Testing schedule performance and reliability for train stations
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Journal of the Operational Research Society (2000) 51, 666±682 #2000 Operational Research Society Ltd. All rights reserved. 0160-5682/00 $15.00
www.stockton-press.co.uk/jors
Testing schedule performance and reliability for train
stations
M Carey* and S Carville
University of Ulster, Northern Ireland
On busy congested rail networks, randomdelays of trains are prevalent, and these delays have knock-on effects which
result in a signi®cant or substantial proportion of scheduled services being delayed or rescheduled. Here we develop and
experiment with a simulation model to predict the probability distributions of these knock-on delays at stations, when
faced with typical patterns of on-the-day exogenous delays. These methods can be used to test and compare the reliability
of proposed schedules, or schedule changes, before adopting them. They can also be used to explore how schedule
reliability may be affected by proposed changes in operating policies, for example, changes in minimum headways or
dwell times, or changes in the infrastructure such as, layout of lines, platforms or signals. This model generates a
reliability analysis for each train type, line and platform. We can also use the model to explore some policy issues, and to
show how punctuality and reliability are affected by changes in the distributions of exogenous delays.
Keywords: rail transport; timetabling; simulation; reliability
Introduction between 5% and 20% of trains arrive or depart late at
typical busy stations.
Busy complex railway stations having dozens of platforms
In view of this, the present paper is concerned with
and subplatforms with more than several hundred trains per
developing methods for illustrating and quantifying the
day arriving and departing are common in Europe and Asia.
behavior of train station schedules when faced with typical
Trains of different types and speeds arrive and depart on
patterns of on-the-day exogenous delays. In considering the
multiple con¯icting lines and are subject to restrictions or
reliability of a schedule we can take the exogenous delays as
preferences concerning which lines and platforms they can
given, so that a reliable schedule is then one in which
use. They also have various dwell time and headway
exogenous delays cause the least knock-on delays. We there-
requirements, typically have desired or preferred arrival
fore introduce typical patterns of exogenous delays and use a
and departure times, and have various costs or penalties for
simulation approach to obtain the distributions of knock-on
deviating from these times. To ensure that all of these
delays. We explore how delays, platform allocations and
constraints are met, detailed schedules are usually
reliability are affected by increasing the average size of
constructed months in advance, and the timetable is usually
exogenous delays, or increasing the number of trains affected
published.
by such exogenous delays, or by some scheduling rules.
To generate good feasible schedules for such busy
Previous work on train scheduling has been mainly
stations, various methods have been developed.1 However,
developing deterministic methods, methods and algorithms
these are deterministic methodsÐand do not indicate how
for constructing schedules.1±9 But deterministic scheduling
the schedule will perform when faced with the delays
is not our concern here. Previous stochastic simulation
which typically occur. When the schedule is implemented,
models of train movements trains have been mainly
on-the-day (deviations from the schedule) are common, due
concerned with simulating rail freight movementsÐparti-
to passengers boarding and alighting, operating delays,
cularly through marshalling yards, or simulating trains
failures of equipment or rolling stock, weather, accidents,
meeting and passing on single-track lines, and generally
etc. These delays in-turn cause further knock-on delays, for
assume the trains are not timetabled. They are not
example a train arriving or departing late may block and
concerned with scheduled traf®c at busy complex stations,
delay the arrival or departure of other trains. It is important
which is the concern here Halloway and Harker10 describe
to keep delays down to a low level, otherwise knock-on
an interesting simulation for scheduled traf®c but deal with
delays can quickly escalate. In Britain, for example,
trains meeting and passing on tracks without stations. Chen
and Harker11 estimate delays for scheduled trains, but
*Correspondence: Prof M Carey, Faculty of Business and Management,
University of Ulster, BT37 0QB, Northern Ireland.
again for meeting and passing on lines rather than at
E-mail: m.carey@ulst.ac.uk stations. One reason for this is that in North AmericaM Carey and S CarvilleÐTesting schedule performance and reliability for train stations 667 busy multi-platform rail stations are almost nonexistent, the full complexity of the rail industry but are simulating a whereas in Europe and Asia they are common. component of it, namely a busy station. The general train There are also some deterministic simulation models of planning problem involves a much wider range of con¯ict- train movements, including movements through junctions ing objectives and constraints. It includes, matching train and stations, but these do not consider the effects of the services to travel demands, minimising journey times, random delays which are prevalent in train services. Such avoiding trains con¯icting not just at one station but at deterministic models are useful for feasibility testing of all stations, junctions and track over which they pass, schedules, which is not the purpose of this paper. We planning for different speed and stopping patterns, and assume throughout that the schedule for which we wish producing schedules that make ef®cient use of rolling to estimate reliability is already a feasible schedule. By a stock and train crews. We do not deal with these issues feasible schedule we mean that if there are no unscheduled in this paper. Nor do we discuss the sophisticated systems delays of any kind to any of the trains in the schedule, then and decision processes that are used in signalling and there are no con¯icts between the times (arrival or departure control nor the role of train planners, signalmen and train times, dwell times, platform occupation times, etc.) of any of controllers who make very complex decisions based on the trains in the schedule, so that all trains in the schedule years of experience, and without which the system could can run exactly according to the schedule. Furthermore, we not operate. assume that a feasible schedule satis®es all minimum head- The research in this paper involved cooperation with ways which are required between trains arriving or depart- many people in the rail industry. It was carried out over ing, connecting, dwelling at platforms, etc. several years, dating back to a few years before the rail Train operators or planners often use words other than industry in Britain was privatised in the mid-1990s. This feasible to refer to a feasible schedule. In British rail involved meetings and discussions with train planners, and operations such a schedule is referred to as a `proven' operators and managers at all levels in British Rail and schedule, and the process of checking the schedule for then, after privatisation, in Railtrack and Train Operating feasibility is called proving the schedule. Deterministic Companies. Because of this time-span, and personnel simulation programmes are sometimes used for detailed changes in the industry, our contacts changed over time proving of a schedule, that is, for detailed checking for and some have moved within the industry and some moved feasibility, and this is sometimes referred