This is the unformatted version of:
Fonzone, A., Schmöcker, J.-D. and Liu, R. (2015). A model of bus bunching under reliability-based passenger
arrival patterns. Transportation Research C. Available from http://dx.doi.org/10.1016/j.trc.2015.05.020. Also
available from Selected Proceedings of the 21th International Symposium on Transportation and Traffic Theory(ISTTT). Kobe, Japan, August 2015.

A model of bus bunching under reliability-based passenger arrival
patterns
Achille Fonzonea*, Jan-Dirk Schmöckerb, Ronghui LiucaTransport Research Institute, Edinburgh Napier University, Edinburgh, U.K.
Kyoto University, Department of Urban Management, C1-2-436, Katsura Nishikyo-ku, Kyoto 615-8540, Japanc
Institute for Transport Studies, Leeds University, Leeds LS2 9JT, U.K.

b

Abstract
If bus service departure times are not completely unknown to the passengers, non-uniform passenger arrival patterns can be
expected. We propose that passengers decide their arrival time at stops based on a continuous logit model that considers therisk of missing services. Expected passenger waiting times are derived in a bus system that allows also for overtaking between
bus services. We then propose an algorithm to derive the dwell time of subsequent buses serving a stop in order to illustrate
when bus bunching might occur. We show that non-uniform arrival patterns can significantly influence the bus bunchingprocess. With case studies we find that, even without exogenous delay, bunching can arise when the boarding rate is insufficient
given the level of overall demand. Further, in case of exogenous delay, non-uniform arrivals can either worsen or improve thebunching conditions, depending on the level of delay. We conclude that therefore such effects should be considered whenservice control measures are discussed.

1. Introduction
Service irregularities increase passenger waiting times, decreasing the attractiveness of public transport. The
more the passengers can trust the service schedule, the better they can time their arrival at stops. Whereas undercompletely random service arrivals the passengers can also do no better than “randomly” arrive at stops, in manycases at least some coherence of the actual arrivals with the service schedule might be expected. Therefore, even
if the schedule might not be known to all passengers and uncertainties in the access time to the stop are considered,
non-uniform passenger arrival patterns can be expected. With a few exceptions, the effect of such non-uniformity
on bus loads has been largely ignored in the literature and is the topic of this contribution.
We propose a “mixed behaviour”: Passengers consider the likely service departure times and leave some safetymargins in order to ensure that they do not have to wait too long for a bus but also minimise the chances of missing
a service. Such a behaviour seems reasonable for passengers in cities with fairly good bus services. As a motivating
example familiar to the authors, consider the bus stop in front of Kyoto University. The stop is close to the officebuildings and the most frequent service arrives around every 15 minutes during the evening hours. Some
passengers, possibly those without knowledge of the schedule, will arrive randomly though the bulk of passengers
will time their arrival to 2-3 minutes before scheduled service arrival. Some “risky” passengers, or those delayedby for example waiting times at the elevator, will arrive even closer to the scheduled departure.
*

Corresponding author. Tel.: +44-131-455-2898.
E-mail address: a.fonzone@napier.ac.uk

In general, it is well known that passenger arrival is influenced by service characteristics, such as average valueof headways and headway deviations. In particular, it is commonly accepted that passengers tend to arrive closer
to the scheduled departure time (i.e., their arrivals are not uniform) when the headways are large. This behaviour
is oftern termed as scheduled-based arrival behaviour. Bowman and Turnquist (1981) provide a model ofpassenger arrival behaviour, which links the arrival distribution to the characteristics of the service, including its
reliability.
Therefore a good understanding of arrival patterns is the foundation to modelling boarding demand. Deriving
bus loads are important to estimate potential capacity bottlenecks and possibly revenue splits among bus operators.
Furthermore, bus loads and bus service dwell times at stops are closely correlated, and unexpected high loads canlead to the well-known “bus bunching” process. The seminal work of Newell and Potts (1964) presents a simplified
model of the phenomenon, which casts light on some causes. However, their model does not provide a realistic
representation of bunching as they neglect aspects such as en-route service perturbations, transport operator
policies concerning holding and overtaking as well as complex network features such as the presence of “commonlines” among which some of the passengers at a stop might choose. Some of these issues have been dealt with inlater literature as reviewed in more detail in the next section.
Newell and Potts further assume uniform passenger arrival. In the above Kyoto bus stop example this might
overestimate the bunching phenomena as only a few additional passengers arrive in the time interval between thescheduled and actual service departure and hence delayed buses have to board fewer additional passengers than
predicted with uniform arrival. Furthermore, Newell and Potts do not capture the effect of severe bunching wherebuses might be overtaken.
The contributions of this paper are twofold. Firstly, a model of passenger arrival extending the approach ofBowman and Turnquist (1981) to allow for overtaking between buses at a stop. We refer to our model as the“reliability-based arrival pattern model” in line with the above example. Secondly, we include these passenger
arrival patterns in a model of bus propagation, highlighting causes of bunching which are not identified by Newell
and Potts.
In line with above discussion, our model will be mainly applicable to situations in which passengers consider
timetables in deciding their arrival at stops. It is conventionally accepted that timetables influence passengers
decisions for services with expected headways of more than 10 minutes and that, instead, if service headways areshorter, uniform passenger arrival patterns can be expected. Actually the threshold between schedule-dependentand uniform passenger arrival can be lower than the conventional one. A review of existing studies on the relation
between service headway and passenger arrival at stops is provided by Luethi et al. (2006). Interestingly, this study
finds that passengers consult schedules even when the headway is 5 minutes. We consider the topic discussed inthis paper especially of topical importance due to the increasing presence of service schedule information topassengers before arrival at a stop even for passengers unfamiliar with the network due to online availability ofjourney planners. More and more cities now provide real-time information (RTI) for passengers. RTI changes the“visibility of the network” and hence passenger behaviour. For instance, it is reasonable to expect that ubiquitous
RTI on departure time (accessed by internet and/or mobile phone apps) induces non-uniform passenger arrivals
also for short headways and irregular services.
The reminder of this paper is organized as follows: Section 2 provides a more detailed review of the two key
references for this paper, Newell and Potts (1964) and Bowman and Turnquist (1981) as well as further related
and newer literature. Section 3 then introduces the notation that is utilized in later sections. Section 4 describes thepassenger arrival model and Section 5 the bus propagation model. Section 6 illustrates both models through casestudy applications before Section 7 concludes this paper.
2. Literature Review
Bus bunching is generally defined as the effect of two successive services of a single line arriving at stops withshorter than scheduled headways. The effect occurs by the first service being delayed at previous stops due tounplanned long boarding times, or being delayed en-route by unforeseen traffic congestion. The subsequent
service then has to pick up fewer passengers at that stop and departs earlier than scheduled. At downstream stops,
the effect is further emphasised as the initial delay to the first vehicle and the early arrival of the subsequent
service result in increasingly longer dwell times for the first bus and increasingly shorter dwell times for thesecond bus.
Bus bunching has a direct negative impact on the passengers as it leads to, on average, longer dwell-times.
Quarmby (1967) already found that transit passengers value their time waiting two to three times more than theirtime on board travelling. Subsequent literature has confirmed this, sometimes reporting even higher disutility
associated with waiting. Using a stated-preference survey, Hollander and Liu (2008) found that bus passengers

