Delay Management in Regional Railway Transport

Abstract

This article is focused on the reliability of transfer connections in regional railway transport. The reliability of the transportation chain in public transport is an essential element for functional, attractive, and long-term sustainable public transport. This article discusses the causes and consequences of railway traffic disruption and related impacts on passenger transfer connections. To reduce the negative impacts of common operational disruptions, the authors present an original approach for determining transfer waiting times between delayed trains based on a modified critical path method (CPM). In addition, an example of the implementation of this method in regional railway transport in the Vysočina Region of the Czech Republic is provided.

Full text

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Citation: Vávra, R.; Janoš, V. Delay
Management in Regional Railway
Transport. Appl. Sci. 2022,12, 457.
https://doi.org/10.3390/
app12010457
Academic Editor: Roland
Jachimowski
Received: 11 November 2021
Accepted: 22 December 2021
Published: 4 January 2022
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4.0/).
applied
sciences
Article
Delay Management in Regional Railway Transport
Rudolf Vávra * and Vít Janoš
Department of Logistics and Management of Transport, Faculty of Transportation Sciences, Czech Technical
University in Prague, 110 00 Prague, Czech Republic; janos@fd.cvut.cz
*Correspondence: vavrarud@fd.cvut.cz
Abstract:
This article is focused on the reliability of transfer connections in regional railway transport.
The reliability of the transportation chain in public transport is an essential element for functional,
attractive, and long-term sustainable public transport. This article discusses the causes and con-
sequences of railway traffic disruption and related impacts on passenger transfer connections. To
reduce the negative impacts of common operational disruptions, the authors present an original
approach for determining transfer waiting times between delayed trains based on a modified critical
path method (CPM). In addition, an example of the implementation of this method in regional railway
transport in the Vysoˇcina Region of the Czech Republic is provided.
Keywords:
public transport service quality; railway operation; transfer waiting times; minimum
transfer times; timetable stability; CPM; disposal measures; timetabling; network planning; public
transport modeling
1. Introduction
For public service functionality (which includes accuracy and reliability), it is necessary
to maintain the integrity of the transportation chain in passenger transport as part of
the network for public transport planning. The integrity of the transportation chain is
often disrupted by negative consequences for passengers (e.g., delay and loss of transfer
connections) as they experience operational disruptions. Accordingly, it is necessary to
ensure an acceptable level of operation regarding stable timetables so that transfer waiting
times and planned transfer bindings may be achieved with tolerable levels of delay.
The necessity of waiting for services due to delays in other services increases with
growing intervals between connections. While waiting of the continuing services for
delayed connecting services is not urgent during short intervals (i.e., up to 30 min) because
the waiting time for the next service is relatively short, the significance of waiting grows up
with longer intervals (i.e., 60–120 min) because waiting for the next service is longer (often
longer than the travel time itself).
This article deals with transfer waiting times, especially in passenger railway transport.
The authors develop an original approach for determining transfer waiting times, based on
a modified critical path method (CPM), which is understood to be an effective tool for the
management of delays that are due to common operational disruptions (i.e., disruptions
with only relatively small deviations from the timetable, not extraordinary interruptions
causing, e.g., traffic suspensions). The main contribution of this approach (compared to
existing approaches examined by the authors) is the determination of transfer waiting times
by taking into consideration the operational influences between trains, without the need
for railway operation simulation (which is often much too time-consuming). The CPM
approach easily applied in cases where there are not many relevant disposal measures and
their combinations are used; however, this is also one of the limitations of the approach.
The anticipated impact of applying this approach in practice is that waiting times would
not be merely estimated (which, unfortunately, is the common Czech praxis); rather, they
would be calculated, so that delays caused by waiting for connecting trains do not interrupt
Appl. Sci. 2022,12, 457. https://doi.org/10.3390/app12010457 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022,12, 457 2 of 19
relevant transfer bindings and the transportation chain remains compact. It is necessary,
however, to mention that this approach does not include the passenger-time consideration.
2. Materials and Methods
2.1. Background: State-of-the-Art
The authors’ research in determining appropriate transfer waiting times and delays is
based not only on their innovative approach, but also on previously published works on
the topics of train scheduling, traffic control, dispatching, and delay management.
¸Sahin [
1
] analyzed the dispatcher’s decision-making process in a single-track rail-
way and developed a heuristic algorithm for rescheduling trains by modifying existing
meet/pass plans in conflicting situations. He determined dynamic priorities and their
weights, based on them the algorithm chooses the best alternative resolution that results in
less total consequential delay in the system.
Caimi [
2
] developed a binary linear optimization model for discrete time rescheduling
in complex central railway station areas. The approach was successfully applied to an
operational day at the central railway station in Berne, Switzerland.
Dollevoet [
3
] proposed an optimization approach based on a macroscopic delay
management model that determines which transfer connections to maintain. The resulting
disposition timetable was then validated microscopically in a bottleneck station within
the network.
Samà[
4
] developed a multi-criteria decision support methodology for dispatchers, in-
corporating safety regulations and considering key performance indicators. Mixed-integer
linear programming (MILP) formulations were proposed and solved via a commercial
approach, wherein a well-established non-parametric benchmarking technique provided
an efficient/inefficient classification of the best solutions for each instance. Computa-
tional experiments were held on a Dutch railway network with mixed traffic and multiple
disturbances.
Luan [
5
] dealt with the integration of real-time traffic management and train control.
He developed three innovative integrated optimization approaches to deliver a solution for
train dispatching and train control, using mixed-integer nonlinear programming (MINLP)
and mixed-integer linear programming (MILP).
Corman [
6
] focused on two streams of research from the perspectives of operations
managers and passengers. He developed microscopic passenger-centric models and pro-
posed several fast heuristic methods based on alternative decompositions of the model. He
also proposed a tabu search scheme [
7
] consisting of effective rescheduling algorithms and
local rerouting strategies. A fast heuristic method and a truncated branch-and-bound algo-
rithm were combined to compute the optimal solution in small instances and a relatively
accurate solution for large situations with complex disturbances in short time periods.
D’Ariano [
8
] developed a branch-and-bound algorithm that included rules to enable
faster computation. D’Ariano combined this algorithm with Corman’s tabu search in the
AGLIBRARY system [
9
]. The procedure computed a first-feasible schedule, in which each
train followed its default route. Then, the procedure iteratively looked for better solutions,
in terms of delay minimization, by changing the routes for some trains. The AGLIBRARY
system was tested on a railway line near London, and the data computed to better solutions
than those of an MILP formulation using the same computing time.
Nedeliakova [
10
] applied lean philosophy in risk management to the impact of train
delays in railway passenger transport. For data assessment, she used multiple statistical
tests. The research focused on highlighting the need for a systematic approach to train
delays and avoiding delays via risk management.
Meng [
11
] added time margins (i.e., headway buffers and running time supplements)
to the planning process for timetables. His research relocated time margins to proper
positions to minimize the sum of train delays. For this task, he treated the train timetable
as a directed arc graph (DAG) and proposed a heuristic technique, critical time margins
allocation (CTMA), which is based on CPM.