to as testing the out. From train planners we collected detailed data on train reliability of the schedule. However, such schedule proving operations, preferences, station layout, etc, (see below and packages are deterministicÐthey do not consider exogen- Carey and Carville1 and Carey12) for a number of stations, ous unscheduled delays or disturbances. In contrast, in this in particular Leeds, Manchester and York. The experiments paper we are concerned with testing the reliability of reported here are based on a Leeds station since we had the already proven feasible schedules, by considering how most complete data set for that station. Much of this data reliable they are in the presence of typical patterns of was not available in published or printed form. To verify delays and disturbances. the train scheduling rules that we used, we took a draft We use the phrase `exogenous delays' or `initial delays' annual timetable produced by train planners at British Rail to refer to the dozens of causes of delay recorded daily on (BR), using their existing, partially manual, timetabling rail networksÐdelays due to breakdown or underperfor- methods, and showed that our scheduling=simulation mance of rolling stock, points failures, crew lateness, line programme could generate almost exactly the same time- maintenance, obstacles on lines, delays in passenger board- table. We checked any differences between our timetable ing or alighting, etc. These exogenous delays frequently and theirs and found that any differences were due to their cause knock-on delays to other trains. For example, if a having made exceptions to their own stated timetabling train is late leaving a station platform this may delay the rules. However, it should also be noted that the results of arrival of the next train scheduled to use the platform, some of the main experiments in this paper could not be which may in turn delay further trains. Or, if a train arrives veri®ed by comparing with observed results from BR or late its scheduled platform may be already occupied, so that Railtrack, since there are no comparable observed results. the train has to be sent to a different platform which may For example, we experimented (see below) with varying delay trains scheduled for that platform. In this paper we the distribution of exogenous delays, varying the percen- are concerned with knock-on delays caused at a single tage of trains that suffer exogenous delays, and with not station. In view of this, the delays incurred by trains prior to allowing platform changes on-the-day. These are experi- arriving at the station are treated as exogenous delays, even ments that the train companies would not wish to conduct though they may be due to knock-on delays incurred at with actual trains. However, we discussed the results of earlier stations. Therefore knock-on delays here means only these experiments with train planners, and they found the the knock-on delays caused at the current station. results were consistent with their expectations. In some Finally, we should note the limitations of this paper, and cases they had no ®rm prior expectations or the results were what is not covered in it. We are not attempting to model considered somewhat different than they may have
668 Journal of the Operational Research Society Vol. 51, No. 6
expected, for example, average knock-on delays were (ii) The minimum dwell time required for each train (which
larger or smaller than expected. However, even in these depends on the train type, and on whether it is a
cases the results were found equally interesting and infor- through train or terminating train).
mative. Finally, we should note that this was not a project (iii) The station layout. This includes: the numbers of in-
commissioned within the rail industry and the initiative for lines and out-lines, number of through platforms and
conducting it came from the authors. terminating platforms and sub-platforms, which lines
The simulation approaches illustrated in this paper may are connected to which platforms, which lines con¯ict
be used in various ways. They may be used by train (intersect or share a portion of track, or track circuit
planners, managers or operators in testing the reliability section).
of proposed station schedules before adopting them. They (iv) A draft timetable. That is, a list of trains indicating the
may also be used in exploring how schedule reliability train type, the lines on which the train is expected to
would be affected by proposed changes in operating poli- arrive and depart respectively, and an approximate or
cies, for example, changes in minimum headways or dwell desired arrival and departure time for each train.
times; or proposed changes in train services, for example, (v) Platform preferences. For example, for each train type,
numbers of times of trains; or proposed changes in the there may be a different cost or penalty depending on
infrastructure, for example, layout of lines, platforms, the platform or subplatform to which a trains is
signals, etc; or major incidents or accidents, etc. To assigned.
assess the effects of proposed changes, run the simulation (vi) Train delay costs. For example, for each train or train
programme with and without the proposed changes, to type, there may be a different cost or penalty for each
generate distributions of knock-on delays for the before minute by which the scheduled arrival or departure time
and after scenarios, and compare these. The distributions to deviates from the desired times.
compare, and the costs or bene®ts of changes in these
The actual data values that we used in the present paper are
distributions, will depend on the context and on the change
given in Carey and Carville1 and Carey.12 Using the above
being considered. Furthermore, the decision for or against
input data the ATTPS model generates the following
any change may be affected by the issues referred to in the
outputs:
previous paragraph. As examples we consider how relia-
bility is affected by permitting, or not permitting, platform (1) A scheduled arrival time, departure time and platform
assignments to be changed on-the-day, or by allowing late for each train. These scheduled times and platforms
trains to depart after less than their normal minimum dwell satisfy all of the data requirements and constraints in
times. (i)±(vi) above.
(2) An analysis of the solution given in (1), including
tables, graphs and distributions of:
Outline of the rescheduling or dispatching model (a) deviations of train arrival, dwell and departure times
To estimate how exogenous delays affect the train schedule from their desired times.
(the train timetable and platform allocation) at a station, we (b) changes of train from their most preferred plat-
used the ATTPS (automatic train timetabling and platform- forms.
ing system) programme,1 which is outlined below. Before (c) times for which each platform is occupied.
we can introduce exogenous delays we also need an initial
The ATTPS algorithms are set out in Reference 1. A
schedule to which to apply the exogenous delays. To
basic version of the algorithm operates as follows. Consider
generate this initial schedule we again used the ATTPS
the trains one at a time in a prespeci®ed order, for example,
programme.1 (As a further test of our results, we also re-ran
order of importance (business or revenue class) and=or
all of the experiments using as the initial schedule the BR
chronological order of desired arrival times. For each train
published schedule for the station. The results which we
t, consider assigning the train to each feasible platform in
obtained when starting from this initial schedule were
turn. Assigning the train to a trial platform involves
similar to those reported here based on starting from our
checking for all con¯icts which this may incur with already
ATTPS based schedule.)
