figure 5.1 Longitudinal progression of metrics 1.

1 Note that these are averages of the 30 routes, and not of all of the individual buses.

0%

1%

2%

3%

4%

5%

6%

7%

8%

16.5%

17.0%

17.5%

18.0%

18.5%

19.0%

19.5%

20.0%

20.5%

7th Feb 15th Feb 23rd Feb 3rd Mar 11th Mar

µ

h 0.25

H 60

H 30

Plotted on right-hand axis


32

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

05

:45

-05

:59

06

:30

-06

:44

07

:15

-07

:29

08

:00

-08

:14

08

:45

-08

:59

09

:30

-09

:44

10

:15

-10

:29

11

:00

-11

:14

11

:45

-11

:59

12

:30

-12

:44

13

:15

-13

:29

14

:00

-14

:14

14

:45

-14

:59

15

:30

-15

:44

16

:15

-16

:29

17

:00

-17

:14

17

:45

-17

:59

18

:30

-18

:44

19

:15

-19

:29

20

:00

-20

:14

20

:45

-20

:59

21

:30

-21

:44

22

:15

-22

:29

23

:00

-23

:14

23

:45

-23

:59

µ

Time of day

5.2 Bunching by time of day Figure 5.2 shows µ by time of day, averaged across all routes over all five days. There are several interesting things to note about this graph. Firstly, there are fairly clear peaks in the morning and the evening, with the morning peak being split into two smaller peaks, and the afternoon peak being spread over several hours. The first morning peak occurs between 07:30 and 07:44 (23.4%), and the second between 09:15 and 09:29 (25.8%). The afternoon peak starts at around 16:00, and lasts until around 18:14, and reaches a maximum value of 25.5%. This peak echoes the findings of Strathman et al. (1999), in which extreme bunching is observed in the evening peak in bus services in Portland, Oregon. It is interesting to speculate about whether the slight peak around lunchtime might be a result of the route controllers and/or the bus drivers having a lunch break.

Figure 5.2. µ by time of day averaged across all fi ve days.

As the next two graphs show, these peaks are not coincident with the peaks in bus usage (Figure 5.3) and road traffic flows (Figure 5.4). This is slightly unexpected: one might have predicted that bunching would be at its most extreme during periods of intense bus use, and very busy road conditions. However, recall Adebisi’s predictions about headway variance (Adebisi, 1986). He suggested that headway variance (which is closely related to bunching) is at its highest at times of abrupt change in demand. Intuitively, this makes sense. For example, one can imagine that if a following bus experiences much lower levels of demand than a leading bus, it will quickly catch up and bunching will occur. Looking closely at Figure 5.3, it can be seen that the morning peak in bus usage falls almost exactly between the two peaks in bunching. Thus, there is a peak in bunching in the transition from pre-peak to peak demand, followed by a second peak in bunching during transition from peak to post-peak demand. This could be coincidence, but it certainly seems to count as evidence in favour of Adebisi’s predictions. The afternoon peak is not quite so well behaved with respect to bus usage, but the afternoon peak in traffic volumes falls between the two afternoon sub-peaks, and one can imagine a similar effect on bunching of abrupt changes in traffic conditions, to that from abrupt changes in demand.


33

Figure 5.3. µ by time of day with bus usage superim posed. Source: TfL 2007, Chart 3.3.5.

Figure 5.4. µ by time of day with road traffic volu mes superimposed. Source: DfT 2011, Table TRA0307.

As stated in Section 4.2, the third of the five days selected for analysis (23rd February 2011) fell in a school half-term. Figures 5.5 and 5.6 show a comparison of

0

100

200

300

400

500

600

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

06

:00

-06

:14

07

:00

-07

:14

08

:00

-08

:14

09

:00

-09

:14

10

:00

-10

:14

11

:00

-11

:14

12

:00

-12

:14

13

:00

-13

:14

14

:00

-14

:14

15

:00

-15

:14

16

:00

-16

:14

17

:00

-17

:14

18

:00

-18

:14

19

:00

-19

:14

20

:00

-20

:14

21

:00

-21

:14

22

:00

-22

:14

23

:00

-23

:14

Bu

s u

sag

e i

n t

ho

usa

nd

s o

f b

us

jou

rne

y s

tag

es

µ

Time of day

µ

Bus usage

-20

30

80

130

180

230

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

06

:00

-06

:14

07

:00

-07

:14

08

:00

-08

:14

09

:00

-09

:14

10

:00

-10

:14

11

:00

-11

:14

12

:00

-12

:14

13

:00

-13

:14

14

:00

-14

:14

15

:00

-15

:14

16

:00

-16

:14

17

:00

-17

:14

18

:00

-18

:14

19

:00

-19

:14

20

:00

-20

:14

21

:00

-21

:14

22

:00

-22

:14

23

:00

-23

:14

Tra

ffic

vo

lum

es

ind

ex

ed

to

av

era

ge

we

ek

da

y h

ou

r =

10

0

µ

Time of day

µ

Traffic

volumes


34

bunching throughout the day during term, and in half term. The differences between the two graphs are intriguing.

Figure 5.5. µ by time of day averaged over the term -time days.

Figure 5.6. µ by time of day during half term.

Firstly, the second peak in bunching in the morning seems to almost totally disappear in half term (see “1” above). It may be that without schoolchildren using the buses, the transition from peak to post-peak is less abrupt. If most schools start

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

05

:45

-05

:59

06

:30

-06

:44

07

:15

-07

:29

08

:00

-08

:14

08

:45

-08

:59

09

:30

-09

:44

10

:15

-10

:29

11

:00

-11

:14

11

:45

-11

:59

12

:30

-12

:44

13

:15

-13

:29

14

:00

-14

:14

14

:45

-14

:59

15

:30

-15

:44

16

:15

-16

:29

17

:00

-17

:14

17

:45

-17

:59

18

:30

-18

:44

19

:15

-19

:29

20

:00

-20

:14

20

:45

-20

:59

21

:30

-21

:44

22

:15

-22

:29

23

:00

-23

:14

23

:45

-23

:59

µ

Time of day

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

05

:45

-05

:59

06

:30

-06

:44

07

:15

-07

:29

08

:00

-08

:14

08

:45

-08

:59

09

:30

-09

:44

10

:15

-10

:29

11

:00

-11

:14

11

:45

-11

:59

12

:30

-12

:44

13

:15

-13

:29

14

:00

-14

:14

14

:45

-14

:59

15

:30

-15

:44

16

:15

-16

:29

17

:00

-17

:14

17

:45

-17

:59

18

:30

-18

:44

19

:15

-19

:29

20

:00

-20

:14

20

:45

-20

:59

21

:30

-21

:44

22

:15

-22

:29

23

:00

-23

:14

23

:45

-23

:59

µ

Time of day

1 3 2

1 3 2


35

within around half an hour of each other, then school travel demand might fall very sharply after a particular cut-off point, causing bunching. The second difference is a slight decrease in bunching between the hours of around 16:15 and 18:00 in half term (see “2”). This fits in reasonably well with what one would expect: a decrease in demand resulting in a decrease in bunching. The third difference, in the evening peak (see “3”), is rather more perplexing. Why should a school holiday be associated with an increase in bunching between around 18:15 and 20:00? Possible explanations might be an overabundance of buses on the route, or perhaps the opposite problem: over compensation by route planners resulting in insufficient buses to meet demand. Either of these might plausibly result in bunching. It is also interesting to consider the difference in µ by time of day for routes in Inner London (Figure 5.7) and Outer London (Figure 5.8). Comparing the two graphs, the first thing to note is that, as one would expect, bunching tends to be more pronounced in Inner than Outer London. The second striking difference is that the structure of the Outer London graph is significantly more pronounced than that of the graph for Inner London: the peaks and the troughs are more defined. This may reflect the fact that traffic levels are more variable throughout the day in Outer London compared with Inner London (see TfL, 2010a, Figure 4.1, p. 88). Alternatively, or additionally, it may be the case that demand levels in Outer London vary more throughout the day than in Inner London, though it has not been possible to find data to confirm this. The fact that the Outer London graph has a double peak in the evening, with the first maximum at 16:00 – 16:14, leads one to conclude that the use of public buses by school children may play a more significant role in bunching behaviour in Outer London than Inner London. One final intriguing difference is that the fall off in µ in the evening is far more pronounced in Outer than Inner London. Indeed, µ begins to rise in Inner London after a low at 21:15 – 21:29. This may reflect a reduction in service control, as it is carried out by fewer staff as the evening progresses2, and services become less regulated. In Outer London, where demand is low, and the feedback mechanism therefore relatively weak, this may not be a significant problem. However, in Inner London where demand remains higher into the late evenings, it may be the case that the fall off in service control causes an increase in µ as the evening goes on, as the positive feedback mechanism is less mitigated.

2 This is confirmed by Butler, 2011.


36

Figure 5.7. µ by time of day for Inner London route s3.

Figure 5.8. µ by time of day for Outer London route s4.

3 I.e. routes 205, 388, 153, 29, 63, 12, 171, 24, 94, and 87. (See Figures 4.1 – 4.5.) 4 I.e. routes 41, 279, 184, 147, 109, 54, 472, 487, H37, 232.