Appl. Sci. 2022,12, 457 3 of 19
Van Thielen [
12
] examined the updating of railway timetables to reduce delays in cases
of unexpected events. He examined the dynamic impact zone that was created and used
in the retiming/reordering heuristic technique, as part of a conflict prevention strategy
(CPS) aimed at assisting dispatchers in considering the relevant parts of the network and
the traffic when addressing conflicts.
Ghaemi [
13
] used three models to obtain reliable disruption-length estimation, which
can potentially reduce negative impacts caused by disruptions: a disruption-length model,
a short-turning model, and a passenger assignment model. The results showed the effects
of short (optimistic) and long (pessimistic) estimates on affected passengers, the generalized
travel time, and the number of passengers rerouting and transferring.
From this background, it is evident that much of the research has dealt with delay
management, operational traffic control, dispatching, etc., because these topics are impor-
tant for transport services. The higher the quality of such services, the more reliable they
are, leading to enhanced attractiveness and sustainability. Delay management also includes
the determination of transfer waiting times in the timetable planning process. Determining
transfer waiting times properly can significantly influence the stability and attractiveness of
a timetable. However, no previous research has used CPM for determining transfer waiting
times. Meng alone [
11
] used CPM in considering delay management, but for the purpose
of proper time margin position planning rather than for determining transfer waiting times.
In this article, the authors used CPM for dealing with this task.
2.2. Waiting Time Issue
The reason for waiting times is to maintain the integrity of a transport chain during
delays in connecting services. In the cases of isolated lines (or isolated operations, such as
subways) and short-interval lines (e.g., S-Bahn daily services and public transport in big
cities), it is not generally necessary to deal with waiting for services because the subsequent
services are managed in a relatively short time; waiting due to delayed services would
bring more negative influences for non-transfer passengers than positive influences for
transfer passengers. However, in the Czech Republic, long intervals in railway transport are
common. Many lines (generally long-distance lines or regional lines outside metropolitan
areas) are operated in 2-h intervals, at least during off-peak hours, so waiting times for
transfer connections in the Czech Republic are relatively high.
The basic thesis for determining transfer waiting times is that delays and anomalies
that are spread across a network should have limited impact and predetermined maximum
allowable waiting times, affecting as few passengers as possible.
2.2.1. Typical Causes of Operational Disruptions
Typical causes of operational disruptions that result in train delays or interruptions
in operational processes, thereby thwarting the operation of timetables, often include
operational failures [14] such as the following:
•vehicle technical problems;
•
fixed installations’ technical or construction problems (e.g., railway switches or safety
system failures);
•electric supply interruptions;
•staff absences or inabilities;
•passengers’ influence (e.g., long dwell times due to unusual passenger frequency);
•outside influences (e.g., weather influences, lower adhesion, etc.).
Operational failures, as well as technological mistakes in timetable construction, can
lead to scheduling conflicts or clashes between the scheduling of two or more trains. Their
real-time mutual path requirements may not be achievable, taking into consideration
issues pertaining to infrastructure, vehicles, crews, sequence of trains, and connections.
Occupancy conflicts, vehicle circulation conflicts, and transfer connection conflicts may be
separate and distinct. Every conflict has two basic solutions, as set out in Table 1, which
shows that the second solution causes delay. On the other hand, the first solution does not

Appl. Sci. 2022,12, 457 4 of 19
cause delay, but can cause difficulty with respect to resources or extending transportation
times for transferring passengers.
Table 1.
The basic solution for occupancy conflict, vehicle circulation conflict, and transfer connection
conflict [15].
Occupancy Conflict Vehicle Circulation Conflict Transfer Connection Conflict
Basic solution 1 One of two trains uses other parts
of the infrastructure.
Cancellation of a circulation binding
(operational standby necessity)
Cancellation of a transfer
binding (connection is lost)
Basic solution 2
One of the trains will be managed
via conflict place later (sequence of
train paths may be changed)
The second train will be managed via conflict place later (sequence of
train paths cannot be changed)
The following disposal measures are commonly used in rescheduling, to achieve
stabilization of railway operation, i.e., for determining timetables [16,17]:
•exchange of train sequence;
•run time or dwell time extension;
•transfer connections’ unbinding;
•alternative paths (run routes) utilization;
•crossing or overtaking relocation/cancellation;
•changes in train stopping patterns (i.e., additional stops or cancellations of stops);
•premature vehicle reversion;
•canceling services, creating additional services;
•substituting train sets or services (e.g., buses); and/or
•exclusion timetables.
Delays on single-track railway line without sufficient buffer times (i.e., margin times)
converge to the value of the most-delayed train on the line/route, until any of the relevant
disposal measures mentioned above are applied. When delay differences of opposite-
direction trains on single-track railway lines are small (considering the time distance of
neighboring crossing stations), the scheme of the crossing is maintained. When there are
larger differences in the delays of opposite-direction trains, it is appropriate to shift the
crossing to another suitable station, particularly for the maintenance of transfer connections
for the train with less delay; however, such a shift of crossing can lead to further delay, and
increase the delay of the train with higher delay
2.2.2. Bindings between Waiting Times, Infrastructure Capacity, and Timetable Stability
In determining transfer waiting times, it is necessary to consider that the waiting
causes partial delay transmission for connecting trains. Such delays are transmitted from
one train to another due to trains’ mutual interactions [
18
]. The number of mutual interac-
tions increases with the number of trains on the corresponding railway line and with the
growing heterogeneity of the timetable [
19
]. The more interactions there are, and the more
heterogeneous a timetable is, the more potential conflicts there will be between trains, with
a rise in and potential sources of delay. Considering the number of interactions between
trains, partially segregated railways in locations where regional transport is managed
separately from long-distance transport (e.g., the German S-Bahn system) can provide an
intermediate resolution. However, in the Czech Republic, long-distance transport and
regional transport are operated on the same railway lines, resulting in delay transmissions
among a large number of mutually interacting trains. Generally, the dependencies per-
taining to the number of trains, timetable heterogeneity, the average speed of trains, and
timetable stability are described by the capacity balance [
20
,
21
] shown in Figure 1. The
perimeter of the quadrangle, which limits the capacity balance, is constant for the railway
with unchanging technical parameters. Therefore, an increase in the value of one of the
four represented parameters leads to a decrease in the value at least one of the remaining
three parameters.