value service reliability four times higher than they do to mean travel time. Hollander et al. (2007) further
demonstrate that bus unreliability has a significant impact on passengers’ response in their departure-times.
Bus bunching is a common feature in urban public transport, and a long-standing problem facing the bus/transit
service providers and academic researchers alike. The bunching effect on a single line has been first analyticallydescribed by Newell and Potts (1964). Assuming that travel times between stops are identical and that passenger
loads are constant, Newell and Potts show that if the passenger arrival rate at a stop is larger than half the loading
rate of buses the bunching effect occurs for small perturbations in the original schedule. If the ratio (referred to
below as ρ-ratio) is smaller, the system can recover from perturbations. Furthermore, bus bunching is more notedin high frequency services, where the headway between buses is small and the delay to headway ratio is more
likely exceeding the threshold for a localised bus bunching to amplify (rather than being damped) further down
the route. Recently, Schmöcker et al. (2015) extended the bunching research to a route section served by two lines
with high frequency so that some passengers might board “whichever line comes first”. They show therefore howbunching can spread to initially unaffected lines but also how a line serving similar stops can help with service
recovery on the initially affected line.
There has been a significant body of research designing operational strategies to address the bus bunchingproblem. In particular, holding strategies of early buses as well as strategies to keep minimum distances between
subsequent services have been analysed and shown to be successfully applied in literature. The holding strategies
are implemented through building slacks in the schedule at key timing points and holding buses at these points tokeep them to schedule (e.g. Osuna and Newell, 1971; Newell, 1974; Hickman, 2001; Xuan et al., 2011; Cats etal., 2012).
Due to the complexity of the problem, most of the analytical studies involved having just one controlled timing
point. Hickman (2001) developed a simulation approach to solve the bus bunching problem through optimal
holding points. These are static holding solutions, which do not respond to dynamical changes in the actual bus
performances on the day. Eberlein et al. (2001) developed a model for dynamical bus holding which take realtime information on bus headways into consideration and achieves to minimise passenger waiting time. The model
assumes deterministic passenger demand and bus travel times between stops. Daganzo (2009) developed a more
systematic approach for dynamical holding analysis with real-time bus performance information. Daganzo’s
method is able to consider holding at multiple timing points, therefore providing opportunity for return to schedule
for long bus routes. In addition, the model takes into account random effects, such as the random variations in bus
travel time, bus dwell time and passenger demand, making it resemble more realistically to real-life situations.
Daganzo and Pilachowski (2011) proposed an adaptive bus control scheme based on a two-way bus-to-bus
cooperation, where a bus adjusts its speed to both its front and rear headways. They show that the scheme yields
significant improvements in bus headways and bus travel time. Moving away from the traditional ideal of scheduleand a prior target headway, Bartholdi and Eisenstein (2012) proposed a self-coordinating method to equalise bus
headway, while Pilachowski (2009) proposed to use GPS data to counteract the cause of the bunching directly by
allowing the buses to cooperate with each other and to determine their speed based on relative position. Using a
dynamic simulation approach to model the stochastic nature of the transport networks, Hadas and Ceder (2010)
examined a range of operational tactics to improve bus service reliability. Muñoz et al. (2013) compared theperformance of dynamic bus control strategies under different scenarios of bus capacity, service frequency andpassenger load profiles. Most of the existing studies are concerned with a simplified bus system, notably with asingle line, fixed service frequency, constant passenger flows, and no bus overtaking. Newell and Potts (1964),
for instance, assume fixed frequency, constant dwell times, equal-distance stops and equal-travel time between
stops, and that buses cannot overtake. In real-life situations these simplifying assumptions obviously might not
hold. Boyd (1983) presented empirical evidence that demonstrated the impact of variability in bus journey timeon bunching. Nagatani (2001) shows a strong relationship between bus delay and the passenger number on bus,
and proposes skipping a bus stop as a way of keeping to schedule. Liu et al (2013) studied the bus stop-skipping
scheme considering random travel time. Recently, Nesheli and Ceder (2014) introduced the concept of skip-asegment in addition to skip a stop. In general, bus corridors operating multiple lines that interact through sharing
passengers and bus stops are considered complex scenarios for most of the existing analytical studies (Daganzo
and Pilachowski, 2011).
Another significant simplification in the existing studies that was mentioned in the introduction is theassumption of a uniform passenger demand distribution over time and space. This hypothesis of random passenger
arrivals is common in models of public transport. When the randomness of the bus arrival at stops is taken into
consideration, a model of passenger arrivals is needed to compute the waiting time and so the overall journey
time. In networks with frequent transit services and no ubiquitous real time information, models based on the
random incidence assumption can be justified: in fact passengers may not consider schedules if they know that
their waiting time for the next service will be low in any case. As shown for instance in Larson and Odoni (1981),
under the assumption of passenger random arrival, the waiting time expectation can be easily derived from thefirst and second moment of the bus arrival distribution. Random incidence is assumed in most of the literature on
frequency-based route choice and assignment, notably in the seminal work of Spiess and Florian (1989) when
deriving “optimal strategies”. The assumption can be still retained when passengers have access to real time
information but only once they have reached a stop, as considered by Gentile et al. (2005). Conversely, passenger
arrival distributions can be expected not to be uniform when real time information is available to passengers even
before they go to a stop. With the increasing diffusion of internet-based information and smart phones, this
scenario is becoming more and more common in many cities around the world. It can be anticipated thatpassengers who know the predicted bus departure times in advanced will try to coordinate their arrival with thatof the next attractive bus. Watkins et al. (2011) observed a reduction of the actual waiting time from 11 to 9
minutes for passengers of King County Metro using information disseminated by internet. Clearly real time
information is not perfectly reliable: bus arrival prediction normally entails information about the current positionof transit vehicles and the use of models to predict the travel time to downstream stops. Both these elements areaffected by uncertainty (see for instance Crout, 2007).
Liu and Sinha (2007) have further confirmed the non-uniformity of demand. They collected data on bus travel
time, dwell time, and passenger boarding and alighting along a commuter bus route in the city of York, in England.
Significant variations of demand (boarding and alighting) were detected both across bus stops and over time.
Building their observed data into a microsimulation model of the bus corridor, Sorratini et al. (2008) showed thatthe variability of the passenger demand distribution has the most significant impact on bus reliability measures.
Based on this empirical evidence therefore it appears reasonable to argue that passenger decision making
contemplates the overall system reliability deriving from the combination of the transport service reliability andthe information system accuracy. A model in which some “schedule-aware” passengers choose their arrival
considering system reliability was put forward by Bowman and Turnquist (1981). In their model, the passenger
arrival pattern at a bus stop is described by a continuous logit function. The utility function depends on the
expected waiting time calculated by averaging waiting times over all possible bus arrival times for a specificpassenger arrival time, i.e.
𝐸(𝑊(𝑡)) = [1 − 𝑃(𝑡)]𝑊(𝑡) + 𝑃(𝑡)𝑊 ′ (𝑡)where 𝐸(𝑊(𝑡)) is the expected waiting time, 𝑃(𝑡) the probability that the intended bus is missed by a passenger
arriving at 𝑡, 𝑊(𝑡) the expected waiting time in case the intended bus arrives after 𝑡 and 𝑊 ′ (𝑡) the expectedwaiting time in case the bus has departed before 𝑡. Bowman and Turnquist provided no details regarding the
calculation of the probabilities and expected values. Considering the context of their paper, it seems that they do
not consider overtaking. This assumption can be valid for stops served by a single line, when severe bunching is
not likely or when overtaking is prohibited by the operator or not feasible due to road space. It has to be relaxedthough when bus bunching is such that the presence of more than one bus at the same stop can be an issue, and/or
when more attractive lines serve the same stop and their services can overlap. Bowman and Turnquist showedthat their model predicts the actual arrival patterns very well, and that their model can cover both the headway
ranges normally modelled with a frequency-based approach and those modelled by a scheduled-based one.
Furthermore, they demonstrated that passengers are more sensitive to schedule reliability than to service
frequency. A limitation of their model, stressed by the authors themselves, is the use of the continuous logit model.
GEV models like the logit one are commonly used for discrete choices within the framework of random utility
maximization. The application of the logit model is constrained by the underpinning hypothesis of the
independence of errors across alternatives. The validity of this assumption becomes particularly debatable when
the decision dimension is continuous. The problem has been addressed in the literature concerning the topic ofthe departure-time choice of travellers using private transport. Choices in a continuous setting instead have been
often described by models based on hazard functions; but such an approach is criticized because it lacks a proper
behavioural support. Recently Lemp et al. (2010) suggested a continuous cross-nested logit model, which retains
the behavioural justification of the logit models and allows for correlation across alternatives.
In summary, most literature related to bus bunching since Newell and Potts (1964) has focused on deriving
control strategies to minimize the bunching effect. There is a separate set of literature discussing arrival patterns
and the value of waiting times but, to the best of our knowledge, the results of the latter set of literature have notyet been included in bus bunching models. We propose that considering more advanced passenger arrival timemodels seems necessary to build a realistic model of bus bunching, especially in the era of ubiquitous (real-time)
information.