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

05

:45

-05

:59

06

:30

-06

:44

07

:15

-07

:29

08

:00

-08

:14

08

:45

-08

:59

09

:30

-09

:44

10

:15

-10

:29

11

:00

-11

:14

11

:45

-11

:59

12

:30

-12

:44

13

:15

-13

:29

14

:00

-14

:14

14

:45

-14

:59

15

:30

-15

:44

16

:15

-16

:29

17

:00

-17

:14

17

:45

-17

:59

18

:30

-18

:44

19

:15

-19

:29

20

:00

-20

:14

20

:45

-20

:59

21

:30

-21

:44

22

:15

-22

:29

23

:00

-23

:14

23

:45

-23

:59

µ

Time of day

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

05

:45

-05

:59

06

:30

-06

:44

07

:15

-07

:29

08

:00

-08

:14

08

:45

-08

:59

09

:30

-09

:44

10

:15

-10

:29

11

:00

-11

:14

11

:45

-11

:59

12

:30

-12

:44

13

:15

-13

:29

14

:00

-14

:14

14

:45

-14

:59

15

:30

-15

:44

16

:15

-16

:29

17

:00

-17

:14

17

:45

-17

:59

18

:30

-18

:44

19

:15

-19

:29

20

:00

-20

:14

20

:45

-20

:59

21

:30

-21

:44

22

:15

-22

:29

23

:00

-23

:14

23

:45

-23

:59

µ

Time of day


37

5.3 Bunching along the route Plotting µ against distance downstream for each route in each direction yields some very interesting graphs, which will be called µ-curves. Three major groups emerge. The first (21 out of 60 routes) is the group for which the curve slopes up more or less continuously. Figures 5.9 and 5.10 are particularly characteristic examples of this group.

Figure 5.9.

Figure 5.10.

5%

7%

9%

11%

13%

15%

17%

19%

21%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

µ

Stop number

54 towards Elmers End

5%

7%

9%

11%

13%

15%

17%

19%

21%

23%

25%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

µ

Stop number

341 towards Tottenham


38

The second group, with 8 members, is characterised by curves that rise steeply initially, and then flatten off for the rest of the route. Figures 5.11 and 5.12 show clear examples of this group.

Figure 5.11.

Figure 5.12.

15%

17%

19%

21%

23%

25%

27%

29%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

µ

Stop number

63 towards Kings Cross

10%

12%

14%

16%

18%

20%

22%

1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353637

µ

Stop number

109 towards Brixton


39

The third group, with 22 members, is in a sense midway between the other two. Curves of graphs in this group rise steeply initially, and then less steeply as the route progresses. Figures 5.13 and 5.14 are good illustrations of this group.

Figure 5.13.

Figure 5.14.

9 routes out of 60 do not fit into any of these groups, and can be considered anomalous.

10%

15%

20%

25%

30%

35%

40%

45%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

µ

Stop number

18 towards Euston

12%

14%

16%

18%

20%

22%

24%

26%

28%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

µ

Stop number

29 towards Trafalgar Sq.


40

It is interesting to consider what might be the cause of the difference in the bunching behaviour of London buses, as represented by the differences in their µ-curves. It seems that most of the routes in the first group are rather less central than others, and that the maximum values of µ tend to be rather small. Those in the second and third group tend to be more central, and of higher frequency. However, it is not possible to split the graphs neatly into the groups by frequency or centrality, and indeed some routes are split into two groups depending on their direction. What this may suggest is that there is some limit to bunching beyond which it tends not to exacerbate. Each route seems to have its own characteristic maximum, which may depend on route factors (frequency, traffic density, etc.) or perhaps on the attitude of the route controllers, who might be content with low-level bunching, but take action when it reaches a certain level. Routes in the first grouping of µ-curves may simply have not reached their characteristic maximum before the end of the run, and it is therefore not apparent. What the functional difference between the second and third groups is less clear. It might perhaps be the techniques employed by controllers, or simply that in the second group the later part of the route is less conducive to bunching than the first. 5.3.1 Anomalies Several of the 60 µ-curves, while fitting into one of the groups identified above, show a distinct and rapid fall off in µ at the end of a route. Figure 5.15 shows an example of this in Route 171. These anomalies prompted further investigation, and it seems that this feature of the curves is caused by route controllers curtailing bunched services at a particular point (between stops 51 and 52 in this example). Turning back services will result in fewer buses running the final portion of the route, which

Figure 5.15.

must result in longer mean headways, and therefore lower values of µ. Figure 5.16, which shows Figure 5.15 in addition to a plot of observed buses at each stop, confirms this. Note that there is a drop in the number of buses observed at the

0%

5%

10%

15%

20%

25%

30%

35%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

µ

Stop number

171 towards Catford


41

beginning of the run, as curtailment is also in operation in the other direction of the 171 – the µ-curve in the other direction also shows the characteristic fall at the end of the route. This feature was observed in around a quarter of the 60 µ-curves, perhaps a surprisingly small proportion, given the supposed prevalence of curtailment as a service control strategy.

Figure 5.16.

90

95

100

105

110

115

120

125

0%

5%

10%

15%

20%

25%

30%

35%

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55

Da

ily

nu

mb

er

of

ob

serv

ed

bu

ses

µ

Stop number

171 towards Catford

µ

Observed

buses


42

As has already been mentioned, 9 of the 60 µ-curves do not fit into the groups defined above. All of these, with the exception of the 205 towards Bow, are low-frequency routes, i.e. have modal headways of 12 minutes or more. This leads one to the conclusion that bunching is most organised and predictable in high-frequency routes. Some of these anomalous curves have interesting features, and will be discussed. It can be seen from Figure 5.17 that the µ-curves of the 419 in either direction both display a spike at the same stop, number 17 in one direction, and number 3 in the other. This may simply be an anomaly in the data collection. It may also be related to the fact that in each case, the stop in question immediately precedes a hail and ride section. Whether this hail and ride section is affecting the behaviour of drivers or of the data acquisition is not clear.

Figure 5.17.

0%

2%

4%

6%

8%

10%

12%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

µ

Stop number

419 towards Hammersmith

0%

2%

4%

6%

8%

10%

12%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

µ

Stop number

419 towards Richmond


43

Figure 5.18 shows a sharp discontinuity in bunching in the P13 towards Streatham, between stops 6 and 8. Unlike the previous example, this feature is not mirrored in the opposite direction. Stop 7 in this direction is Peckham Bus Station, so there may be an issue with the bus station in this direction. This could be a design issue – perhaps a problem over accessing a stand. Alternatively, if this is a driver change-over point, this could be an operational issue. This example shows the potential of using µ-curves to analyse how well routes are functioning.

Figure 5.18.

0%

2%

4%

6%

8%

10%

12%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

µ

Stop number

P13 towards Streatham


44

While the µ-curves shown in Figures 5.17 and 5.18 might, if it were not for their unusual features, be classified into one of the three groups, 4 of the curves defy grouping altogether. Figure 5.19 show the µ-curves for two low-frequency routes. It might be possible to interpret these curves, but the simplest explanation seems to be that the bunching behaviour along these routes is chaotic. The fact that bunching does not become progressively more severe on these routes would appear to confirm Hill’s prediction (Hill, 2003) that low frequency buses would have to serve hundreds of stops before bunching due to the positive feedback effect would be observed (assuming that the feedback mechanism is responsible for keeping the gradient of the µ-curves positive and relatively steep – see discussion and simulations below).

Figure 5.19.

6%

7%

8%

9%

10%

11%

12%

13%

14%

15%

16%

1 3 5 7 9 1113151719212325272931333537394143454749515355

µ

Stop number

232 towards Neasden

5%

6%

7%

8%

9%

10%

11%

12%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

µ

Stop number

487 towards Willesden


45

Finally, Figure 5.20 shows two curves which should perhaps have their own group. These two curves rise towards the centre, and fall off towards the end – particularly clearly in the case of the 337 towards Richmond. To be clear, this corresponds to buses, on average, evening themselves out as they move from the middle to the end of their routes. Given the propensity of buses to bunch, as detailed in Section 2.1, the most likely explanation of this seems to be that controllers of these two routes take action to mitigate bunching as it happens towards the centre of the route, and are so successful in doing so that it gradually eases towards the end. It may be that because these routes are low-frequency, it is more straightforward for controllers to prevent bunching, than for those controlling busier routes.

Figure 5.20.

8%

9%

10%

11%

12%

13%

14%

15%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

µ

Stop number

337 towards Richmond

6.0%

6.5%

7.0%

7.5%

8.0%

8.5%

9.0%

9.5%

10.0%

10.5%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

µ

Stop number

P13 towards New Cross


46

5.3.2 Comparison with results/predictions of other studies The general rising form of the non-anomalous µ-curves would seem to be at odds with one of the main findings of Peng et al. (2008), viz. that bunching did not propagate down the route of the bus service in that study. With a headway of 7 minutes, one would have expected that the route in that study might have had a µ-curve in one of the three main groups as detailed above. There may be some feature of North American bus routes (e.g. wider roads) that makes them less likely to display bunching behaviour. However, the different methodologies used in this dissertation, and the Peng et al. study mean that it is difficult to compare results directly. It would be interesting to see what form the µ-curve of the Route 20 in Chicago would take. Chapter 2 also detailed a simulation study by Abkowitz et al. (1986) that produced a model predicting bunching behaviour downstream. The curve produced was depicted in Figure 2.2, reproduced here for convenience. Despite the fact that the dependent variable is the headway variance (called variation here) as opposed to µ, the left hand curve above bears a striking resemblance to the µ-curves derived in this dissertation, particularly those in the third group. This lends credence to µ as a metric, and also seems to substantiate the theoretical work of Abkowitz et al..