Appl. Sci. 2022,12, 457 5 of 19
Figure 1. Capacity balance according to UIC-Codex 406 [20,21].
The number and size of reserves and spaces in a timetable determine timetable stability.
The reserve means time supplements (i.e., margin times) in technological times, which
influence transportation times [
18
,
19
,
21
–
26
]. Reserves can be in run times, dwell times,
transfer times, etc. Reserves in run times arise by using additional time supplemental to
the technical run time. These additional time supplements can have either point character
or linear character (depending on the run time or the distance). The sum of the technical
run time and the time supplement is the regular (scheduled) run time, which is then used
for timetable construction.
In contrast, spaces (or headway time margins) indicate the empty time windows
between individual trains’ paths, which limit primary delay transmission from one train to
another. The margin time between two train paths can be deleted, and trains can be moved
to each other’s timetable without influencing either of them [
18
,
19
,
21
–
26
]. The spaces are
not parts of transportation times; rather, they are important margin times, which influence
transportation times and quality. They are buffer times in minimum headways, train set
reversion times, etc.
Although the reserves and spaces reduce infrastructure capacity, they are important in
achieving a stable timetable [
19
,
27
–
29
]. The stability of a timetable can be assessed through
time for the total elimination of an input delay of a certain amount [
23
,
25
,
26
]. In the case
of periodic (or interval) timetables, this time may be related to the interval/period [
25
].
Equation (1) provides a calculation of a stability coefficient
cstab
that reports how many
timetable periods are needed for the total elimination of input delay
td,input
, due to the
existence of spaces and reserves in the timetable. (
∑tspaces,period
is the sum of spaces in one
timetable period; ∑treser ves,period is the sum of reserves in one timetable period).
cstab =td,input
∑tspaces,period +∑treserves,period
(1)
Timetable stability cannot be marginalized while assigning transfer waiting times.
Every delayed train disrupts the ride of all waiting trains, and the delay tends to spread
through the net [
18
]. The greater the timetable interconnectedness, the greater the influence
will be upon waiting times for delayed trains, because the number of affected trains
increases disproportionally with the growing number of bindings in the timetable and the
delay elimination time. For example, if there is a delay, it is transmitted from each train
to
v
trains (where the intersection of the sets of trains to which the delay is transmitted
is an empty set), there are
n
degrees until the delay is eliminated, then the total number
of trains affected by the input delay can be described by a power function of order
n
,
as shown in Equation (2). The dependence of the number of affected trains on timetable

Appl. Sci. 2022,12, 457 6 of 19
interconnectedness with 5 degrees of delay transmission is graphically displayed in Figure 2.
Figure 3describes the transmission of delay between trains and the gradual elimination of
delay in the case of a stable timetable. The sum of the delays of all trains that were delayed
(directly or indirectly) by the ride of delayed train A, with input delay
tinput,A
, is described
in the chart in Figure 4.
N=
n
∑
i=0
vi(2)
Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 21
influence will be upon waiting times for delayed trains, because the number of affected
trains increases disproportionally with the growing number of bindings in the timetable
and the delay elimination time. For example, if there is a delay, it is transmitted from each
train to 𝑣 trains (where the intersection of the sets of trains to which the delay is trans-
mitted is an empty set), there are 𝑛 degrees until the delay is eliminated, then the total
number of trains affected by the input delay can be described by a power function of order
𝑛, as shown in Equation (2). The dependence of the number of affected trains on timetable
interconnectedness with 5 degrees of delay transmission is graphically displayed in Fig-
ure 2. Figure 3 describes the transmission of delay between trains and the gradual elimi-
nation of delay in the case of a stable timetable. The sum of the delays of all trains that
were delayed (directly or indirectly) by the ride of delayed train A, with input delay
𝑡,, is described in the chart in Figure 4.
𝑁=𝑣

 (2)
Figure 2. Dependence of the number of trains affected by an input delay on the number of bindings
of every train transmitting the delay when the delay is spread in 5 degrees. Source: the authors.
Figure 2.
Dependence of the number of trains affected by an input delay on the number of bindings
of every train transmitting the delay when the delay is spread in 5 degrees. Source: the authors.
Figure 3.
Delay transmission between trains and gradual elimination of the delays. Source: the authors.

Appl. Sci. 2022,12, 457 7 of 19
Figure 4.
The course of the delay sum of trains caused by delayed train A with input delay t
input,A
in
the case of the stable timetable. In the delay spread phase, the increase of delay caused by mutual
interactions between trains is at a higher level than the gradual elimination of those delays. At the
stagnation stage, both of these factors are at approximately the same level. In the delay elimination
phase, the level of delay elimination is higher than the ongoing delay spread. After expiring c
stab
-
multiple of the timetable period from the time of the input delay origin (T
0
), there comes a total
elimination of that delay and all delays caused by it. It is clear from the chart that local anomalies are
possible. Source: the authors.
For the reasons described above, transfer waiting times should have, at a maximum,
such values that maintain the timetable’s stability (e.g., reaching no more than a certain
value of stability coefficient). This means that it should be possible to eliminate delay that
is caused by waiting for a delayed connecting train, for an adequate amount of time using
the reserves and spaces in the timetable.
2.2.3. Introduction to the Czech Praxis
In Czech railway transport, the basic terms related to transfer waiting times are defined
in the timetable manual, “Transfer connections between passenger trains” [
30
], which is
published by Czech Infrastructure Manager Správa železnic (“IM SŽ”).
According to this timetable manual, transfer connection applies to a pair of trains,
where there are at least 2 min and at most 60 min between the scheduled arrival time of the
first (connecting) train and the scheduled departure time of the second (continuing) train.
Different transfer times are mentioned in the manual for every station.
Non-connection refers to pairs of trains for which the transfer connection will not be
ensured, despite the in-time ride of the trains, for example:
•
when trains are traveling in opposite directions on the same railway line without
regard to stopping patterns (e.g., when a connecting train arrives from the same
neighboring station that a continuing train departs from, regardless of whether the
trains stop there); exceptions are cases when the neighboring station is a turning
station without a stop;
•
when trains are traveling in the same direction on the same railway line and both
trains have the same scheduled stop;
•
when there is a longer transfer time at a station that is not reached, between the
connecting train’s arrival time and the continuing train’s departure time (this scenario
is mentioned for every station in the manual “Transfer connections between passenger
trains,“ and in the timetable for passengers);