schedule trains, and ®nding all adjustments which would
The simplest and most useful way to describe the ATTPS
be needed to resolve these con¯icts. For each trial platform
model is to state what takes as input data, what it produces
for train t, the algorithm computes the set of train delays,
as outputs and to outline how the outputs are obtained from
platform preference costs, etc., which would be incurred if
the input data. The input data for ATTPS consists of:
train t was sent to that trial platform. By comparing these
(i) The minimum headways required between trains delays, costs and penalties for each trial platform, the
(which depend on whether their paths intersect, the algorithm chooses a best platform for train t.
train types, whether each train is arriving or departing, Having assigned train t to a platform, the algorithm
and from which platforms). proceeds to the next train in the list, and so on. OtherM Carey and S CarvilleÐTesting schedule performance and reliability for train stations 669
features of the algorithm include: in which order to uniformly randomly distributed ranging from ÿ2 to 20
consider platforms, how to choose among subplatforms of minutes. (Since the range of delays varies widely in
the same main platform, how delay costs and platform costs practice, in Experiments 3 and 4 below we consider
are combined or traded-off, looking ahead before con®rm- various ranges, ÿ2 to 2, ÿ2 to 4, and so on up to
ing a platform choice, etc. ÿ2 to 60 minutes.)
In one version of the ATTPS model we required that all Dwell delays: half as many trains experience exogenous
trains go to their already scheduled platformsÐcall this the dwell time delays as arrival time delays, that is, dwell
®xed platform model. In another version we allow trains to delays for a random 10% of trains. Also, exogenous dwell
be sent to different platforms if this would reduce lateness delays are uniformly randomly distributed from 0 to 10
or other penaltiesÐcall this the ¯exible platform model. minutes. (In Experiments 3 and 4 below we consider
various ranges.)
Schedule performance, with uniform distributions Minimum dwell time: as well as a scheduled dwell time
of exogenous delays there may be an absolute minimum dwell time which can
be used if the train is running late. We introduce this in
There is very little published concerning distributions of
Experiment 2 but not in Experiment 1. If a train would
train arrival, dwell or departure delays at stations, or which
otherwise depart late we allow the scheduled dwell time
forms of distributions ®t best. Also, the parameters of these
to be reduced to not less than a minimum dwell time,
delay distributions may vary from station to station, vary
which we assume is fraction, for example, 0.8, 0.6, etc. of
over time, and depend on the train types. In view of this, for
the scheduled dwell time.
illustrative purposes, we here use two different types of
simple distributions, namely uniform distributions in the The simulation experiments and their results are set out in
present section and beta distributions in the following more detail below.
section. We choose parameters for these distributions so
that the mean delays, and the percentages delayed more Experiment 1 Distributions of knock-on delays and total
than 0, 5, 10 or 15 minutes, are consistent with delays delays.
patterns recorded at various train stations in Britain. In
To estimate the distributions of knock-on delays caused
view of this, we are concerned here with illustrating what
by given distributions of exogenous delays we proceeded as
can be done, rather than with ®nding de®nitive numerical
follows.
results for a particular station.
The experiments and simulations in this section and the (i) Choose the arrival time and dwell time delay distribu-
next are different, for example, in this section we simulate tions, and the percentages of trains to subject to these
how knock-on delays vary as the range of exogenous delays delays, as set out above.
increases or decreases, while holding ®xed the percentage (ii) Using the delay distributions stated above, (a) to the
of trains having exogenous delays. To vary the range of scheduled arrivals time of each train add a delay drawn
exogenous delays we simply vary the bounds of the at random from the distribution of exogenous arrival
uniform distribution. In contrast, in the next section we time delays; (b) similarly, to the scheduled dwell time
simulate how knock-on delays vary as the percentage of of each train add a delay drawn at random from the
trains having exogenous delays varies. In doing this we distribution of exogenous dwell time delays.
keep the distribution of exogenous delays for each train (iii) Simulate running this perturbed timetable for one day,
unchanged. and record all delays (exogenous and knock-on)Ðsee
In this section we introduce exogenous delays drawn Figure 1(a).
from a uniform distribution, by adding a sample of such (iv) Repeat steps (ii)±(iii) 1,000 times, to simulate 1,000
delays to the arrival and dwell times of a sample of trains. daysÐsee Figure 1(b).
In practice only a certain percentage of trains experience (v) Compute descriptive statistics for the distribution of
exogenous delays, and the distribution of these delays can delays obtained in (iv). Typical descriptive statistics
vary between stations and train types. We experimented used to measure transport reliability or punctuality are:
with various percentages and distributions. For the results (a) mean, median, mode, standard deviation, etc., of
below we assume mean delay values ranging from zero up delays.
to greater than or equal to those which are typical in (b) the percentage of trains having knock-on delays of
practice at, for example, various BR stations. In the less (or more) than 0, 5, 10, etc., minutes.
experiments reported below we used the following data
In Step (iii), instead of listing all delays we can save storage
and assumptions.
space by recording only the numbers of delays between 0
20% of trains, selected at random, experience exogenous and 5 mins, 5 and 10 mins, etc. Also, we may wish to record
delays to arrival times. (In the next Section we experiment a separate distribution of delays for each type of train, for
with different percentages.) These exogenous delays are example, express, inter-city, local, freight.670 Journal of the Operational Research Society Vol. 51, No. 6
Frequency distributions or pdfs of delays obtained in similar, but the differences between them show that the
Step (iv) above are shown in Figures 1 and 2. We also averages contained in Figure 1(b) conceal signi®cant daily
computed the statistics in (v) (a)±(b) for knock-on delays differences.
and for total delays (exogenous delays knock-on delays). The percentages of trains delayed more than 0, 5, 15,
etc., minutes can be read directly from the Figure 1. These
are percentages used by train operators, planners, and the
general public, as measures of reliability or performance
Reliability measures from cumulative delay distributions
for trains. Public transport operators are frequently required
Figures 1(a) and 1(b) shows the cumulative distributions to publish a selection these percentages, for example the
which our simulations yielded for ®ve measures of delay. percentages of trains arriving or departing more than 0, 5 or
Figure 1(a) is obtained by running the simulation for a 15 minutes late. In Figure 1 the percentages of trains
single day ((iii) above), and 1(b) obtained by running the delayed more than 0, 5, 15, etc., minutes are consistent
simulation for 1,000 days ((iv) above). The shapes and with those recorded in practice for BR stations of this size
relative positions of the lines in Figures 1(a) and 1(b) are and complexity.