Figure 5.21. Source: Abkowitz et al., 1986. Finally, the work of Adebisi (1986) was also cited in Chapter 2. His paper provides a simple numerical example of his model, and the figures provided were used to plot the graph in Figure 5.22. Again, the measure being used is different, this time the coefficient of variation of the headway5. Even so, the resemblance of the curve produced to the µ-curves is again remarkable. This time, it is most similar to the first group of curves, but the steep initial incline followed by a shallower gradient is common to the majority of the curves.

5 i.e. the standard deviation of the headway divided by the mean headway.


47

Figure 5.22.

5.3.3 Discussion The fact that the µ-curves shown above almost always have a positive gradient does seem to constitute evidence that the positive feedback mechanism is at play in London’s bus system. However, it is not in itself conclusive. Consider a bus route on which buses are despatched with perfectly regular headways. In any system, there will be some variability in running time across any stage in the journey (e.g. due to a signalised junction). This variability will cause the perfectly spaced vehicles to become less regular, and an increase in µ will occur. Indeed, it can be seen that even in a system where buses are not dispatched at perfectly regular intervals, some increase in µ may occur as buses progress down the route, simply from a variation in journey time across any given stage. Thus, the forms of the µ-curves shown above could be in evidence even in a system in which there is no positive feedback mechanism in operation. This consideration was the motivation behind carrying out simulations in which there is no positive feedback mechanism. This allows a comparison of reality with a situation without positive feedback, in order that comment is possible on Aim 4 of this dissertation. Note that on an infinitely long route in which feedback is at play, all the buses would eventually end up bunched together (if there were no controllers). The headways would all be less than 30 seconds, and µ would tend to 100%. Without feedback, the buses would eventually sort themselves according to speed and spread out: µ would then fall to zero. However, the time scales involved before µ would reach these limits are likely to be far longer than the 50 stop scenarios in these simulations would allow for – and indeed µ showed no sign of beginning to fall off in any of the simulations run. Simulations were run for four different mean headways of bus service: 5 minute, 8 minute, 12 minute and 20 minute (see Section 4.6). The forms of the µ-curves in the simulations were similar to those observed in reality, particularly in the first group of steadily rising curves. Figure 5.23 shows an example of a µ-curve for a simulation of a 5 minute headway service.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

He

ad

wa

y's

co

eff

icie

nt

of

va

ria

tio

n

Stop number


48

Figure 5.23. Simulation of a service with a 5 minute headway.

The simulations of services with longer headways yielded similar results, with the gradient of the µ-curves becoming shallower as the headway increases: see Figure 5.24. At 20 minute headways, the µ-curves of the simulations become chaotic in nature, with no discernable form.

Figure 5.24. Simulations of services with different headways.

This similarity of form between simulation and reality might call into question whether the positive feedback mechanism is indeed an important explanatory factor of bunching in London buses. However, the scale of the simulated µ-curves is rather different to that of the real µ-curves. Figure 5.25 shows the curve from Figure 5.23 on the same pair of axes as that of a typical real 5 minute headway service. It can be seen that although the two curves roughly share the initial value of µ, they diverge as the route progresses. This would seem to indicate that the positive feedback mechanism does indeed have a significant impact on the development of bunching down the route of a bus service. However, the limitations of the simulation should be borne in mind (see Section 4.6), so this result should be treated with some caution. Further, more realistic simulations are required to provide more evidence that the real µ-curves are steeper in gradient than would be the case without the positive feedback mechanism.

12%

13%

14%

15%

16%

17%

18%

19%

20%

21%

22%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

µ

Stop number

7%

9%

11%

13%

15%

17%

19%

21%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

µ

Stop number

5 mins 8 mins 12 mins 20 mins


49

Figure 5.25. Comparison of simulated 5 minute headw ay service with an equivalent real service – the 24

towards Hampstead. The approximately linear form of these simulated µ-curves is rather different from the form of the headway variance curves derived by Abkowitz et al. (1986), as shown in Figure 5.21. The Monte Carlo techniques used by this team were rather more sophisticated than the spreadsheet techniques in this dissertation, using non-normal probability distribution functions for stage speeds. It is not clear from the paper whether or not the positive feedback mechanism forms part of their model. If it does, it may be the case that it is the bunching mechanism which causes the steeper increase in bunching at the beginning of the route, becoming less important further downstream as bunching becomes more severe. This would make intuitive sense. It should be noted that the Abkowitz et al. study is concerned with headway variance, which is of course not identical to µ, though the two measures are closely related. 5.4 Regression analysis In order to carry out the regression analysis, values for the independent variables to be used were derived; these are shown in Appendix 1. Before moving on to the analysis of these variables, there are two values that are interesting in themselves. The average speed (averaged over the 30 routes, not over all of the buses) was found to be around 14.3 km/h or 8.9 mph. This is roughly two to three times walking speed. TfL6 estimates that average traffic speed in Central London is around 14 km/h, and in Inner London around 20 km/h. Average cycling speeds in London are considered by TfL7 to be 15 km/h. On all these comparisons, the bus seems to be performing well. Average stop dwell times were found to be around 10 seconds (again, averaged over the 30 routes). As mentioned in Section 4.1, this value could be used for α in the calculation of µ. It is unlikely that taking α as 10 rather than 30 would have a significant impact on any of the results under discussion.

6 TfL (2010a) p. 88. 7 TfL (2010b) p. 14.

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

32%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

µ

Stop number

24 Simulation


50

5.4.1 Correlations between the intended independent variables As detailed in Section 4.5.3, it was not possible to carry out a multiple regression as has originally been intended, because of the large number of colinearities among the independent variables. However, the correlation matrix (shown in Appendix 2) is in itself worth discussing, as it throws up some interesting relationships.

• As one would expect, route length is positively correlated with mean journey time (.787**)8 and the standard deviation of the journey time (.743**). However, there is no significant correlation between route length and the coefficient of variation of the headway, which is a little surprising.

• The percentage of route in bus lanes is negatively correlated with the distance between stops (-.473*). This is what one would expect, as the former would increase with centrality of route, and the latter decrease. Bus lane percentage is further correlated with almost all of the other variables, generally indicating that it is a measure of centrality of the route (e.g. passenger numbers (.694**), number of traffic signals (.594**), etc.). Interestingly, the bus lane percentage is negatively correlated with mean speed (-.630**), which might lead one to conclude that bus lanes slow buses down! However, one must conclude that again, the underlying variable of centrality is at play. An encouraging result for transport planners is that bus lane percentage is negatively correlated with the coefficient of variation of journey time (-.535**). This would seem to indicate that the introduction bus lanes can lead to reductions in journey time variability, an especially impressive result given the centrality factor discussed above. This confirms the outcomes of the simulations in Turnquist, 1981 – see Section 2.1.

• Average stop distance is negatively correlated with daily passenger numbers (-.443*) and daily passenger numbers per route-km (-.558**), which is what one would hope for from a route planning perspective. It is also positively correlated with mean speed (.623**), again an expected result. A more unexpected result is that average stop distance is negatively correlated with stop dwell time (-.506**): the longer the distance between stops, the shorter the average stop time. One might have expected that the longer the distance between stops, the more time there is for passengers to build up, and hence a positive correlation would result. However, route planners will take density of demand into account, and plan longer stop spacings on routes with lower demand. On reflection therefore, this relationship is not so surprising.

• Number of traffic signals, like bus lane percentage, seems to be a measure of centrality of route, with a large number of highly significant correlations. One surprising result is that number of traffic signals is not significantly correlated with coefficient of variation of headway. One would have expected that there would be a fairly strong positive correlation between the two. This may indicate that TfL’s signal priority system is functioning well (see Section 3.2).

• Daily passenger numbers is, as one would hope, positively correlated with scheduled passenger place-km (.909**) and negatively correlated with modal headway (-.779**).

• Modal headway is positively correlated with the coefficient of variation of journey time (.465*). This means that as the headway between buses increases, run times become more variable. This may be because there is more time for passengers to build up at stops when longer headways are

8 These relationships are presented with the Pearson correlation coefficient, followed by one or two asterisks. Two asterisks after a relationship are used to denote a relationship significant to the 1% level, one asterisk to denote significance to the 5% level.


51

used. It might also indicate that there is a stabilising effect on buses due to other buses being close by (see Section 5.4.2)

• A rather unexpected correlation is that the coefficient of variation of journey time is negatively correlated with average stop dwell time (-.418*). It seems difficult to see a causal link here, so perhaps there is some underlying cause (e.g. bus lane percentage). However, the relationship does seem to undermine somewhat the theoretical case for bunching (see Section 2.1).

• An interesting relationship given the Mayor’s current stance on articulated buses, is that average stop time per passenger is negatively correlated with the articulated dummy variable (-.541**). Running a regression test on this basis indicates that articulated buses save around 1.9 seconds per passenger in boarding times. However, it should be noted that the dwell time per passenger figure is based on passenger volumes, and it should be held in mind that passenger volumes for articulated bus routes were crude approximations (see Section 4.5.1). This figure, therefore, should be treated with circumspection.