(1)3. Notation
The following notation is used throughout the paper.
Table 1 Notation

𝑖
𝑛
𝑎
𝑑
𝜏𝑖𝑛
, 𝜏𝑖𝑛

Stop number. Stop 0 is the bus depot, where no boarding is allowed.
Bus index or bus run (concisely, “bus”) designated by index 𝑛 (the meaning will be clarified
by the context)

T𝑖𝑛

Arrival, departure time of bus 𝑛 from stop 𝑖. Superscript 𝑑 is omitted when redundant.
Perceived probability distribution function of 𝜏𝑖𝑛 ; perceived cumulative distribution functionof 𝜏𝑖𝑛 . These and all the remaining probabilities mentioned in the paper have to be considered
as describing the perception of the travellers rather than the actual characteristics of theservice.
Interval of perceived possible departure times of bus 𝑚 from stop 𝑖, i.e. Τ𝑖𝑛 = {𝜏𝑖𝑛 |𝑓𝑖𝑛 (𝜏𝑖𝑛 ) >0}𝐵𝑖Set of buses serving stop 𝑖

𝐵𝑖 (𝑡)

Set of buses available at stop 𝑖 after 𝑡, i.e. 𝐵𝑖 (𝑡) = {n ∈ 𝐵𝑖 | sup Τ𝑖𝑛 ≥ 𝑡}

𝑓𝑖𝑛 ; 𝐹𝑖𝑛

𝐴𝑖𝑛 (𝑡)𝐴−𝑖𝑛 (𝑡)𝒫(𝐴−𝑖𝑛 (𝑡))𝑃𝑖𝑛 (𝑡); 𝑃̅𝑖𝑛 (𝑡)Set of buses alternative to the focal bus serving stop 𝑖 after 𝑡, i.e. 𝐴𝑖𝑛 (𝑡) = {𝑚 ∈ 𝐵𝑖 (𝑡), 𝑚 ≠𝑛}
Set of alternative buses which can depart from stop 𝑖 before 𝑡 , i.e. 𝐴−𝑖𝑛 (𝑡) = {𝑚 ∈𝐴𝑖𝑛 (𝑡)| inf Τ𝑖𝑚 ≤ 𝑡}
Power set of 𝐴−𝑖𝑛 (𝑡) , i.e. is set of all subsets of 𝐴−𝑖𝑛 (𝑡) , including the empty set and𝐴−𝑖𝑛 (𝑡) itself.
Generic element of 𝒫(𝐴−𝑖𝑛 (𝑡)); set of elements of 𝐴𝑖𝑛 (𝑡) not included in 𝑃𝑖𝑛 (𝑡), i.e. 𝑃̅𝑖𝑛 (𝑡) ={𝑆 ∈ 𝐴𝑖𝑛 (𝑡)|𝑆 ∉ 𝑃𝑖𝑛 (𝑡)}Π𝑖𝑛 (𝑡)Probability distribution function of the event “bus 𝑛 departs from stop 𝑖 at 𝜏𝑖𝑛 > 𝑡 and before
any other bus in the set 𝑆”.
Probability that 𝑛 departs from stop 𝑖 after 𝑡 and before any bus in the set 𝑆
Probability that all buses in the set 𝑆 have departed from stop 𝑖 before 𝑡
Probability distribution function of the event “passenger arriving at stop 𝑖 at 𝑡 and boarding
bus 𝑛 at 𝜏”
Probability that a passenger arriving at stop 𝑖 at 𝑡 boards bus 𝑛𝐸(𝑤𝑖 (𝑡))Expected waiting time of a passenger arriving at stop 𝑖 at 𝑡

𝐸(𝑤𝑖𝑛 (𝑡))Expected waiting time for bus 𝑛 of a passenger arriving at stop 𝑖 at 𝑡

φ𝑖 (𝑡, 𝜏𝑖𝑛 , 𝑆)
Φ𝑖𝑛 (𝑡, 𝑆)
Ξ𝑖 (𝑡, 𝑆)
π𝑖𝑛 (𝑡, τ)

𝑊(𝑡)
𝛼, 𝛽
𝑞𝑖 (𝑡)

Anticipated risk-averse waiting time of a passenger arriving at stop 𝑖 at 𝑡
Parameters measuring aversion of passengers to wait
Passenger arrival rate at stop 𝑖𝑏𝑖 (𝑡), 𝑏
𝜌
𝑣𝑖𝑛
𝐷𝑖𝑛
𝛿𝑖𝑛

Time-dependent boarding rate at stop 𝑖; time and stop independent boarding rate
Ratio between arrival and boarding rate, referred to later as the saturation rate
Travel time of bus 𝑛 from stop 𝑖 to stop 𝑖 + 1
Dwell time of bus 𝑛 at stop 𝑖
Exogenous delay of bus 𝑛 at stop 𝑖

4. Reliability-based passenger arrival
4.1. Assumptions and model limitations
To explore the impact of non-uniform passenger arrivals on bus bunching, in this section we derive a model ofpassenger arrival at bus stops based on the assumption that travellers aim to reduce their expected waiting time
considering that they may miss some or all of their intended buses because of service irregularity. Our passenger

utility function is based on an anticipated risk-averse waiting time. By “anticipated” we mean that passengers
decide their arrival time based on their perceived probability of bus departure times in a non-perfectly reliable
system, i.e. in a system in which the actual departure times can differ from the scheduled ones. By “risk-averse”
we mean that passengers attach a cost to the possibility of missing their last service. Our utility function is basedon a behavioural model similar to that considered by Bowman and Turnquist, but we extend their approach byallowing for the existence of an indefinite number of buses and for the possibility of overtaking among buses.
Different specifications of the utility functions are discussed below. The model presumes that passengers have apriori information on the scheduled departure time of buses and on the accuracy of such information. The first
hypothesis holds in the case of relatively low-frequency services for which a timetable is published and known by
the travellers. In the case of services for which schedules are not published, it may still be valid if the day-by-day
service regularity is high. The accuracy of the information considered in the decision-making process depends
both on the punctuality of the service and on the passenger’s knowledge of it: passengers who have used a given
transit system only few times, might not trust the information they have and hence consider a large variability ofdeparture times even when the system is in fact very reliable. In the following, by system reliability we mean thereliability accounted for in the decision-making process, as a consequence of the perceived information accuracy.
Both the expected waiting times and the derived utility function are deterministic. However, we consider thatthe perception of the value of each arrival time varies among users. In particular, following Bowman andTurnquist, we assume that the passenger arrival distribution can be described by a continuous logit model. Not
much work has been done concerning passenger arrival patterns after the contribution of Bowman and Turnquist(1981). Since our focus here is on the derivation of the utility function, we use the continuous logit model as it has
been proven in the Bowman and Turnquist paper to be able to describe the actual behaviour of public transport
users for their scenario assumptions despite its theoretical limitations. In our model, passengers always get to abus stop at their intended arrival time. In reality, this might not happen because passengers cannot always
anticipate their walking time exactly. This phenomenon exceeds the scope of this study.
It is assumed that passengers board the bus that departs first after their arrival. Such an assumption is realistic
if we assume that only one bus line serves the stop or that all bus lines travel the same route downstream from theboarding point. In these cases passengers do not face the “common line issue” where the choice set might dependon the time spent waiting at the stop already (see Noekel and Weckek (2009) for elaboration of such cases). The“board the first departing bus” behaviour is further justified by the empirical findings that passengers put morecost on wait time than in-vehicle travel time (e.g. Quarmby, 1967). However, we acknowledge that there are cases
in which “board the first bus” strategy is not realistic, for instance when crowding on buses affects thedecision/possibility to board, when passengers might anticipate variations in downstream travel times between
buses, or when passengers prefer express services. Our model does not cover these cases. Statistical independence
is assumed for both the departure times of different buses at the same stop and the departure times of the same bus
at consecutive stops. This assumption is clearly an over-simplified representation of reality as departures of buses
from the same stop can be correlated. Our assumption does not hold if, for example, particular operational policies
exist, such as a FIFO rule for buses departing from the stop, or, if the stop layout does not allow more than one
bus to board passengers at a time.
We further assume that the passenger arrival distributions at each stop are independent from what happens atother stops and therefore in this section the subscript representing the stop is redundant and will be omitted for thesake of simplicity. We add the subscript in Section 5 when considering the (obviously correlated) departures ofthe same bus from subsequent stops. This stop-independence assumption in the behavioural model implicitly
assumes that real time information are not available but that passengers only know the scheduled information (plustheir perception of the service reliability and the variance in their stop access time) when deciding their stop arrivaltime.
We hypothesize that a bus leaves a stop as soon as all waiting passengers have boarded, i.e. we do not consider
any holding policy. This assumption implies that we consider bus stops as on the main road, rather than in a bus
bay. Meng and Qu (2013) show that bus dwell time at bus bays possesses a high degree of uncertainty caused by
the fact that buses having to merge onto the main road and that the merging process is dependent on the level oftraffic on the shoulder lane. In our model, as in most works in literature, the dwell time depends only on theboarding process. This is realistic when the boarding time is longer than the alighting time, which can occur
because (a) either passengers who board are generally less than those who alight, and/or (b) the boarding time perpassenger is considerably longer than the alighting one, due for instance to the fact that tickets are issued whenboarding. In addition, no influence between boarding and alighting is considered. This happens when buses are
not crowded, and/or boarding and alighting take place at different doors. Finally, we note that we also do notinclude capacity effects in this model; thus assume all waiting passengers can board the first departing bus.