Figure 1 (a) Cumulative distributions of delays from a single day simulation. (b) Cumulative distributions of delays from a 1,000-day
simulation.M Carey and S CarvilleÐTesting schedule performance and reliability for train stations 671
We assumed only 20% of trains experience exogenous in Figure 2, the former being the integral of the latter. If we
arrival delays (U ÿ2 to 20)) and 20% experience exogen- divide the vertical scale in Figure 2 by 100 the curves in the
ous delay (U (0±10)). To illustrate the effects of this in the ®gure become pdfs (probability distribution functions), and
simulation consider train delays of 5 minutes or more, for brevity we may refer to them as pdfs. One difference
as shown by ordinates in Figure 1(b) corresponding to 5 on between these Figures 1(b) and 2 is that in the former we
the horizontal axis. We ®nd 13.1% with exogenous arrival can see the percentage of trains which have zero delays,
delays (55 mins) and 7.2% with knock-on arrival delays which is a majority of the trains. This percentage is now
(55 mins), which yields 20.3% in total with arrival delays shown in Figure 2, but could be shown, for each of the
55 minutes. These 20.3% with arrival delays, plus 9.1% relative frequency curves, as a point mass at delay 0. The
with exogenous dwell delays (55 mins), caused a total of curves in Figure 2 represent only the tail of the pdf
25.3% to have departure delays 55 minutes. Recall that in corresponding to trains that are actually delayed, and the
the simulation we did not allow scheduled dwell times to be area under each curve is the % of trains actually delayed.
reduced, hence every train arriving late automatically Figure 2 shows the pdfs for ®ve types of delays. By
departed late. These knock-on departure delays could be de®nition (exogenous delays of arrivals) (knock-on
reduced by allowing dwell times of late trains to be delays of arrivals) (total delay of arrivals). The pdfs of
reduced, as discussed later. Also, the percentage of trains the sum of two random variables is the convolution of the
with knock-on delays would be signi®cantly reduced if we component pdfs, see for example Reference 13 for con-
counted only delays greater than say 10 minutes. volutions). There are no other such direct relationships
We have also generated graphs (not shown here) similar between the ®ve pdfs. Their relative shapes are consistent
to Figure 1 for all trains using a particular platform at the with a large busy station such as we simulated but can be
station, or all trains of a particular type, or all trains arriving quite different for other types of stations. For example, if
or departing on a particular line. This helps identify the the station had very little traf®c then knock-on delays of
problem platforms, trains or lines, and focuses train plan- arrivals could be negligible even if the exogenous delays
ners attention on these. For example, if a particular train of arrivals were large. Conversely, if the station was
type is less punctual than others perhaps the minimum very congested then knock-on delays of arrivals could be
headways for this train type should be increased. Also, very large even if the exogenous delays of arrivals were
there may be quite different punctuality targets for different negligible.
train types, for example, for intercity express trains, local Knock-on delays of arrivals or departures can be caused
stopping trains and freight trains. by any or all of the ®ve types of delay shown in Figure 1
(note that knock-on delays can be caused by other knock-on
delays). To see this, recall that exogenous delays of arrivals
Typical patterns of relative frequency distributions (or pdfs) or of dwell times can cause train time con¯icts which then
of delay delay (knock-on) the arrival and=or departure of that or
other trains. And these arrival or departure delays can cause
The data in the cumulative distributions in Figure 1(b) can further con¯icts which delay the arrival or departure of later
instead be presented as the relative frequency distributions trains. Hence the curves (in each Figure, 1 and 2) are all
Figure 2 Relative frequency distributions (or pdfs) of delays from a 1,000-day simulation.672 Journal of the Operational Research Society Vol. 51, No. 6
interdependent in a very complex way which is captured daily delay will vary from day to day and will keep varying
only by the full scale simulation. Each curve (except the no matter how many days we consider. However, train
exogenous delay curves) depends on the other four. operators are also interested in d, the mean of the `mean
Slight ¯uctuations in curves in Figures 1(b) and 2. We
daily delays'. The latter (d) will converge as we increase the
observe that the curves in Figures 1(b) and 2 are fairly number of days in the sample, and the con®dence intervals
smooth but they have some bumpiness or unevenness, will become narrower (the estimates more accurate) the
except for the exogenous delay curves. This unevenness is larger is the sample of days in the simulation. Since the
not caused by the inherent randomness in the sample of days days in the sample are independent of each other, the central
in the simulation hence does not go away if we take a larger limit theorem applies, hence we expect the standardpdevia-
sample. The simulation covered 1,000 days and even when tion of the mean of the `mean daily delays' to be n ÿ 1
we simulated far more (10,000) or fewer days the shapes of times smaller than the standard deviation of the `mean daily
the curves, including the bumps, remained almost exactly delay' given above, where n is the sample size. Therefore
the same. The slight bumpiness in the curves is due to the with a sample size of 1,000 days the 95% con®dence
fact that (before we add random disturbances) the daily intervals for the mean of the `mean daily delays' are
p
timetable, like all timetables, is a set of ®xed times, which 1000 ÿ 1 31:6 times smaller than those in the table
p
inevitably have certain patterns. For example, many sched- above. That is (1:96s= n ÿ 1), hence 0.0222, 0.0093 and
uled dwell times tend to be 2, 4, 10, etc., minutes, and many 0.0242 minutes respectively, which is less than 2 seconds.
scheduled arrival and departure times tend to be on-the-hour
or half hour, or 10, 20, etc., minutes after. This can cause Experiment 2 Punctuality improvement or deterioration
some patterns in the lengths of knock-on delays. If a train at a stationÐgetting back on schedule.