5.4.2 Linear regression of individual independent variables

Dependent variable

Independent variable

µ� ∆µ ∆µ/10N

Route length No No No

Bus lane percentage 0.327** 0.115* No

Total traffic signals 0.403** 0.125* No

Mean stop distance 0.149* No No

Passenger numbers 0.704** 0.541** 0.117*

Passengers per route-km 0.707** 0.486** 0.213**

Scheduled bus-km 0.729** 0.618** 0.121*

Modal headway 0.745** 0.483** 0.377**

Mean journey time 0.125* No No

Standard deviation of JT No No No

Coeff. of variation of JT 0.164* No No

Mean speed 0.226** No No

Average stop dwell time 0.376** 0.305** No

Av. dwell time per passenger No No No

K 0.729** 0.637** 0.303**

Articulated No No No

Operators:

Arriva No No No

GAG No No No

Stagecoach No No No

RATP No No No Table 5.2.

As stated in Section 4.5.3, two further measures, ∆µ and ∆µ/10N were used as dependent variables in the regression analyses. Table 5.2 summarises the results of these linear regression analyses, with ‘No’ indicating that the independent variable does not predict the dependent variable with significance, ‘*’ indicating that the model predicts the independent variable with 5% significance, and ‘**’ indicating that the model is successful at the 1% level. The number before the asterisks gives the adjusted R2 value for the model, a measure of its predictive power. It can be seen


52

that µ� is the most ‘successful’ dependent variable, i.e. has the most significant models predicting it from the various independent variables. With this in mind, and also the fact that to discuss all 20 independent variables with respect to each of the 3 dependent variables would be impractical, only the results for µ� will be discussed. Firstly considering the non-significant relationships:

• Route length . It is somewhat surprising that this should not be significantly correlated with µ�. One might have expected that a long route would provide more opportunity for bunching to occur. However, this may be because less central routes tend to be longer than more central routes (route length is very close to being significantly related to mean speed: Pearson correlation coefficient = .364, significance = .062), and less central routes have lower values of µ� than more central ones.

• Standard deviation of journey time . It is also surprising that the standard deviation of the journey time is not significantly correlated to bunching, given that mean journey time is (although the relationship is a fairly weak one – see Table 5.3).

• Average dwell time per passenger . This is arguably the most surprising result of these regressions. This variable had a p value of 0.987, so was almost perfectly uncorrelated with µ�! Note that the inverse of dwell time per passenger, boarding rate, was also not significantly correlated. Given the emphasis placed on boarding rate by the literature, this seemed highly likely to be significant but was not. This may indicate that the positive feedback mechanism is not at play in the London bus system (but see K below).

• Articulated . Given that only 3 members of the sample were articulated buses, it is not surprising that this variable did not produce a significant result. It would be interesting to carry out further study in which the sample is chosen to include all remaining articulated routes, in order to compare their performance with double-deckers. Alternatively, one could carry out a longitudinal study to examine the change in µ� on the conversion of an articulated service to a double-decker service.

• Operators . Again, given the small number of data used, it is not surprising that these dichotomous variables did not produce significant results. If one carried out a study using all of the routes in London, it might be possible to perceive a difference in performance, which would be interesting from the point of view of service quality analysis. Another possibility would be to group the routes by the control centres that regulate them, to see whether this has a significant impact on bunching.

Table 5.3 summarises the relevant statistics for the statistically significant linear regression models with respect to µ�, ranking them according to the adjusted R2

value. Note that this value indicates what proportion of the variation in the dependent variable is predicted by the variation in the independent variable. For example, a value of 0.547 for the Adjusted R2 would imply that the independent variable in question predicts 55% of the variation of the dependent variable.


53

Statistic


Adjusted R 2 B (coefficient) C (constant) p

Modal headway (mins) 0.745 - 1.292 30.807 < 0.001

Scheduled bus-km (1000s) 0.729 3.452 6.456 < 0.001

K 0.729 552.060 9.046 < 0.001

Passengers per route-km 0.707 0.011 9.819 < 0.001

Passenger numbers (1000s) 0.704 0.373 10.653 < 0.001

Total traffic signals 0.403 0.202 10.863 < 0.001

Average stop dwell time (s) 0.376 1.481 6.004 < 0.001

Bus lane percentage 0.327 0.343 12.522 0.001

Mean speed (km/h) 0.226 - 1.473 40.046 0.007

Coeff. of variation of JT 0.164 - 118.693 38.511 0.021

Mean stop distance (m) 0.149 - 0.070 41.323 0.026

Mean journey time (mins) 0.125 0.199 8.134 0.039 Table 5.3.

Taking the independent variables one by one:

• Modal headway . This is the most successful predictor of µ�, predicting 75% of the variation. Given the definition of µ, this is not surprising: a delay (e.g. due to the red phase of a traffic signal) on a route with a short headway will cause more of an increase in µ than an equivalent delay on a less frequent route. In a sense therefore, this result is almost tautological. The value of B implies that increasing the headway by one minute is predicted to result in a reduction in µ� of around 1.3% on average.

• Scheduled bus -km. Given the close relationship between this and the modal headway, it is no surprise that it too is a good predictor of µ�. This model predicts around 73% of the variation in µ�, and the coefficient indicates that an increase of 1000 daily bus-km would result in a 3.5% increase in µ�.

• K. The fact that K is such a good predictor of bunching, accounting for 73% of the variation in µ�, is evidence in favour of the theoretical analyses presented in 2.1 that cited its importance. The high predictive power of K seems to indicate that a positive feedback effect may indeed be at play in the London Bus network. However, it should be observed that K is also closely related to the modal headway (r = - 0.821), so this result should be treated with some caution. This is particularly so given the lack of predictive power of the average dwell time per passenger. It should be noted that the maximum value of K is 0.047 (for Route 18), and this is a relatively small value – one might not therefore expect the effects of the positive feedback effect to be large.

• Passengers per route-km. This is a measure of density of demand along the route, and as one would expect, it is a good predictor of bunching, predicting 71% of the variation in µ�. It should be noted that this variable is also very closely correlated to modal headway (r = - 0.849), so once again it is difficult to say how much of the predictive power can be ascribed to this variable itself. The model predicts that a daily increase of 100 passengers per route-km would result in a 1.1% increase in µ�.

• Passenger numbers. Comments as with the previous variable. The model predicts that an increase of 1000 daily passengers would result in a 0.4% increase in µ�.

• Total traffic signals. This is, as expected, a good predictor of bunching, accounting for 40% of the variation in µ�. The model is, like all of the above, highly significant (p < 0.001). The prediction is that, on average, adding an extra set of traffic lights would lead to a 0.2% increase in bunching.


54

• Bus lane percentage. This is a somewhat bizarre result: adding bus lanes would seem to increase bunching! However, this is presumably because of the strong correlation between bus lane percentage and other highly predictive variables such as modal frequency and number of traffic signals.

• Mean speed. This is shown to have some predictive power of bunching. Again, one wonders how much of this is causal, and how much the result of mean speed acting as a proxy for other factors. The model predicts that an increase of speed of 1 km/h would result in a decrease in µ� of 1.5%.

• Coefficient of variation of journey time. This is another rather perplexing result: decreasing bunching is associated with increasing journey time variability. Perhaps the causal arrow is pointing the other way here. It may be that bunching has some kind of stabilising effect on bus runs (see Section 5.4.1, bullet point concerning modal headway and coefficient of variation of journey time). The mechanism for this could be as follows. Bus A experiences a delay, and becomes late. If the following bus, Bus B, is relatively close behind, i.e. the two are somewhat bunched, Bus B will eventually catch up Bus A and overtake. Once this has happened, Bus B will be able to pick up some of the excess passengers along the route, and Bus A will be less delayed than it would have been had it not been overtaken. Thus, having Bus B running close by could have a stabilising effect on Bus A’s journey times. Of course this is just speculation, and there could be no causality between these variables, with an underlying cause being responsible for their correlation.

• Mean stop distance. This relatively weak relationship is likely to be based on the correlation between mean stop distance and passenger volumes.

• Mean journey time. As mean journey time is not strongly related to the variables near the top of Table 5.3, this relationship would seem to reflect a different effect from that underlying modal frequency. However, the relationship is not extremely strong, with p = 0.039. The model predicts that bunching increases with journey time, with a 5 minute increase in journey time responsible for a 1% increase in µ�. It is interesting that there should be a relationship between µ� and mean journey time, but not µ� and route length.

All of the above models are linear. There is one obvious theoretical limitation of linear models predicting a variable such as µ�, and that is that there is a maximum possible value of µ� (100%) which will at some point be violated by a straight line. The models should, therefore, be treated with some caution, and are likely to become less accurate as their independent variables increase in magnitude. On the basis of scatter plots of each variable against µ�, it was not thought worthwhile to run non-linear regressions on many of the variables, with two exceptions. Modal headway displayed a clear non-linear tendency. Several different models were tried, but the most successful was the following exponential model:

µ� = 40.3 × 0.913�, where H is the modal headway. This model has an R2 value of 0.838, the most successful of all the models. Figure 5.26 shows a scatter plot of µ� versus modal headway, with the exponential model shown as a solid line: the fit is remarkably good. An exponential model is sensible: one would not expect that adding a minute to the headway at a headway of 20 minutes would have the same bunching reduction effect that adding a minute would have at a headway of 5 minutes (for example).


55

Figure 5.26.

Daily passenger numbers too, displayed a tendency which appeared logarithmic. A logarithmic analysis produced a model with a better predictive ability than the linear model, with R2 = 0.765. The model is given by:

µ� = 7.07 × ln ! − 49.50, where N is the total daily passenger numbers. Figure 5.27 shows this model with the scatter plot of passenger numbers against µ�. The three articulated routes are labelled. From this evidence it does not appear that the crude scaling up of passenger numbers on these routes was too problematic (see Section 4.5.1). The logarithmic model is, once again, unsurprising. One would expect that adding passengers at low volumes would have a larger effect on bunching than doing so at high volumes – hence the levelling off curve.