4.2. Perceived probability of boarding a bus
Clearly if 𝐵(𝑡) = {𝑛} (i.e. if bus 𝑛 is the only departing after time t), the probability that a passenger who
arrives at the stop at time t boards bus 𝑛 is equal to the probability that 𝑛 departs after 𝑡. Consider now that thepassenger arrives at 𝑡 but 𝐵(𝑡) = {𝑛, 𝑛 + 1} and inf Τ𝑛+1 ≤ 𝑡. Still the passenger can only board 𝑛 if it departs
after 𝑡. In this case he will actually board 𝑛 if 𝑛 + 1 has already departed or it has not departed yet and actuallyleaves after 𝑛. In the remaining scenario, in which 𝑛 + 1 has not departed yet but it leaves before 𝑛, the passenger
boards 𝑛 + 1. In other words, the overall probability of boarding 𝑛 is equal to the sum of the probability ofboarding it in each of two cases of 𝐴−𝑛 (𝑡) ∈ 𝒫(𝐴−𝑛 (𝑡)) = {∅, {𝑛 + 1}}:Π𝑛 (𝑡) = Pr(𝜏𝑛 > 𝑡 ∩ 𝜏𝑛+1 < 𝑡) + Pr(𝜏𝑛 > 𝑡 ∩ 𝜏𝑛+1 > 𝜏𝑛 )(2)

In the general case in which 𝑐 = |𝒫(𝐴−𝑛 (𝑡))| ≥ 2, i.e. considering the possibility of more than one alternative
to 𝑛 which can depart before 𝑡, 𝑛 is boarded in all the cases in which some of the alternatives have departed before𝑡 and all others depart after 𝑛. The number of the scenarios to consider in the calculation of Π𝑛 (𝑡) is equal to thenumber of combinations of buses in 𝐴−𝑖𝑛 (𝑡), i.e.
Π𝑛 (𝑡) =∑Pr ((𝜏𝑛 > 𝑡) ∩ (𝜏𝑙 < 𝑡 ∀𝑙 ∈ 𝑃𝑛 (𝑡)) ∩ (𝜏𝑚 > 𝜏𝑛 ∀𝑚 ∈ 𝑃̅𝑛 (𝑡)))(3)

𝑃𝑛 (𝑡)∈𝒫(𝐴−𝑛 (𝑡))

𝑓𝑖𝑚

time
t'

t

Fig. 1. Example of bus arrival probability distributions functions at a stop

To illustrate (3), consider the case in Fig. 1, in which lines of different colours represent the perceived
probability distribution functions of different buses (in the explanation of the example, buses are indicated withtheinitialletterof
theircolour).
Followingthedefinitions
inSection3, 𝐵 ={𝑃𝑢𝑟𝑝𝑙𝑒, 𝑌𝑒𝑙𝑙𝑜𝑤, 𝑅𝑒𝑑, 𝐺𝑟𝑒𝑒𝑛, 𝐶𝑦𝑎𝑛}, 𝐵(𝑡) = {𝑌, 𝑅, 𝐺, 𝐶}, 𝐴𝑅 (𝑡) = {𝑌, 𝐺, 𝐶}, 𝐴−𝑅 (𝑡) = {𝑌, 𝐺}, 𝒫(𝐴−𝑃 (𝑡)) ={∅, {𝑌}, {𝐺}, {𝑌, 𝐺}}. Bus 𝑃 has surely departed at 𝑡, therefore 𝑃 ∉ 𝐴𝑅 (𝑡) and it does not influence the probabilityof boarding 𝑛. Note that it does influence such probability if the passenger arrives at 𝑡 ′ . There are four possible
circumstances (note that |𝒫(𝐴−𝑛 (𝑡))| = 4) under which 𝑅 is boarded by a passenger arriving at 𝑡:
 No bus has departed before 𝑡, and 𝑅 departs before all alternative buses: 𝑃𝑅 (𝑡) = ∅, 𝑃̅𝑅 (𝑡) = 𝐴𝑅 (𝑡) ={𝑌, 𝐺, 𝐶} 𝑌 has departed before 𝑡, and 𝑅 has not departed yet and departs before 𝐺 and 𝑃: 𝑃𝑅 (𝑡) = {𝑌}, 𝑃̅𝑅 (𝑡) ={𝐺, 𝐶}
 𝐺 has departed before 𝑡, and 𝑅 has not departed yet and departs before 𝑌 and 𝑃: 𝑃𝑅 (𝑡) = {𝐺}, 𝑃̅𝑅 (𝑡) ={𝑌, 𝐶}
 𝐺 and 𝑌 have departed before 𝑡 , and 𝑅 has not departed yet and departs before 𝑃 : 𝑃𝑅 (𝑡) ={𝐺, 𝑌}, 𝑃̅𝑅 (𝑡) = {𝐶}The four cases are mutually exclusive; therefore, the overall probability of boarding 𝑅 is equal to the sum of
the probability of each case, as indicated by (3).
To calculate the probabilities in the summation in (3) we need to derive the probabilities that a bus leaves before
or after a set of other buses. As to the former, consider a set of buses 𝑆 alternative to 𝑛 and assume without loss ofgenerality 𝑆 = {1, … , 𝑘}. In general, the probability that 𝑛 departs after 𝑡 and before any other bus in 𝑆 can becalculated from the joint distribution of departure time probability asΦ𝑛 (𝑡, 𝑆) =𝑓(𝜏𝑛 , 𝜏1 , … , 𝜏𝑘 ) 𝑑𝜏𝑛 𝑑𝜏1 … 𝑑𝜏𝑘∫(4)𝜏𝑛 ≥𝑡;𝜏𝑗 >𝜏𝑛 ,𝑗=1,…,𝑘

Under the assumption that passengers perceive the departure times of different buses as statisticallyindependent, (4) becomes1+∞

Φ𝑛 (𝑡, 𝑆) = ∫ 𝑓𝑛 (𝜏𝑛 ) ∙ ∏(1 − 𝐹𝑚 (𝜏𝑛 )) 𝑑𝜏𝑛

(5)

𝑚∈𝑆

𝑡

The integrand function used to calculate Φ𝑛 (𝑡, 𝑆) can be interpreted as the pdf of the event “bus 𝑛 departs at
𝜏𝑛 > 𝑡 and before any other bus in the set 𝑆”, i.e.
φ𝑛 (𝑡, 𝜏𝑛 , S) = 𝑓𝑛 (𝜏𝑛 ) ∙ ∏(1 − 𝐹𝑚 (𝜏𝑛 ))

(6)

𝑚∈𝑆

Note that, as expected, the probability of bus 𝑛 departing before any other bus in the set 𝑆 decreases when𝑡 increases. In fact
Proposition 1: Φ𝑛 (𝑡, 𝑆) is a non-increasing function of 𝑡
Proof: φ𝑛 (𝑡, 𝜏𝑛 , S) ≥ 0 because it is a product of pdf and CDF. Applying the Leibniz’s rule for integral
differentiation𝑏(𝜃)𝑑 𝑏(𝜃)∫ 𝑓(𝑥, 𝜃)𝑑𝑥 = ∫ 𝜕𝜃 𝑓(𝑥, 𝜃)𝑑𝑥 + 𝑓(𝑏(𝜃), 𝜃)𝑏 ′ (𝜃) − 𝑓(𝑎(𝜃), 𝜃)𝑎′ (𝜃)𝑑𝜃 𝑎(𝜃)𝑎(𝜃)it follows
𝑡 ̅ >𝑡𝜕Φ𝑛 (𝑡, 𝑆)= ∫ 𝜕𝑡 𝑓𝑛 (𝜏𝑛 ) ∙ ∏(1 − 𝐹𝑚 (𝜏𝑛 )) 𝑑𝜏𝑛 +φ𝑛 (t,̅ 𝜏𝑛 , S)∙0-φ𝑛 (t, 𝜏𝑛 , S)∙1 = −φ𝑛 (t, 𝜏𝑛 , S) ≤ 0
𝜕𝑡
𝑚∈𝑆

𝑡

which shows that Φ𝑛 (𝑡, 𝑆) is a non-increasing function of 𝑡. QED.
In general the probability that all buses in the set 𝑆 have departed earlier than 𝑡 isΞ(𝑡, 𝑆) =𝑓(𝜏𝑛 , 𝜏1 , … , 𝜏𝑘 ) 𝑑𝜏𝑛 𝑑𝜏1 … 𝑑𝜏𝑘∫(7)

𝜏𝑗 <𝑡,𝑗=1,…,𝑘

Under the assumption of independence of the departures𝑡Ξ(𝑡, 𝑆) =∫𝑡𝑓(𝜏1 , … , 𝜏𝑘 ) 𝑑𝜏1 … 𝑑𝜏𝑘 = ∫ 𝑓(𝜏1 ) … ∫ 𝑓(𝜏𝑘 ) 𝑑𝜏𝑘 … 𝑑𝜏1 = ∏ 𝐹𝑚 (𝑡)𝜏𝑗 <𝑡,𝑗=1,…,𝑘

−∞

−∞

(8)

𝑚∈𝑆

Intuitively such probability has to increase in time and in fact
Proposition 2: Ξ(𝑡, 𝑆) is a non-decreasing function of 𝑡
Proof: The proposition follows immediately by the fact that 𝐹𝑚 (𝑡) are non-decreasing functions of 𝑡.
Let E and L denote two disjoint sets of buses not including n. Using (5) and (8) the joint probability that all
buses in 𝐸 depart before 𝑡, and 𝑛 departs after 𝑡 and before all buses in 𝐿 isΞ(𝑡, 𝐸) ∙ Φ𝑛 (𝑡, 𝐿)
Using this result, (3) becomes1

In the following, we stick to the usual convention of considering an empty summation, i.e. a summation with no addend, equal to 0 and an
empty product, i.e. a product with no factor, equal to 1.