Appl. Sci. 2022,12, 457 8 of 19
•
when the manual “Transfer connections between passenger trains,“ as well as the
timetable for passengers, mention that there is no transfer connection between particu-
lar trains;
•
when trains are from different operating companies, subject to exceptions mentioned
in the manual.
The transfer waiting time means the maximum possible departure delay (compared to
the regular or exclusion timetable) of the continuing train from the transfer station/junction,
influenced by waiting for a delayed connecting train. Transfer time is also included in
transfer waiting time. Waiting time cannot be exceeded without special instruction from
the train’s operating company (i.e., via the dispatcher). Basic transfer waiting times in
the network of IM SŽ are determined uniformly for all stations: 5 min for express trains
(category “Ex”) and 10 min for fast trains (category “R”), fast regional trains (category
“Sp”), and regional trains (category “Os”). Basic transfer waiting times are valid for all
continuing trains if there is no exception for an individual station and/or for pairs of trains.
Continuing trains do not have to wait for a connecting train to set the transfer waiting
time if there is no transferring frequency of passengers for the continuing train, if this is
announced by train staff to operation employees.
In the contemporary Czech praxis, the train’s operating company evaluates transfer
waiting times. For PSO-services, the carrier does this after a discussion with the public
purchasers of the service. Unfortunately, in most cases, the values of transfer waiting
times are only estimations, arrived at without deep traffic analysis or any supporting tools
(e.g., mathematics or simulations).
In addition to the basic considerations mentioned above, the manual “Transfer con-
nections between passenger trains” introduces a method for assigning transfer times. In
the table section of the manual are exceptions for basic transfer times for each station, for
connection trains using the transfer connection definition, and for basic transfer waiting
times. Figure 5provides an example of such a table where these exceptions are stated for
the Protivín station and valid for the timetable for 2019–2020.
Regulation D7 (Regulation for operational traffic control [
31
] of IM SŽ) applies to the
operational management of services and provides principles for organizing railway traffic
during the operational management of services. Implementation procedures for Regulation
D7 (“PND7”) are published for the validity period of the timetable, comprised of parts A
and B.
Part A of PND7 describes the traffic control structure and activities of individual
operational sections of IM SŽ, which correspond to the valid timetable, and for the determi-
nation of mutual duties and responsibilities. In addition, it contains regulations that are
permanently valid for the whole net, as well as temporary regulations that are valid for one
period of the timetable, related only to certain central dispatcher districts.
Part B of PND7 determines the rules of railway traffic control for individual railway
lines and provides additional information about train operating companies, intended for
railway traffic control employees. The rules are valid for one period of the timetable. No
timetable of the trains of other train operating companies can be changed if the companies
do not agree. PND7 provides manuals for operational traffic control on railway lines, where
applicable, as well as tolerable anomalies from the general traffic control rules. In part B,
there are usually instructions related to the following:
•
cases of an increase in the tolerance of a higher category train’s delay, influenced by a
lower category train ride;
•
changes of a train sequence, depending on the level of delay (crossing, overtaking
relocation, etc.);
•
transfer binding relocation to another traffic point, when certain operational disrup-
tions occur;
•
cancellation of a pair of trains in a section when certain operational disruptions occur
(e.g., disruptions due to elimination of a delay transmitted during train set reversion);

Appl. Sci. 2022,12, 457 9 of 19
•
different circulation bindings of the train sets, when certain operational disruptions
occur (e.g., disruptions due to delay elimination);
•
additional stops for arranging connections for services that would otherwise be can-
celed (or the travel time disproportionally extended) due to certain operational dis-
ruptions;
•
the introduction of substitute trains for arranging connections, where service would
otherwise be canceled (or the travel time disproportionally extended) due to certain
operational disruptions;
•transfer waiting times between train and bus services.
Measures mentioned in PND7 can influence transfer waiting times values; therefore, it
is necessary to assign transfer waiting times simultaneously, ideally, using the manual to
solve common operational disruptions that are stated in PND7.
Figure 5.
Transfer waiting times table for the Protivín station, timeTable 2019–2020 [
30
] Translation
by the authors.