Figure 5.27.

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0 5 10 15 20

µ

Modal headway (mins)

12 25

29

0%

5%

10%

15%

20%

25%

30%

35%

µ

Passenger numbers

(values omitted for reasons of commercial sensitivity)


56

5.5 Factor analysis Factor analysis was carried out in an attempt to discern underlying factors that might be at play (see Appendix 3). The Keiser-Meyer-Olkin measure of sampling adequacy was 0.591 – less than desirable, but within acceptable limits. Bartlett’s test of sphericity was significant (0.000). The determinant of the correlation matrix was 2.84 x 10-8, indicating that the factors are orthogonal to a good approximation. Three orthogonal (i.e. uncorrelated) factors were derived with eigenvalues greater than 1. After rotation, the first factor explained 39% of the variance, the second 21%, and the third 20%. As a whole, therefore, the three factors explained about 80% of the variance of µ�. Table 5.4 shows the rotated component matrix from the analysis. The primary variables for each factor have been circled in red. The first component seems to index density of demand. Once again, it is surprising to see the coefficient of variation of journey time with a negative sign, as one would expect that as centrality/density of demand increase, journey time variability would also. Interestingly, K has the highest factor weight for this component. The second component seems to index the centrality of route, being negatively dependent on mean speed and stop distance, and positively dependent on bus lane percentage. The third component seems to approximate a temporal route factor.

Rotated Component Matrix a

Variable

Component

1 2 3

K .868

Modal headway -.831

Daily passengers per

route-km

.796 .420

Total daily scheduled

bus-km

.777 .573

Total daily passenger

numbers

.767 .521

Coefficient of variation of

journey times

-.716

Average stop dwell time .659 .413

Mean speed -.912

Average distance

between stops

-.778

Percentage of route in

bus lanes

.546 .578

Mean journey time .937

Total traffic signals on

route

.424 .742

Extraction Method: Principal Component Analysis.

Rotation Method: Varimax with Kaiser Normalization.

a. Rotation converged in 5 iterations.

Table 5.4.


57

The factor analysis therefore seems to be suggesting that three underlying factors are at play:

1. Density of demand 2. Centrality of route 3. A route time factor, dependent on the journey time and number of traffic lights.

5.6 Prediction of µ� for the hold out samples The most successful model derived above was the exponential model using the single independent variable of modal headway. It is a little disappointing that all of the above analysis should result in such a simple predictive model using only one independent variable, but this does not alter the fact that this model has the most predictive power. Table 5.5 shows the values of µ� predicted by the model, compared with their actual values.

Route

Predicted value of µ�

Actual value of µ�

Percentage error

41 25.57 23.38 8.55%

63 23.34 26.89 -15.20%

341 16.22 16.71 -3.03% Table 5.5.

The model seems to be functioning well, within reasonable margins of error – around 9% on average. Figure 5.28 shows the scatter plot of modal headway against µ�, this time with the three hold out plots included in red. It can be seen that they fit into the pattern well. Their inclusion in the derivation of the model would have had little impact on the calibration of the model.

Figure 5.28.

41

63

341

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0 5 10 15 20 25

µ


58

Chapter 6 – Conclusion 6.1 Summary of results The results of the analysis will be summarised in this Section, with respect to the Aims established in Section 1.2. Aim 1 . To produce a satisfactory metric with which to measure bunching. Several metrics were considered, but the favoured one was µ, defined as

µ = 1 −� − �

�� − � ,

where H is the headway, Hs is the scheduled headway, and α is the cut-off point at which buses are considered 100% bunched - set at 30 seconds in this dissertation. The main advantage of µ over the other metrics considered is that it is continuous in nature, taking any value from 0 to 1, whereas some other metrics were dichotomous. The continuous nature of µ allows for more fine-grained analysis, while retaining the features one would intuitively wish for in a bunching metric.

Chapter 6 - Conclusion

59

Aim 2 . To use iBus data to establish the severity of bunching on the routes being studied, according to the metric derived for Aim 1. These results were presented in Table 5.1, reproduced here as Table 6.1.

Metric µ �� .��

Route Headway (mins)

Value (%) Rank Value (%) Rank Value (%) Rank Value (%) Rank

18 4 32.16 1 15.59 1 8.87 1 14.56 1

94 5 29.43 2 11.65 2 6.19 2 13.90 2

253 5 27.73 3 10.19 3 5.90 3 13.15 3

25* 5 27.53 4 7.64 8 4.08 8 11.31 6

12* 5 27.21 5 8.20 5 4.35 7 11.14 7

254 5 27.07 6 8.79 4 5.51 4 12.36 4

63 6 26.89 7 8.02 6 4.69 5 11.89 5

279 6 24.67 8 7.77 7 4.61 6 10.33 8

41 5 23.38 9 6.99 9 3.68 9 8.92 9

29* 5 23.11 10 4.84 12 2.45 12 7.83 11

24 5 22.64 11 5.74 10 2.98 11 7.78 12

205 8 21.93 12 3.94 14 2.31 13 8.48 10

171 8 21.18 13 3.53 15 1.85 15 7.56 13

472 6 20.53 14 5.30 11 2.98 10 7.41 14

H37 6 19.93 15 4.05 13 2.11 14 6.31 15

87 6 19.32 16 2.35 18 1.21 20 4.52 19

109 6 19.09 17 2.58 16 1.26 18 4.44 20

147 8 17.52 18 2.42 17 1.38 17 4.78 17

388 10 17.47 19 1.98 20 1.26 19 4.75 18

153 12 16.98 20 1.98 19 1.38 16 5.78 16

341 10 16.71 21 1.87 21 1.08 21 4.05 21

54 12 14.44 22 1.40 22 0.81 22 3.34 22

184 10 14.18 23 1.35 23 0.79 23 3.00 23

333 10 13.55 24 0.56 26 0.29 27 2.12 26

337 12 12.15 25 0.85 24 0.50 24 2.45 24

232 20 11.76 26 0.67 25 0.39 25 2.44 25

487 15 9.66 27 0.24 29 0.14 29 0.73 29

272 15 9.15 28 0.24 30 0.13 30 0.89 28

P13 20 9.07 29 0.39 27 0.38 26 0.91 27

419 15 7.01 30 0.25 28 0.22 28 0.43 30 Table 6.1. Bus routes with their metrics in percent ages and ranks, sorted by µ rank. * indicates that the

route is served by articulated buses.


60

Aim 3 . To use the metric to investigate how bunching varies with time of day, and how it develops down a route. Figure 5.3 showed the value of µ by time of day, with bus usage superimposed, and is reproduced here for convenience as Figure 6.1. This graph showed how the peaks in µ lag peaks in bus usage by around an hour. A similar effect was observed with respect to traffic volumes (see Figure 5.4). This time-lag may be explained by Adebisi (1986). He suggested that buses might be expected to bunch together at times of abrupt demand change, for example a following bus experiencing lower levels of demand than a leading bus. Alternatively it may be that the peak in bunching is a delayed reaction from the peak in demand.

Figure 6.1. µ by time of day with bus usage superim posed. Source: TfL 2007, Chart 3.3.5.

The form taken by the graph of µ by time of day on the day in half term (see Figures 5.5 and 5.6) suggested that the extensive use of London buses by school children may be having a significant impact on bunching, particularly in the morning peak. Comparison of Figures 5.7 and 5.8 suggested that the effect of school travel on bunching may be more pronounced in Outer London. Several curves of µ by stop number (called µ-curves) were presented (See Figures 5.9 to 5.20). Although these showed several different characteristics, the majority of curves – 51 out of 60 – belonged to one of three major groups, represented by the three examples in Figures 6.2 to 6.4. In the first group, µ increased steadily throughout the route. In the second group, an initially fast increase was followed by a marked levelling off of the curve. In the third group, the curve levelled off more gradually, apparently approaching an asymptote. It was not clear what caused these differences in form of the µ-curves, with some curves representing the two directions of one route belonging to different groups. It may be that the behaviour of route controllers is important, or perhaps the physical characteristics of routes, such as road conditions. The apparent limit on the value of µ (assuming there is a limit for the first group, which has not been reached by the end of the route) seemed likely to be a result of controllers breaking up bunched pairs of buses.

0

100

200

300

400

500

600

0%

5%

10%

15%

20%

25%

30%

05

:00

-05

:14

06

:00

-06

:14

07

:00

-07

:14

08

:00

-08

:14

09

:00

-09

:14

10

:00

-10

:14

11

:00

-11

:14

12

:00

-12

:14

13

:00

-13

:14

14

:00

-14

:14

15

:00

-15

:14

16

:00

-16

:14

17

:00

-17

:14

18

:00

-18

:14

19

:00

-19

:14

20

:00

-20

:14

21

:00

-21

:14

22

:00

-22

:14

23

:00

-23

:14

Bu

s u

sag

e i

n t

ho

usa

nd

s o

f b

us

jou

rne

y s

tag

es

µ

Time of day

µ

Bus usage


61

Figure 6.2. Example of the first group of µ-curves.

Figure 6.3. Example of the second group of µ-curves.

Figure 6.4. Example of the third group of µ-curves.