(9)Π𝑛 (𝑡) =Ξ(𝑡, 𝑃𝑛 (𝑡)) ∙ Φ𝑛 (𝑡, 𝑃̅𝑛 (𝑡))

∑

(10)

𝑃𝑛(𝑡)∈𝒫(𝐴−𝑛 (𝑡))The pdf of the event “a passenger arriving at 𝑡 departs at 𝜏 with bus 𝑛” is
π𝑛 (𝑡, 𝜏) =Ξ(𝑡, 𝑃𝑛 (𝑡)) ∙ φ𝑛 (𝑡, 𝜏, 𝑃̅𝑛 (𝑡))

∑

(11)

𝑃𝑛(𝑡)∈𝒫(𝐴−𝑛 (𝑡))

Proposition 3: The probability of boarding a bus 𝑛 is not influenced by the presence of services which candepart only after bus 𝑛 has surely left the stop.
Proof: It is sufficient to prove that π𝑛 (𝑡, 𝜏) is not influenced by a bus 𝑚 that surely departs after 𝑛. In fact,
Π𝑛 (𝑡) is the integral of π𝑛 (𝑡, 𝜏) over 𝜏.Therefore if π𝑛 (𝑡, 𝜏) is not affected by 𝑚, neither is Π𝑛 (𝑡). We show now
that the presence of 𝑚 affects neither Ξ(𝑡, 𝑃𝑛 (𝑡)) nor φ𝑛 (𝑡, 𝑃̅𝑛 (𝑡)).
Case 1: 𝑡 > max 𝑇𝑛 . Since 𝑡 > max 𝑇𝑛 ⇔ 𝑓𝑛 (𝜏𝑛 |𝜏𝑛 ≥ 𝑡) = 0 , it turns out that+∞+∞𝜑𝑛 (𝑡, S) = ∫ 𝑓𝑛 (𝜏𝑛 ) ∏(1 − 𝐹𝑘 (𝜏𝑛 )) 𝑑𝜏𝑛 = ∫ 0 ∙ ∏(1 − 𝐹𝑘 (𝜏𝑛 )) 𝑑𝜏𝑛 = 0𝑘∈𝑆𝑡

Independent from 𝐹𝑚 (𝜏𝑛 ).

𝑘∈𝑆𝑡Case 2: 𝑡 ≤ max 𝑇𝑛 . The hypothesis that a bus 𝑚 can depart only after bus 𝑛 has surely left the stop implies
that min 𝑇𝑚 < max 𝑇𝑛 . It follows that 𝐹𝑚 (𝜏𝑚 |𝜏𝑚 ≤ min 𝑇𝑚 < max 𝑇𝑛 ) = 0 . Therefore+∞

𝜑𝑛 (𝑡, S) = ∫ 𝑓𝑛 (𝜏𝑛 ) ∏(1 − 𝐹𝑚 (𝜏𝑛 )) 𝑑𝜏𝑛𝑘∈𝑆max Τ𝑛𝑡= ∫max Τ𝑛𝑓𝑛 (𝜏𝑛 ) ∏(1 − 𝐹𝑘 (𝜏𝑛 )) 𝑑𝜏𝑛 = ∫
𝑘∈𝑆

𝑡

𝑡

𝑓𝑛 (𝜏𝑛 ) ∏ (1 − 𝐹𝑘 (𝜏𝑛 )) 𝑑𝜏𝑛𝑘∈𝑆∖𝑚

The previous holds for any S so also for any 𝑃̅𝑛 (𝑡). Since 𝑚 cannot be part of 𝑃𝑛 (𝑡) for any 𝑡 ≤ max 𝑇𝑛 , thepresence of 𝑚 does not affect Ξ(𝑡, 𝑃𝑛 (𝑡)). Since 𝑚 does not influence 𝜑𝑛 (𝑡, 𝑃̅𝑛 (𝑡)) and Ξ(𝑡, 𝑃𝑛 (𝑡)), it does not
influence their product π𝑛 (𝑡, 𝜏). QED.

4.3. Expected waiting time
The waiting time of a passenger arriving at 𝑡 and departing at 𝜏 is 𝑤 = 𝜏 − 𝑡. Since boarding different buses
are mutually exclusive events, the probability that a passenger arriving at 𝑡 departs at 𝜏 is equal to the sum of theprobabilities that he departs with any of the available buses. Therefore the expected waiting time for a passenger
arriving at 𝑡 is+∞𝐸(𝑤(𝑡)) = ∫ (𝜏 − 𝑡) ∑ π𝑛 (𝑡, 𝜏) 𝑑𝜏(12)n∈𝐵(𝑡)𝑡

and the expected waiting time corresponding to bus 𝑛+∞𝐸(𝑤𝑛 (𝑡)) = ∫ (𝜏 − 𝑡) π𝑛 (𝑡, 𝜏)𝑑𝜏

(13)

𝑡

Proposition 4: The overall expected waiting time is equal to the sum of the expected waiting times of allavailable services
Proof:+∞+∞

𝐸(𝑤(𝑡)) = ∫ (𝜏 − 𝑡) ∑ π𝑛 (𝑡, 𝜏) 𝑑𝜏 = ∑ ∫ (𝜏 − 𝑡) π𝑛 (𝑡, 𝜏) 𝑑𝜏 = ∑ 𝐸(𝑤𝑛 (𝑡))𝑡n∈𝐵(𝑡)

n∈𝐵(𝑡) 𝑡n∈𝐵(𝑡)

QED.
4.4. Passenger arrival distribution
The expected waiting time obtained from (12) cannot be used to specify the utility function of passengers
deciding their arrival time at a stop as we need to consider the possibility of missing all buses after a certain 𝑡̅, i.e.
if ∃𝑡̅|𝐵(𝑡) = ∅∀𝑡 ≥ 𝑡 or, more commonly, the possibility that the passenger aims to board before a time 𝑡̅ andattaches higher disutilities to later departures. To solve this shortcoming we consider
𝑊(𝑡) = 𝐸(𝑤(𝑡)) + (1 − ∑ Π𝑛 (𝑡)) c(14)𝑛∈𝐵(𝑡)

where 𝑐 is a constant representing the aversion to the possibility of missing the last attractive boarding
opportunity. We further assume that the utility function has the form𝑈(𝑡) = 𝛼(𝑊(𝑡))

𝛽

(15)

and that the arrival time choices can be described by the continuous logit model𝑞(𝑡) = −𝑒 𝑈(𝑡)
𝑡2∫𝑡 𝑒 𝑈(𝑡) 𝑑𝑡

(16)

1

Where [𝑡1 ; 𝑡2 ] is an interval spanning all the possible departure times of the buses considered by passengers.
We note that different specifications of the anticipated risk-averse expected value are possible. For instance, an
optimistic passenger may choose the arrival time with
𝑊𝑜𝑝𝑡 (𝑡) = min{𝐸𝑛 (𝑤(𝑡))|1 − 𝐹𝑛 (𝑡) > 0}(17)i.e. considering the bus with the minimum waiting time among the buses that can depart after 𝑡. Instead, the utility
function of a pessimist could be based on the maximum waiting time among those buses that can be surely boarded,i.e.
𝑊𝑝𝑒𝑠 (𝑡) = max{𝐸𝑛 (𝑤(𝑡))|1 − 𝐹𝑛 (𝑡) > 0}
Our reliability-based passenger arrival model is illustrated in Fig. 2 for the case of three buses with the triangular
pdf as in (a). We assume that the modal value is the scheduled departure time of each bus (in general, the modal
value may differ from the scheduled departure time, for instance when passengers think that the buses are
systematically delayed). Note that perceived departure times earlier than the scheduled ones, although normallyunlikely, cannot be completely ruled out, for instance because bus drivers may not comply with the schedule, or
because of discrepancies between the time of the passenger and the time of the bus system.