Appl. Sci. 2022,12, 457 10 of 19
2.3. The Maximum Transfer Waiting Times Principle Based on Modified CPM
In the Czech praxis, one of the normal methods for estimating transfer waiting times
is maximization, so that a certain subset of transfer bindings set out in a timetable can
be maintained and further marked (as crucial transfer bindings), while considering the
mutual interactions between trains (i.e., crossing, post-rides, vehicle reversions, etc.) and
using disposal measures (see Section 2.2.1). By that method, it is common to progress
from the highest segment trains to the lowest segment trains. Transfer waiting times for
higher segment trains serve as input for assigning the transfer waiting times for lower
segment trains.
When choosing crucial transfer bindings, it is necessary to consider the volume of
passengers of a corresponding transfer binding and the frequency of continuing trains
(i.e., how long passengers will wait for the following connection when the continuing train
is missed). It is also possible to reduce transfer waiting times based on the assessment of
these factors.
Net-planning methods may be used to calculate transfer waiting times, rather than
simply estimating the times, via a modified critical path method (CPM). In doing so, the task
may be expressed as an “operational/technological processes graph” that represents train
arrivals, departures, and passages, as well as the bindings between the trains (transfers,
operational bindings, circulation bindings, etc.). The graph contains:
•vertices at,i, which express the arrival of train tto traffic point i;
•
vertices
dt,i
, which express the departure or passage of train
t
from/trough traffic
point i;
•
oriented edges
rt,i−j=dt,i;at,j
or
rt,i−j=dt,i;dt,j
, which introduce the run of the
train
t
from traffic point
i
to traffic point
j
and potential stops at intermediate traffic
points between traffic points
i
and
j
(if the edge goes into a vertex
at,j
, train
t
stops at
traffic point j; if the edge goes into a vertex di,j, train tpasses through traffic point j);
•oriented edges dwt,i=[at,i;dt,i], which introduce the dwell of train tat traffic point i;
•
oriented edges
tt−u,i=[at,i;du,i]
, which introduce the transfer binding from connecting
train tto continuing train uat traffic point i(while t6=u);
•
oriented edges
ht,i−u,j=at,i;au,j
or
ht,i−u,j=at,i;du,j
or
ht,i−u,j=dt,i;au,j
or
ht,i−u,j=dt,i;du,j
, which introduce operational interactions between the trains (while
t6=u); and
•
oriented edges
ct−u,i=[at,i;du,i]
, which introduce the circulation binding between
trains tand u(while t6=u) at traffic point i(e.g., the train set reversion).
Vertices at,iand dt,ieach have three evaluations:
•s(at,i)
or
s(dt,i)
, which means the scheduled arrival, departure, or passage time of train
tat traffic point i;
•e(at,i)
or
e(dt,i)
, which means the earliest possible arrival, departure, or passage time
of train tat traffic point i; and
•l(at,i)
or
l(dt,i)
, which means the latest possible arrival, departure, or passage time of
train tat traffic point i.
The vertices
dt,i
, whose highest tolerable waiting times are known for a certain combi-
nation of train
t
and traffic point
i
and which are used as an input for assigning the other
trains’ waiting times, have one more evaluation
lIN(dt,i)
, which means the latest possible
departure time of train tat traffic point iwhile waiting for delayed connecting trains.
All edges of the graph have evaluations
mtt
, which mean the minimum value of
corresponding technological times:
•mttrt,i−j
means the minimum run time of train
t
from traffic point
i
to traffic point
j
, increased by the minimum dwell times of this train at intermediate stops between
traffic points iand j;
•mtt(dwt,i)
means the minimum dwell time of train
t
at traffic point
i
(whether due to
transportation or traffic reasons);

Appl. Sci. 2022,12, 457 11 of 19
•mtt(tt−u,i)means the minimum transfer time between trains tand uat traffic point i;
•mttht,i−u,j
means the minimum headway between train
t
at traffic point
i
and train
u
at traffic point
j
(when
i=j
, it is a minimum station headway, a minimum crossing
headway, a minimum departure headway, or a minimum arrival headway; when
i6=j
,
it is a minimum track headway);
•mtt(ct−u,i)
means the minimum reversion time of train sets between train
t
a train
u
at traffic point
i
, or the minimum technological time of another circulation binding
(e.g., the time of train sets coupling or decoupling where a part of train set switches to
another train).
We designed the operational/technological processes graph so that it would reflect
transportation and operational interactions between the trains, as established by the
timetable, or the interaction between the trains while considering used disposal mea-
sures. Assuming that delays caused by waiting for delayed connecting trains from the end
of an operational day do not transmit to trains at the beginning of the following operational
day, we formed the operational/technological processes graph for one operational day. If
this assumption was not applicable, we formed the graph for two operational days in a
row, eventually for the number of operational days until the transmitted delays from the
first operational day are expected to be eliminated. In the case of the integrated periodic
timetable (IPT) [
32
,
33
], it was sufficient to form the operational/technological processes
graph only for the number of periods in a row, so that all operations and transportation
periodically repeated during interactions between trains in a solved subnet were taken
into consideration. The resulting transfer waiting times for the certain combination of line,
direction, and traffic point were assigned as the minimum of the calculated transfer waiting
times through all vertices of the corresponding combination in the graph. When there were
different timetables for different operational days, it was necessary to form a graph for each
type of operational day, as each graph is intended for calculating the transfer waiting times
for corresponding operational days.
In forming the operational/technological processes graph, it was necessary to consider
that the interaction of a train passing through a traffic point with another train that is not
in time (i.e., it was delayed or ran earlier) can lead to a forced stop of the passing train for
traffic reasons and/or to extend some of the minimum technological times.
The algorithm for assigning the evaluation of the operational/technological processes
graph vertices, and for the following calculation of the train transfer waiting times, is
comprised of the following steps:
1.
For all combinations of train
t
and traffic point
i
for which there are known the
maximum transfer waiting times
wIN,t,i
(which are the inputs for assigning other
trains’ transfer waiting times), we pasted the corresponding vertex
dt,i
onto set
Iw
. For
all elements of set
Iw
, the latest necessary departure time
lIN(dt,i)
was calculated by
Equation (3).
lIN(dt,i)=s(dt,i)+wIN,t,i(3)
2.
If
m[x,y]
is an oriented path from vertex
x
to vertex
y
and
ed ∈m
is the total of oriented
edges contained in the path, for all vertices
v/∈Iw
of operational/technological
processes graph:
a.
if there is at least one oriented path
m[v,dt∗,i∗]
, where
dt∗,i∗∈Iw
, the latest
necessary arrival, departure, or passage time l(v)is assigned by Equation (4);
l(v)=min
∀dt∗,i∗∈Iw

lIN(dt∗,i∗)−max
∀m[v,dt∗,i∗]

∑
ed∈[v,dt∗,i∗]
mmt[ed]

(4)
b.
if there is no oriented path
m[v,dt∗,i∗]
, where
dt∗,i∗∈Iw
, the evaluation
l(v)
stays undetermined.