5%

7%

9%

11%

13%

15%

17%

19%

21%

1 3 5 7 9 111315171921232527293133353739414345474951

µ

Stop number

54 towards Elmers End

15%

17%

19%

21%

23%

25%

27%

29%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

µ

Stop number

63 towards Kings Cross

10%

15%

20%

25%

30%

35%

40%

45%

1 3 5 7 9 1113151719212325272931333537394143454749

µ

Stop number

18 towards Euston


62

Some µ-curves displayed chaotic behaviour, which seemed to render them impossible to analyse. This may indicate that the positive feedback mechanism is not at play on these routes. Since all of these routes were low demand, low frequency routes, this was in apparent harmony with the theoretical analyses predicting that the feedback mechanism increases in magnitude with demand. It was postulated that more detailed analysis of individual µ-curves might lead to helpful insights into route performance. An example of this was given in Section 5.3.1, in which analysis of the P13 towards Streatham revealed a sharp increase in µ in the vicinity of Peckham Bus Station. This may reflect underlying operational issues at this busy interchange point. Further analysis of the µ curves of other routes could provide similar insights. Aim 4 . To ascertain whether bunching occurs because of the positive feedback effect discussed in Section 1.1, or whether it is simply a result of natural variation in bus speeds. There were three pieces of evidence that suggested that the positive feedback loop is indeed an important factor in bunching behaviour in London buses. Firstly, the µ- curves produced were far steeper than it appeared would be the case without feedback, as evidenced by the results of the simulations carried out in this dissertation. Figure 5.25, reproduced here for convenience as Figure 6.5, showed the difference between a typical real µ-curve, and that for a simulation without the positive feedback effect. As could be seen, the real µ-curve is significantly steeper than the simulated one. This may count as evidence that the positive feedback effect causes bunching to develop at increased rates as buses move down their routes. However, it should be noted that there were differences between the model and reality other than simply the presence or absence of the feedback effect. The weaknesses of the simulations may be significant, and were presented in Section 4.6.

Figure 6.5. Comparison of simulated 5 minute headwa y service with an equivalent real service – the 24

towards Hampstead. The second piece of evidence that indicated that the positive feedback effect may be exacerbating bunching was the fact that the parameter K (defined as the ratio of

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

32%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

µ

Stop number

24 Simulation


63

passenger arrival rate to passenger boarding rate) was a highly significant predictor of µ. In various theoretical studies, K was considered to be the crucial factor in determining the magnitude of the feedback effect. The fact that K was closely correlated with µ, such that buses with high values of K tended to have high values of µ, therefore seemed to indicate that the positive feedback effect is an important factor in bunching behaviour. However, it should be noted that there was a close relationship between K and other predictive variables such as modal headway, so the predictive power of K could in part be due to this. The factor analysis grouped K with 6 other variables in a component described as ‘density of demand’, indicating that it was closely related to all of these variables. Finally, the fact that bunching fell off rapidly in the evenings in Outer London, but increased in Inner London, may be significant (see Figures 5.7 and 5.8). A possible explanation for this is that as routes are less closely controlled in the evenings, the positive feedback mechanism causes progressively more bunching to occur in Inner London (where demand is stronger) as the evening progresses. Aim 5 . To use statistical tests to assess which factors of a route are most closely associated with bunching. Table 5.3, reproduced here as Table 6.2, listed the factors of a route which were most closely correlated to µ (i.e. µ averaged over the 5 sample days and the whole length and both directions of a particular route). As can be seen, the most successful predictor of µ as a linear relationship was the modal headway, followed by scheduled bus-km and K. 9 other factors were also significant linear predictors of µ.

Statistic


Adjusted R 2 B (coefficient) C (constant) p

Modal headway (mins) 0.745 - 1.292 30.807 < 0.001

Scheduled bus-km (1000s) 0.729 3.452 6.456 < 0.001

K 0.729 552.060 9.046 < 0.001

Passengers per route-km 0.707 0.011 9.819 < 0.001

Passenger numbers (1000s) 0.704 0.373 10.653 < 0.001

Total traffic signals 0.403 0.202 10.863 < 0.001

Average stop dwell time (s) 0.376 1.481 6.004 < 0.001

Bus lane percentage 0.327 0.343 12.522 0.001

Mean speed (km/h) 0.226 - 1.473 40.046 0.007

Coeff. of variation of JT 0.164 - 118.693 38.511 0.021

Mean stop distance (m) 0.149 - 0.070 41.323 0.026

Mean journey time (mins) 0.125 0.199 8.134 0.039 Table 6.2. Summary of linear regression models.

Aim 6 . To construct an empirical model using the iBus data that uses the factors from Aim 5 to predict how severe bunching will be on a given route. Although it had been intended to construct a multiple regression model, the multiple colinearities between the intended independent variables made this impossible. Instead, a model based upon the most successful predictor of µ, modal headway, was derived. This was of the form

µ = 40.3 × 0.913�,


64

where H is the modal headway. This model had an R2 value of 0.838, indicating that it predicts around 84% of the variation in µ using H. Aim 7 . To test this model against one or more hold-out samples to ascertain how successful it is at predicting bunching. The model was tested using the three hold-out samples, and was reasonably successful, with an average error of approximately 9%. Figure 5.28, reproduced here as Figure 6.6, showed the curve of the model equation, with the hold out samples in red. The model performed well for modal headways of up to 15 minutes, but might underestimate µ for bus services with modal headways of 20 minutes or more.

Figure 6.6.

Aim 8 . To compare the results obtained in this dissertation with those of earlier studies, summarised in the literature review. Although the use of the metric µ as a measure of bunching was not encountered in the literature review, there were several instances in which the work carried out in this dissertation was comparable to earlier work. In particular the µ-curves presented in Section 5.3 bore a striking resemblance to the form of two different measures of bunching put forward by Abkowitz et al. (1986) and Adebisi (1986). The curves suggested by their analyses (see Figures 5.21 and 5.22) were suggestive of the third and second groups respectively of the µ-curves in this dissertation. However, the form of the µ-curves seems to contradict the findings of Peng et al. (2008), that bunching does not propagate down a bus route (though the methodology of that study was rather different from this one). The fact that µ-curves for some of the lower frequency routes appeared to be chaotic in nature seems to confirm Hill’s prediction that positive feedback-induced bunching would take hundreds of stops to appear in low frequency routes (Hill, 2003). The graphs showing µ by time of day also substantiated the work of other researchers. For example, Strathman et al. (1999) found that bunching is at its most

41

63

341

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

0 5 10 15 20 25

µ



65

severe in the late afternoon. The position of the peaks in bunching relative to bus demand also substantiated Adebisi’s hypothesis that bunching is sensitive to abrupt changes in demand (Adebisi, 1986). 6.2 Future research There are several ways in which the work undertaken in this dissertation might be extended to gain a clearer understanding of bunching behaviour in buses.

• An obvious extension would be to carry out similar analysis to that presented here, but using all of the routes under the administration of London Buses. This would allow further study of the various forms of µ-curves, and greater insight into the relationships between the independent variables and µ.

• As only three of the 31 routes originally selected were served by articulated buses, no conclusions were reached as to the relative merits of articulated and double decker buses in reducing bunching, though the fact that modal headway is such a crucial factor in bunching behaviour might lead one to predict that a higher frequency double-decker service would be more bunched than an articulated service offering the same capacity. It would be interesting to look into this question in more detail. This could be done by selecting a sample of routes deliberately to include all of the articulated routes in London. Alternatively, a longitudinal study could be performed in which analysis of bunching on a given route is carried out first when it is served by articulated buses, and subsequently by double-deckers. There is certainly scope for this currently, as TfL plans to convert all existing articulated routes to double-deckers by the end of 2011. Results could provide an interesting new dimension to the articulated versus double-decker debate.

• It would be interesting to carry out regression analysis within routes, rather than just between them. This would require more fine-grained data about the routes themselves, but might aid more detailed analysis of µ-curves, and explain why the gradients of the curves take particular values at particular points. It might also allow clearer analysis of the relationships between the independent variables and µ.

• It would be interesting to run regression analyses of the independent variables against the form of a route’s µ-curve, where a dummy variable is ascribed according to which of the three major groups the curve belongs to. This could give some insight into why different routes have differently shaped µ-curves.

• Once TfL has integrated its BREMS database with iBus (see Section 3.2), it will be possible to know how many boardings take place at each stop event. This will allow detailed analysis of the relationship between boardings and bunching.

• As discussed in Section 4.6, the simulations carried out in this dissertation were somewhat limited in nature. These could be extended in various ways. In particular, the probability distributions used for the variation of journey times on stages, and the initial variations of headways, were assumed to be normal. This may not be reflected in reality, and could be investigated further. In addition, the findings of Strathman et al. (2003) found that driver identity was a significant fact in journey time variation. This variability in speeds between drivers could also be built into future simulations.

• A more detailed factor analysis could be carried out to look in finer detail at underlying relationships between the independent variables used in the regression analysis.


66

• Anecdotally, there seems to be a perception among bus users that buses tend to arrive in pairs. It would be interesting to carry out survey work to ascertain whether this is indeed a prevalent perception, and whether its strength increases with µ for a given route. This could be important given the likely effect of bunching on ridership.