(18)

(a)

(b)

(c)

(d)

Fig. 2. Illustration of the reliability-based passenger arrival model

In (a) it can be seen that bus 1 is scheduled to depart before bus 2, but it can be overtaken by bus 2. Instead bus
3 departs surely after buses 1 and 2. Fig. 2(b) shows that, because of the possibility that bus 1 is later than bus 2,
the perceived probability of boarding bus 1 (and analogously that of bus 2) is always smaller than 1. The
probability of boarding 1 and 2 are constant while the pdf of bus 1 is nil, i.e. until the first possible departure ofbus 1. After that, the chances of missing bus 1 increase, so the probability of boarding it decreases and instead theprobability of boarding bus 2 increases. Analogously, the perceived probability of boarding bus 3 is nil until the
moment when the passenger is sure to be able to board at least bus 2, then it starts increasing. Different from buses
1 and 2, the probability of boarding bus 3 reaches 1, because between times 7 and 8 it has not departed for sureand it cannot be surpassed by any other bus. The overall probability of boarding at least one bus is equal 1 until
the possibility of missing bus 3 arises. Fig. 2(c) shows the expected waiting times of each bus and thecorresponding overall expected waiting time. After time 7, the probability of catching a bus coincides with theprobability of boarding bus 3 and therefore the overall waiting time is equal to the expected waiting time of bus
3. Such expected waiting time decreases when the probability of boarding bus 3 decreases. Hence, without thecorrection suggested in (14), the perceived expected waiting time of a passenger arriving at 𝑡 > 8 would decrease
with the paradoxical consequence that the probability of arrival would increase when the chances of missing thelast service increase. In Fig. 2(d) it is evident that the arrivals when service reliability and risk aversion are
considered are far from being uniform as commonly assumed for bus load models.
5. Bus propagation model
In the following, we develop a bus bunching model consistent with the non-uniform arrival process. Consider
a common scenario of a route served by buses of different lines which serve the same destinations. To avoid theissue of having to consider different passenger destinations and hence different attractive sets, we assume that thebuses might originate from different terminals, then merge in the city centre and all are destined for the same
terminal. As in Newell and Potts (1964), the undisturbed trajectories of the buses are described by the following
bus propagation equations

{

𝑎
𝑑
𝜏𝑖𝑛
= 𝜏𝑖−1𝑛+ 𝑣𝑖−1𝑛

(19)

𝑑
𝑎
𝜏𝑖𝑛
= 𝜏𝑖𝑛
+ 𝐷𝑖𝑛 + 𝛿𝑖𝑛𝑑Assuming that 𝑣𝑖𝑛 and the scheduled departure times of the buses from the first stop (𝜏0𝑛) are known, (19) can𝑑𝑎be solved recursively if a formulation of 𝐷𝑖𝑛 is available as a function of 𝜏𝑗𝑙 and 𝜏𝑘𝑙 , with 𝑗 < 𝑖; 𝑘 ≤ 𝑖; 𝑙 ≤ 𝑛.
Remember that 𝛿𝑖𝑛 is an exogenously determined delay, and so it is considered known in (19).

5.1. Newell & Potts model
In their bus propagation model, Newell and Potts assume that passenger arrival and boarding rate are timeindependent, and that the boarding process follows the law𝑑𝑑𝐷𝑖𝑛 = 𝜌(𝜏𝑖𝑛− 𝜏𝑖𝑛−1)(20)

where 𝜌 is the (time-independent) ratio between the passenger arrival and boarding rates. That is, they consider
boarding as a stationary unsaturated deterministic queuing process with constant degree of saturation. They also
assume that the bus departs as soon as the queue of waiting passengers vanishes, independent of any schedule or
holding policy.
5.2. Our model
We retain the hypotheses that boarding can be represented as a stationary unsaturated deterministic queue, andthat buses depart immediately after boarding all waiting passengers. In line with Section 4 we relax though theassumption that the degree of saturation is constant. Eq. (20) can be regarded as a particular case of the generalrelationship (21) regulating an unsaturated deterministic boarding process with time-dependent arrival andboarding rates:
𝑑
𝜏𝑖𝑛

𝑑
𝜏𝑖𝑛

𝑑
𝜏𝑖𝑛

∫ [𝑞𝑖 (𝑡) − 𝑏(𝑡)]𝑑𝑡 = ∫ 𝑞𝑖 (𝑡)𝑑𝑡 − ∫ 𝑏𝑖 (𝑡)𝑑𝑡 = 0
𝑑
𝜏𝑖𝑛−1

𝑑
𝜏𝑖𝑛−1

(21)

𝑎
𝜏𝑖𝑛

𝑑
𝑎
Note that evidently 𝑏𝑖 (𝑡) = 0 between 𝜏𝑖𝑛−1and 𝜏𝑖𝑛. We consider that passengers do not arrive uniformly but
they follow the distribution derived from (16). We further assume that the boarding rate is constant which appears
to be a reasonable assumption since the rate is mainly linked to vehicle characteristics and the ticketing system.
Under our hypotheses, (21) simplifies to
𝑎
𝜏𝑖𝑛
+𝐷𝑖𝑛

∫

𝑞𝑖 (𝑡)𝑑𝑡 − 𝑏𝐷𝑖𝑛 = 0

𝑑
𝜏𝑖𝑛−1

In our formulation, 𝑞𝑖 (𝑡) does not depend on the actual bus arrival and departure times but only on passenger
perception of the pdf of the bus departure times. Therefore it is exogenous to the bus propagation model, i.e. it is𝑑known in (22). If 𝜏𝑛𝑎 and 𝜏𝑛−1
are known, (22) can be easily solved with the iterative approach described in Table𝑎
2. Note that the algorithm works also if the boarding rate is not constant but it can be known once 𝜏𝑗𝑙𝑑 and 𝜏𝑘𝑙, with𝑑𝑎𝑗 < 𝑖; 𝑘 ≤ 𝑖; 𝑙 ≤ 𝑛 are known. Since 𝐷𝑖𝑛 in (22) is a function of 𝜏𝑖𝑛−1and 𝜏𝑖𝑛, (19) and (22) can be easily used to𝑑derive the bus trajectories from 𝜏0𝑛.

(22)

Table 2 Algorithm to calculate 𝐷𝑖𝑚

𝛿𝑡 ← small time step𝑚←1

Initialisation

𝑎
𝜏𝑖𝑛
+𝑛 𝛿𝑡

Main
𝑄←

∫

𝑞𝑖 (𝑡)𝑑𝑡

Obtain number of passengers who arrived𝑑𝑎
between 𝜏𝑖𝑛−1and 𝜏𝑖𝑛
+ 𝑚𝛿𝑡

𝑑
𝜏𝑖𝑛−1
𝑎
𝜏𝑖𝑛
+𝑚 𝛿𝑡

𝐵←

∫

𝑏𝑖 (𝑡)𝑑𝑡 = 𝑏𝑚𝛿𝑡

𝑑
𝜏𝑖𝑛−1

Obtain number of passengers who boarded𝑑𝑎between 𝜏𝑖𝑛−1and 𝜏𝑖𝑛+ 𝑚𝛿𝑡

If 𝑄 ≤ 𝐵 then 𝐷𝑖𝑛 ← 𝑚𝛿𝑡 otherwise 𝑚 ← 𝑚 +1 and repeat Main

6. Bus bunching with passenger reliability-based behaviour
To illustrate the effects of reliability-based passenger arrivals on bunching, we compare the solutions provided
by our model of bus propagation to those given by the Newell and Potts model in an analogous situation. The
exemplification is carried out by varying some parameters of the basic scenario described in Table 3.
Table 3 Case study, base parameter settings

Bus service
 6 buses, 10 stops (including the depot)𝑑𝑑 Scheduled headway: 𝜏0𝑛+1− 𝜏0𝑛= 10min, 𝑛 = 1, … ,5
 Travel time: 𝑣𝑖𝑛 = 3min, 𝑖 = 0, … ,8, 𝑛 = 1, … ,6
 𝛿𝑖𝑛 = 0min, 𝑖 = 0, … ,9, 𝑛 = 1, … ,6
Reliability-based arrival model Perceived pdf of departure time at all stops: triangular function with min Τ𝑛 = 𝜏̅𝑛 − 1min, max Τ𝑛 =𝜏̅𝑛 + 2min where 𝜏̅𝑛 is the scheduled departure time. The scheduled departure time coincides with thedeparture time predicted by the Newell and Potts model when no delay occurs.
 To avoid the influence of boundary conditions and to have periodic demand, the considered six buses
do not include the first and the last service along the route. The passenger arrival probability functionin the basic scenario is shown in Fig. 3.
 𝛼 = −1, 𝛽 = 0.55 as in Bowman and Turnquist (1981)
 At each stopo Passengers start arriving one scheduled headway (i.e. 10min) before the scheduled
departure of the first bus from the stop (so that there are passengers waiting for the first bus
at the stop);o All passengers have arrived before the schedule departure of the last bus;o Therefore passenger arrival simulation time: 𝒯 = number of buses ∗scheduled headway = 60min Total demand: 𝑄 = 100pax when not specified differently