misses a time slot the alternative slots occur after certain
®xed intervals, not randomly. This makes some knock-on A question of interest to transport operators and users is:
delay durations more likely than others. It is perhaps `Will the delays encountered at a station mean that trains are
surprising that the curves in Figures 1(b) and 2 are as even further behind schedule when they depart from the
smooth as they are. station than when they arrived?' At ®rst sight it may seem
Con®dence intervals for the mean of the delays occurring that the distribution of delays must be worse on departure
on any one day. For each of the 1,000 simulated days we than on arrival. However, there is a mechanism for helping
computed the mean delay d for arrival, dwell and departure get trains back on time if they arrive late. If a train is late it
knock-on delays. We found that the distributions of these can depart after a minimum required dwell time rather than
daily means are approximately normal. Assuming the mean adhering to the original scheduled dwell time. (Suppose a
daily delay d is normally distributed, an estimate of the 95% train has a scheduled dwell time of say 10 minutes and a
con®dence intervals for d is: (d ÿ 1:96s) to (d 1:96s), minimum required dwell time of 7 minutes. If it arrives 4
where d and s are the mean and standard deviation minutes late it is ready to leave in the 4 7 11th minute,
respectively of the 1,000 values of d from the 1,000 day that is, only 1 minute late instead of four minutes late. On
simulation. the other hand, if the minimum required dwell time is 4
minutes, then it is ready to depart in the 4 4 8th minute,
Example Assume 20% of trains experience exogenous but of course it is not allowed to leave until its scheduled
arrival delays distributed U ÿ2 to 30) exogenous dwell time, hence it leaves on time in the 10th minute.)
delays distributed U (0 to 15). From the 1,000 day simula- To investigate this, let,
tion we obtained the standard deviations and hence con®-
dence intervals shown in the following table. r
scheduled dwell time ÿ minimum required dwell time
95% con®dence interval
Random variable d s.d. of d for d 1:96s scheduled dwell time
Mean over a day of knock-on 0.36 0.702 mins and refer to this as the `maximum dwell reduction ratio' or
delays to arrivals
simply the `dwell reduction ratio'. This dwell reduction
Mean over a day of knock-on 0.15 0.294 mins
delays to dwells ratio may be different for different trains: for some trains
Mean over a day of knock-on 0.39 0.764 mins the scheduled dwell time may already be close to its
delays to departures minimum and for others it may not. However, for simpli-
city we will assume here that the maximum dwell reduction
r is the same for all trains. In the rest of this section, that is,
Con®dence intervals for the mean delay (over all days). Experiments 1, 3 and 4 and we assume that r is zero. In the
Above we computed the con®dence interval for the `mean following table and in Figure 3 we show the delay distribu-
daily delay' d. This spread of mean daily delays is not tions which result from letting the maximum dwell reduc-
reduced by taking a larger sample of days, since the mean tion r be 0.0, 0.2, 0.4 and 0.8 respectively. To generateM Carey and S CarvilleÐTesting schedule performance and reliability for train stations 673
these results we ran a separate 1,000 day simulation for lines between them. The graph of exogenous arrival lateness
each value of r. or punctuality is the straight line in Figure 3. Comparing this
straight line (arrival punctuality) with the curves (departure
punctuality) in Figure 3 shows that for larger values of r the
Maximum dwell reduction ratio, 0.0 0.2 0.4 0.8
deterioration in punctuality is less, or the improvement in
r
punctuality is greater. However, we note that in this exam-
% of trains with exog arr. delay 85.7 85.7 85.7 85.7 ple, for all levels of r, the percentage of trains having zero
less than 5 mins arrival delays is always greater than the percentage having
% of trains departing less than 72.0 78.0 80.6 82.7
5 mins late zero departure delays. This could be different in another
Difference ÿ 13.4 ÿ 7.7 ÿ 5.1 ÿ 3.0 example. For example, if the exogenous dwell delays were
% of trains with exog arr. delay 91.2 91.2 91.2 91.2 smaller that would decrease the departure delays more than
less than 15 mins the arrival delays.
% of trains departing less than 89.4 92.2 93.7 95.4 Comparing the straight line (arrival punctuality) with the
15 mins late
Difference ÿ 1.8 1.0 2.5 4.2 lowest of the four curves in Figure 3 shows that if the dwell
reduction ration is r 0 then train punctuality on departure
is always less than on arrival. Now consider say r 0:4.
The `difference' rows in the table shows the differ- For this compare the straight line (the arrival punctuality)
ences between the percentage of trains arriving late and the with the third up (r 0:4) of the curves in Figure 3. This
percentage departing late. shows that if we are concerned with lateness up to about 11
In Figure 3 the vertical axis gives the percentage of trains minutes, then departure punctuality is worse than arrival
less than x minutes late, where x is corresponding lateness punctuality. However, if we are concerned with lateness
value on the horizontal axis. Hence, the higher the distribu- greater than about 11 minutes, then departure punctuality is
tion curve in Figure 3, the lower the percentage of late better than arrival punctuality. Similar remarks apply to the
trains, or the higher the train punctuality or performance. We curves for other values of r.
see that, as expected, the larger the dwell reduction ration In view of the above, train operators can use the ratio r as
r the higher the distribution curve, hence the higher the a policy instrument in designing more reliable schedules.
percentage of trains departing on time. By better management of resources at stations, operators
As discussed above, we wish to see if train punctuality is may be able to cut minimum dwell times hence cut r and
better or worse when trains are departing from the station substantially improve punctuality of departures.
than when they arrived. For this we compare the percen- Of course it is not only at stations that trains can get back
tages of trains departing late with the percentages having on schedule. In practice, the scheduled trip times between
exogenous arrival lateness: since we are here considering stations are sometimes set slightly larger than the minimum
only one station, the exogenous arrival lateness implicitly time needed. This extra time or `recovery' time allows late
include all delays incurred at all previous stations or on the trains to reduce their lateness. The distribution of arrival
Figure 3 Effect of dwell reduction ratios on total departure delays.674 Journal of the Operational Research Society Vol. 51, No. 6
lateness at the next station may then be `better' than the An interesting feature of the curves is that, except for the
distribution of departure lateness at the present station. lowest curve, they start off ¯at or near to ¯at. For example,
From a passengers perspective arrival punctuality matters the curve showing the percentage of trains with knock-on
more than departure punctuality. delays of `430 minutes' is ¯at up until the maximum
exogenous delay UL is 30 minutes. This indicated there are
Experiment 3 How knock-on delays vary with size of no knock-on delays greater than 30 minutes unless there are
exogenous delays. some exogenous delays greater than 30 minutes. Similarly,
there are almost no knock-on delays greater than 10, 20,
In Experiment 3 we simulated 1,000 days to obtain the etc. minutes unless there are some exogenous delays
distribution of knock-on delays (and distribution of total greater than 10, 20, etc. minutes respectively.