67

References Abkowitz, M.D. & Engelstein, I. 1983. Factors affecting running time on transit routes. Transportation Research Part A: General, vol. 17, no. 2, pp. 107-113. Abkowitz, M., Eiger. A. & Engelstein, I. 1986. Optimal control of headway variation on transit routes. Journal of Advanced Transportation, vol. 20, no. 1, pp. 73-88. Abkowitz, M. & Tozzi. J. 1987. Research contributions to managing transit service reliability. Journal of Advanced Transportation, vol. 21, no. 1, pp. 47-65. Adebisi, O. 1986. A mathematical model for headway variance of fixed-route buses. Transportation Research Part B: Methodological, vol. 20, no. 1, pp. 59-70. Balcombe, R., et al., 2004. The demand for public transport: a practical guide. TRL report TRL593 [Online]. Available at <http://www.demandforpublictransport.co.uk/TRL593.pdf> [Accessed 5th August 2011]. Bellei, G. and Gkoumas, K., 2010. Transit vehicles’ headway distribution and service irregularity. Public Transport, vol. 24, no.7, pp. 1-21. [online] Available from: <http://dx.doi.org/10.1007/s12469-010-0024-7> [Accessed 31st August 2011]. Bly, P., and Jackson, R., 1974. Evaluation of bus control strategies by simulation. TRRL Laboratory Report No. 637. Crawthorne : TRL. Butler, T., 2011. Discussion on service control. [Conversation] (Personal communication, 24th June 2011). Cham, L.C., 2006. Understanding bus service reliability: a practical framework using AVL/APC data. MSc. Massachusetts Institute of Technology. Cohen, N., 2008. Bendy buses aren't fair to London's fare-payers. The London Evening Standard, 2nd Jan 2008. Daganzo, C.F. 2009. A headway-based approach to eliminate bus bunching: Systematic analysis and comparisons. Transportation Research Part B: Methodological, vol. 43, no. 10, pp. 913-921. DfT, 2009. Road vehicle emission factors 2009. Bus length categories - mass/axle count/passenger capacity. [Online] Available at <http://www.dft.gov.uk/publications/road-vehicle-emission-factors-2009> [Accessed 24th August 2011]. Feng, W., and Figliozzi, M., 2011. Using archived AVL/APC bus data to identify spatial-temporal causes of bus bunching. Transportation Research Board Annual Meeting 2011 (Paper #11-4262). London Bus Routes, 2011. Operational details of bus routes forming the core London bus route network. [Online]. Available at <http://www.londonbusroutes.net/details.htm> [Accessed 7th July 2011]. Hill, S.A. 2003. Numerical analysis of a time-headway bus route model. Physica A: Statistical Mechanics and its Applications, vol. 328, no. 1-2, pp. 261-273.

References

68

Kisiel, R., 2009. Bag yourself a bendy bargain: Boris Johnson sells off hated buses for £80,000 each (o.n.o). The Daily Mail, 21st May 2009. Kulash, D., 1970. Routing and Scheduling in Public Transportation Systems. PhD. Massachusetts Institute of Technology. Newell, G. & Potts, R. 1964. Maintaining a bus schedule. Proceedings of the 2nd Australian Road Research Board, vol. 2, pp. 388-393. Peng, Z., Lynde, E. & Chen, W. 2009. Improving Service Restoration Using Automatic Vehicle Location. Midwest Regional University Transportation Centre. Strathman, J.G., Dueker, K.J., Kimpel, T., Gerhart, R.L., Turner, K., Taylor, P., Callas, S. & Griffin, D., 1999. Service reliability impacts of computer-aided dispatching and automatic vehicle location technology: A Tri-Met case study. Portland: Portland State University. Strathman, J.G., Kimpel, T.J., Dueker, K.J., Gerhart, R.L. & Callas, S. 2002. Evaluation of transit operations: data applications of Tri-Met's automated Bus Dispatching System. Transportation, vol. 29, no. 3, pp. 321-345. TfL, 2009. London’s bus contracting and tendering process. [Online]. Available at <http://www.tfl.gov.uk/tfl/businessandpartners/buses/tenderresults/lbsl-tendering-and-contracting-feb-09.pdf> [Accessed 21st July 2011]. TfL, 2010a. Travel in London – Report 3. [Online]. Available at <http://www.tfl.gov.uk/assets/downloads/corporate/travel-in-london-report-3.pdf> [Accessed 20th July 2011]. TfL, 2010b. Analysis of Cycling Potential. [Online]. Available at <http://www.tfl.gov.uk/assets/downloads/roadusers/Cycling/analysis-of-cycling-potential.pdf.pdf> [Accessed 20th July 2011]. TfL, 2010c. London Buses factsheet. [Online]. Available at <http://www.tfl.gov.uk/assets/downloads/corporate/london-buses-factsheet.pdf> [Accessed 21st July 2011]. TfL, 2010d. Transport for London factsheet. [Online]. Available at <http://www.tfl.gov.uk/assets/downloads/corporate/transport-for-london-factsheet(1).pdf> [Accessed 21st July 2011]. TfL, 2011a. Service Control Project – Good Practice Guidelines [Internal document]. London: TfL. TfL, 2011b. iBus. External Presentation March 2011. [Internal document]. Transportation Research Board, 2003. Transit Capacity and Quality of Service Manual – 2nd ed. Washington, D.C. : TRB. Trompet, M., Liu, X., Graham, D.J., 2011. Development of a Key Performance Indicator to Compare Regularity of Service between Urban Bus Operators. Forthcoming in Transportation Research Record. Available at <https://workspace.imperial.ac.uk/rtsc/Public/Trompet_Liu_Graham%20Regularity%20F%20for%20TRB%20Compendium%20of%20papers.pdf> [Accessed 8th July 2011]. Turnquist, M.A. 1981. Strategies for improving reliability of bus transit service. Transportation Research Record, , no. 818, pp. 7-13.

ROUTE

Route length

Bus lane %

Total signals

Mean stop

distance

Passenger

numbers

Passengers

per route-km

Scheduled

daily bus-km

Modal

headway

Mean

Journey time

(mins)

Standard dev.

of JT (secs)

Coefficient of

var. of JT

Mean speed

(kph)