Fig. 3. Arrival patterns under basic demand conditions

In the following, to allow for easy comparison of our model with Newell and Potts model, in the latter weassume that the 𝑄 passengers arrive uniformly during the simulation period 𝒯 and so we set
𝑄( ⁄𝒯 )⁄
𝑞 𝑁𝑃⁄
𝜌=
=
𝑏
𝑏
In the figures showing bus trajectories, red lines represent the results of the Newell and Potts model (with“uniform” arrival of passengers, blue lines the results of our model (the arrival distribution is indicated in thelegend of each figure).
6.1. Effect of boarding rate
In Fig. 4 we present the results provided by the two bunching models with different boarding rate values,
namely b=11.11 2 and b=2.78 pax/min.
Since no random delay is considered, the Newell and Potts model predicts no bunching. When a variable
passenger arrival rate is considered, a regular service is provided for high values of boarding rates (Fig. 4(a)). But
when the boarding rate is low, the service can be severely disrupted also without exogenous delays (Fig. 4(b)).
Note that our bus propagation model, as well as the Newell and Potts one, is not adjusted to deal with cases inwhich overtaking occurs. In case of low saturation rates 𝜌 (as mostly expected in reality) this does not lead to large
errors as the overtaking and overtaken buses will remain bunched, however, for large 𝜌 this might lead to somemore significant errors in the trajectories of, both, overtaken and overtaking buses.
To avoid the issue and exclude overtaking, one could introduce a condition where buses do not depart prior to
the departure of the previous bus in the main loop in Table 2. This adjustment can be easily implemented. To
explicitly deal with overtaking, further adjustments are needed in (21) and the main loop in Table 2 In the present
paper, we decided to avoid both adjustments to allow for comparability with the Newell and Potts model and toremain the focus of this paper.

(a)

(b)

Fig. 4. Effect of boarding rate; no exogenous delay

The trajectories in Fig. 4(b) can be explained by looking at the cumulative passenger arrival and boarding
functions. Fig. 5 shows cumulative arrivals and boarding for two different values of boarding rates. The two
scenarios share the same scheduled headway (10 min). Since we assume the schedule equal to that predicted by
the N&P model with no delay, in Fig. 5(b) the first scheduled arrival occurs at minute 4 and the following ones
every 10 minutes after; each bus leaves the stop after 6 minutes dwell time (corresponding to ρ=0.6). In (b), with
ρ=0.15, the first scheduled arrival is at minute 8.5 and the following ones every 10 minutes after; the dwell timeof each bus is 1.5 minutes.
Both the Newell and Potts model and our model assume that a bus leaves the stop as soon as all the waiting
passengers have boarded, i.e. when the cumulative boarding function intersects the cumulative arrival function.
As illustrated in both Fig. 5(a) and 5(b), in the case of uniform passenger arrival the first departure happens at time
2 We developed our case study by setting the value of ρ, and then deriving the boarding rate corresponding to the assumed total demand. This

explains the “unusual” value of b. Although it has not been possible to retrieve literature on the boarding rates, 11.11 pax/min seems realistic
for passengers using travel cards.

10 and then every 10 minutes (when the red line crosses the blue one. In Fig. 5(a) the cyan and red lines overlap),
i.e. the service operates according to schedule. Conversely, if passengers arrive considering the schedule as well
as system unreliability (basic scenario) and the boarding rate is low (Fig. 5(b)), at each instant the cumulative
number of passenger arrivals is smaller than in the case of uniform arrivals. Therefore, when the first bus arrives
at time 4, it has to board fewer passengers and it leaves earlier, in the figure at the time in which the cyan line
crosses the green one. This earlier departure triggers bunching.

(a)

(b)

Fig. 5. Queue processes under different scenarios

(a)

(b)

Fig. 6. Effect of small positive (a) and negative (b) deviations from 𝑏crit ; no exogenous delay

Note that the case in Fig. 5(b) is a case of serious bunching. In fact, assuming that all waiting passengers board
the bus arriving at time 14, this bus would leave at 26, i.e. after the third bus is due at the stop. Of course, thepossibility for passengers to choose between two buses can alter the queuing process shown in Fig. 5(b) but thisclearly exceeds the scope of the analysis here. Fig. 5(a) shows also that bunching is not triggered by non-uniform
arrival if 𝑏 is high enough, In other words, there is a critical value 𝑏crit such that non-uniform passenger arrivals
generate instability for 𝑏 ≤ 𝑏crit whereas they do not compromise stability for 𝑏 > 𝑏crit . Even a small negative
deviation from 𝑏crit can induce large perturbations: For example in Fig. 6 the difference between 𝑏 in (a) and (b)is only 0.05pax/min. The existence of the critical value is clearly linked to the shape of the passenger arrival
function: suffice it to notice that if the green line was above the blue one in Fig. 5(b), the cyan line would notintersect it before time 10 and bunching would not arise.

6.2. Effect of total demand
In the Newell and Potts model the boarding rate does not appear explicitly but is linked to the arrival rate via
the degree of saturation 𝜌. It follows that an increase of the total demand 𝑄 results in a lower saturation rate,
assuming that the boarding rate stays constant. Therefore higher total demand does not lead to bunching in the
Newell and Potts model unless the saturation rate is assumed to be bus specific. In our model, the boarding andarrival rates are represented explicitly and enter the model separately. The consequence is that, as illustrated inFig. 7, bus bunching can arise in case the demand is different from the one planned for by the operator, even if theboarding rate remains constant (b=11.11pax/min) and there is no exogenous delays. Note that bunching arises forvalues of total demand both lower (Q=75pax, Fig. 7(a)) and higher (Q=125pax, Fig. 7(b)) than the “design” onei.e. that used to derive the scheduled departure times. In the former case, bunching is triggered by an early departureof the first bus, in the latter by a delay of the same bus. Again the phenomenon can be explained by considering
the cumulative arrival and boarding functions. Increasing the total demand while the boarding rate stays constantmeans that, in Fig. 5(a), the cyan line does not change whereas at each instant the green line is below (if thedemand is smaller) or above (if the demand is greater) that in the figure. Therefore they can intersect before (incase of smaller demand) or after (in the opposite case) the first scheduled departure time.

(a)

(b)

Fig. 7. Effect of total demand; no exogenous delay

6.3. Effect of exogenous delay
In the Newell and Potts formulation, bunching arises only when an exogenous delay occurs at any stop. In thefollowing we study what happens when the second bus is delayed at stop 1, with b=11.11pax/min and Q=100pax(Fig. 8). Fig. 8(c) to (f) show system performance measures for two values of delay. The bars in Fig. 8(c) and 8(d)represent the mean of the absolute deviations of the headways from the scheduled ones calculated over all buses
at each stop. The bars in Fig. 8(e) and 8(f) instead show the mean of the absolute headway deviations from theschedule calculated over all stops for pairs of subsequent buses. Crosses hatch the bars when not all buses (stops)
are used in the calculations because the models predict overtaking and so the results from the moment of theovertaking on are not fully comparable. For the sake of comparison, the same subsets of buses (stops) are used to
calculate the means for both passenger arrival processes. It is evident that larger initial delays give rise to larger
deviations, and that bunching becomes a more significant problem at stops further from the depot. The
performance of the system is more or less similar under the two arrival processes at the initial stops. In fact, theschedule deviations are slightly lower if passengers consider service reliability. However, the headway deviations
in case of reliability-based passenger arrivals are greater than those occurring in case of uniform arrival for stops
further downstream. The differences between the two processes diminish for larger initial delay. In Fig. 8(e) and8(f) it can be observed that irregularities at the beginning increase but then tend to reduce. In both cases, the system
performs better under reliability-based passenger arrivals for the first runs. The better performance of non-uniform
passenger arrivals in the case of the first delayed bus can be easily explained by considering the cumulative numberof passengers arriving at bus stops shown in Fig. 5. At each instant the cumulative passenger number in the basic
scenario is smaller than that under the uniform arrival assumption, except at time 0 and time 100. Therefore, whenbus 2 experiences a delay at stop 1, the additional passengers waiting to board at stop 2 will be fewer in the caseof reliability-based arrivals than in the case of uniform arrivals, and so bus 2 is less delayed along its route. The

difference of performance for later runs between the two arrival processes is attenuated for the larger delay in thiscase as well. For later runs, the headways tend to be less different from the scheduled ones if passengers arrive
uniformly. Note that deviation peaks for intermediate runs, in particular, in the case of 𝛿12 = 1.5, the maximum
headway deviation is between the second and the third bus, when the performance with non-uniform arrivals is
still better than with uniform ones. In the latter scenario, the deviation between bus 2 and 3 is so large that
overtaking occurs, i.e. the system experiences a very serious distortion at stop 8. In contrast, there is no overtaking
if the arrivals are non-uniform. This shows that despite the system underperforming on average with non-uniform
arrivals, consideration of system reliability in passenger arrival times avoids the occurrence of extreme bunchingconditions in the presence of large delays.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 8. Effect of exogenous delay6.4. Effect of passenger perceptions and preferences
The passenger arrival distribution depends on the perceived pdf of the bus departure times, linked to the system
reliability, and to the passenger risk-aversion, measured by parameters 𝛼 and 𝛽.