Appl. Sci. 2022,12, 457 12 of 19
3.
For all vertices
v
of the operational/technological processes graph, evaluation
e(v)
was determined (i.e., the earliest possible arrival, departure, and passage times at
traffic points). The following factors must be considered:
◦the scheduled arrival, departure, and passage times at traffic points;
◦
the possibility of scheduled technological times shortening in light of timetable
reserves and spaces utilization (however, they cannot be shortened below their
minimum value, therefore
e(y)≥e(x)+mmt[x,y]
is valid for each oriented
edge [x,y]of the operational/technological processes graph);
◦
the unacceptability of a train’s departure before the scheduled departure time
(e.g., departure at arrival time, departure immediately after passenger board-
ing, or departure if the train stops for traffic reasons).
4.
If at least one vertex
v
exists for which
e(v)>l(v)
is valid, it was not possible to en-
sure at least one transfer binding with used input waiting times; this situation should
not occur when no disposal measure is used. Let
me>l[x,dt∗,i∗]
, where
dt∗,i∗∈Iw
,
be an oriented path with unfeasible transfer binding if for all edges
[f,g]∈me>l
both evaluations
l(f)
and
l(g)
are determined,
e(f)>l(f)
and
e(g)>l(g)
are
valid, and there concurrently exists no oriented edge
[y,x]
where
e(y)>l(y)
and
e(dt∗,i∗)>lIN(dt∗,i∗)
. For each oriented path with unfeasible transfer binding
me>l[x,dt∗,i∗]
, it is necessary to drop at least one transfer binding edge
tu1−u2∈me>l[x,dt∗,i∗]
; this step means resignation from the corresponding trans-
fer binding. If at least one edge was dropped, the algorithm must be performed again
from the second step; otherwise, the algorithm should be continued to the fifth step.
5. For all vertices dt∗,i∗∈Iw, the evaluation l(dt∗,i∗)was adapted by Formula (5):
l(dt∗,i∗)=min{l(dt∗,i∗);lIN (dt∗,i∗)}(5)
6.
For all vertices
dt,i
, which are entered by at least one oriented edge
tu−t,i
, the transfer
waiting time wt,iof train tat traffic point iwas determined:
◦
if evaluation
l(dt,i)
was determined, the transfer waiting time was calculated
by Formula (6):
wt,i=l(dt,i)−s(dt,i)(6)
◦
if evaluation
l(dt,i)
was not determined, the transfer waiting time
wt,i
was not
restricted by the input transfer waiting times and it was necessary to deter-
mine the transfer waiting time by another method or by an expert estimate,
e.g., according to the transportation effects on passengers in train
t
and in
trains that directly or indirectly interact with train t.
7. If mlim [x,y]is a limiting (oriented) path if for all edges [f,g]∈ml im both evaluations
l(f)
and
l(g)
are determined and
l(g)−l(f)=mmt[f,g]
is valid. Such a path limits
the maximum value of at least one transfer waiting time. To increaseof the transfer
waiting time
wt,i
, for which evaluation
l(dt,i)
is determined, it is necessary, for each
limiting path
mlim [dt,i,dt∗,i∗]
, where
dt∗,i∗∈Iw
and
l(dt∗,i∗)=lIN(dt∗,i∗)
, to drop
from the operation/technological processes graph at least one transfer binding edge
tu1−u2,j∈mlim [dt,i,dt∗,i∗]
. If vertex
dt,i∈Iw
, the increasing of
wt,i
is only possible if
l(dt,i)<lIN(dt,i)
. Dropping of an edge means resignation from the corresponding
transfer binding when a connecting train
u1
is so delayed at arrival to traffic point
j
that continuing train
u2
does not wait any longer for connecting train
u1
. After
dropping corresponding edges, it is necessary to repeat the algorithm from step 2.
If no edge was dropped, the algorithm is finished and
wt,i
are the resulting transfer
waiting times.
The disadvantage of this algorithm could be its usage for assigning the transfer waiting
times while having a high number of different usable disposal measures, especially if they
are mutually combined. A combinatorically difficult (NP-complete) task then arises, where
it is necessary to form additional different operational/technological processes graphs for

Appl. Sci. 2022,12, 457 13 of 19
one task, which must be assessed. It can be limited to using only those disposal measures
that are relevant to those delay values, that are expected and assumed considering the
assigned input transfer waiting times, or that enable the maximization of the transfer
waiting times. In more complex nets, the only solution for coping with the difficultness of
the task and the high number of usable disposal measure combinations might be railway
operation simulation [
17
,
26
], because a microsimulation proceeds autonomously (eventu-
ally based on assigned input suppositions) to use appropriate disposal measures and their
combination according to the delay amounts of particular trains. Then, the result is not the
determination of optimal transfer waiting times, but the takeover of tolerable delay values
from performed microsimulation, which still ensures convergence of the operation to the
scheduled timetable.
The principle of maximum transfer waiting times described in this section does not
solve the passenger transportation issue, which can sometimes lead to:
•
transfer waiting times that are too short (e.g., for trains where almost all passengers
are transferring from connecting trains),
•
transfer waiting times that are too long (e.g., for trains where most of the passengers
are not transferring from other trains).
However, dealing with the transportation aspect is not a goal of this research.
3. Results
3.1. Practical Implementation of the Maximum Waiting Time Principle to a New Operational
Concept in the Vysoˇcina Region
The new innovative principle for determining transfer waiting times, described in
Section 2.3, was implemented by the authors as a new operational concept for regional
railway transport in the Vysoˇcina Region, and launched with timetable for 2019/2020. This
operational concept reflects transportation demand in the region [
34
] and is based on the
IPT-principles. The scheme is introduced in Figure 6as the net graph [
35
] (weekend schedule).
In the description of the solution, the following designation of the lines is used:
•line R9: fast trains, Praha–Kolín–Havlíˇck˚uv Brod–Brno;
•line R11: fast trains, Plze ˇn– ˇ
CeskéBudˇejovice–Jihlava–Brno;
•line Sp: fast regional trains, Havlíˇck˚uv Brod–Jihlava–Tˇrešt’–Telˇc–Daˇcice–Slavonice;
•line Os1: regional trains, Kolín–Havlíˇck˚uv Brod–Žd’ár nad Sázavou;
•line Os2: regional trains, Havlíˇck˚uv Brod–Jihlava mˇesto;
•line Os3: regional trains, Jihlava–Tˇrebíˇc;
•line Os4: regional trains, Jihlava–HorníCerekev–Pelhˇrimov–Tábor.
The input boundary condition for solving this task was compliance with transfer
waiting times of lines R9, R11, and Os1. For lines R9 and Os1 at Havlíˇck ˚uv Brod station
and line R11 at Jihlava station, it was calculated with a basic 10 min waiting time.
The following parts of railway lines that proved to be decisive for determining trans-
fer waiting times at junctions Havlíˇck ˚uv Brod and Jihlava were involved in the opera-
tional/technological processes graph:
•railway line 225 in the Havlíˇck ˚uv Brod–Kostelec u Jihlavy section;
•railway line 227 in the Kostelec u Jihlavy–Daˇcice section;
•railway line 240 in the Jihlava–Tˇrebíˇc section;
and other related or continuing trains at the Havlíˇck ˚uv Brod station from railway lines 230
(Kolín–Havlíˇck ˚uv Brod) and 250 (Havlíˇck˚uv Brod–Tišnov). Except for railway lines 230
and 250, all railway lines in the solved area are single-track lines.
The operational/technological processes graph was assembled for two 2-h periods
in a row, in which all operational and transportation periodically repeating interactions
between trains in the solved area were taken into consideration.