delays) when the exogenous delays are from a uniform
distribution with bounds 0 and UL. We repeated this Experiment 4 Effect of allowing platform changes on-
simulation experiment for 25 different values of UL, start- the- day.
ing at UL 0 minutes and increasing in steps of 2 minutes
up to UL 48. Also, we assumed exogenous dwell delays The most dramatic aspect of the simulation results is the
are on average about half as long as exogenous arrival effect of allowing or not allowing trains to change from their
delays, i.e., if the exogenous arrival delay is ÿ2 to 20 the scheduled platforms. If a train arrives later than scheduled
exogenous dwell delay is 0 to 10, since negative dwell its scheduled platform may be already taken by a later train.
delays are not allowed. In that case we could hold the late train until its scheduled
For each of these 25 different simulation experiments we platform is free, or send it to some other platform if one will
computed various statistics, for example, the percentage of be free sooner. Similarly, if a train departs later than
trains more than 0, 5, 10, etc., minutes late. Here we graph scheduled, the next train scheduled to go to that platform
some of the results (Figures 4(a)±(b)), to show how the may either wait until the platform is free, or go to another
knock-on delays (and total delays) increase as UL the platform if one is free sooner. Note that any of these on-the-
maximum exogenous delays increases. Since the exogen- day changes of train times or platforms may cause yet
ous delays is uniformly distributed from ÿ2 to UL, the further knock-on changes to following trains.
expected exogenous delay is UL 2=2, which increases To explore the effect of allowing on-the-day changes of
as UL increases. We illustrate the results mainly for platforms we ran the experiments twice. Firstly we requir-
departures delays but the results for arrival and dwell ing that all trains go to their scheduled platforms. We refer
time delays are similar. It can be seen that the knock-on to this as the `®xed platform model'. Secondly we ran all
delays increase fairly smoothly as the exogenous delays UL the experiments while allowing trains to be sent to
increase. For example, from the lowest curve in Figure 4(a) a different platform if this would reduce lateness or other
we see that if UL is say 10 minutes then 65% of trains penalties (platform desirability). We refer to this as the
experience no knock-on delay, and if UL is say 20 minutes `¯exible platform model'. The results are illustrated in
then 55% of trains experience no knock-on delay. Figure 5.
The characteristic shapes of the curves in Figures 4(a)± We found that allowing platforms to be changed in
(b) can be explained as follows. Consider the lowest curve response to on-the-day lateness caused a dramatic reduction
in Figure 4(a). In this curve the number of knock-on delays in knock-on delays. When exogenous delays are large, for
increases sharply at ®rst and then much more slowly. Each example, up to 60 minutes, the mean size of knock-on
increase in the exogenous delays UL increases the like- arrival delays is reduced by about 90% and the mean size of
lihood that trains will loose their scheduled time slot and=or knock-on departure delays is reduced by about 40%. This
platforms. That is, if a train is late another train may have large reduction in the size and number of knock-on delays
arrived at the platform, so that the late trains has to wait or is perhaps larger than appears to be expected by rail
go to a different platform. However, if a train is so late that operators. It has relevance for the design and operation of
it has already lost its scheduled time slot then any further train stations. It suggests it is important that platforms be
lateness may have less effect on how soon it can ®nd a new feasible for as many of the various train types as possible.
slot, and on whether this causes further knock-on delays. This involves layout of lines and signals, but it also
This causes the curve in Figure 4(a) to ¯atten out. Some- traveller information systems, and ensuring that travellers
what similar remarks apply to the next curve in Figure 4(a), can easily walk from one platform to another. Some
but less so to the other curves. The reason is that the latter stations are designed so that changing trains from their
are caused by larger exogenous delays. With larger exogen- scheduled platform is very inconvenient for passengers,
ous delays the trains have already lost their initial sched- involving long walks up and down stairs perhaps with
uled time slot, hence any further increases in the size of the luggage. On the other hand, some stations are designed
exogenous delay will simply cause a proportionate increase so that all platforms are quickly accessible from a
in knock-on delays. convenient central waiting area. In that case the platformM Carey and S CarvilleÐTesting schedule performance and reliability for train stations 675
Figure 4 (a) Train performance decreases with the size of exogenous delays: knock-on departure delays. (b) Train performance decreases
with the size of exogenous delays: total departure delays.676 Journal of the Operational Research Society Vol. 51, No. 6
Figure 5 Effect on mean size of delays of allowing platform changes on the day.
allocation for each train need not be announced until different distributions and perform different simulation
shortly before its arrival or departure. Indeed, the platform experiments.
schedule need not be published in advance. This is the The pdfs of arrival, dwell and departure delays for
custom for the main multi-platform terminal stations scheduled transport services typically have a ®nite range,
around London. However, even in this case it is generally are unimodal and skewed bell-shaped with a longer tail of
desirable to have an unannounced planned platform for lateness than earliness. This is typically true for train delays
each train so that on-the-day train controllers and operators including those on the BR network. One reason for this
need worry only about deviations from this schedule. skewness is that if a train is running earlier than scheduled
it can get back on schedule by slowing down, whereas if it
is running late it may not get back on schedule as it has to
Fitting equations to graphs of simulation results respect prespeci®ed maximum speeds and accelerations.