Mean stop

dwell time

Dwell per

passenger

K

Artic

Arriva

GAG

Stagecoach

RATP

µ�

∆µ

∆µ/10N

12

2

4.4

3

2

75

3

22

*

* 4

,94

9

5

61

.78

5

65

0

.15

24

1

1.5

3

6.3

2

3.1

6

0.0

16

59

5

1

0

1

0

0

27

.21

%

7.7

1%

2

.02

%

18

2

9.7

3

3

57

3

00

*

* 7

,64

9

4

64

.04

5

12

0

.13

33

1

3.8

1

13

.46

5

.85

0

.04

73

22

0

0

0

0

0

3

2.1

6%

2

2.9

1%

4

.63

%

24

2

3.7

1

7

72

3

04

*

* 4

,30

8

5

60

.39

5

22

0

.14

40

1

1.1

5

11

.25

5

.43

0

.02

88

04

0

0

1

0

0

2

2.6

4%

1

4.6

1%

3

.73

%

25

3

6.9

2

2

88

3

05

*

* 7

,58

2

5

92

.73

9

49

0

.17

05

1

2.1

2

10

.62

3

.91

0

.02

71

96

1

0

0

1

0

2

7.5

3%

1

4.1

6%

2

.34

%

29

2

4.2

3

8

64

2

78

*

* 4

,43

2

5

59

.76

5

03

0

.14

03

1

2.0

3

7.4

3

2.1

8

0.0

18

57

4

1

1

0

0

0

23

.11

%

10

.41

%

2.3

9%

41

1

6.9

1

1

25

3

13

*

* 3

,29

6

5

36

.37

3

97

0

.18

20

1

3.7

7

11

.20

4

.27

0

.02

91

97

0

1

0

0

0

2

3.3

8%

1

2.2

6%

4

.54

%

54

3

4.1

1

4

24

3

32

*

* 3

,28

1

12

5

8.8

4

58

4

0.1

65

4

17

.34

1

0.0

8

5.8

6

0.0

13

13

2

0

0

1

0

0

14

.44

%

10

.68

%

2.0

7%

63

2

4.3

2

4

47

3

00

*

* 4

,32

8

6

54

.91

4

99

0

.15

14

1

3.2

0

10

.29

4

.81

0

.02

43

48

0

0

1

0

0

2

6.8

9%

6

.90

%

1.7

0%

87

2

1.1

3

5

43

3

30

*

* 3

,30

9

6

48

.16

3

67

0

.12

69

1

3.0

0

12

.16

5

.10

0

.02

70

95

0

0

1

0

0

1

9.3

2%

8

.81

%

2.7

8%

94

2

1.6

1

3

68

3

32

*

* 4

,46

1

5

52

.09

5

35

0

.17

13

1

2.3

6

9.9

1

4.6

1

0.0

28

59

2

0

0

0

0

1

29

.43

%

12

.21

%

3.8

7%

10

9

22

.9

37

3

4

30

2

* *

3,9

40

6

5

2.8

3

46

4

0.1

46

3

12

.90

1

2.6

1

4.7

7

0.0

30

35

3

0

1

0

0

0

19

.09

%

9.4

4%

2

.48

%

14

7

17

.2

9

24

2

15

*

* 2

,94

2

8

49

.94

5

27

0

.17

59

1

3.1

5

11

.20

4

.92

0

.01

99

37

0

0

0

1

0

1

7.5

2%

1

0.6

8%

2

.69

%

15

3

17

.3

16

2

8

26

6

* *

1,5

00

1

2

41

.54

4

48

0

.17

96

1

2.0

4

7.6

9

6.4

0

0.0

08

71

1

0

0

0

0

0

16

.98

%

5.5

4%

1

.77

%

17

1

35

.0

32

6

7

31

3

* *

4,5

50

8

7

3.2

9

69

8

0.1

58

7

14

.24

9

.37

4

.84

0

.01

63

66

0

0

1

0

0

2

1.1

8%

1

2.2

8%

2

.20

%

18

4

32

.6

0

11

3

32

*

* 3

,76

7

10

5

6.3

1

72

0

0.2

13

1

18

.12

5

.25

4

.71

0

.00

75

08

0

1

0

0

0

1

4.1

8%

1

1.3

3%

2

.31

%

20

5

28

.6

31

8

3

28

9

* *

3,9

42

8

7

5.1

7

72

1

0.1

59

8

11

.32

1

3.7

6

5.4

4

0.0

24

42

3

0

0

0

1

0

21

.93

%

10

.42

%

2.1

3%

23

2

43

.9

3

28

3

99

*

* 2

,58

0

20

7

7.8

5

96

6

0.2

06

8

17

.44

6

.69

7

.39

0

.00

55

00

0

0

0

0

0

1

1.7

6%

5

.90

%

1.0

8%

25

3

24

.2

31

3

8

27

6

* *

4,7

35

5

5

1.1

4

43

4

0.1

41

5

14

.11

1

0.6

8

4.7

6

0.0

28

48

8

0

1

0

0

0

27

.73

%

13

.20

%

3.0

0%

25

4

26

.2

30

4

4

27

3

* *

4,9

99

5

5

9.3

5

53

2

0.1

49

4

13

.14

1

0.7

5

4.8

5

0.0

28

25

0

0

1

0

0

0

27

.07

%

13

.82

%

2.8

9%

27

2

19

.3

7

17

3

33

*

* 1

,24

1

15

4

1.3

0

52

3

0.2

11

0

13

.88

5

.55

4

.26

0

.00

53

48

0

0

0

0

1

9

.15

%

7.2

9%

2

.54

%

27

9

29

.3

24

2

6

30

2

* *

5,5

72

6

6

2.6

1

63

9

0.1

70

0

14

.00

1

0.7

0

4.8

9

0.0

26

82

4

0

1

0

0

0

24

.67

%

10

.59

%

2.1

8%

33

3

23

.6

30

3

9

26

8

* *

2,7

58

1

0

51

.89

4

71

0

.15

12

1

3.5

7

10

.06

5

.20

0

.01

57

57

0

0

1

0

0

1

3.5

5%

6

.95

%

1.5

8%

33

7

24

.4

16

2

9

35

4

* *

2,0

88

1

2

44

.06

5

02

0

.18

97

1

5.0

5

9.1

9

5.3

3

0.0

11

74

6

0

1

0

0

0

12

.15

%

4.5

1%

1

.29

%

34

1

36

.2

16

4

6

29

9

* *

4,2

86

1

0

80

.58

7

79

0

.16

10

1

3.4

1

7.6

9

4.3

1

0.0

12

37

4

0

1

0

0

0

16

.71

%

12

.32

%

2.0

3%

38

8

18

.2

13

3

9

31

4

* *

1,9

22

1

0

45

.49

4

83

0

.17

70

1

1.8

5

8.6

1

5.9

6

0.0

12

45

6

0

0

0

0

0

17

.47

%

4.7

4%

1

.60

%

41

9

17

.2

8

19

4

20

*

* 1

,14

1

15

2

9.2

1

33

6

0.1

91

7

17

.50

2

.69

2

.15

0

.00

24

07

0

0

0

0

1

7

.01

%

5.1

8%

2

.59

%

47

2

29

.5

4

21

3

60

*

* 5

,18

2

6

43

.63

2

86

0

.10

92

2

0.1

4

8.7

0

5.2

1

0.0

20

42

7

0

0

0

1

0

20

.53

%

9.3

0%

2

.27

%

48

7

28

.9

3

27

3

57

*

* 2

,14

8

15

5

0.1

2

45

1

0.1

49

9

17

.29

5

.23

4

.66

0

.00

54

12

0

0

0

0

0

9

.66

%

3.6

3%

0

.87

%

H3

7

16

.3

11

1

4

32

6

* *

2,6

37

6

3

1.4

7

34

8

0.1

84

5

15

.29

6

.03

3

.96

0

.01

36

15

0

0

0

0

1

1

9.9

3%

1

3.6

6%

5

.52

%

* T

hese

dat

a ar

e co

mm

erci

ally

sen

sitiv

e, a

nd h

ave

ther

efor

e be

en o

mitt

ed fr

om th

is a

ppen

dix.

The

y ar

e av

aila

ble

in th

e al

tern

ativ

e, u

npub

lishe

d ve

rsio

n of

this

dis

sert

atio

n.

App

endi

x 1

– In

depe

nden

t var

iabl

es

69

70

Appendix 2 – Correlations between independent variables

Appendix 2 - Correlations

71

Appendix 2 - Correlations

72

73

Appendix 3 – SPSS output of factor analysis

FACTOR /VARIABLES Headway Schedkm K PaxPerkm Passengers Signals StopTime Buslane Speed CoeffVAr StopDist JT /MISSING LISTWISE /ANALYSIS Headway Schedkm K PaxPerkm Passengers Signals StopTime Buslane Speed CoeffVAr StopDist JT /PRINT UNIVARIATE INITIAL CORRELATION DET KMO EXTRACTION ROTATION /FORMAT SORT BLANK(.4) /PLOT EIGEN ROTATION /CRITERIA FACTORS(3) ITERATE(25) /EXTRACTION PC /CRITERIA ITERATE(25) /ROTATION VARIMAX /METHOD=CORRELATION.

Factor Analysis [DataSet1] C:\Users\Ruth & Dave\Documents\Dave\MSc\Dissertation\Analysis\Regression\Regression model.sav

Descriptive Statistics

Mean Std. Deviation Analysis N

Modal headway 9.04 4.661 27

Total daily scheduled bus-km 3671.15 1727.395 27

Passenger arrival rate divided

by passenger boarding rate

.01826422 .010802775 27

Daily passngers per route-km 883.22 557.461 27

Total daily passenger numbers 22731.48 15729.187 27

Total traffic signals on route 41.00 22.445 27

Average stop dwell time 8.861974 2.9613368 27

Proportion of route in bus lanes

as a percentage

19.26 12.005 27

Mean speed 14.2041 2.38288 27

Coefficient of variation

(standard deviation over mean)

of journey times

.163300 .0258871 27

Average distance between

stops

315.00 41.994 27

Mean journey time 3321.5481 835.28213 27


74

Correlation Matrix a

a. Determinant = 2.84E-008

[The full table has been omitted as it replicates the correlations in Appendix 2]

KMO and Bartlett's Test

Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .591

Bartlett's Test of Sphericity Approx. Chi-Square 367.776

df 66

Sig. .000

Communalities

Initial Extraction

Modal headway 1.000 .812

Total daily scheduled bus-km 1.000 .934

Passenger arrival rate divided

by passenger boarding rate

1.000 .895

Daily passngers per route-km 1.000 .898

Total daily passenger numbers 1.000 .927

Total traffic signals on route 1.000 .814

Average stop dwell time 1.000 .659

Proportion of route in bus

lanes as a percentage

1.000 .694

Mean speed 1.000 .919


(standard deviation over

mean) of journey times

1.000 .522


stops

1.000 .694

Mean journey time 1.000 .891



75

Total Variance Explained

Component

Initial Eigenvalues

Extraction Sums of Squared

Loadings

Rotation Sums of Squared

Loadings

Total

% of

Variance

Cumulative

% Total

% of

Variance

Cumulative

% Total

% of

Variance

Cumulative

%

dimension0

1 7.135 59.458 59.458 7.135 59.458 59.458 4.709 39.241 39.241

2 1.301 10.838 70.295 1.301 10.838 70.295 2.562 21.352 60.592

3 1.222 10.182 80.478 1.222 10.182 80.478 2.386 19.885 80.478

4 .713 5.943 86.420

5 .573 4.777 91.197

6 .457 3.811 95.008

7 .313 2.607 97.615

8 .143 1.194 98.809

9 .078 .646 99.455

10 .038 .318 99.773

11 .023 .190 99.963

12 .004 .037 100.000



76

Component Matrix a

Component

1 2 3

Daily passngers per route-

km

.938


numbers

.938

Passenger arrival rate

divided by passenger

boarding rate

.907

Modal headway -.841

Total daily scheduled bus-

km

.831 .453

Average stop dwell time .801



.798

Total traffic signals on route .744 .454

Mean speed -.632 .448 -.565


stops

-.576 .567




-.574 .434

Mean journey time .512 .625 .487


a. 3 components extracted.


77

Rotated Component Matrix a

Component

1 2 3

Passenger arrival rate

divided by passenger

boarding rate

.868

Modal headway -.831

Daily passngers per route-

km

.796 .420

Total daily scheduled bus-

km

.777 .573


numbers

.767 .521




-.716

Average stop dwell time .659 .413

Mean speed -.912


stops

-.778



.546 .578

Mean journey time .937

Total traffic signals on route .424 .742


Rotation Method: Varimax with Kaiser Normalization.

a. Rotation converged in 5 iterations.

Component Transformation Matrix

Component 1 2 3

dimension0

1 .768 .469 .436

2 .032 -.708 .705

3 -.640 .528 .559


Rotation Method: Varimax with Kaiser

Normalization.