Appl. Sci. 2022,12, 457 14 of 19
15 15 08
44 44 50
14 01 59
14 45 01 59 59 01
05 04 45 47 59 01
03 04 21
56 52 57
04 07 03
45
15
24 37
34 23
40 37
42 40 20 37 23
46 17 45 18 42 21 33 08 07 51 49
31 14 27 15 24 18 12 27 00 58 35 52 35 53 19 06 07
30 33 36 47 59 01 23 24 41
58
02
00 01 47
59 59 44
00
00 59
59
14
47
01
00
57
01
33
25
24
34 dep. arr.
arr. dep.
01
56
54
57
25
27
Tábor ←
Figure S1: Net graph of the new operational concept in Vysočina Region – weekends
→ Brno hl.n.
Třešť
Telč
Dačice
Slavonice
Line style
interval 120 minutes
Line colour
line of regional trains
line of regional fast
trains
→ Plzeň hl.n.
direction where
the line continues
Other
Node description
Timetable notations
Node name
Třebíč
Luka nad Jihlavou
Havlíčkův Brod
Jihlava
Okříšky
→ Znojmo
Plzeň hl.n. ←
Kostelec u Jihlavy
Světlá nad Sázavou
Jihlava město
→ Praha hl.n.
→ Kolín
Humpolec
→ Hlinsko v Čechách
→ Brno hl.n.
→ Žďár nad Sázavou
Ledeč nad Sázavou ←
line of fast trains
59
even hour, minute 59
59
odd hour, minute 59 (italics)
Figure 6.
Net graph of the new operational concept in Vysoˇcina Region weekends. Source: the authors.
As the authors did not have any passenger number data to assess the transportation
significance of individual transfer connections (nor any passenger flow information from
previous operational concepts, which would not have been usable anyway because of the
differences between the old and the new ones), all transfer connections were considered
as crucial (i.e., no transfer connection was left out of the operational/technological pro-
cesses graph) and maximal waiting times were determined so that no transfer connections
were broken.
There were assumed to be 4% scheduled linear run time supplements to the technical
run times. In the case of minimum run times, 2% linear run time supplements to the
technical run times were used. Minimum headways were estimated by the authors consid-
eration of the types of station, the track signaling systems, and the track configurations of
every station.

Appl. Sci. 2022,12, 457 15 of 19
The following disposal measures were used:
•
the exchange of arrival order of even-direction trains on line Sp and even-direction
trains on line R11 at the Jihlava station past the odd hours, because this arrival order
is more probable in the range of delays corresponding to anticipated waiting times (as
the Sp line has a greater ability to shorten the delays in the Jihlava–Tˇrešt’ section than
does the R11 line in the Jihlava–Krahulov section);
•
the exchange of train sequence order on lines Os2 and R11 in the Jihlava–Jihlava mˇesto
section in both directions, as it enables the extension of line Os3 waiting times at the
Jihlava station and removes restrictive influences for R11 line trains reversion, because
during the process of locomotive change, there is one train that is less restricting with
respect to locomotive shunting (i.e., the train of line Os2).
The assembled operational/technological processes graph in Supplementary Figure S1
is displayed in cut-out form in Figure 7. The vertices are represented by:
•brown if it is a vertex of at,itype (train arrival); or
•black if it is a vertex of dt,itype (train departure or passage).
The corresponding name of the traffic point
i
is introduced in bold type in the grey
field in which the vertex is placed. The vertex is divided into 4 to 5 rows:
•
in the first row, the sign of train
t
is introduced (the line sign, behind which there
is an ordinal number of the train in brackets that grows with the period ordinal
number; even numbers are used for trains of even direction, odd numbers for trains of
odd direction);
•in the second row, the scheduled arrival/departure/passage time sis introduced;
•
in the third row, the earliest possible arrival/departure/passage time
e
is introduced;
•in the fourth row, the latest possible arrival/departure/passage time lis introduced;
•alternatively, in the fifth row, the latest possible departure time lIN is introduced.
In time details in the second, third, fourth, and fifth rows, “S” means an even hour,
while “L” means an odd hour.
Oriented edges are represented by:
•blue if it is an edge of rtype (train runs);
•red if it is an edge of dw type (train dwells);
•green if it is an edge of ttype (transfer bindings between trains);
•orange if it is an edge of htype (operational interactions between trains); or
•purple if it is an edge of ctype (circulation bindings between trains).
The evaluation
mtt
is introduced in minutes near the edge by the same color as the edge’s.
The transfer waiting times for regional transport trains, using the new operational
concept in the Vysoˇcina Region for the basic weekend IPT model, are summarized in
Table 2.
Table 2.
Determined transfer waiting times of regional transport trains using the new operational
concept in the Vysoˇcina Region for the basic weekend IPT model. Source: the authors.
Line Traffic Point Direction Maximum Waiting Time
Sp Havlíˇck˚uv Brod odd (>Jihlava > Telˇc > Slavonice) 12 min
Jihlava odd (>Tˇrešt’ > Telˇc > Slavonice) 11 min
Jihlava even (>Havlíˇck˚uv Brod) 10 min
Os2 Havlíˇck˚uv Brod odd (> Jihlava mˇesto) 13 min
Jihlava even (>Havlíˇck˚uv Brod) 15 min
Os3 Jihlava odd (>Tˇrebíˇc) 22 min
Os4 Jihlava odd (>HorníCerekev > Tábor) 16 min
Transfer waiting times for the Sp line, determined by the authors’ approach, were
implemented by the train operating company ( ˇ
Ceskédráhy) and by IM SŽ for regular

Appl. Sci. 2022,12, 457 16 of 19
operational control in this area. Other transfer times, as determined, were moderately
adjusted to take into account the transportation (passenger time) factor.
Reached results of train delays and achieving transfer connections for the first half-
year of operation of the new concept showed that qualified steps and projected values of
transfer waiting times were calculated correctly. If unplanned and unexpected incidents,
operational disruptions, vehicles breakouts, and signaling system breakouts do not occur,
the operation functions stably. Transfer connections are functional in the maximum possible
volume, and a low delay in operations leads to compliance with the planned timetable.
Figure 7.
Cut-out of the operational/technological processes graph for solved part of Vysoˇcina Region
weekends. Source: the authors.
4. Discussion
The application of the proposed method confirmed the expected limitations of the
method that were mentioned in the introduction.