The beta distribution pdf has all the above characteristics
The graphs of the simulation results in Figures 1 to 5 are all
of a typical pdf of delays, hence it is often appropriate for
fairly smooth. When we initially used small samples of
modelling transportation delays and we use it here. Also, it
days, say 10 or 20, the curves were much more jagged and
has four parameters (a, b, T min and T max de®ned below),
¯uctuated randomly about the curves which were obtained.
which gives it more ¯exibility in ®tting empirical data than
We increased the sample sizes until we obtained curves
a pdf such as the normal or exponential which have fewer
which are almost identical in repeated simulations, and
parameters.
hence have narrow con®dence intervals. The smoothness of
The beta distribution has a pdf f de®ned on the interval
the curves is of course not only due to averaging over large
[0, 1] by,
numbers of days, but also to the inherent regularity in the
system being simulated.
xaÿ1 1 ÿ xbÿ1
f x ; 1
B a; b
Schedule performance, with beta distributions
of exogenous delays
where a; b > 0 and B a; b is the beta function (hence the
In the previous Section we used uniform distributions name of the distribution). An example of a beta density is
for a set of simulations and experiments. Here we use given in Figure 6. The beta density can be rescaled andM Carey and S CarvilleÐTesting schedule performance and reliability for train stations 677
shifted to be de®ned on any ®nite interval, say [T min , T max ]. then simulated all train arrivals and departures for thou-
The pdf (1) then becomes, sands of days using the ATTPS package,1 and kept a record
of all knock-on delays and changes of platforms caused by
x ÿ T min aÿ1 T max ÿ xbÿ1
f x : 2 the exogenous beta distribution delays.
T max ÿ T min abÿ1 B a; b These experiments and their results are set out below.
We used shape parameters a 2, b 4, minimum delay
T min ÿ2, and maximum delay T max 20 minutes, which Experiment 5 Distribution of knock-on delays.
gives the beta distribution in Figure 6. This implies 63% of
delayed trains are more than 5 minutes late, 19% are more To estimate the distributions of knock-on delays caused
than 10 minutes late, and 2% are more than 15 minutes late, by given distributions of exogenous delays we proceeded as
which is typical of the pattern of exogenous delay for follows. Steps (iii)±(v) are the same as in Experiment 1, and
delayed trains in Britain. the comments made there following Steps (iii)±(v) also
If we apply this distribution of exogenous delays to all apply here.
BR trains then, when we add in the resulting knock-on
(i) Choose percentages of train to experience exogenous
delays, the total delays would be far in excess of the typical
delays of arrival times and dwell times. We initially
pattern of delays for BR. However, in practice only a
chose p1 20% for arrivals and p2 10% for dwell
percentage p of trains experience exogenous delay and
times.
this percentage is often different for different train types
(ii) Select p1 % of trains at random and to the arrival time of
and parts of the rail network. In Experiment 5 we assumed
each of these trains add a delay drawn at random from
p 20% of trains chosen at random experience exogenous
the above beta distribution of delays. Similarly for train
delays, with a beta distribution, and we simulate this
dwell times. For dwell time delays we used a beta
occurring every day for a 1,000 days.
distribution with parameters T min ÿ2 to T max 20
In Experiment 6 we experiment with different percen-
minutes.
tages p, starting at p 0% and increasing in steps of 2% to
(iii) Simulate running this perturbed timetable for one day,
50%. We choose a cut-off of 50% as it is unlikely that on
and record all train delays (exogenous and knock-on)
any one day more than 50% of trains arriving would be
for the dayÐsee Figure 7(a).
subject to exogenous delay (as opposed to a knock-on
(iv) Repeat steps (ii)±(iii) 1,000 times, to simulate 1,000
delay). We conducted these experiments to show how the
daysÐsee Figure 7(b).
number of exogenous delays affects the number and size of
(v) Compute descriptive statistics (mean, median, standard
knock-on delays. This is important to train operators seek-
deviation, etc.) for the distribution of delays in (iv).
ing to reduce delays. In particular, we can ®nd the level of
exogenous delays at which the total delays (exogenous plus An example of the frequency distribution (or pdf) of delays
knock-on) will exceed the punctuality targets set for train over 1,000 days, as obtained from (iv), is given in Figure 7.
operators. For comparison, the Figure also shows the pdf of exogenous
For each day simulated in each of the above experiments delays and the pdf of total delays (exogenous delay
we applied the beta distribution of delays to the train knock-on delays).
timetable for a busy station (Leeds). To train arrival
times and station dwell times we added an exogenous
Con®dence intervals for (parameters of) the distribution
delay drawn at random from the beta distribution. We
of delays
These can be computed in exactly the same way as in
Experiment 1 above. Again, as in Experiment 1 the widths
of the con®dence intervals are all so small as to be
negligible, indicating that the statistics obtained from the
1,000 day simulation, for example, the mean, median,
percentiles, etc. of delays are accurate simulation estimates.
Experiment 6 How knock-on delays vary with % of
trains subject to exogenous delays.
In the above experiment we simulated 1,000 days to obtain
the distribution of knock-on delays (and distribution of total
delays) when a ®xed percentage ( p) of trains are subject to
Figure 6 Beta probability density on interval [0, 1] with shape exogenous beta distribution delays. We repeated this simu-
parameters a 2, b 4. lation experiment for 25 different values of p, starting at678 Journal of the Operational Research Society Vol. 51, No. 6
Figure 7 (a) Frequency distributions of delays from a single day simulation. (b) Frequency distributions of delays from a 1,000-day
simulation.
p 0 and increasing in steps of 2% up to 48%. For each of the results, in Figures 8 and 9, to show how the knock-on
these 25 different simulation experiments we computed delays (and total delays) change as the number of trains
various statistics, for example, the percentage of trains subject to exogenous delays increases. As in the previous
more than 0, 5, 10, etc., minutes late. Here we illustrate section, the sample of days is so large that the graphs changeM Carey and S CarvilleÐTesting schedule performance and reliability for train stations 679
Figure 8 (a) Train performance decreases with the size of exogenous delays: knock-on departure delays. (b) Train performance decreases
with the size of exogenous delays: total departure delays.680 Journal of the Operational Research Society Vol. 51, No. 6
little if we repeat the experiment with a different sample. platform if this would reduce lateness or other penalties (the
Hence the con®dence intervals are extremely small or `¯exible platform model'). The results are illustrated in
negligible for each plotted point on these graphs. Figure 9.