Fig. 9. Arrival processes for different values of system reliability and risk-aversion

In the following we analyse the effects of three different arrival profiles (the cumulative arrival distributions
under the different scenarios are represented in Fig. 9 for one scheduled headway), namely







Unreliable system: The perceived unreliability is higher than in the basic scenario, i.e.
triangular pdf functions are considered by the passengers with min Τ𝑛 = 𝜏̅𝑛 − 2, max Τ𝑛 =𝜏̅𝑛 + 4. Therefore passengers anticipate that bus arrivals can occur along a wider interval,
therefore their arrivals tend to be more spread as well and the distribution is more similar tothat predicted under the uniform arrival assumption.
Wait averse: The aversion to wait ( 𝛼 = 1.55) is higher than in the basic scenario. High
expected wait times have higher disutilities under this scenario than under the basic one. The
opposite holds for low expected wait times. Therefore passengers tend to avoid both very
early arrivals – because they might give rise to long waits for the intended buses – and verylate arrivals – because the might wait long for the next bus, in case the intended one is missed.
The consequence is that passenger arrivals tend to be concentrated around a specific time.
Early departures: In this scenario we consider passengers who fear the possibility of very early
departures but not of late departures, so that we set min Τ𝑛 = 𝜏̅𝑛 − 8, max Τ𝑛 = 𝜏̅𝑛 . Becauseof high aversion to long wait, arrivals tend to be concentrated as in “wait averse”. Given the
pdf of the bus departure times, the arrival time is earlier than in “wait averse”. As a result,
there are times in which the cumulative arrivals are higher than under uniform arrivals. This
scenario is unlikely in reality, where normally buses cannot leave long before the scheduled
departure time. We add this scenario mainly in order to illustrate the effect of a partly concave
cumulative arrival function.

Fig. 10 shows the bus trajectories arising from the three profiles assuming the same exogenous delays of bus 2
at stop 1 (and same b and Q) considered in Fig. 8. Since the Newell and Potts results depend only on the value ofthe exogenous delay, the red trajectories are the same in (a), (c), (e) and in (b), (d), (f), and they coincide withthose in Fig. 8(a) and (b) respectively. Under the “unreliable system” scenario the behaviour of the system ((a)
and (b)) is more similar to that with uniform arrivals than that illustrated in Fig. 8, as it can be anticipated by thefact the arrival distributions tend to be more uniform.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 10. Effect of passenger perceptions and preferences

The worst performance of the system occurs under “wait averse” arrival patterns, i.e. when passenger arrivals
are concentrated near the scheduled departure time. In this case, even a small delay gives rise to large headway
deviation (Fig. 11(a)).

(a)

(b)

Fig. 11. System performance under different values of system reliability and risk-aversion

When real time information on bus arrival/departure is available, one can expect that most passengers using theinformation arrive at stops very close to the advised time. Given the result obtained for “wait averse”, the impactof this behaviour in the presence of unpredicted bus delays should be carefully examined. Under “early departure”
the system is only little disturbed, because in the first instants of the scheduled headway there is almost no
passenger arrival and therefore a delayed bus faces a boarding situation very similar to the case with no delay. In
this case, the system performs even better than with uniform passenger arrivals (Fig. 11 (b)).
7. Conclusions
In this paper we discussed the relationship between passenger bus stop arrival patterns and dwell times of buses
that might lead to bunching effects. We advanced the Bowman and Turnquist (1981) arrival model by developinga “reliability-based arrival pattern model” in which possibly overlapping perceived departure time probability
density functions for several buses are considered. Passengers hence consider the schedule as well as possible
early or late departures in order to minimize their expected waiting time. We then implemented these behaviouralassumptions into a bus propagation model similar to that of Newell and Potts (1964) but in which passenger arrivaland boarding rates enter the model separately. In our case study we compared our model to the assumption of
uniform arrival as commonly done in the literature on bus bunching.
Throughout the paper we assume that after the release of the buses from the depot the buses are uncontrolled.
That is, buses leave stops whenever there are no more waiting passengers. Holding points or headway equalizing
strategies are not considered. This allows us to illustrate how a mismatch between the operators’ perception ofservice demand and actual demand can lead to bunching effects even without exogenous delays. Remarkably
bunching occurs also for values of demand lower than the planned one. Bunching generated by unexpected demandcannot be derived with previous models, which assumed uniform demand. As all holding related control measures
introduce inefficiency for the operator into the service we therefore suggest that our model shows the importancefor operators to correctly assess the demand and resulting dwell time at stops.
Another main finding is that non-uniform arrival patterns can lead to more severe bunching effects over time.
This highlights the need for control measures. We believe therefore that the model developed here could be used
to test the effectiveness of various control strategies, i.e. that it is worth revisiting literature discussed in our review
section with consideration of the behavioural issues discussed in this paper. An alternative helpful implicationfrom this research for operators might be to aim to smoothen the arrival pattern of passengers if on-time arrivals
cannot be guaranteed. That is, if exogenous delays are feared and boarding rates are substantial it might be worth
for operators to consider making notes to schedules that slightly early or late departures are possible in order toencourage a more spread arrival pattern of passengers.
Our approach is based on the assumption that passengers are aware of a “static” service schedule information.
In the introduction we already mentioned the importance of real time information (RTI) to our topic as it willencourage even more peaked arrival time patterns also for services with short headways. Modelling the effect of
upstream delays on downstream arrival patterns we leave though as further work. Not only due to the modelingchallenges (consideration of RTI requires a dynamic modelling framework) but also since it is not clear in how
far RTI is (and should be) trusted by passengers from available literature. Possibly, a simple Bayesian approach
could be investigated to model behaviour once passengers obtain information on the next departure(s).

In this paper, we have considered an unsaturated boarding process, assuming that buses can always
accommodate all passengers waiting to board. This might not be true in very busy bus systems where capacity
constraints affect system operations. On the supply side, consideration of bus capacity is equivalent to constrain
the dwell time. In addition, the interaction between boarding and on-board passengers may reduce the boarding
speed at high levels of bus occupancy. Therefore, system performance forecast requires knowledge about origindestination matrices, so that the number of on-board passengers can be derived in each bus trip leg. On the demandside, one can expect that at least some categories of passengers will try to avoid demand peaks; hence, risk aversion
may change in time in the presence of time-dependent demand. Finally, the utility of a given arrival time should
include the level of comfort, which may vary considerably as a function of crowdedness.
We note further the effect of electronic ticketing. Through smart card technologies, boarding has become faster
which possibly reduces the bunching effect (see a review in Pelletier et al., 2011). At the same time though more
efforts are made by bus operators to cater also for the needs of population groups with special needs such as
wheelchair users. These require additional boarding times, leading to more variability in the boarding rates.
Therefore, one possible further work direction is to consider non-constant boarding rates b where boarding perpassenger in off-peak times might take longer when less passengers with travel cards and proportionally morepassengers with special needs travel (Khoo, 2013). More generally, one might consider different passenger classes
with different boarding rates as also considered in Bowman and Turnquist (1981) who distinguish a group of
randomly arriving passenger and a group of optimizing passengers.
There are various other directions in which this work could be extended, partially already mentioned withinprevious sections but we believe worth summarising here. Firstly, developing a bus propagation model that,different from Newell and Potts (1964), gives a realistic representation of the system operations after once
overtaking between buses has taken place. Secondly, more complex network structures with common lines should
be considered. In combination with the work presented here, this could lead to conclusions in how far network
structure and information to passengers could avoid the need for or change the characteristics of holding strategies.
Thirdly, better understanding of passenger behaviour is needed. We take the parameters 𝛼 and 𝛽 in our modelfrom Bowman and Turnquist (1981) and showed that these values affect the service quality which might justify
further calibration efforts. Furthermore, currently we assume that passengers consider all the possible departuresfrom the stop, but some passenger groups might be interested only in a subset of the services as indicated in
Schmöcker et al. (2013). Related to this, the value of the cost of missing the last desired service has been simply
set to a large value in our model. Furthermore, also the shape of the perceived departure probabilities needs
calibration. Finally, obviously the bus choice model could be made more complex by considering crowdedness on
buses and expectations on residual travel times for different lines (similar to Gentile et al., 2005).
Acknowledgements
This paper is partly supported by the UK Rail Safety and Standards Board Project RSSB-T1071, DITTO. In
addition, we would like to thank the editor and reviewers for their critical comments and constructive suggestions.
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