Appl. Sci. 2022,12, 457 17 of 19
A relatively low number of possible disposal measures, which were assessed in the
practical application in Section 3.1, leads to a higher number of their combinations. The
application was simplified by using only those disposal measures that are relevant in
the expected delay values, and which lead to maximization of the transfer waiting times.
This rationalization finally led to only one combination of disposal measures, and thus to
one graph for operational/technological processes. If this rationalization is not made, the
number of graphs would be higher, and the task would become more time-consuming.
As stated in the introduction, the passenger transportation aspect was not a subject
of this research. It is obvious from the achieved results, especially the transfer waiting
times of trains on line Os3 in Jihlava (22 min), that the transfer waiting time is too long for
passengers who do not transfer from other trains. Such transfer waiting times apply to
most of the passengers in the trains of this line. The passenger transportation aspect, using
the authors’ approach, might be a subject for further research, leading to a rationalization
of the transfer waiting time values of passenger trains (which have all the bindings while
the other trains are relatively loose).
On the other hand, the research method satisfactorily addressed all the technological
bindings between trains. The method is likely to be successful in its application to other
practical cases of determining transfer waiting times.
5. Conclusions
This article dealt with the topic of delay management in the timetable planning process,
especially in determining transfer waiting times. First, state-of-the-art train scheduling,
traffic control, dispatching, and delay management were examined. The authors found that
the critical path method is commonly used in some of the tasks carried out for operational
traffic control and/or by dispatchers (e.g., for proper time margin positioning), but CPM
has not been used for determining transfer waiting times, which was the authors’ primary
challenge in this research.
Next, the article described the basic relationship between line periods/intervals and
length of transfer waiting times. With increasing intervals between connections, transfer
waiting times grows, because when is lost to a continuing train, real waiting time cor-
responds with waiting for the following service (and the travel time for passengers is,
accordingly, a line-interval longer).
The article then described the relationship between transfer waiting times and timetable
stability. Timetable stability must also be considered when determining transfer waiting
times. The more interactions there are between trains and the more heterogeneous a
timetable is, the more potential conflicts between trains arise, with potential sources of
delay transmission or formation. Delays tend to spread due to mutual interactions between
trains; the number of affected trains increases disproportionately with the number of bind-
ings in the timetable and the delay elimination time. The delay elimination time alone,
which can occur in the case of a periodic timetable described by the so-called stability coef-
ficient, is one of the possible indicators of timetable stability. Thus, in the case of marginal
connections (e.g., during evening or night), transfer waiting times are generally longer
during other times of the day, due to the lower number of interactions. The transmission
of delays does not usually take place at the same rate during evenings or night as it does
during the day.
To determine transfer waiting times, it is necessary to correctly identify the critical
section of a railway line, which is decisive with respect to time supplements, reserves,
spaces, synchronization times, and the length of interstation sections in cases of crossing
or overtaking relocation. The possibility of train crossing and overtaking relocation, or
utilization of other disposition measures, also affects the length of transfer waiting times.
Therefore, it is common Czech practice to determine transfer waiting times at the same time
as compiling instructions (i.e., dispositional measures) for resolving operational disruptions
(which are included in the “D7–Regulation for operational traffic control” of IM SŽ).

Appl. Sci. 2022,12, 457 18 of 19
The researched then described an approach for determining transfer waiting times.
This approach, called the maximum transfer waiting times principle, consists of deter-
mining transfer waiting times to their maximum value so that none of the crucial transfer
connections are disrupted. Using this approach, an algorithm was determined to convert
this task into an oriented graph (called the technological-operational processes graph) and
usea modified CPM (the critical path method) to determine transfer waiting times. This
method takes into account operational/technological interactions between trains, but does
not consider the transportation (passenger time) aspect.
The method was applied as a new operational concept in the Vysoˇcina Region, where
it has been implemented since the beginning of the timetable for 2019/20. Based on the
waiting times for the long-distance trains Praha–Havlíˇck ˚uv Brod–Brno and Plzeˇn– ˇ
Ceské
Budˇejovice–Jihlava–Brno and the regional trains Kolín–Havlíˇck ˚uv Brod–Žd’ár nad Sázavou,
transfer waiting times were proposed for relevant regional trains lines at the Havlíˇck ˚uv
Brod and Jihlava junctions.
Using this practical application, the limitations of the method mentioned in the intro-
duction were confirmed, especially the fact that if many relevant disposal measures are
used, the number of technological/operational processes, as constructed and calculated,
increases with the number of their combinations, which can be time-consuming. However,
when the number of relevant disposal measures is low or when a rationalization can be
made, the research method satisfactorily coped with all the technological bindings between
trains and is likely to be successful when applied to other practical cases of transfer waiting
times determination.
Another limitation of the method is that it does not deal with passenger time, so some
transfer waiting times can be too long or too short from that perspective. Including this
aspect could be a possible area for further research.
The correct way to determine transfer waiting times is a small piece in the whole
mosaic of functional public transport and network planning. It is a supporting tool to
improve the reliability, efficiency, and sustainability of public transport systems.
Supplementary Materials:
The following supporting information can be downloaded at: https:
//www.mdpi.com/article/10.3390/app12010457/s1, Figure S1: Operational/technological processes
graph–solved part of Vysoˇcina Region weekends.
Author Contributions:
Conceptualization, R.V. and V.J.; methodology, R.V.; validation, R.V. and V.J.;
formal analysis, R.V.; investigation, R.V. and V.J.; resources, R.V. and V.J.; data curation, R.V.; writing—
original draft preparation, R.V.; writing—review and editing, V.J.; visualization, R.V.; supervision, V.J.
All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: All data used in the study are presented in the manuscript.
Conflicts of Interest: The authors declare that they have no conflict of interest.
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