In Figure 8 the curves showing the percentage of trains Allowing platforms to be changed in response to on-the-
having knock-on delays of `410 minutes' are approxi- day lateness again causes a substantial reduction in knock-
mately linear functions of the percentage of trains having on delays, but not as much as in the case of uniform delays
exogenous delays. This seems signi®cantly different from (previous Section).
Figure 4, but this ®gure is not really comparable with
Figure 4. In Figure 4 we increased the (average) sizes of
Fitting equations to graphs of simulation results
the delays for a ®xed number of trains, whereas here we do
the reverse. As explained earlier for Figure 3, if we increase The graphs of the simulation results in the above ®gures are
the sizes of the exogenous delays, a stage is soon reached all remarkably smooth curves, and the same comments
where these trains have lost their scheduled slots, and have apply as in the previous section. Fitting a quadratic equa-
to ®nd new slots. Any further exogenous delay has less tion to the points in Figure 9 gives an R2 0:999. This
additional impact. In contrast, if we increase the number smoothness implies that almost the same curves would be
and proportion of trains that have exogenous delays, then obtained by generating only say 5 simulation points instead
each additional delayed train may have a similar impact at of all 25 in the ®gure. This is useful since it indicates that
least until a high proportion of trains are delayed. we do not need to simulate so many points.
Experiment 7 Effect of allowing on-the-day platform
Concluding remarks
changes.
In this paper we set out a simulation approach to estimating
The discussion, experiments and results here are similar reliability for proposed schedules or schedule changes for
to those in Experiment 4, where the distributions of exogen- a typical busy complex station. To do this, we simulated
ous delays was assumed to be uniform. To explore the effect running the schedules for hundreds of days, subjected to
of allowing on-the-day changes of platforms, we again ran distributions of exogenous random delays, incidents, etc.,
the above experiments twice, ®rst requiring that all trains which are typical of those occurring in practice. When
must go to their scheduled platform (the `®xed platform con¯icts of train times or platforms occur we resolve the
model') and second allowing trains to be sent to a different con¯icts by rescheduling trains in a manner typical of those
Figure 9 The effect on departure knock-on delays of allowing platform changes on the day.M Carey and S CarvilleÐTesting schedule performance and reliability for train stations 681
used in practice. The simulations are useful to develop trains to depart after less than their usual minimum dwell
understanding of the behavior of knock-on delays at a times. We ®nd that both these policies can have dramatic
station. effects in improving punctuality.
We use the simulations to generate distributions of A further use of the detailed simulation approach of this
knock-on delays and total (knock-on plus exogenous) paper is in evaluating and validating heuristic measures of
delays, including numbers and sizes of delays. From reliability that are widely used in practice. This arises as
these distributions we can immediately read off measures follows: A disadvantage of the simulation approach is that
of punctuality that are of direct interest to train operators, it is time consuming and requires a substantial amount of
for example, the predicted percentage of trains more than data. An alternative that is often adopted in practice is to
0, 5, 10 or 15 minutes late. Operators in Britain are now use a heuristic or `rule of thumb' ex ante measures
required to record and publish such measures, and to reliability or punctuality. For example, the percentage of
display them on public notices in stations. And they are trains that are operating at minimum headways, or the
subject to penalties if they fail to meet certain punctuality number of alternative platforms available. Though these
targets. It is therefore of interest to operators to be able to measures are easier and faster to apply, their usefulness
estimate such measures in advance for proposed timetables. for predicting schedule reliability has not been system-
New or revised timetables are produced at regular intervals, atically tested. In ongoing research we are testing such
for example approximately every six months in Britain for heuristic measures by comparing their predictions with
the whole rail network. As well as the published timetables, those obtained from the detailed accurate simulation
operators also produce timetables to be used in emergen- approach in this paper. We apply both approaches separately
cies, for example, if a certain track is out of operation. to rank the reliability of a set of proposed schedules for a
The simulations are also useful in identifying and target- station. If a heuristic measure gives the same ranking of the
ing reliability problems or bottlenecks in the system. To do schedules as the simulation approach, it can be taken as
that, the predicted delay distributions from the simulation potentially useful, at least for that station or context. This of
are automatically broken down by train service, train type, course does not prove it will be always or everywhere
platform, line, etc. accurate or useful, but it increases con®dence in its use.
The simulation model can also be used to estimate what Therefore, even when heuristic rather than simulation
effect proposed changes in operating rules or infrastructure measures of reliability are used in practice, the detailed
will have on punctuality. There are a large number of simulation approach is useful as a way of periodically
possible changes for which operators may wish to estimate testing or validating existing or proposed heuristic
and evaluate the effect on reliability. For example, as measures.
regards train operating rules, suppose it is proposed to
reduce minimum headways for some trains, or reduce
minimum dwell times, or dedicate certain platforms to AcknowledgementsÐWe wish to thank two anonymous referees for their
one train operating company, or restrict one of the thoughtful and helpful comments. This research was supported by Engi-
`through' platforms to one-way traf®c. As regards infra- neering and Physical Sciences Research Council (EPSRC) grants
structure, suppose it is proposed to eliminate a platform, or GR=H=48033 and GR=K=75798, which are gratefully acknowledged. The
authors also wish to thank various individuals in Railtrack, its predecessor
add a platform, or lengthen a platform so that it could take
British Rail, and the railway industry generally, for their cooperation, advice
intercity trains, or change the track layout so that certain and comments. Some of this dates back to the three years Professor Carey
lines no longer con¯ict. For each of these changes we can spent at Oxford University as British Rail=Fellowship of Engineering
use the simulation model in this paper to evaluate how the Senior Research Fellow. However, it is emphasised that none of these
change will affect reliability. To do that we simply run the bodies or people assume any responsibility for any of the opinions, results
or data in this paper. All of these are the responsibility of the authors.
simulation without the proposed change, then with the
proposed change and compare the results. These reliability
changes may be suf®cient to justify, or to rule out, the
proposed changes. Alternatively, they may have to be References
balanced against other costs and bene®ts of the proposed
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Complex Stations: Models, Algorithms and Results. Research
to the changes in reliability. There is an extensive literature Report, Faculty of Business and Management, University of
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of some policy or parameter changes. For example, we use Sciences Working Paper 89-11-02, Decision Sciences Depart-
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